NBSIR 76-1089

An Analysis of Inertial

Seisdmeter-Galvanorneter

Combinations

D. P. Johnson and H. Matheson

Mechanics Division

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

June 1976

Final

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU Of STANDARDS

•

TABLE OF CONTENTS

Page

SECTION 1 INTRODUCTION

1 . 1 Background ^ 1.2 Scope ^ 1.3 Introductory Details 2

SECTION 2 GENERAL EQUATIONS OF MOTION OF AN ELECTROMAGNETIC

INERTIAL SEISMOMETER 3

2.1 Dynaniical Theory 2

2.1.1 Mechanics ^ 2.1.2 Electrodynamics 5

2.2 Choice of Coordinates ^ 2.2.1 The Coordinate of Earth Motion 7

2.2.2 The Coordinate of Bob Motion 7 2.2.3 The Electrical Coordinate 7 2.2.4 The Magnetic Coordinate 3

2.3 Condition of Constraint: Riagnet 3

2.4 The Lagrangian Function • 2.4.1 Mechanical Kinetic Energy 10

2.4.2 Electrokinetic and Electropotential Energy H

2.4.3 Gravitational Potential Energy 12

2.4.4 Mechanical Potential Energy 13

2.5 Tne General Equations of Motion i4

2.6 The Linearized Equations of Motion

2.7 Philosophical Notes and Interpretations 18

2.8 Extension to Two Coil Systems 18

2.8.1 General

2.8.2 Equations of Motion -^9 2.8.3 Reciprocity Calibration Using

the Basic Instrument 21

2.8.4 Reciprocity Calibration Using

Auxiliary Calibrating Coils . 23

2.9 Application to Other Electromagnetic

Transducers -

SECTION 3 RESPONSE CFARACTERISTICS OF A SEISMOMETER

WITH A- RESISTIVE LOAD

3.1 Introduction

3.2 Seismometer with Resistive Load
II

TABLE GF CONTENTS

Page

3.2.1 Steady State Response 27

3. 2. 1.1 Steady State Response to

Earth Motion, Shake Table 27

3.2. 1.2 Steady State Response to

Bob Force, Calibrator 29

3. 2. 1.3 Steady State Response to

Electrical Excitation 31

3.2,2

Transient Response 32

3.2.2. 1 Solution for the Weight-Lift

Calibration 35

3. 2. 2. 2 Transient Response Boundaries,

Loci and Damping 39

3.2.3 Transient Response Curves,

Analog Computer 42

3.2.4 Electrical Analog of a Seismometer 44

SECTION 4 D'ARSONVAL GALVANOMETER CONSIDERATIONS 46

•

' 4.1 General 46

4.2 D'Arsonval Galvanometer, Equations of Motion 46

4.3 Galvanometer with a Resistive Load 48

4.3.1 Transient Response 48

4.3.2 Response to Sinusoidal Electrical

Excitation 49

4.4 Determination of Galvanometer Parameters 54

4.4.1 Galyanometer-Photoelectric

Amplifier Combination 54

4.4.2 Galvanometer Measured as a Two

Terminal Network 56

SECTION 5 SEISMaiETER, RESISTIVE T PAD AND GALVANOMETER 59

5.1 General 59

5.2 Seismometer with Bilateral Coupling

to the Galvanometer 59

5.2.1 Influence of Variable Attenuation on

the Over-all Response-Frequency

Characteristics 63

5.2.2 Direct Coupling of Seismometer and

Galvanometer 64
SECTION 6

SECTION 7

SECTION 8

III

TABLE OF CONTENTS

SEISMOMETER- GALVANOMETER SYSTEMS WITH NEGLIGIBLE

INDUCTAIJCE

6.1 General

6.2 Seismoneter , Resistive T Pad, and

Galvanometer

6.3 Direct Coupled Seismometer-Galvanometer System

6.3.1 Direct Coupled System, "Constant

Velocity" Response

6.3.2 Direct Coupled System, "Constant

Amplitude" Response

6.3.3 Direct Coupled System, "Constant

Acceleration" Response

6.4 Further Remarks on the T Pad Problem

SPECIAL response-frequency CHARACTERISTICS

7.1 General

7.2 The Notch Filter

7.3 Multiple Frequency Response Systems

7.4 The Telemetering Problem

7.4.1 A "Flat Velocity Sixth" Order Response

7.4.2 A "Flat Amplitude" and a "Flat

Acceleration Sixth" Order Response

SENSITIVITY LIMITS IN PRINCIPLE AND IN PRACTICE

8 . 1 General

8.2 Brownian Motion

8.3 Quantum Effects

8.4 Spurious Voltages in the Electrical Circuit

8.5 Air Buoyancy

8.6 Air Convection

8.7 Light Pressure and the "Radiometer Effect

8.8 Magnetic Noise

8.9 External Magnetic Fields . 8.9.1 Field Uniform in Space, but

Varying in Time

8.9.2 Field Non-Uniform in Both

Space and Time

Page

68

68

68

72

74

82

87

88

93

93

93

99

100

101

108

112

112

112

115

117

120

123

124

126

127

128

128
IV

TABLE CF CONTENTS

Page

SECTION 9 CALIBRATION 129

9.1 General 129

9.2 Shake Table 129

9.3 Semi-Direct Calibration 129

9.3.1 Weight Lift 130

9.3.2 Pneumatic Calibrator 130

9.3.3 Electromechanical Calibrator 131

9.3.4 Electromagnetic Calibrator 131

9.3.5 Electrodynamic Calibrator 132

9.3.6 Injected Voltage 133

9. 3. 6.1 Amplitude of Series Injected Voltage 133

9. 3. 6.Bridge Methods 134

9. 3. 6.

3

Phase Measurements 134

9.4 Indirect Methods 136

9.4.1 Recapitulation 137

9.4.2 Special Tests 137

SECTION 10 NOMENCLATURE AND PERTINENT REFERENCES 142

10.1 General 142

10.2 Nomenclature 143

10.3 References 162
V

TABLES

Table 3.1 Constants for Figures 3.3 through 3.9

Benioff Variable Reluctance Seismometer

Table 0.1 Computed 3 db Points for Direct and T

Pad Linked Fourth Order Flat Velocity

Systems

Table 7.1 Notch Width as a Function of Filter

Damping (For galvanometer having negligible

internal losses)

Table 7.2 Phase Lead of the Recorder Voltage with • Respect to Earth Velocity for the Sixth

Order Direct-Coupled Cons'tant Velocity

System

Table 8.1 Thermal Voltage Equivalent to One Millimicron Earth Displacement

Page

43

92

96

107

121
VI

FIGURES

Page

F igure 2.1 Magnetizing Force -Induct ion 9 Figure 3.1a Graph for Obtaining Real Roots of an

Algebraic Cubic Appendix

Figure 3.1b Plot of a and D against fg and p Appendix

Figure 3:2 Character of the Transient Response Appendix

Figure 3.3 Analog Computer Response to Simulated

Weight Lift Appendix

Figure 3.4 Analog Computer Response to Simulated

weight Lift Appendix

Figure 3.5 Analog Computer Response to Simulated

Weight Lift Appendix

Figure 3.6 Analog Computer Response to Simulated

Weight Lift Appendix

Figure 3.7 Analog Computer P^esponse to Simulated

Weight Lift Appendix

Figure 3.8 Analog Computer Response to Simulated

Weight Lift Appendix

Figure 3.9 Analog Computer Response to Simulated

Weight Lift Appendix

Figure 3.10 Electrical Analog of Electro-Mechanical

Transducer 44

F igure 3.11 Electrical Analog of Complete Seismometer 45

Figure 4.1 Electrical Analog of Galvanometer 50“

Figure 4.2 Test Circuits for Galvanometer 57

Figure 4.3 Test Circuits for Galvanometer 58'

Figure 5.1 Seismometer, T Pad, and Galvanometer 59

Figure 5.2 Direct-Coupled Seismometer and Galvanometer 64

Figure 6.1 Fourth Order Flat Velocity Appendix

Figure 6.2 "Flat Fourth" Order Response Curve

with Asymptotes 79

Figure 6.3 A Special Case, Amplitude-Frequency Response

Curve, Direct Coupled Instruments Appendix

Figure 6.4 Damping Required for Flat Amplitude with

Direct Coupled Instruments Appendix

Figure 6.5 Constant Amplitude Direct Coupled Instruments Appendix

Figure 7.1 Galvanometer as a Circuit Element 94

Figure 7.2 Seismometer and Galvanometer with

Notch Filter 96

Figure 7.3 A Possible Broad-Band Direct-Coupled System

Using a Galvanometer as a Microseism Filter Appendix

Figure 7.4 Equipment at the Vault and Recording Site 101

Figure 8.1 Equivalent Circuit Showing Generators of

Spurious Voltages 117

Figure 9.1 Determination of Motor Constants 138

Figure 9.2 Measurement of Ratio of Motor Constants 141

USCOMM-Ni-5-iJC
AN ANALYSIS OF INERTIAL

SEISMOMETER-GALVANOMETER COMBINATIONS

Section 1

INTRODUCTION

1.1 Background

Previous intei*est at the National Bureau of Standards in seismic

Instrumentation seems* to be limited entirely to the work of the late

Dr. Frank Wanner whose realization of the Interaction between an electro

magnetic inertial instrument and its associated galvanometer led him to

develop his horizontal instriment.

In the fall of 1954 the authors became interested in seismic, instru

mentation with particular interest in a vertical component device having

the least possible instrumental noise^ consistent' with minimijm weight,

with a pass-band of roughly 1 to 5 cps and designed for field use by

relatively unskilled personnel.

The study began with a theoretical and practical examination of the

system composed of a Benioff Variable Reluctance Seismometer and a d'Arsonval galvanometer and phototube amplifier. Alternative detection

methods were considered. It became apparent that an Inertial instrument

with an electromagnetic sensing element has an inherent instrumental

noise level which, at least in principle, is that due to the thermal

agitation of the inertial mass. It also became apparent that the instru

mental noise of a galvanometer can be as low as the thermal agitation of

its (Toil. We therefore concluded that the combination of an electromagnetic pickup and galvanometer has, in principle, a minimum noise level

which may perhaps be equalled by other position sensing arrangements but

no other arrangement can exceed this combination as to ultimate sensitivity.

1.2 Scope

In order properly to understand the scope and limitations of a theoretical treatment of the equations of motion for seismographgalvanometer combinations and the amplitude-frequency response curves

derived from them, the authors foxind it necessary to start from first
2

principles. Where assumptions have been made in the development there

has been an attempt to state them explicitly. }

I The reader will note that there is practically no reference to the

work of others in the main body of this report. The authors are not so | naive as to believe that all^ or even most, of the conclusions in this i report are new. The work was deliberately done without reference to the

literature in the hope that: 1) an Independent development would result

in a more complete understanding on our part than would result from a j study of the work of others, and 2) a fresh treatment, unbiased by the

lines of attack and assumptions used by others, might reveal relations

not previously noted. An obvious disadvantage is that another system of |i nomenclature, more or less unrelated to the many already in the litera- ^ ture, is introduced. However, an attempt has been made to Introduce only

symbols having physical significance and which are measurable, at least

in principle. The reader should not infer that our conclusions are, in j all cases, new to the art or that we are attempting to claim credit for

the work of others.

We have collected all the symbols used in the text together with

their definitions as the first part of Section 10. The second part of ' that section gives a few references to pertinent previous work with lists I of symbols used and their equivalents in out notation, where a correspondence was found.

It was tempting to include a discussion of each reference as it

applied to, or was confirmed by, this report, however such a development j would have increased the length of this report excessively and was there-*

fore abandoned.

t 1.3 Introductory Details t

Equations have been numbered using a "Dewey decimal system", the j; first digit signifying the section and the second the subsection in which

the equation is developed. ^

ij Throughout the course of the work we have noted relations which \[ could be used for evaluating specific instruments or which seemed especially useful to the designer of a seismic instrument or system. itj

I Despite the length of this report no attempt has been made to exhaust

the subject as there remain many interesting facets to be explored.

In numerical calculations based on the equations in this report the I] authors follow the Systeme International (MKS) since the practical elec- i trical units may be used with it directly. Use of the CGS system results J in the appearance of various conversion factors in unobvious places.
- 3

-

Section 2

GENERAL EQUATIONS OE MOTION OF M ELECTROMAGNETIC

INERTIAL SEISMOMETER

2.1 Dynamical Theory

2.1.1 Mechanics

A general formulation of Newtonian mechanics was made by Lagrange.

A modern treatment may be found in Whittaker. Clark Maxwell gives a statement in physicist's language.

A mechanical system can be regarded as a black box with push rods

each of which can move along a straight line. The number of push rods

is equal to the nuxmber of independent variables required to define the

configuration of the svstem. Tae equations of motion define relations

between the positions and velocities of the rods, and the forces applied

to them, without regard to the mechanism inside the black box.

Let q denote the position, q^,

iC the velocity, and the external force applied to the push rod numbered k. If the configuration at

any time is specified by the positions of the push rods, the motion can

be specified in terms of the velocities q^ . In Newtonian mecbanics,

the kinetic energy -T of the system is the sum of terms each of which

contains the square of a velocity q^^ , or a product of a pair of velocities q q , with a coefficient which may depend on the coordinates.

There may List within the black box a potential energy V which is a function of the coordinates. A- function (known as the Lagrangian

function) is defined as the difference between the kinetic and potential

energy, that is

= T - V ( 2 . 1 . 1)

Then the general equation of motion is.

6q

ir

'5k dt

or

ST

^

®“k

( 2 . 1 . 2)
- 4 -

As developed by Lagrange these equations apply to systems which are

conservative. Energy losses, such as those due to electrical resistance,

viscosity, hysteresis, etc., may be handled by introducing retarding

forces with the external forces.

The Lagrangian’ equations of motion do not apply directly to systems

which contain internal constraints, in which the configuration is not

uniquely defined by the positions of the push rods accessible from the

outside. A familiar example is that of a ball, which is constrained to

roll on a table, without sliding. If the ball is rolled around a closed

path, it may be returned to the starting point in an orientation differing from that at the start. Additional coordinates will be required to

specify the configuration. The constraints Impose relations between the } velocities consistent with the linkage. For example, the condition that j the ball must rotate as it moves so that there 1s no relative motion at

the point of contact, is expressed by two relations between translational

and rotational velocities. '

The' extension of Lagrange's equations' to such systems is discussed

by Whittaker in Article 87, Chapter VIII, essentially as follows: i

Each constraint within the system may be represented by a relation ' of the form: »

A^bqj^ + A^Sq^ ... + A^6t = 0 (2.1.3)

or

Aj^q^ + A2^2 ... + - 0 (2.1.3a)

The constraint may be associated with a set of additional forces

Q*i #

• Q'o which compel the system to move in a direction consistent i with the relations between velocities. These forces do no work in an ^ Instantaneous displacement consistent with the conditions of the con- L straint. That is: •

+ Q' 2^'12 •••'

Q’n^'^n “ ° (2.1.4)

j

!

for an instantaneous displacement (5t “ 0) such that ^

A,8qi + A^Bq^ ... + A^5q^ - 0 (2.1.5)
5

These eqixations are consistent if

( 2 . 1 . 6)

A sufficient set of equations of motion for the system consists of

the Lagrangian equations with the addition of the sec of forces Q'^ ^

Q' , etc. Equations 2.1.4, 2.1.5 and 2.1.6 represent the conditions on

the displacements and forces associated with each constraint.

2.1.2 Electrodynamics The extension of dynamical theory to electromagnetism.

Maxwell showed that the Information then known about electricity

and the magnetic fields due to currents could be brought into the scope

of the dynamical theory already developed for mechanics. The energy in

electrostatic fields is added to the mechanical potential energy so that

^ “ ^mech ^electrostatic (2.1.7)

electrostatic

d (Vol) ( 2 . 1 . 8)

all space

where ^ is the electric field strength and S) the electric displace'

ment.

The energy of a magnetic field is added to the mechanical kinetic

energy, that is:

T t- + T

mech mag

(2.1.9)

T

mag

d (Vo.l) ( 2 . 1 . 10)

all space
6 -

where H la the intensity of the magnetic field and ^ the magnetic

induction.

In setting up his field equations, Maxwell regarded the quantity of

electricity as a coordinate, the electrical current as a velocity, the

electromotive force as a force, and the magnetic vector potential as the

"electroklnetic momentum”.

When the attempt was made to apply these general methods to the

problem of the variable reluctance transducer difficulties were encountered. The similarities between electric and magnetic fields may trap

one into considering the field of a permanent magnet as the gradient of

a scalar potential function, and the energy of a permanent magnet as potential energy, and treating the forces in the magnetic gap on that

basis. The Inductive effects of a current were clearly kinetic in nature, but in analyzing this problem, the terms which gave rise to the

motor and generator constants were hybrid, neither kinetic nor potential,

and had no part in the Lagrangian theory.

Maxwell considered this matter at length and pointed out that the

magnetic field of a permanent magnet can be duplicated by a suitable distribution of electrical current. He then stated that a consistent theory

could not be obtained iinless the energy of all magnetic fields, whether

due to currents or to permanent magnets was regarded as a portion of the

kinetic energy.

The electric current which simulates a permanent magnet differs from

other electric currents in the pick-up since the magnitude is not subject

to arbitrary independent variation by application of an external emf.

It therefore corresponds to the angular velocity of a rolling ball which

cannot be varied independently of the translational velocity. It must

be treated as an internal constraint of the system, by methods such as those outlined above.

2,2 Choice of Coordinates

Lagrange attempted to set up equations which were valid regardless

of the parameters chosen as coordinates. That the end result is independent of the choice of coordinates constitutes a partial check of the

method. However, the coordinates must be independent and must be sufficient in number completely to define the configuration of the system.

It is easy to overlook Internal constraints, which are associated with

coordinates in excess of the number of degrees of freedom.

There remains considerable freedom in the choice of the coordinates

to be used in the general equations. Here the coordinates will be chosen

as to lead to equations in terms of quantities which are measurable, at

least in principle.
7

At least four coordinates with associated velocities, momenta and

forces are required to describe the electromagnetic seismometer.

2.2.1

The coordirate of earth motion

The seismometer as a whole moves as a result of the motion of its

: support. In use this motion should be that of the earth; in calibration

i it may be that of a shake table. For the vertical component instrument

j the coordinate y will be the displacement (positive upward) of the

(rigid) frame of the instrument with respect to a stationary point. As

a result of the change of y there will be a change in the gravitational

.potential energy V

gray

The mechanical kinetic energy of the system

will depend in part on the earth velocity y.

The force Q

=» Y is the force (positive upward) exerted on the

instrumen^t frame by its support.

2.2.2

The coordinate of bob motion

The mechanical coordinate x will be taken as’ the linear displacement of a point on the principal seismic mass or bob with respect to t’tie frame of the instrument, parallel to the motional axis of the instrument.

For a magnetically symmetrical Instrument it will be measured from the

point of symmetry. In a vertical component instrument it will be taken

as positive if the mass is displaced upward. The force acts on the

mass along the axis of the Instrument. It is taken as positive in the

direction to increase x. In addition to a retarding force there

may be forces X exerted on the bob by auxiliary calibrating devices,

or by extraneous disturbances. Hence Q = X - r x .

It is assumed throughout this work that r^ is a constant, i.e.

that retarding forces are proportional to the velocity. A small hysteresis loss will be replaced with a linear retarding force which will produce an equivalent energy loss. Ignoring the high order effects.

2.2.3

The electrical coordinate

In a coil connected to an external circuit the electrical coordinate

will be taken as the quantity of electricity. The current I^ will be

the generalized velocity and the electromotive force will be the

generalized force. As indicated by the names this is the conventional

choice for the electrical coordinate.
- 8 -

With the external emf will be Included a term - to

represent the effect of losses due to the electrical resistance of the

coils.

As stated above. Maxwell identifies the vector potential of the magnetic induction associated with a current as its "electroklnetlc momentum". When integrated over a coll its value is n<> where n is the

number of turns and the magnetic flux linked with the coll.

2.2.4 The magnetic coordinate

As mentioned before, the external magnetic field of the permanent

magnet may. be simulated by an electric current. For the sake of symmetry

this "phantom" current will be assumed to flow in a simulated coil

of the same number of turns n as the real coil associated with the

electrical coordinate. Sec. 2.2.3. The electromagnetic momentum of this

circuit will be n4> o

2.3 Condition of Constraint: Magnet

The magnetic coordinate differs from the electrical coordinate in

that there is no "push-rod" by which an arbitrary displacement can be

produced. In this respect it resembles the mechanical coordinates which

are associated with constraints.

The magnetic constraint may be found in the B-H curve of Fig. 2.1

which defines the relation between the induction and field in an element

of a magnet. During the process of charging, the magnetic material is

carried from the demagnetized state 0, along the curve 01. In the course of the stabilizing process a reverse field is applied to carry it

to the point 2, from which it "springs back" along the line 23. The

operating region is shown as the solid portion of the line. The operating curve is really a very narrow loop, but can be represented for our

purpose as a segment of the straight line 23 the equation of which is:

B

^

- (40) - H » 0 (2.3.1)

|i m

where B is the induction ana H is the magnetizing force in the ® m

o o aB

magnet. The slope of this line is the "reversible permeability"

m

of the magnetic material.
Hm

MA&MSnZING FORCE

Figure 2.1

Let the permanent magnet be simulated by. an electromagnet having

a coil wound on a core of the same external dimensions as the magnet.

For convenience, this coil will be taken as having the same number

of turns n as the coil connected to the external circuit. Let i

m

be the length and the cross-sectional area of the permanent magnet, Pq its pe'nneability, and be the” current in the coil, so that
10

the mean magnetomotive force per unit length of the magnet will be

4«nl —

—

+ H and the flux will be given by

/ m

m

4jtnlo

i

m

+ H

m A p. mo

(2.3.2)

Comparison with eq. 2.3.1 suggests setting

<t» » B A

o mm (2.3.3)

and

/»

4«nl = £ (40), a constant

o m

' (2.3.4)

The flux -4> and H depend on all the currents and on the diso m , placement x. Equation 2.3.4 shows that the simulating electromagnet

must have associate^ with it a potential E^‘ which varies with all the

currents and displacements in such a manner that is constant. This

is the condition of constraint. Since is the only current in eq.

2.3.4 will be the only supplementary emf required. The significance

of and |i^ are burled deep in the theory of ferromagnetism, which

is beyond the scope of this discussion.

This mathematical artifice may be applied to all the materials in

the magnetic structure so that a linear relation between B & H may be

said to hold. The assumption of zero hysteresis would be justified in a practical case if the user of a device observed negligible damping

chargeable to the magnetic material.

2.4 The Lagrangian Function

2.4.1 Mechanical kinetic energy

The velocity of any elementary mass dm of the seismometer may be

divided into an axial and a transverse component. The axial velocity is
11

the sum of the earth velocity y and some fraction, a, of the velocity

i of the bob relative to the instrument frame. The transverse velocity

is some fraction, b, of the velocity of the bob. In general, the coeffi

cients a and b are functions of x. The kinetic energy is then

mech

IMif n O j dm

(2.4.1)

• 2 .2

+ b^) dm + xy / adm + / dm

Let = / (a^ + b^) dm

M »= / adm (2.4.2)

M “ / dm.

y

An exact evaluation of these integrals requires a detailed study

of each piece of the Instrument and the nature of its attachment to all

other pieces^.

Then ^

"^mech

+ M xy +

xy

(2.4.3)

2.4.2 Electrokinetic and electropotential energy

The energy of the magnetic field is given by the Integral of eq.

2.1.10, over all space. In terms of the currents it can be put in the

fora:

The ratio of M ^ to M enters the equations for instrument performance. It may be made very nearly unity by careful design. This ratio

has been accounted for in the work of others by introducing a concept

termed "the reduced length".
12 -

.T

mag

W I- + W , I I- z oo o ol o 1 11 ^? (2.4.4)

Where W , W , and W, , aep^nd on the geometry of the magnetic clroo ol 11

cult and therefore may be functions of the coordinates.

An alternative expression Is. obtained by noting that the electromagnetic mcomenta are given by

n<> o W I oo o W ,1, ol 1

n4>. 5T

mag

SI,

W .1

ol o

(2.4.5)

(2.4.6)

J

E

With these substitutions, . the electromagnetic energy Is given by:

T

mag

I o o (2.4.7) i

Here the energy Is expressed In terms of the currents In the colls and I the flux linking each of them. I

I, Electrostatic energy, such as that associated with capacities in 1 the transducer. Is potential in form and ignored In this development. |

2.4.3 Gravitational potential energy ,

In the vertical component seismometer the change 6V in gravita-^

grav

tional potential energy due to small displacements 5x and 6y is equal

to the integral over the mass of the axial displacement of the mass due

to these displacements. ’

I
- 13 -

6V

gray / (a6x + 5y) gdm. (2.4.8)

Prom eq. 2.4.1 this may be Integrated over the mass to give:

5V

gray

gM 5x + gM 5y

xy y

(2.4.9)

Integration with respect to x and y gives;

V

gray

X y . (2.4.10)

The reader will note that here g is taken to be the constant upward

acceleration of the whole instrument required to replace the downward

attraction due to gravity.

2,4.4 Mechanical potential energy

The energy V is that stored in the elastic distortion of the pri3

mary and other springs. Since tlie springs connect, in one way or another,

the bob and the frame of the instrtnnent, their potential energy is

a function of x only. If they are linear the function is a quadratic.

If they are non-linear, other powers than the second will be required to

represent the potential energy.

In general, a practical instrument will respond to transverse and

rotational motions. It will also have multiple and distributed masses

and elastic elements. In a good mechanical design these effects are made

negligible. Their consideration at this time would lead to complications

of the analysis which would obscure the points to be developed. Therefore

we will restrict this discussion to a uniaxial device with a minimum number of degrees of freedom.

It should be repeated that the tractive forces of the permanent magnet enter the Lagrangian with the electrokinetic energy. These forces

are analogous to the centrifugal forces on the bobs of a governor or on

the arms of a spinning skater. They may resemble forces derived from the

potential energy in some respects, but must be treated as kinetic in nature. That is, the negative stiffness Introduced into an instrument by a magnetic structure does not appear in the potential energy terms of the

Lagrangian function.
14 -

2.5 The General Equations of Motion

It l 3 now possible to write the Lagranglan function of the Vertical

Seismometer.

M -i- M xy + ^ M

2 X xy 2 y'

+ “ W + W , 1. 1 + W, , if 2 oo o ol 1 o 2 11 1

• / M gdx - / M gdy - V

xy •' y° 3

(2.5.1)

Note^^that the Lagranglan function as written here contains cross

product terms between the mechanical velocities x and y and between

the currents I and I, , but no cross product between a mechanical

o 1'

velocity and an electrical current. Cross products between mechanical

velocities and electrical currents are associated with such subtleties

as the Hall effect, and the gyromagnetic effect. The first applies to

the current distribution within conductors. The second relates the

change in magnetization to a torque on the coll core. Such effects are

ignored in this development.

From eq. 2.3.4 the magnetic constraint is given by

I ** a constant,

o (2.5.2)

The general equations of motion are obtained by performing the

Lagranglan operation, eq. 2.1.2, on the Lagranglan function, eq. 2.5.1

to give:

Y (M X + M y) + gM dt xy y' ^ ^ y

(2.5.3)
15

d dV dM X-rx-~(Mic+M y)+gM +

X dt X xy" ^ xy dx 2 dx

(2.5.4)

- xy

dM 1 dW dW — XX - 1 t2 —2.2 . T T dx 2 o dx o 1 dx 2 dx

1-1 dw.

,

ol — ^

E, - R I = -j- (W .1 + W,,I.)

1 cl dt ol o 11 1' (2.5.5)

E

o ^ (W I dt oo o (2.5.6)

2.6 The Linearized Equations of Motion

The following manipulations will make these equations easier to

handle:

First: Let the rest point x = x^ be defined as the equilibrium

position of the bob in the absence of earth motion y. external forces

X and emf E^. Equation 2.5.4 then gives

gM +

xy

dV (x ) s o dx 2 ^ 2 odW (x ) 2 OO O

dx ( 2 . 6 . 1 )

The rest point may be regarded as fixed in considering the response

of the instrument to a signal. It will drift relatively slowly as the

ambient conditions change over intervals of hours or days. For small

excursions in the neighborhood of the rest point, squares, products, and

terms of higher power in the excursion, x - x^, the velocities x and

y and the current will be assumed small with respect to the linear

terms in these variables'.
16

Second: The quantities which are functions of x may be developed

as Taylor series in the neighborhood of the rest point x^. For example

dV (x ) (x - X d^V (x ) 9 O O 9 o' V (x) *= V (x ) + (x - X ) + r

= etc.

s

' s o o dx 2. ,2 dx

(2.6.2)

For the linearized equations terms as high as the second order will be

retained for V and W , and terms of the first order for M , W , a oo' xy’ ol

and W, ,

11 ' Third: In performing the indicated differentiation with respect to

time

dW dW

oo . oo

, note that —;— = x

dt

and

dW ,

. dW , ol . ol ax dt dx

Equations 2.5.3 through 2.5.6 now reduce to:

Y » M y .+ M X + M g y' xy y

(2.6.3)

X-M y=»M g +

xy xy

dV (x ) , dW (x ) S

' o' 1 .2 oo ^ o'

dx

“ I

2 o dx

+ M X + r X

X X

d^V , d^W^„ dM

s -1 ,2T OO XV 4. L dx'

2 0 dx' dx

(2.6.4)

dW

(x - X ) - I

°

1 . o o dx 1

dW

E - I ic + W-,i, -h R I.

1 odx 111 cl (2.6.5)
17 -

E

o

Io

dW

oo dx X + W T. ol 1

( 2 . 6 . 6 )

Equation 2.6.3 gives the force applied to the Instrument as a whole. It

would be used In considering the flexure of a pier, in shake table calibrations, or tests on an isolating block ("stable table").

Equation 2.6.6 defines the Internal behavior of the permanent magnet, or the back emf in the simulating coll.

Equations 2.6.4 and 2.6.5 define the relation between earth motion,

bob displacement and current in the seismometer.

Fourth: It is now possible to identify certain coefficients in eqs,

2.6.4 and 2.6.5 as the physical quantities S, G and L;

d^V

dx' — I^

2 ^o

d^W

oo dM

dx'

dx (2.6.7)

is the resultant spring stiffness of the system.

G Io

dW

ol

dx ( 2 . 6 . 8)

is the "motor constant" in eq. 2,6.4 and the "generator constant" in

eq. 2,6.6 and

L » W

11 (2.6.9)

is the coll inductance.

These quantities will be regarded as constants over the period of

any signal to which the seismometer responds. They are functions of the

rest point x^ and therefore change with any drift of the rest point.

After subtracting eq. 2.6.1 from 2.6.4 and substituting S, G and

L, eqs. 2.6.4 and 2.6.5 reduce to:
18 -

X

« « * * - M y xy

M X + r ic + S (x - X ) XX - GI

o

( 2 . 6 . 10)

E, - Gx + LI, + R I, . 1

• 1 c 1

( 2 . 6 . 11 )

2.7 Philosophical Notes and Interpretations

The identification of the coefficient as the self inductance

of the coil suggests an interpretation of and . The coefficient W may be interpreted as the self inductance of the coil simuoo lating the magnet. The coefficient may be interpreted as the

mutual inductance of these two coils. These interpretations are consistent with eqs. 2.4.5 and 2.4.6.

The motor "constant"^ G in eq. 2.6.10 is equal to the generator

"constant" G in eq. 2.6.11, but enters a term of opposite sign. In

the electrostatic transducer the corresponding terms are of the same

sign. The difference is associated with the difference in the way electrostatic and electromagnetic energies enter the Lagrangian function.

|

I Electrostatic energy is considered as potential energy, and is a function of the coordinates, conventionally displacement and charge.

j The motor constant would come from differentTation of the interaction ' energy with respect to displacement, and the generator constant from dif- j ferentiation with respect charge. Both_ are of the same sign.

As emphasized above, the electromagnetic energy enters the Lagrangian

with the kinetic energy. The motor constant comes from a differentiation [ with respect to x, and has a negative sign. The generator constant

' comes from a differentiation with respect to the current I and has a positive sign.

I 2.8 Extension to Two Coil Systems ;

2.8.1 General

I In a multicoil instrument the coils may be divided into groups con- p nected into separate electrical circuits. The division may be done in
19

several ways. (a) In a detailed analysis of a symmetrical magnetic

circuit it is necessary to consider the upper and lower cores and their

colls separately. (b) In service, there is the option of connecting

coils on the upper and lover cores in series, or in parallel. The relative merits of these connections must be considered. (c) Use has been

made of a damping circuit electrically distinct from the galvanometer

connection. (d) There is a possible method of reciprocity calibration,

based on measurements in which the coils are divided into two groups

which, for convenience, may be made as nearly alike as possible.

2,8.2 Equations of motion

For such arrangements the Lagrangian function may be set up as

before, with the obvious additional terms;

• P £ M x= X

+ M, xy + - M y'

+ I W I^ + W ,I I, + W--I I_

2 oo o ol o 1 02 o 2

+^ —IpW I +W II +—'IpW

2 11-^1 12

-^ l-^2 2 22-^2

( 2 . 8 . 1 )

- / M gdx - / M gdy - V

xy® > s

The following axixlliary quantities analogous to those of section 2.6

may be defined:

d^V

^ ^

S = - — I" + -• g is the resultant stiffness as before,

dx= 2 o dx »

I

o

dW , ol

dx is the motor-generator constant of coil 1
20

dWo2

G„ - I —:— is the motor-generator constant of coil 2,

2 o dx

is the self-inductance of coil 1 as before,

^2

" ^22 self-inductance of coil 2 ,

Mj

^2 “ mutual inductance between coil 1 and coll 2.

Equations 2.4.5 and 2.4.6 relating the flxixes <1) , in each

0 1.^ core with the currents I , , I., become:

o ^ 1

^ 2

n«}> = W I + W T + W

o oo o ol 1 o2 2

( 2 . 8 . 2)

' “ol^o “lin "'12I2 (2.8.3)

n<t., — W + W,

2 o2 o iZ 1 + W I 22 2 (2.8.4)

The linearized equations of motion become:

Y=M x+My + gM xy y . y

(2.8.5)

- m;x + r^x + S (k - x^) - - Gjij (2.8.6)

E, - G^x +

(2.S.7)
21

^2 “ ^

2^ ^12^1 ^2^2 ^2^2 ( 2 . 8 . 8)

dW

oo

(2.8.9)

2.8.3 Reciprocity calibration uslns the basic Instrument

Suppose the colls are divided into two groups as symmetrically as

possible. Then ~ ^1 ~ ^2 * mutual inductance be

measured with the mass /blocked, so that x =» 0 , and with sinusoidal

excitation, as by measuring the voltage on open circuit (I

2 = 0)

and currjint . Then, from eq. 2.8.8, with x = 0 and = 0

The experiment is then repeated with the mass free to move. From

eqs. 2.8.6 and 2.8.8, with “ 0 and y

«= 0:

«12 “ ( 2 . 8 . 10)

0 = (- + joor + S) (x - X ) - G, I,

' x"^x '^ o'^ 11 ( 2 . 8 . 11 )

E^ *= joiG2(x - x^) + ( 2 . 8 . 12)

whence

(- + S)

(2.8. 13)
22

or

GJG

CD CD 2

SM (-2 - —

)

X CD CD^ + r"

where cdo X

(2.8.13a)

Thus from measurements at various frequencies one can compute

r ^ 1^0 X ,12 S/M , x and

12 The mass can be computed from weigh ^ ’ X

M

X

ings. ^1’ 4’ and may be measured electrically.

If, in service, the two groups be connected in parallel, then t he

effective motor-generator constant G will very nearly equal

and the inductance L will very nearly equal + ^12^*

determines all the constants in the equations of motion.

If the two groups are connected in series, the effective motorgenerator constant, G^ ^2’ very nearly 2 and the inductance

will be approximately 2 + ^12^'

If it is inconvenient to divide the two colls into identical groups,

El

the ratio G /G is equal to the ratio —^ of open circuit voltages

1 L . when the bob is moved by mechanical means, as by a weight lift, or earth

|

motion. Given the product G^

quantities can be determined.

motion. Given the product G^G^ and the ratio ^

1 /^

2

’ individual
23 -

2.8.4 Reciprocity calibration using auxiliary calibrating coils

If tha rnstrinnent be equipped v/ith an auxiliary calibrating coil

and magnet then the nathod o£ calibration outlined in 2.8.3 is simplified since M. - is zero.

i.2

As presently used the motor constant of the calibrator coil where

one is used is generally given with the coil, rather than determined

during the calibration. However for a seismometer on open circuit Eq.

2.8.13 still applies.

2.9 Application to Other Electromagnetic Transducers

Although the equations of motion have been developed v/ith particular reference to the vertical component Benioff Seismometer, the results

may be applied to other electromagnetic transducers.

Equations 2.6.10 and 2.6.11 also apply to the d'Arsonval galvanometer, the coordinate x being regarded as an angle and M and M

. X xy

as moments and products of inertias. If. there is no m.agnetic material

in the armature, is a constant. Usually the inductance of the coil

is small enough to be neglected. The d'Arsonval galvanometer is considered in detail in Section 4.

In the dynamic loudspeaker, if x is the displacement of the coil,

X is the force with which the air column reacts on the coil. Here also

W is usually constant, and the inductance of the coil may often be

neglected.

The treatment of Section 2.8 applies to the differential transforme

in which case I^ is the current in the exciting coil. The self induct

ance of the exciting coil is W , the mutual inductances between the ° oo exciting coil and the pick-up coils are and •

In some of the above applications the neglect of electropotential

energy (Sec. 2.4.2) may not be valid.
24 -

Section 3

RESPONSE CHARACTERISTICS OF A SEISMOMETER

WITH A RESISTIVE LOAD

3.1 Introduction

The cver-all response characteristics of the seismometer are complicated by a number of minor parameters which can obscure an understanding of the behavior of the instrument. For this reason a number of

special cases will be discussed. In the special cases non-linearities

and viscous damping will be neglected. When a factor which is normally

small becomes significant (as in strong resonances) its effect will be

noted. A qualitative discussion of the effects of the minor parameters

is included in one of the more general cases. In general the mass M^,

relating to the kinetic energy of the bob, differs from M , relating

xy

to the potential energy of the bob, because the masses of the spring and

other suspension members are involved differently for motion and position. In many practical instruments M^ and M differ by a fraction

of a percent. Here they will be taken as equal and be represented by M.

If the distinction need be made,' the correction factor M /m may be

- xy' X

applied to the earth motion y. The rest point x^ in the expression

(x - x^) will be suppressed, with the realization that the parameters of

the system may be functions of the rest point.

The more important parameters of the seismometer are: the motorgenerator constant G, the mass M of the bob, the resultant spring

stiffness S, and the inductance L and resistance R of the electrical

circuit.

3.2 Seismometer With Resistive Load

The equations of motion, as developed in Section 2.6 (Eq. 2.6.10

and 2.6.11) may be written as:

X M + rX

dx - dt

+ Sx GI (3.2.1)

E L

dt

+• RI (3.2.2)
25 -

where I is the current in the electrical circuit. It is taken as positive when positive current flows from the external circuit

into the Instrument et the positive terminal.

L is the inductance of the circuit, including the coil and any

external series inductance.

R is the resistance of the circuit, including the coil and any

external resistance.

In the general discussion of Section 2, was defined as the

.potential difference measured at the terminals of the seismometer coil,

and R and L were the resistance and inductance of the coil. If the

c seismometer is connected in series with a pure resistance, (or a resistance and inductance in series), these external impedances can be combined

with tho^e of the coil. When the external circiiit is closed, then E

is the voltage across such zero impedance generators as may be in the

external circuit. The other symbols have already been defined.

Before proceeding with the solution of these equations, it will be

convenient to introduce the following quantities:

impedance at the open circuit frequency.

7 = G/ n/Ts - G/o>^ 'flM is the normalized motorgenerator constant.

T = oi t is the normalized time variable. When

o measured in these units, the open circuit angular frequency of the seismometer is one or the

open circuit period is 2rf.

D = d/dr = d/co^dt is the normalized time derivative operator.

vS/M is the angular frequency of the seismometer on open circuit, as previously defined in

paragraph 2.8.3

f = normalized frequency

p = R/Lo) is the ratio of resistive to inductive

o
26

P

X

r /biJD is the normalized internal damping

x' o coefficient.

« 9-2 ^ this will be shown to be the "short

s circuit frequency".

With these substitutions, the equations of motion can be put into

normalized form:

|-D^= (D^ + pD + 1) X - 7I 'Jljs (3.2.3)

o X

E/cd^L = TDx 4T/l + (D + p) I (3.2.4)

The quantity l/S is the ratio, in consistent units, of the energy

stored in the inductance by unit current, to the energy stored in the

spring system by unit displacement. These equations may be solved for

the current I by eliminating the bob displacement x, neglecting the

mechanical damping, i.e. p^ = 0, to give

yD (D^y - X/S) + (D^ + 1)

CO

(3.2.5)

[D^ + PD^ + f^ D + p] I s/l/S

Mathematically, 7 and p are independent variables and may, at

least in principle,' take on any values whatsoever, however, as indicated

below, the instrumental adjustments available to the user of a vertical

component Benioff variable reluctance seismometer affect both parameters.

An increase of tension in the horizontal tension strips increases

co^ and therefore reduces both 7 and p in the same proportion.
27

An Increase of magnet charge, without other change, will increase

y since the generator constant will be increased and the effective

spring stiffness will be decreased. At the same time p will be increased since oj is decreased,

o

The user may, within limits, change 7 without changing p by

(a) simultaneously changing the magnet charge and horizontal tension

strips to keep the free period of the instrument constant or (b) simultaneously changing the horizontal tension strips and external resistance

to keep the ratio of total circuit resistance to free period constant.

There is a minimum value of p, nearly independent of coil configuration, which for this instrument is about 1.5 times the free period in

seconds. This valtie of p corresponds to zero external resistance and

is increased by increasing the external resistance. On the other hand,

any value of 7 is nearly independent of coil configuration and is completely independent of external resistance.

3.2.1 Steady state response

3. 2. 1.1 Steady state response to earth motion, shake table.

Let the instruraent be connected, to a resistive load, and subjected

to sinusoidal excitation at angular frequency oj, as by earth motion or by

a shake table with displacement y. Let X and E both be zero. Replacing the operator D by jf, Eq. 3.2.5 becomes:

7f^y = L(f^ - f?) f - jP (f^ * i)l In/lTs (3.2.6)

S T,

The phase angle by which the current leads the earth motion is

given by

tan

y

=

p

“ 1)

f (f2 - fj)

The ratio of the absolute magnitudes of the current and the earth displacement is given by

V S_/'L ^ ^ (f"
28 -

The Nyqulst diagram, the phase- frequency curve, and the amplitudefrequency curve for the steady state response can be plotted from these

equations. These equations have the following properties:

(a) At the open circuit frequency f “ 1,

1

CD OJ 1 ^/I7i and the current is 180“ out of

y

phase with the earth displacement, or in phase with the

acceleration;

(b) At low frequencies f « 1, — vL/S | ^

q> - j “ f • That is, as the frequency is decreased the phase angle

approaches 270°. Its cotangent approaches (f^f/p);

(c) At the frequency f + 1 )

- n/l/S

y CD = CO = j

2 3/2 - V. = if /p^ the current

p6 -'s' or

is in phase with the velocity;

(d) On the high frequency asymptote, ^

-Jl/S 7

CD -> 00 the phase angle approaches zero and its tangent

approaches p/f .

It should be noted that at- high frequencies and at f = 1, the . seismometer has an output current which is independent of p, and therefore of the load resistance. Accordingly, the instrument can be regarded

as a high impedance generator, or a constant current source, at these

frequencies.

At low frequencies and at f = f^ the output current is inversely

proportional to p or to the circuit resistance R. At these frequencies the seismometer can be regarded as approximating a low impedance

generator, or a’ constant voltage source.

So far, the mechanical losses have been neglected (p^ = 0). This

approximation is not valid if the electrical losses are comparably small,

as occurs when p is very small or very large. The general expression

for p^ and p of the same order is messy, but the two limiting cases,

p « 0 and p = 00 will illustrate the point.
29

For the case of short circuited superconducting coils p

•=> 0,

E = 0. Equation 3.2.4 then gives:

7 Dx'X Vs/l -i- DI = 0 whence I « - yx n/s/l

substituting for I in Eq. 3.2.3 then gives;

I

- D^y = (D^ + p^D + 1 + 7^) x (3.2.7)

showing a strong resonance at o) ~ od vl + 7^ or f ~ f , For this reao s son f is called the "short circuit frequency".

s

f may also be thought of as that frequency at which the spring3 mass combination reflects into the electrical circuit as a capacitative

reactance (cf. paragraph 3.2,4) which is in series resonance with the

electrical inductance of the coil. It is easily measured^ cf. paragraph

3.2.1.3(c) and Figs. 4. 2 and 4.3.

On open circuit p

»= 00 and therefore the current is zero. Equation

3.2.3, with 1=0 shows a strong resonance in the motion of the bob at or a)C;£. CD^', that is x will be large. Then, from 3.2.4, the

open circuit voltage E will also be very large at f ~ 1. This behavior is consistent with the definition of o)^' as the open circuit angular

frequency.

3. 2. 1.2 Steady state response to bob force, calibrator

Using an auxiliary coil and magnet, not linked electrically or magnetically to the reluctance transducer, or a light spring and eccentric

it is possible to apply a controlled variable force directly to the bob

of the seismometer. Such arrangements are used as remote calibrators

since the force applied to the bob may be made quite linearly proportional

to the current in the auxiliary coil or to the total indicated runout of

the eccentric.

For this case y and E are both zero. Assuming sinusoidal excitation the operator D in £q. 3.2.5 may be replaced by jf to give Eq.

3.2.8, wherein X becomes tne product of the motor constant of the calibrator and the calibrating current cr the change in force applied by the

calibrator spring.
30 -

yt X/S - [f (f^ - ff) - JP (f^ - 1)] In/l/s (3.2.8)

The phase angle by which the seismometer current leads force exerted

by the calibrator is given by

tan „

p - 1), f (f^ - ff)

that is, the phase relations are the same for excitation by a calibrator

as for earth motion.

The absolute magnitude of the ratio of the seismometer current and

calibrator force is given by

'Ll

sTsl [f^ (f^ - (f^

From these equations the Nyquist diagram, the phase- frequency curve,

and the amplitude- frequency curve for the steady state response can be

plotted. These equations have the following properties;

(a)

(b)

At the open circuit frequency f » 1,

- 1

CO *= CJO and the seismometer current is 180'

o ^ "/sL

out of phase with the calibrator force.

At low frequencies f « 1, — — ' X ' CO - 0

phase angle approaches 270°.

At the frequency f »= f = v/7^~+~l, ^

^ and the

And the current is in quadrature with the force.

(c) jfg/p7 JIl .
31

(d) On the high frequency asymptote.

I ^

G

and its tangent approaches

The phase angle approaches zero

)/f.

While this treatment has neglected mechanical losses (o assumed

X

zero) it will be essentially correct at any reasonably small values of

Px3. 2. 1.3 Steady state response to electrical excitation

Let the seismometejr be excited by a sinusoidal voltage applied in

the electrical network, with no ground motion or externally applied

forces on the bob. Then, setting D = jf, Eq. 3.2.5 takes the form:

A

o o

f 7

= R + jfaj L (-^

) (3.2.9)

• • i - f .

By inspection, it can be seen that:

(a) At f « 1, measurement of the applied voltage and

resulting current determines the parameter R.

(b) In the neighborhood of f = 1 the relative phase of

applied voltage and resulting current changes very

rapidly.

(c) At f = f the applied voltage and resulting current

s are in phase.

(d) At f » f^- the ratio of the applied voltage and.

resulting current is dominated by the inductance.

The above treatment assumes p * 0; however, when a seismometer is

operated so that « p « oo measurements made as suggested by these

conclusions should adequately determine all significant parameters of

the seisaiometer. The equipment required would include a u-c olmmicter, a signal generator, an oscilloscope, and an impedance bridge operating at

a frequency well above f^.
32

3.2.2 Transient response

So far, this chapter has considered only the response of the

seismometer to sinusoidal excitation, i.e., particular solutions of Eq.

3.2.5. In the general solution additional terms, containing arbitrary

constants, represent the transient response of the instrument to changes

in the excitation. With step function excitation, as by sudden removal

of a small weight from the bob, the transient terins represent the whole

solution.

The transient response (or complementary function) is the sum of

terms each of which is a solution of the differential Eq. 3.2.5 with the

forcing function (left hand side) set to zero, i.e.,

0 » + PD^ + (7^ + 1) D + p] I n/l7s (3.2.10)

It can be verified by repeated differentiation and substitution in Eq.

3.2.10 that a term of the form

Be , or Be o

is a solution, provided that p is a root of the algebraic equation:

0 «= - + pP^ - (7^ + 1 ) p + p . (3.2.10a)

The most general solution of a linear differential equation of the

third order will contain three such exponential terms, with three independent constants B^, B^, and B^ whose values are chosen to fit the

Initial or other boundary conditions.

Numerical evaluations of p, for various values of p and 7, to

obtain solutions for Eq. 3.2.10 are readily performed using a nomogram.

Fig. 3.1, constructed as follows:

Equation 3.2.10a may be rearranged to read

7^p

P^ + 1 p *= P + (3.2.10b)
- 33

Let W = — (p - p) (3.2. 10c)

Then W «= (3.2. lOd)

+ 1

In Fig. 3*1. a W is plotted as ordinate against p as abscissa.

Eqxxation 3.2.10c gives a family of straight lines [such as (a)^ (b),

(c), (d) ] each member of v/hich represents a unique pair of values for

7 and p. For each member the intercept on the abscissa is p and the

slope is - 1 /t^ .

The curve on Fig3*l,ais a plot of Eq. 3.2.i0d which^ it will be

noted, is independent of both 7 and p.

The intersections of the curve and each member of the family of

straight lines give the real values of p which are solutions of Eq.

3.2.10a and therefore Eq. 3.2.10.

For any given instrument and coil configuration, 7 is dependent

almost entirely on magnet charge. The weaker the magnet the greater the

slope of an appropriate straight line drawn in Fig. 3*1, a while for increasing magnet strength the straight line approaches more nearly the

horizontal.

For any given instrument and coil configuration, p is linearly

related to the resistance external to the instrument: that is, as the

external resistance is changed the straight line is translated parallel

to itself. Increased external resistance represents a position further

from the origin.

In order that the transient response of the instrument shall be

aperiodic, i.e., contain no periodic term, it is necessary and sufficient

that the three roots of Eq. 3.2.10 be real. Such a condition is shown on Fig.3*l,ahy line (a). Note, that for this value of 7 there is a minimum

value of p, or- of the circuit resistance, below which there is but one

intersection with the curve and therefore only one real root of Eq.

3,2.10a. The other two roots are complex and the transient response of

the system contains a damped periodic term. This value of p corresponds to (b) of Fig. 3*1. a. There is also a maximum value of p for this

value of 7, above which the transient response contains a periodic term.

This boundary corresponds to line (c) of Fig. 3.1. a As the magnet charge

of an instrument is lowered (7 is lowered) the range of p for which

Eq. 3.2.10a has three real roots is decreased and there is a minimum
- 34 -

I

I

I

value below which only one aperiodic solution for Eq. 3.2.10 can be found.

y

The corresponding straight line is tangent to the curve at the Inflection y point (curve d of Fig. .

^

If ^ ^ , and are the roots of Eq. 3.2.10a the operator on 1^2 3

the right hand side of Eq. 3.2.10 may be rewritten in factored form as

(D + pp (D + P

2

> (D + ^

3

) (3.2.11a)

or

(3.2.11b)

Comparison of coefficients between 3.2.11b and 3.2.10 shows that

p = Pt + P + = P, P«P l'"2'"3 (3.2.11c)

and

7^ + 1 = Pj^P2 + ^2^3 ^3^1 (3. 2. lid)

It may be convenient (as for a check on calculations) to use the relation 1

(3. 2. lie)

I

?

Using p and f^ as coordinates, Fig. 3.2 presents the field of varia- I

tion of the parameters of the system, divided into regions on the basis

of the character of the transient response. Combinations of parameters

are marked to correspond to the lines in the nomogram (Fig. 3.1a) and to ;
response curves in Figs. 3.3 through 3.9, the preparation of which is

discussed in Section 3.2.3.

3. 2. 2.1 Solution for the weight-lift calibration

Since the arbitrary constants B

2

and B^ of the transient

terms depend entirely upon the boundary (usually initial) conditions,

the general method will be illustrated by a detailed evaluation of the

seismometer response to the step function frequently used for calibration, sometimes referred to as calibration by "weight-lift".

For this calibration, the initial conditions are:

at t = 0, I 0, x = mg;

t > 0, X = 0:

where m is the mass suddenly removed from the bob at t = 0. For this

case we take y

= 0 and E = 0, with assumed «p. These boundary

conditions substituted in Eq. 3.2.2 give the further condition that at

t ^ 0, dl/dt = 0 and, when substituted in Eq. 3.2.1 for t < 0,

- mg = Sx or X = - ^

For 0 = y

= E at all values of t, and X = 0 for t > 0 Eq. 3.2.1

becomes

0 + Sx GI

At t *» 0 and by application of the boundary conditions.
36 -

Dlfferentiacing 3.2.2 with respect to time under the above

restrictions

d^i

dt

and substituting the above boundary conditions it follows that.

at t 0 , dfl ^ Gmg

dt‘

LM

Now, from the constants of the instrumental set-up, f^ and p

may be evaluated. Using the nomogram. Fig. 3.1a, and possibly

and are evaluated. Assuming that ^

3

* are real and

distinct,

t ’P^ t

‘ -P-O) t then I = B^e + 826 + B^e (3.2.12)

is a solution of Eq. 3.2.10. Knowing from the boundary conditions that

at t *= 0, I = 0 it follows that

0 = Bj^ + B^ + B^

Taking the time derivative of Eq. 3.2.12 and knowing that at t = 0,

dl /dt =0 it follows that

0 =

^1 Bi + B2 + Pj B3

The second time derivative of Eq. 3.2.12 is, by the boundary conditions, equal to - mg G/ML from which

- mg G/m1io^ = P^ B^ + PJ‘b2 + P

3

B

3
37 -

By algebraic manipulation of these three equations B

2

and

may be expressed in terms of ^

3

* Substitution in Eq.

3.2.12 gives

<^2 - ^3> ^ ° <Pl - P2> ^

' “ «%1L (Pi - P2> (P2 - P

3

> (P3 - Pl>

o (3.2.13)

which is the desired relation between current and time, for the weightlift calibration.

It may happen that the calculated values of f and p are approS prlate to a boundary of the aperiodic region; i.e., that two roots of

the polynomial, say and are equal. In the nomogram (Fig. 3.1a)

the straight line is tangent to the curve (lines b and c). The solution

for the current is the limit of Eq. 3.2.13 as ^2 ^ ^

2

’ Equation 3.2.13

with ^2 ^3 takes the indeterminate form O/O.

The desired equation may be obtained either by the above process of

evaluating B^, B

2

and B^ in a solution of the form

-p,ao t ^ . •, ^ T. lo.T, 2o,„ ^ z o

I = B,e + B«e + B^o) te

1 2 Jo

-P2®ct

or by obtaining the limit of Eq. 3.2.13 as P3

« Either process

leads 'to

mgG

r" ^ ^ t~,'

I t -p 03 t '^2 o ! 2 o 1 o 03 te

- e MW

:L (Pi - P2>^

+

02 - pp j

(3.2.14)

On the left (low resistance) side of the aperiodic region (Fig. 3.2),

right (high resistance) side of the aperiodic region

Pi > .
38 -

At the downward pointing cusp of the boundary of the aperiodic

region (p = , f^ - 3) the three roots are equal

( 3

^ 3^ 3^ -v3) and the solution reduces to:

I 2ML

OJ t o (3.2.15)

Again note that the condition for three equal roots can only be obtained

for a particular value of ^ (magnet charge and period) and a particular

value of p (resistance, inductance, and period).

» Outside the aperiodic region, two of the roots are complex conjugates. There will always be one real root of the cubic, obtainable from

Fig. 3. fa. The complex roots may be written as cr + jv and P - jv , and evaluated from Eq. 3.2.11c, d and/or e, or estimated from Fig.

3.1b.

The solution may be written:

Bie

-OOi

+ e

t / jVU3 t

> B,e

°

- jvoi

+ 838

or

I = B.e + e (Bl cos voi t + Bi sin vco t)

1 2 o 3 o

Evaluation of B^, B

2

and B^ from the boundary conditions as before gives the' equation for I as a function of t which may be

written:

-Pl^o^t -oui t o

e cos voj’^o ^) ^^1 '

-OUi t o

e sin voj t o OJ-ML

o

V - cr)^ + v'

(3.2.16)
39

3. 2. 2. 2 Transient response boundaries, loci and damping

The boundary between periodic and aperiodic response has been

discussed earlier in Section 3. 2. 2.1 and is shown on Fig. 3.2.

The value ® 1 is also a boundary of the periodic region (corresponding to zero magnet charge). In principle the periodic region is

also bounded by p

>= zero, although for a passive system the resistance

of the colls sets a minimum value of p (about 1.5 in the Benioff variable reluctance instrument).

Another boundary of interest on Fig. 3.2 is the dotted line extending toward the origin from the cusp of the aperiodic region. This line

is the locus of those values of f and p for which 6, = a, or

s 1

’

p = 3^j^, and o2 + = 3. For this special case Eq. 3.2.16 may be

written

t - mgG -2 ^ ^

I = — V e (1 - cos vq t)

oXi o

This equation has the interesting property that for ail positive

values of t the current I is < zero. The negative sign of the

current arose in setting up the- Lagrangian equations of motion when Hq.

2.4.4 for the magnetic energy was written with a positive sign for the

cross-product term. The point at issue is that a plot of current vs

time will show no zero crossings. It will be a damped sinusoid, tangent

to zero for all values of which are multiples of 2s:, c. f . Fig.

3.3, curves 2. For values of and p above and to the left of the

dotted line in Fig. 3.2 there will be no zero crossings of the current

resulting from a weight-lift test (c. f. Fig. 3.7, cumves 10) while below

and to the right, zero crossings will be observed.

Outside the aperiodic region the current resulting ftov;'. a woich;.-

lift test, as a function of time, may be looked upon as the sum of an exponentially dccayir.g current and a damped oscillatory current, Ttiis

may be shown by rewriting Eq, 3.2.16 as:
- 40 -

SL

/ -P,uj t -tJ:o t ha ^ ° ° [0 COB - 0(P^ - a)^ +v'

. ypt] ]t (3.2.16a)

( J

which i3 of the form

‘total

I + I exp 03C

While the concept of damping as applied to Eq. 3.2.16 is somewhat

obscure^ it is feasible to consider the decrement of the oscillatory

term as reflected in the excursions of the total current above and below

its exponential component.

The current I will be a maximum at some time t, , it will again

osc 1^

be a maximum at t„ where voi (t„ - t.) * 2«.

2. O i. L

The ratio of the contributions of the oscillatory term at times

t^ and t^ is:

*:h—

2m/v “ e = e

' ^2

and is the decrement of the oscillatory term, or

In decrement = 2jtcj/v

The cuTTve on Fig. 3.2 enclosing the aperiodic region is the locus

of values of f^ and p for which a = v; for the oscillatory term the

logarithmic decrement = 2rt, the decrement is 535 and the "overshoot" is

1/ v^535 or 4. 1% .

Obviously this line is only one of the family of loci of constant

decrement. For decrements greater than 535 the curves hug the aperiodic

region more closely than the curve shown, while for lesser decrements the curves are nearer the axes of the graph.
- 41

From this it follows that, having selected a mass, magnet charge,

coil configuration and open circuit angular frequency (predetermined

0)^), adjustment of the load resistor to a predetermined decrement may

occur at two values of external resistor (two values of p). It may well

happen that the lower value of p is not physically realizable with a passive external resistor, since the resistance of the instrument coils

sets a lower limit to the obtainable value of p.

If the magnet charge be reduced (as for instance to increase the

"linear" range in the gap of a variable reluctance instrument) then a preselected decrement may not be obtainable by adjustment of the external

resistor* As f decreases, the maximum obtainable decrement also des

^ creases. The locus of this maximum is indicated on Fig. 3.2 as "maximum

damping at constant

From Eq. 3.2.11c, remembering that =* d + jv and = c - yy

Using Eq. 3.2.10b to eliminate p gives

pv (f; - i)‘

2 , 2

* C + V + i) + fg)

When a/v is an extreme. will also be an extreme. Sin

at constant f , an extreme of a/v (maximum or minimum) as p varies

3 is given by ^ (a/v)/Ap = 0 it follows that h

o = C

and since p . is monotonically related to the single real root, ::u of

Eq. 3.2.10a in the periodic region it follows that ^ [ a'^/a^ + (nT)

/ 4 2 3

=» 0. Therefore, a/v is an extreme when = f _ or p = ano

P = f

3/2

is the locus of maximum decrement in the region f 3.
42

The value of the decrement on thi3 locus is obtained from

(y/v) max

r 4

1

3.2.3 Transient response curves, analog computer

In order to make more vivid the curves of currant vs time after a weight-lift^ when a seismometer is operated in the various regions of

Fig. 3.2, an electronic analog computer was wired to solve Eqs. 3.2.1

and 3.2.2.

The curves. Figs. 3.3 through 3.9 show.-the results of weight lifts

for the conditions given in Table 3.1. The numbered crosses on Fig. 3.2

show the appropriate location on this diagram for each group of curves.
- 43 -

Table 3.1

Constants for Figures 3.3 through 3.9

Benioff Variable Reluctance Seismometer

For all curves;

The upper coils are in parallel

The lower coils are in parallel

The two groups of coils are in series

The instrument resistance is 62.5 ohms

The circuit inductance is 6.8 henries

The mechanical damping, r^, is assumed zero

The open circuit frequency is 1 cps

The mass of the bob is 1C7.5 kg

The full scale deflection, for the bob displacement

is 2.4 microns (one gm lifted from 107.5 kg),

for bob velocity 15 microns per second, for

current /'v/ 30 microamperes

Approx. Field

in gap 7^ f

s P

Remarks

see

Fig. 3.2

External

Resistance

ohms

Fig. and curve

650 3 2 1.5 1.3 3.3 (1)

2. 12 27.6 (2)

2.8 Max. 56 (3)

damping

6 192 (4)

1060 8 3 1.5 1.3 3.4 (5)

4.28

N

. = y 119 (6)

5.2 Aperiodic 158 3.5 (7)

6.3 CT = i) 205 (8)

10 3 . 6 (^3)

1450 15 4 1.5 1.3 3. 7 (iO)

5.67 178 (11)

8.0 Middle 111 3.8 (12)

aperiodic

11.48

A A- =S 0 (13)

20 787 3.9 (14)

50 2060 (15)
- 44 -

3.2.4 Electrical analog of a seismometer

Equation 3.2.1 may be rewritten as

M d^x dx , ^ j. d^y

G ^^2 G dt G G G ^^2

or, taking e = G

T * il_ -1 / . ^ dv. , X , S r (e + G -^) + — e + — f --i (3.2.19)

• > It will be noted that the first term on the right side of Eq. 3.2.19

is mathematically identical to the current flowing through a capacitance,

= m/g^, in series with a zero impedance generator of voltage, G dy/dt,

when subjected to a voltage e. The second term may represent the current

flowing through a resistance R^, of magnitude G^/r^, when subjected to

a voltage e. The third term is appropriate to the current flowing through

an inductance, = G^/S, as a result of the voltage e. The fourth term

may be considered as representing the current output, i, of an infinite

impedance (constant current) generator, i = X/g.

Therefore, Eq. 3.2.19 is a valid expression for the currents in the

following network:

Fig. 3.10
- 45 -

and that, insofar as the mechanical portions of the seismometer influence

the current flow in its electrical circuit, the mechanical portions may

be replaced by the above circuit identically, with the directions of current flow and generator potentials taken positive as indicated.

As defined above, e is the open circuit voltage at the terminals of

the seismometer so that, if the whole instrument be replaced by the following equivalent circuit Eq. 3.2.2 will apply in addition to Eq. 3.2.1.

Fig. 3.11

Here is the potential' difference at the terminals of the seismometer.
46

Section 4

D'ARSONVAL GALVANOMETER CONSIDERATIONS

4. 1 General

With the possible exception of several methods for measuring the

critical damping resistance of a galvanometer (subsection 4.4) the material of this chapter is better developed in such standard works as "Electrical Measurements" by Forest K. Harris. It is included here to

show that the usual treatment of the galvanometer represents the special

case of zero inductance in the development of Sections 2 and 3, to

develop the galvanometer nomenclature used throughout the remainder of

this report and to make the mathematical development of this report rea- sonably independent of specific literature references.

The derivations of Sections 2 (cf. subsection 2.9) and 3 are as valid for the d'Arsonval galvanometer as they are for electromagnetic

Inertial seismometers, provided the obvious substitutions of angular

coordinates be made for the corresponding linear coordinates.

The basic equations corresponding to Eq. 3.2.1 and 3.2.2 are,

therefore:

6 = mechanical torque applied directly to the galvanometer coil,

9 ~ angular position of the coil relative to the case,

T) = angular position of the galvanometer case, with

respect to a "fixed" direction in space.

4.2 D'Arsonval Galvanometer, Equations of Motion

(4.2.1)

and

(4.2.2)

where
47 -

K = moment of inertia of the coil,

r =» mechanical (air) damping "constant”,

U

U = stiffne^^s of the coil suspension,

= galvanometer motor/generator constant,

= current in the galvanometer coil, .

= the potential difference at the terminals of the

galvanometer circuit,

= the electrical inductance of the galvanometer coil

plus series inductance (if any),

- the electrical resistance of the galvanometer coil

plus series resistance (if any), in any real case

this will include the resistance of the suspensions.

As a galvanometer is normally used, the equations of motion, insofar

as they relate the terminal voltage and/or current to angular position

of the case or to mechanical torque applied to the. coil, are generally

of little interest. In special cases, a mechanical torque may be applied

to the coil either deliberately or inadvertently, and the effect of 0

must be considered. Some sources of torque will be discussed in Section

8, Limitations in Principle and Practice.

The motor/generator constant, G^, should not be confused with the

"current sensitivity" usually given in catalogues. The latter is, by

convention, related to u/G^.

Equations 4.2.1 and 4.2.2 may be combined to eliminate I^. With

, \ and CO S g g

as

(fi = Gf/2co K, X ^ cf/2R..co K and co = v/u7k

g 3'g'g 3^og g

^

and, L« presumed zero, q constant, and defining^

the result may be written

v/2X CO E„/ vTT =

g g

3^ 3

Jif .1

!_dt-

+ (' + 2X ) CO •J7 + co^

CO K g g dt g| O

Q sIk (4.2.3)
48

It should be pointed out that, given an ideal instrument, i.e., one with no mechanical losses,^ is the critical damping resistance (CDRX > o

! plus coil resistance) which one would measure by determining, for in- ; stance, the maximum circuit resistance adequate to suppression of overshoot'

of the coil whan excited by square waves at a frequency which is low com- > pared to U) . Such measurements on a real instrument will lead to a

o I higher value of circuit resistance than(R/ because the coil motion ‘ * I will be critically damped when r^/2w^K is, when '

plus the fraction of critical damping represented by mechanical losses

equals unity.

4.3 Galvanometer With a Resistive Load

4.3.1 Transient response

In Fig. 3.2 the region of greatest mathematical interest is that ‘ lying around the cusp of the curve bounding the aperiodic region. This ^ is also the region in which the variable reluctance pick-up seismometer

j is generally operated^ as a compromise between linearity in the gap,

generator constant, generator resistance -and generator inductance. In

the d'Arsonval galvanometer somewhat different compromises result in a generator/motor with a much lower inductance and, often a relatively j higher internal resistance^. That is, the value of f for the galva- i ® l: nometer is, in general, much higher than for the seismometer; as is p, j for a given value of circuit res.istance; both reflecting the lowered

value of coil inductance. Consider the ratio

f^ - 1 G? Lo) G? /Q _3 _3_ S = l_

2p 2LU * R 2Kco R R

8 s

This quantity Oi, /R is the ratio of the electrodynamic damping to crit- • I 8

ical. (This appears as the quantity

galvanometer by Harris, cf p.59.)

7-7 in the discussion of the

In Section 3, Fig. 3.8 and 3.9, the curves of groups 13, 14, and 15 are

seen to resemble the response of a galvanometer for X = 0.65, 0.375,

^ j and 0.15 respectively. The most conspicuous difference is the rounding '

These remarks are not applicable to the Thompson moving magnet galvanometer.
- 49 -

of the initial break. For larger values of f the resemblance would be

s even closer. For instance, while a reluctance pick-up may be operated

with f approaching 3 and with p as low as 1.5, a high sensitivity

s d'Arsonval galvanometer will have f 600 and a minimum obtainable value

5

® of P Pi 10 . These values are well beyond the upper right corner of Fig.

3.2, where the difference between the third order differential equation

and the second order is very slight.

4.3.2 Response to sinusoidal electrical excitation

Throughout the remainder of this section we shall be concerned

chiefly wich the motions of the galvanometer mirror in response to sinussoidal electrical excitation and with the influence of the galvanometer

on the electrical circuits to which it is connected.

In line with the discussion of Sections 4.2 and 4.3.1 we will con- sider T] to be constant and 0^ r , and L_ to be zero. Equations 4.2.1

C7 J

and 4.2.2 now take the form

0 K

d^e

dt^

+ ue G3I3 (4.3.1)

and

S' dt

+ V3 (,4.3.2)

Considered as a tv;o terminal electrical network the current-vol cage

relation for a galvanometer may be obtained either from Eqs. 4.3.1 and

4.3.2 above or from Eq. 3.2.9 by letting L -> 0. Either approach yields ~ = R- + jco (4.3.3)

3 (o) “ O) ) K

g

Since the critical damping resistance^

^

= G^/2s/uK (cf. paragraph 4.3.1) Eq. 4. 3 . 3 may

given by

be written
50

^3

T ” ""3 I3

+ 2

00g 00 O; O)g

(4.3.4) ^

Equation 4.3.4 has been derived assuming that mechanical losses may

be neglected. While this assumption will be used in developing the re- ii sponse of an overall system consisting of a seismometer coupled by a

resistive network to a galvanometer, it may not always be realized in

practice. In particular, in measuring the parameters of a galvanometer

a more exact relation may be useful.

By analogy with the equivalent circuit of a seismometer (paragraph

3.2.4) one may sketch an equivalent circuit and assign nomenclature for

a galvanometer, ignoring the generators representing case motion and

mechanical forces applied directly to the coil as well as electrical

inductahc^ associated with the coil. The result is Fig. 4.1.

Figure 4.

The quantity ^ which we have been calling the critical damping

g p—

resistance turns oiit to be half the quantity ^/LQ/C„, a familiar paramC7 U

eter in the theory of tuned circuits.

The impedance, Z^, of the circuit to the left of the dotted line is

given by

+

'
51 -

which by algebraic manipulation, remembering that

LgCg “ (G^/U) (K/G^) = l/o)J and that

may be written /(x> , yields

1 +

R Rg ©

2oi GO /

g 1

2«2 R? - 00^',

1 . ; _s ^ \ 2m m /

*

(4.3.5)

for the impedance of a galvanometer considered as a two terminal electrical network. It should be noted that at go = go the reactive terra g

drops out, giving

GO = GO

= R + R.

g 0

(4.3.6)

It will be convenient in Section 4.4 to have available expressions

for the impedance of series and parallel combinations of a galvanometer

with a capacitor or an inductor. The remainder of this section is

devoted to developing these relations.

If a capacitor be placed in series with a galvanometer the impedance

of the combination is given by
52

where Z is the impedance of the combination and C is the electrical

sc e capacitance of the added capacitor. There is an angular frequency, o) , s o at which the imaginary part of this equation is zero. Again algebraic

manipulation produces the relation

o) C (^ =

sc e^*^ -Sco sc

CO g sc

2co CO g sc

co^ ) sc

(4.3.8)

which, for R >(fl may be solved for^ by successive approximation.

^ g ' 8

If a capacitor is placed in parallel with a galvanometer, there is an angular frequency, ^o^^, at which the imaginary part of the impedance

of the combination is zero. For this frequency the following relation

holds

2 2

CO - CO _£ E£

4co co^ C

g pc e

'R + R_

g 0

2

) [ 1 +/T

2 9

) ] pc e g Rg

(4.3.9)

Note that co is real provided that

pc

R

0

R + R.

g 0

> 2co C R - pc e g

(4.3.9a)
53

The combination of an inductor in series with a already been considered in paragraph 3. 2. 1.3 for the

R- - 00 Cr 0) . The reactive term vanishes at od 0 X 3

galvanometer has

special case of

from which

(4.3.10) I

Inclusion of the mechanical losses as an equivalent shunt resistor,

leads to

R, )

• P

s z

4<jd 00 L

g s

^2 ! 4co^L^

i

i

f

J

(4.3.10a)

Note that neither the coil resistance nor the resistance of the added

inductor appears in this equation. For R. large with respect to to L

y s series expansion of the radical leads to Eq.-4.3.10 so that, within engineering accuracy, for R^ > lOo) L Eq. 4.3.10 is to be preferred. It

y s should also be noted that for oo • to be real it is necessary that

s

R^ > 2oi L

9 s (4.3.10b)

Consider a physical inductor having resistance, R^ , in parallel with a galvanometer. The combination has an impedance whose reactive component

vanishes at an angular frequency, to , for which

irXj

0)-^ - (JO _ PT. g

^9

ji +v'i

R + R , 1 _S §. R. J

/ \

1 iR£ ^

(4.3.11)
54 -

For real values of co it is necessary that

A Lt

(4.3.11a)

4.4 Determination of Galvanometer Parameters

As the galvanometer is commonly used, the angular position of the

mirror is observed with a lamp and scale or a telescope and scale. The

determination of the operating parameters of such a galvanometer by com- bined optical and electrical measurements is not difficult in principle.

In some cases one cannot obser’/e the angular deflection of the galvanometer mirror (as for instance when combined with a photoelectric amplifier)

and the galvanometer parameters must be inferred from measurements on the

input and output electrical’ terminals without direct measurement of the

angular position of the coil.

In subsequent sections it will be shown that, in the equations for

the overall frequency response of seismometer- galvanometer combinations,

the galvanonveter characteristics appear in various combinations, all of

which can be expressed in terms of co , (^ , and K. The determination of

/D 8 8

u) and Ot- is discussed in Sections 4.4.1 and 4.4.2. In an assembled

8 8

galvanometer without optical qr mechanical access to the coil, a procedure

resulting in the evaluation of K, the moment of inertia of the suspended

system, has been sought unsuccessfully. This problem is resolved in the

following paragraph. The measurement of a single mechanical quantity,

such as a torque or deflection, would provide the necessary relation

between electrical and mechanical parameters. For instance, when the

coil can be observed, Eq. 4.2.3 at zero frequency becomes

'</2^iJ oi ~ = -v/k 0 = CO ^/u G (4. 4.1)

8 8 R3 8 8 .

so that if Lq) and co be measured by any means, a single determination

8 8 '

of the coil deflection for a known d-c current serves to evaluate the

moment of inertia, K, and the suspension stiffness, U.

4.4.1 Galvanometer-photoelectric amplifier combination

For one reason or another one may prefer not to record the deflections of the galvanometer directly but rather to convert them to an
55

electrical signal. If one px'esumes that the output of such a device is

linear with mirror rotation, and is not frequency dependent^ then it may

be considered to have the transfer relation E = k0. Here k is a

o constant having the units volts/radian, cps/radian, etc. as may be appropriate.

In the overall system response curves of Sections 5 and 6 and to some extent Section 7 the output has been expressed as 6 which always

appears in a term of the form n/k 0. When a system^ otherwise appropriate to one of these equations, is used with a photo-electric amplifier

then 0 must be replaced by its equal s/K/k.

It is easily shown that the equation corresponding to 4.4.1 which

is appropriate to a galvanometer-photoelectric amplifier combination is,

at zero frequency.

"S g R-: = 03 Vk

(4.4.2)

Therefore as far as the determination of galvanometer moment of

inertia is concerned we find the happy circumstance that, when one is

recording galvanometer deflections directly, the galvanometer is accessible and Eq. 4.4.1 may be used. VThen the galvanometer is not accessible

one is recording some other variable related to its deflection and one

needs to know '^K/k which may be evaluated using Eq. 4.4.2.

Given a galvanometer combined with a photoelectric amplifier whose

output voltage follows the relation = k0, where k is independent

of frequency, one may proceed as follows:

First : Apply a- m.omentary %'-cltage to the tenriinals and record the output

voltage as a function of time with the input circuit open.

From the record obtained, compute the logarithmic decrement.

Depending upon the intended use of the instrument, this may or may not justify the suppression of r^, as has been done in most

of this report.

Determine the time required for a convenient number of

cycles of oscillation of the galvanom.eter . From this, o3 is

readily computed.
56

Second : Establish a known direct current, in the galvanometer input

circuit and measure the resulting voltage, e^, at the output.

If 3 be the galvanometer sensitivity in amp/radian (U/G^) then

G It i^ = 30= Ue^/G^k or = -|- (4.4.3a)

Third : Energize the galvanometer with a sinusoidal voltage e^ at o>^. The value of oi computed above may be used or the voltage

S source may be adjusted to that frequency for which i^, at con- stant e^, is a ’minimum. Measure the. output voltage e^ then:

• P

G~oi Q

3 g

G^oi e /k I 3 g o' ' or

oj G~ = g

3

k

(4.4.3b)

Combining Eq. 4.4.3a and 4.4.3b gives:

e 'oi i_ 'dc o g 3

03 G?/U = 201

g

3' g

(4.4.4)

4.4.2 Galvanometer measured as a two terminal network

An alternative procedure for measuring and 03ginvolves the

use of a high gain cathode ray oscilloscope and quality capacitor or inductor rather than the voltmeter required above. It is more general

in that only the galvanometer input terminals need be accessible.

If the galvanometer under test, a signal generator and a suitable

oscilloscope are connected as shown in Fig. 4.2 with the switch in the

Y position then at one and only one frequency will a straight line pattern appear on the CRT face.
57 — I C4

j Signal

t

i O) f £ ;

1

generator

j Oj

Figure 4.

From Eqs , 4.3

for which o) = cog

loop be expressed

moved to the Y

.5 and 4.3

and

in '.Tiin cr position

. & it can be seen that this frequencs' is t' is. a minimum. Let the tilt of this closed

mv as convenient. The switch is then

and the tilt is again measured. Then

ns u

or + R RY/(Y' - Y) (4.4,3)

where Y' and Y are the numerical values ct th,.- tilts r.ca^.r. d it. the

corresponding switch positions Re. the d-c resist

ometer_, may be measured by an> convenient method.

Addition of a capacitor cr inductor to the galvanom.^.tcr

measurement usinp the. circuit of Fig. 4.2 with the sv.'itch .m tion gives all the inf unra ... ion necessary to compute y . th

.iG 3. hv V pO:.iC V 3 J 1 C 3 I damping resistance, usiiy.' unich-euer of A . 3 . c t'lro.,,^,'.. - . 3 . ^ 1 i- appropriate.
58

Alternatively we have also used

Fig. 4.3 to determine both oo and

8

the bridge circuit sketched in

+ R . We have used an oscilloe g

scope with a differential Y input as detector although, in principle,

any detector with a "floating", or differential, input may be used.

Figure 4.

3

The detector will indicate a null or balance only when cjd = o) and

ri/r2 ' Tj/CRg + Rg).

If the galvanometer sketched above is replaced by a galvanometer

and reactor (vide supra) then balance is obtained only at the frequency

appropriate to a determination ofiJ(/ . The reader should be cautioned

8

that in this case ^2^3^^^! give the correct value for (R^ + R^)

to use elsewhere in this report.
39

Section 5

SEISMOMETER, RESISTIVE T PAD AND GALVANOMETER

• 5.1 General

In this section equations are developed for- the performance of a variable reluctance seismometer connected to a galvanometer by a resistive T pad. The equations are based on the assumption that mechanical

damping of both seismometer and galvanometer may be neglected. Also

implicit in these equations is the assumption that the inductance of the

galvanometer may be neglected.

Attempts to obtain equations for the transient solution analogous

to Eqs. 3.2,13, 3.2.14, and 3.2.15 have not been successful.

We shall discuss limitations on the design of an adjustable T pad

without Aihe formality of explicitly displaying amplitude-frequency or

phase- frequency equations sines their development by the methods used in

Section 3 is s traightfonward.

5.2 Seismometer with Bilateral Coupling

to the Galvanometer

A commonly used arrangement ' consists of a seismometer, a three

terminal network, and a galvanometer. The three terminal network is

usually resistive, arranged to provide adjustable current attenuation;

and some attempt is usually made to present a constant load to both the

seismometer and galvanometer.

Since any three terminal resistive network may be replaced by an

equivalent T network, we may sketch such a network (Fig. 5.1) and write

the equations of motion.

Figure 5.
60

They are:

X = M

^

+ Sx

dt^ dt®

(5.2.1)

dx ‘*’^1 ^ = <=1 IT

+ I- IT hh (5.2.2)

j2g

0 <= K —- + W - G.I- (5.2.3)

dt® ' ^

®“®3dr + V3

E - - (1^ + I

3

) , (5.2.5)

is the sum of the resistance in the input arm of the T and

the resistance of the seismometer coils, R . It therefore has a minimum

c value set by the resistance of the coils themselves.

R^ is the sum of the resistance in the output arm of the T and

the resistance of the galvanometer coil. It therefore has a minimum

value set by the galvanometer coil.
61

R is the shunt resistor of the T pad and may have any positive

3

value.

The other s^onbols have the same significance as previously assigned

them.

Eliminating E, I., and x from these five equations gives

R G?G? 13 _d_ / _2^ _ I M d^vl_

i^s ^1^3 ^

^'3^s

^

\ nS n ^ dt^y

r

A I + / 2

P c '

®1 ^ , ML g

2 t d

+1 03“

dt^ V

°2

p2\

! 03^ +

ML / g I

1 3 _3 d , 2) , R,R + R,R. + R.R • K • dtj , 2

’ “oj Is 13 3s j dt j

(5.2.6)

•=3 d®

R, R + R, Ro "h RoR Is 13 3 s dt- dt'

’!l 03^ + +

o ML i J

~1 A j2

y + (03^ + 03^) + 03^^03'^I V 6

dt^ ° S dt^

A "mean coupling coefficient" may be defined as follows;

With the galvanometer coil blocked^ the fraction of the current generated by the seismometer which passes through the galvanometer coil is

given by R /(R^ + R ). Likewise if the seismometer mass be blocked and

its inductance neglected then the fraction of the current generated by

motions of the galvanometer coil which passes through the seismometer is

given by R /(R, + R ). Each of these may be looked upon as a coupling

S X S coefficient whose value lies between zero and one. We may therefore define a mean coupling coefficient
62

a = R^/y (R^ + Rg) (R3 + Rg), 0 < a < 1

This definition ‘of OL has been based on current attenuation in the

forward and reverse directions with the appropriate mass blocked. One

may equally well define the same mean coupling coefficient in terms of

the forward and reverse voltage attenuations witji the appropriate output

open. Alternatively suppose that a d-c e.m. f. is introduced in either

the seismometer or galvanometer branch of the circuit. The current in

the other branch dissipates a power, in the network. If the total

power put into the network is then the ratio is accord

Ing to either of the above definitions.

From the concept of the short circuit frequency developed in paragraph 3. 2. 1.1 we may, as before, define

CDso== CD "v/l + G?/LS and f = cd /cd 1' 3 s' o

The defining equation for p given in paragraph 3.2 here takes the

form

P =

’‘l R

R-R

3 s n , R^R„ + R,R + R_R

_ I _1 13 Is 3 s

_ + R I CD L

3 ^ o (R + R ) CD L

3 so

where the bracket is the resistance of the circuit as seen by the seismometer when the galvanometer armature is blocked.

The "ratio to critical damping", i.e., \ = (f^ /r, as commonly

used in analyzing circuits involving galvanometers will be used here with

the defining equation

X

g

G

2

3

2 n/uk +

Gf (R^ + R^)

R.R 1 2cd K (R,R„ + R,R

1 s I g

''13 Is + R^R ) 3 s^

The reader will note that this definition is equivalent to defining

X as the fraction of critical damping seen by the galvanometer if the

g

seismometer is replaced by a non-inductive resistor of value equal to the

coil resistance.
63

Since cog

manipulated to

as before and using D = d/a>^dt, Eq. 5.2.6 may be

the alternative forms 5.2.7 and 5.2.8 with f = 00/00

g g o

av/02>. f„ / y ^V S S 7D / — - vM Dy|

CO . I sTs /

o

= [(D^ + PD^ + f^D + p) (D^ + 2\ f D + f^) - a^2\ f (D^ + f^) ] 0

3 g g g g g

^ s^"

(5.2.7)

(D + PD^ + f^D + p) [D^ + 2\ f (1 - a^) D + f^] + q:^2X f pD (D^ + 1) I 0 nTk

I 3 g g g g g

-» ) (5. 2. 8)

5,2.1 Influence of variable attenuation on the over-all responsefrequency characteristics

The over-all gain of a seismometer-galvanometer combination is

usually adjusted by altering the resistive T pad. Unfortunately it is

not possible to do so without some alteration of the shape of the

response- frequency characteristics. This can be seen from Eqs. 5.2.7

and 5.2.8. In both cases^ the coupling coefficient a appears as a factor on the left hand side^ so that Q, for any given excitation is proportional to it. The right hand side of Eq. 5.2.7 contains two terms^

the first of which does not contain a, while the second is proportional

to The first term is the product of two factors, of which one

appears in the equation for the response of a seismometer connected to a resistive load, (Eq. 3.2.5) and the other in the equations for the galvanometer (Eq. 4.2.3). If the T pad maintains constant input and output resistance, then p and \ are constant. The term in will

g

affect the second and fourth derivatives only.

Alternatively a resistive network may be designed to have p and

R- + R constant. Here (1 - 0C~) \ is constant and Eq. 5.2.8 shows

3 s

"

' g

that a affects the first and third derivatives. In neither case are

the constant term and the fifth derivative affected.

In the first case, constant K , the frequencies at which the earth

' g

velocity is in phase, or 180° out of phase, with the output are independent of Q, or the attenuator setting. The relative gain at these
64 -

frequencies Is dependent on a. Conversely the relative gain at the

frequencies of ± 90° phase shift are Independent of a but the frequency at which this phase relation occurs is a dependent.

In the second case, with + R^ constant the frequencies at which

the earth displacement is in phase, or 180° out of phase with the output

are functions of the T pad setting while the relative gain at these

frequencies is not. Conversely, the frequencies at which the output is

90° out of phase with the earth velocity are independent of the attenuator setting, but the relative gain at these frequencies will depend upon

attenuator setting.

As a practical problem one may set tolerance limits on the permissible variations of the response- frequency curves and, for a specific

seismometer and galvanometer, compute the maximum value of CC (and of

system gain) attainable without exceeding the. prescribed limits.

The greatest range of gain settings for a prescribed variation of

the relative response- frequency curves will be obtained using a judicious

.compromise between these two ways of restricting the T pad parameters.

In any case, as R^ -* 0, a -* 0 also and the shape of the responsefrequency curve becomes that expected from the two electromechanical

systems considered as filters, connected by some unilateral network such

as an idealized cathode-follower. The amplitude of the overall system

gain also approaches zero.

5.2.2 Direct coupling of seismometer and galvanometer

Since currents flowing in the shunt arm of a T pad represent an energy loss it becomes of interest to examine the special case of direct

coupling for possible use as a high gain system of minimal size or weight.

In this case the interaction of the seismometer and galvanometer is

greatest since all the current in the transducer also passes through the

galvanometer. The circuit may be sketched as shown in Fig. 5.2 wherein

the inductance L may be distributed between the seismometer and galvanometer in any manner.

Figure 5.2
- 63

The resistor R, (R.^ + of Section 5.2) Includes the coll resistance

of both the selsracmeter and galvanometer as well as any resistance in

the connecting cable.

As R -> c" in Eq. 5.2.6^ the third of the four terms on the right

3 hand side vanishes because it contains R in the denominator only.

s

' The equation can be rearranged to the following form, in which even and

odd orders of the derivative are separated:

i V?

, i„± °3

J M k

\ \'MK

sh ifn

\

dt^/

. d d^ , 2^ ,d^ 2. , ^1 d

, 2v , ^3 d ,d- , ; dt^ ° dt^ S dt- dt^

^

+ R (— 4- o)^) (— + o>;) ] 5

dt^ dt^ ®

(3.2.9)

For sinusoidal excitation, ‘ and with no forces . applied directly to

the mass, one may substitute ' jiu ' for • d/dt- ..Vin.-Eq. 5.2.S to obtain:

' -i- 3 ; M ^

V MK \K 6

(o)^ - lo^) (ar - cn^) (a)^ - cd^) o g g L

a + —i

- v2 , ,2

O) Qj

Li-. — CO

j. O J

“i —

“ j

U u;; - a>")

J

CO

(0.2. 10)

Since the imaginary part of Eq. 5.2.10 vanishes at a> = co

a'* = iO it follows that the galvanometer defiecrlon

S

100“ out of phase) wii.h the earch oisul.

i.s in = lOophase

a nd

a no

V-'’" tccment at a)
66

The sensitivity at may be computed from

(5.2.11)

i: (5.2.12)

i /

Thus the ratio of the system gaias at o) and cu is proportional

O o to the square of the frequencies, proportional to the ratio of inertias,

M and K, and inversely proportional to the squares of the generator

constants.

It is also of interest to note that neither the circuit inductance

nor the circuit resistance appears in Eq. 5.2,11 or 5,2.12. The explanations are that at co^ the seismometer may be regarded as a high impedance

generator, or constant current source (cf. paragraph 3.2.1), so that the

current through the galvanometer is independent of circuit impedance,

while at cn the galvanometer nay be considered a high impedance load so

g - that the current in the electrical circuit approaches rero.

By inspection it can be seen that if G"/M = G^;^/K then Eq. 5.2.9

is unaltered as to form and numerical value by an incerchango of free

angular frequencies as might be obtained by adjustment of S and U

leaving all other parameters unchanged. Considering the engineering difficulties associated with the produccion of long and short period galvanometers and seismometers, serious consideration should be given to this

manner of interconnection for, .if a satisfactory response- frequency curve

can be obtained, then it may well be a matter of less engineering difficulty to make the natural frequency of the seismometer of the order of

say 5 cps and the galvanometer 1 cps than vice versa.

Since the circuit will not permit the inclusion of attenuators for

the control of sensitivity Its use requires considerable dynamic range

in the equipment used to determine the galvanometer deflection.

^ i 0 'on “ ui tk'fE r«K /”a

while at (u the sensitivity is given by

g

^ ! 0 'tu = CDg

S ^/K ! MX

K jM jp2p2 i 2
67

Using the parameters of an available Benloff Variable Reluctance

Seismometer and a commercial galvanometer the authors have not been

successful in determining circuit constants leading to adequate suppression of the resonant peaks, near oo and o) , in the amplitude- frequency

response curve. The'y are of the opinion that the inductance associated

with any variable reluctance transducer of reasonable rest-point stability

and negligible mechanical dainping precludes its use in a direct coupled

system. The possibility of the judicious use of mechanical damping to

suppress these peaks has not been investigated.

• P
Section 6

SEISMOMETER- GALVANOMETER SYSTEMS

WITH NEGLIGIBLE INDUCTANCE

6.1 General

Since seismometers using moving coil transducers can be built with

very little inductance in the electrical circuit it is of interest to

examine the form which the equations of motion take as L - 0.

As in Section 5, the transient solutions of the equations of motion

have not been obtained. However, some success has been obtained in

describing the response- frequency curves for several special cases.

It may be noted that terms of the form n/m/k (y/d) which appear

throughout this report represent a normalized gain expressed in reciprocal form. The reciprocal form has been used since it permits expressing the frequency dependent factors in the equations for system response

to simusoids in algebraic forms which are relatively simple.

6.2 Seismometer, Resistive T Pad, and Galvanometer

A seismometer having negligible inductance may be approximated by

a second order linear differential equation so that it becomes permissible

to speak of a "critical damping resistance": and a "ratio to critical

damping" definable as

’

= G?/2 = G?/2a3 M = oj Gf/2S

O 1 1 O O i

and \ equal to fV divided by the circuit resistance seen by the

o L. o seismometer. These definitions are consistent with those for a galvanometer introduced in Section 4 and 5 and are rigorous under the assumption

of negligible damping other than that due to dissipation in the electrical

circuit. It will also be convenient to use a mean ratio to critical

damping., \ , defined as X H JTx , and the geometric mean of the

m m o g

two natural frequencies, co , defined as oo H Vo) o) . m m o g

Making the appropriate substitutions, letting L-^0, equation 5.2.6 may

be rewritten
69

2a\ o) m m

MI K dt 'M dt‘

2 2

(^ + 2?\. CJO “^ + 03^) (-^ 1- 2X 03 + 03^)

I ^^.2 o o dt '^^2 g g dt g'

V

- Aa X. 03' 2, 2 2 d

2 'I m m ^^2i/

Letting X = 0 and d/dt = jo3 Eq. 6.2.1 may be written as

(6.2.1)

2ak 03 „ m m) K 6

1 = 2X 03 o o

2 2

03 - 03 -_S

03^

+ 2X 03 g g

2 2

03 - 03 O

03

- J03 r - Qi®

I _S

03

03^ - 03^

+ 4 (a^

03^

03 1 ) ~ m203

( 6 . 2 . 2 )

Earth displacement will be in phase (or 180° out of phase) with

galvanometer coil displacement when 03 - satisfies the relation

2o3^ =*03^ + 03^ + 4X^ (1

g o m

' - a'^) 03*" ± / [03^ + 03^ + 4X^ (1 - m -J g o m

a^) 03^1 - 4j3^ m m

(6.2.3)

In the special case of a =» 1 (direct coupling) the two values for

03 in Eq. 6.2.3 are 03 and o3 . It can be shown that for \^ > 0

o g m

(which is the case for any real system containing only passive elements)

and a < 1 these two points lie outside the region between 03^ and 03 .
70

Earth velocity and galvanometer displacement are in phase at

o> X 03 o g

m X O) g g

+ X o) S_2

+ X o> o o (6.2.4)

Equation 6.2.4 shows that this "quadrature” frequency is independent of

a, must always lie between o) and o3 . and for X = X it is at o) . ’ o g o g m

Elementary considerations suggest that is that frequency at

which the LC circuit equivalents of the seismometer and galvanometer show

series resonance. This concept is discussed in detail in paragraph 7.2

and can be used in the calibration of a seismic system (see 9. 3. 5. 3).

The general expression for system gain at the in-phase (and 180°

out of phase) point is algebraically messy, involving, as it does, the

insertion of the value of go from Eq. 6.2.3 into the real pare of Eq.

6.2.2. This statement is also true for the system gain at the quadrature

point, the equations being 6.2.4 and the imaginary part of 6.2.2.

The gain at oi = is given by

_o M

I i I Oi K ' 0 'go g

00

X n) 00 ^ ) + 4X= (a^ - 1)^ X oo 00 m

^ •go S

(6.2.5)

while at oo = oo Eq. 5.2.2 leads to

g

^ Vf

00 K

o ^ I 0 'oo 00

X oo 00 ^

(_0 _ _S

) + 4^2 ^^2

X 00 00 m

o E O

1)^

( 6 . 2 . 6 )

For the special case of X

o of the natural frequencies,

exact value of X and X . o g

•= X

g

the ratio of these gains is the ratio

independent of the value of CL and of the

As before it will be noted that, as the’actenuation of the T pad

increases, not only does ‘the overall system gain decrease but the shapes
71

of the response- frequency curves approach those of a pair of buffered

filters. The possible response- frequency curves which may be obtained

at small values of a for various values of oo , cn , X. and X. have

o g o g

been well considered in the literature. One special case is further

discussed in subsection 6.4.

Also, as in the case of circuits containing inductance, it is not

possible to design a T pad which will alter absolute gain without

simultaneously altering the relative shape of the amplitude and phasefrequency response curves.

Admittedly, within engineering tolerances it is feasible to design

an attenuator for any given seismometer and galvanometer which will give

an adjustable system gain with reasonably stable response- frequency

curves, provided CC is not permitted to become too great.

Several features of Eq. 6.2.1 should be pointed out as being of some

interest to the equipment designer. Having decided on values of co^,

0) , X. and X.^ which will produce desired shapes for the responsefrequency curves at low values of o; and a maximum tolerable value for

a, then the maximum overall system gain,* or magnification depends only

on m/K and the method used to record 9, the galvanometer deflection.

A second point of interest is the fact that the system response--

both transient and steady state-ris independent of whether is the

natural frequency of the seismometer or galvanometer. That is, the

designer is at liberty to select specific values for S, U, G^, and

the values of resistance in the T pad within the restrictions imposed

by a prior selection of cd^, to , X,^, X.

, n/m/k and range of a. In this

sense these systems differ from buffered electronic filters. For instance

the following two systems are, as magic boxes, identical in the response

of the galvanometer coil to earth motion o.r calibration techniques making

use of a force on the bob.
72

Seismaaieter

System A System B

1 kg mass 1 kg

0.01 cps natural frequency 1 cps

1000 ohm critical damping

resistance

100 ohm

50 ohm coil resistance 50 ohm

any T pad same as for Systei

A with input and

output terminals

interchanged

Galvanometer

X 10 ^ kgm^ moment of inertia 1 X 10 ^ kgm^

1 cps natural frequency 0.01 cps

100 ohm critical damping 1000 ohm

resistance

50 ohm coil- resistance 50 ohm

Instrument designers will agree that, at least for vertical componen

devices, system B would probably be more easily fabricated as a practical, stable system.

6.3 Direct Coupled Seismometer-Galvanometer Systems

With the elimination of inductance from the equations of motion it

becomes of interest to re-examine the direct coupled case (R^ -* for

potentially useful response-frequency curves.

As R -» no point is served by attempting to distinguish R,

S 1. from R^. Throughout this section the series loop resistance will simply
73

be called R. It should be noted that as R -* oo the relation

Letting a = 1, Eq. 6,2.2 may be rewritten

TK 00 “ 2?^ CD —^ m m I K 0 o o 2

CD + 2X CD

2 2

CD “CD

O

8 8

(6.3.1)

CD2222 - CD CD “ CD “ JCD

CD •2CD

As mentioned in paragraph 5.2.2, y and Q are in phase (or 180°

out of phase at cd and cd for all values of the series loop resistance

o g

while the quadrature frequency cd^ is given ‘by Eq. 6.2.4.

The system gain for cd = cd^ and cd = cd^ may be written, from

Eqs. 6.2.5 and 6.2.6

M I JZ I K '0 'cd - CD

X CD CD CD

' o g _ _2)

X CD ''CD CD

'' goo g

(6.3.2)

and

M

K

V

0

2

^CD = CD

X CD CD CD s_£ (_2 . _JS)

2

X CD O g

CD CD g

(6.3.3)

The ratio of the system gains at these two frequencies may be written

^CD X CD —S I =

0 'y constant • X cd CD go

(6.3.4)
74 -

From Eq. 6.3.4 it-- can be-- seen that if X. then the system gain g

(amplitude basis) is proportional to the r*tio of the two natural " frequencies. On an earth velocity basis these two gains would be equal.

In the direct coupled case, X =» X is equivalent to stating that the

c g

two instruments havh the sarrie critical damping resistances. This special

case is considered at length in the following paragraph (6.3.1).

If X cjo =“ X oi then the system gain (amplitude basis) is the same

O g o ^ at the two natural frequencies. That is, if the critical damping resistances of the instruments are in direct proportion to their natural frequencies then the system gain at these two frequencies is the same on an amplitude basis. This special case is discussed in paragraph 6.3.2.

A third possible special case of interest is that in which the

quadrature frequency shall be the arithmetic, mean of the in phase and

180° phase shift frequencies. This case is intimately related to the

problem >f dispersion which, in the present state of the art, is too

elaborate a subject for inclusion in this paper.

6.3.1 Direct coupled system, "constant velocity*' response

As indicated above, when a seismometer and galvanometer have equal

critical damping resistances then the system gains at the natural frequencies of the two instruments will be equal on an earth velocity basis.

Throughout this subsection it wfll be convenient to use the notation

y

= dy/dt = jcoy for sinusoidal earth velocity excitation. Since

X = X = X the subscript will be omiitted.

o g m

It will also be convenient to use a normalized angular frequency

defined as 0 3 0 therefore, is always positive, 0 < 1 for

cjt < o)^*, 0 = 1 at the geometric mean of the natural frequencies and

0 > 1 for 03 > CO . From Eq. 6.2.2

m

-L>J^ 1

03 IK a m

+ 03o

03m

) ] j

+ 03O

03m

)(0

1 V

o3„ I K 0 m

03 - 03 03 + 03

^ £ £) ] . j (_a o

03m

03m

)(0 - t) (6.3.5)

or
75

Inspection of Eq. 6.3.5 shows that: (a) The amplitude response, on

an earth velocity basis, when plotted against log <t> (or log cd) is

symmetrical about the point 4>=«1 (00 = 00), (b) At 4> => 1 (00 = oo ^ mm) galvanometer displacement is in phase with earth velocity for all values

of \, (c) At $ = 1 (00 = 00^) the system gain varies directly with

Alternatively, for a given critical damping resistance, the system gain

varies inversely with the loop resistance of the circuit.

Since Eqs. 6.3.2 and 6.3.3 show that the system gains at 00 = 00^ and 00 = 00 are independent of \ (or the circuit loop resistance)

while c) above indicates that the gain at the geometric mean frequency

varies inversely with the circuit loop resistance, it follows that for

high values of loop resistance (X -> 0) the amplitude response curve

must have two maxima and one minimum (three extremes, all real). At low

values of circuit resistance there is only one maximum (three extremes,

one real, two imaginary). From all of the .above reasoning, we see that

there ma/ be a unique value of X representing a triple root, at = 1,

for these extremes.

. 2

If one evaluates d

j ^

| /dln<t>^ for Eq. 6.3.5 and sets the resulting function equal to zero, <{)^ = 1 will be a root corresponding to the

extreme at o) = oi . The other two extremes occur at the roots of

m

00 + CD (1 - 2X^)(-

CDm

These two roots will also occur at <t> = 1 when

- CD - CD - (-2 2

) 2 '‘CD + CD g o (6.3.6)

Note that as the two natural frequencies approach each other X -» 0

while as they are more and more separated X -* lj^2. Substituting the

value of X from Eq. 6.3.6 into Eq. 6.3.5 one finds that the "midband”

gain, at cd = cd^, is given by
76 -

2

e /2OJ

0^m

iri

03 03

^03 03

^ O g

(6.3.7)

while the system gains at the two natural frequencies are given by

o

03g

03m

03 03

O g

(6.3.8)

Comparing 2qs. 6.3.7 and 6.3.8 it can be seen that the response at

the two natural frequencies is down 3 db from the midband gain.

If Eq. 6.3.6 is used to eliminate X from Eq. 6.3.5, one obtains

03

03

03 m g

" ^ rxA • 1 2 03^

- 03^ ^

£

o

I VM V

^

1 _ _s o I + 03 M K 0

^ ^ 03

' BO m

03 - 03

03m

(0 -

or, taking the square of the absolute value

(6.3.9a)

<, 03 - 03 „ _L /_£ 2n M ! 2

2

'^03 + 03 K ' 6

“m S

% 4 03 - 03 4

1 o

- t) + (-^

'

<t> 03m

(6.3. 9b)

The latter is convenient for computing amplitude frequency response

curves.

Several curves for different values of 03 /o3 are shown in ris. 6.1.

o The designer will see that using Fig. 6.1 he can quickly estimate the

ratio of m/K required to have a desired earth velocity correspond to a given trace amplitude, once he has defined the band width and mechanism

for recording galvanometer deflections. We have, for convenience, callv-c

this special case the "flat velocity fourth order direct-coupled case."
77

A discussion of Eq, 6,3.9 vis-a-vis its asymptotes as the octave

separation of and changes is of interest for comparison with

the response curve developed in subsection 6.4.

For <t> » 1 Eq. 6.3.9 is asymptotic to

r« O) - CD r.. il I __S 2 ] 1

CD ' CD + CD ' I K ' 0

m e o

CD - CD (6.3.10)

while for 4> « 1 the asymptote is

f2, CD - CD rr.

I _S 2 I '/H I 2

CD ' CD + CD ' K ' 0

m E o = <t> -2

(6.3.11)

The . intersection of asymptotes is at 1 with an ordinate value

CDm y

' 'M 'CD

CD - CD

CD + CD 8 o (6.3.12)

Parentheticxally it will be noted that as j cd^ " [ -> oo intersection -> nT2 while as CD • /^ 1 this intersection -*• 0.

g/ o

this

It is also of interest to locate the intersections of the curve

with the asymptotes. This is readily done as follows:

Let a

CDm

(6. 3. 13)

a is therefore a measure of the octave separations of cd and co . o g
78 -

Then Eq. 6.3.9b niay be expanded to

2a^ M

a^ + 4 oj^K m

1 |2 e ' 4l>^ + 6 + a

While for the low frequency asymptote, Eq. 6.3.11, one may write

2 a^ M I i 1^ _L

2 , / 2„

' 9 .4

a + 4 oi K <J) m

so that the ratio to the low frequency asymptote may be written. The

curve and its asymptote intersect at those values of 0 for which this

ratio is one, i.e. the intersections are at the roots of

<j)^ [(j) - 4<5> + (6 + a^) - 4] = 0

It will be noted that the bi-cubic in brackets, of this equation,

is in a form suitable for graphically obtaining the real root from Fig.

3.1 with p = 4 and 7

^ = a^* + 5. Doing so, one finds that as a -*• 0,

<!>'-» 2; for a = 1, 0 =1; and for a > 1, 0 < i. The situation is more clearly shovm in the sketches of log-log plots in Fig. 6.2 (a),

(b) and (c).
79 -

Go/oi -!, a-»0

g' o

1 < <J>^ < 2

Oi /go

gv oa

2

± 1.39

1 4> = 1

(a) (b)

GO /co >2

g/ o ±1.39

a > 1 4) < 1

Figure 6.2
80

The frequencies of the intersections of the midband tangent with the

asymptotes can be obtained from the relations;

2 2 ’ Oi 4 , M

I i

I . [ (^ - -2) +

K ' 5 ' 2 0^ a m

from Eq. 6.3.9 —)f-l

1 ' m< high frequency

asymptote

low frequency

asymptote

midband tangent

Calling the intersection of ttie high frequency asymptote with the

midband tangent and the corresponding intersection of the midband

tangent with the low frequency asymptote one may write

U)m

CO - CO -S o

COm

or CO, CO - CO g o and

CO - CO g o

That is, the frequency at which the high frequency asymptote intersects

the. midband tangent is the difference of the natural frequencies while

the intersection of the midband tangent with the low frequency asymptote

occurs at a period equal to the difference of the natural periods of the

instruments.

COm

CO,. a =

CO - CO _S o

COm

or CO,
81

At this point it may be in order to summarize the properties and

limitations of the flat velocity fourth order direct-coupled case.

1. When plotted on a log- log basis the galvanometer

deflection for unit earth velocity will be symmetrical

about the geometric mean of the two natural frequencies.-

2. The response curve is convex upward between and

cn having maximum "flatness" at the geometric mean of

sthe two instrument natural frequencies. The first

three derivatives of the gain are zero at oi = ct . m

3. The asymptotes have slopes of 12 db/octave.

4. The 3 db points are at the two natural frequencies.

5. The knees are the sharpest obtainable, without introduction of a concave upward region in the midband,

for a fourth order system.

6. The curve is independent of interchange of natural

frequencies of the galvanometer and seismometer.

7. For a given value of m/K the greater bandwidth

will have the lesser midband gain.

8. For a given value of m/K the midband gain will be

greater than that obtainable witli a T pad coupled

system.

9. As no gain control can be placed between the seismometer and galvanometer it is necessary for any

given pair of instruments that the recording mechanism be provided with dynamic range and gain controls

adequate to the application.

10. The gain (angle/velocity) of the system, at constant

bandwidth, can only be altered by changing the ratio

of the two instrument inertias and is proportional

to the square root of this ratio.

11. The working gain obviously depends on the means of

recording the deflection of the galvanometer.
82

6.3.2 Direct coupled system, "constant amplitude" response

It was implied ac the end of subsection 6.3 chat, at least

superficially, there seen to be conditions under which a close coupled

seismometer-galvanometer system will approximate a constant amplitude

response in the midband region. A few preliminary remarks on elementary

grounds may be appropriate.

Well above its natural frequency a moving coil seismometer having

negligible inductance has an emf proportional to velocity, and the

problem in developing a "constant amplitude" system is so to adjust the

interaction terms in the amplitude- frequency response equations as most

nearly to correct the inherently rising characteristic with frequency of

the seismometer. It nay not be possible in the general case to arrive

at an acceptable compromise. Another point suggesting difficulty is that

for \ 4 \ there is a range of values for the series resistance which

o ' g

results in one instrument being overdamped while the other is underdamped

and this range becomes greater as the value of ^o^^g from

unity. Another point lo remember is that the low frequency asymptote is

18 db/octave while the high is only 6 so that equations having the simplicity and symmetry of Eq. 6.3.9 cannot be expected.

The mathematical attack might proceed as follows: from Eq. 6.3.4

it follows that the system gain on an amplitude basis will be the same

at the two natural frequencies provided the critical damping resistances

of the two instruments are in the ratio of their natural frequencies.

The in-phase and 180° phase shift frequencies are cn and cu while

the quadrature frequency is, from Eq. 6.2.4,

S

00

200^

00^

00^ + 00^

O g

(6.3.14)

Substituting \ 0) o 8

X 05 g o in Eq. 6.3.1 one may write

05 05

05 05m

05 1 ^ 2 — Ik a 05 ''

05

05 0

- m

05 05 — [ (—

2 \ 05 ^05

m

05

05

) - (

05 - 05 2 £

m

05

) ] m

2
83 -

or

a

1 _

9 2

0)V

o 2 , 2

I //COuu

rn / _v

CO 2X CO m m

/

(6.3.15)

For prescribed values of co and CO , Eq. 6.3.15 represents a single

8

infinity of phase- frequency curves all of whose members have their quadrant points at co * 0, co CO , CO o'

CO , CO v'

COg

and CO = 00. The amplitudefrequency curves of this infinity all pass through the points given by

'co 6c COg

CO CO _2_S_

CO - CO fi

g

(6.3.16)

Inspection of Eq. 6.3.15 shows that if be small then the gain

in the neighborhood of co^ will be small compared to that at co^ and

CO . Conversely, as X is increased the relative gain in the neighborg m

hood of CO increases. That is, at least for small values of X , the

V m'

amplitude- frequency response curve will have three extremes (two maxima

and one minimum). One may attempt arbitrarily to define the "constant

amplitude system" as having that amplitude- frequency response curve for

which these three extremes coincide in frequency, i.e., that value of

X^ (if any real value exists) for which the amplitude- frequency curve is

always convex upward on a log- log plot but most nearly linear in the midband region.

Examination of Eq. 6.3.15 for extremes is messy; however, for small

value of X one would expect two inflection points, lying between

m

CJD and CO , in the amplitude- frequency response curve. Further, the

^ g

"constant amplitude system" might be that for which the two inflection

points occur at the same frequency, and this attack on Eq. 6.3.15 is

algebraically manageable. One obtains
84 -

CO

CO + 2 - 16\"

m \ CO 3co' i

2 16X CO CO m o g

(6.3.17)

The two roots of co^ in Eq. 6.3.17 give the frequencies of these

inflection points for various values of X , and examination shows that

m

a single , real value of X exists only if co^/co^ + co^/co^ = 4. This ^ > m ^ o' g

g' o corresponds to a separation of co and co •• of 0.949 octave with

^

o g

X^ =v6/24 or X— 0.319. For a closer spacing of the two natural fre- m TM

quencies there are two real and distinct values of X^ meeting the

prescribed conditions while for greater separations of the natural frequencies all values of X^ are complex. Figure 6.3 has been computed

for the special case of 0.949 octave separation. The reader will agree

that this curve leaves something to be desired as a "constant amplitude"

response. The physical significance of a complex X as required for

instruments with greater separation of their natural frequencies escapes

the authors. From this we may conclude that a broad-band "constant ampli

tude" response r based on -P lijt.- oo /oo is not obtainable in a close- i, o' g

coupled system free of mechanical damping. The reader will note that the

mathematical difficulties encountered above are quite consistent with the

remarks at the beginning of this subsection.

A more general line of attack is as follows.

For convenience let 2 (Xcd + Xco) =Aa) and 2(Xo) + Xo))^Ba) og go__m 'oo g

g' m

Incidently A/b is, from Eq. 6.2.4, (xi_v' m

Then Eq. 6.3.1 may be

written

2X

m ir. I X.

ccm

Oi

/co y H

O)2,2+03

—in — K O

03 i 4 2 m\ 03 03 + 1 (6.3.18)
85 -

has already been defined as oi/cjo so the absolute value of

m

earth displacement over galvanometer deflection may be written

4\'£ M I m K

+ - 2 /

(XT + aS 1 I

2 , ^

' \^ n r* ol - 1 P2

00m

- 2AB

/ 00^ + 00^ 12

* + /_S 0 ^

m iJ

a £

(6.3.19)

+ -

00^ + 00^

2 -S 2 + <^2 m

Note that the coefficients of the second and third terms on the

right hand side will be zero when

A^flo^ = 4(Xoo + Xoo)^ = 2 (00^ + 00^) and (6.3.20) m ' o g g o' > g o'

2AB = 8(Xoo + \co)

o g go' (\ 00 + X 00 00 g g =- 2+1

far + 00 \

2

£ 1

00m /

(6.3.21)

Explicit solutions of Eqs. 6.3.20 and 6.3.21 for and

are messy. As given they may readily be used for numerical evaluation

of and X^^ for any given bandwidth. Figure 6.4 shows the values

of these quantities plotted against 00 /oo for physically realizable

o g

systems.

When X and X have values consistent with Eqs. 6.3.20 and

o g

6.3.21 then Eq. 6.3.19 takes the form
86

m K

1

e

- + (6.3.22)

where = 2 (cjd^ + B - (8 + A^)/8A. This equation is

plotted in Fig. 6.5 for 2 and 4 octave separations of the instrument's

natural frequencies. may be obtained as

m

40)'=' (o)"' +

m £- “o>

2o) +

m

(o)^ +

g

2n 2 (6.3.23)

As 00 ; CO /oo v' m

2co

r\f/CO and as co CO CO /co V m

(co - CO 2/J~3 ) , . .g p

o V' m O' m g o which is >u) /co or CO /co , i.e. the quadrant point crosses an in-phase ^ _m g m

point. As such a crossing is not physically realizable the minimum

ratio of co /co (or co /co ) which may be used is 4 >/3 or 0.396

o ^ o octave separation.

It will be noted that for a physically realizable system the right

side of Eq. 6.3.22 must be positive for all values of ®. The minimum

value of l/o^ + occurs at 0 = ^and is 4 ^3/3 . Therefore

3 (B^ - A^) + 4 Vs > 0 is necessary. Substituting for A and 3 the ; values indicated above one can show that this restriction is equivalent

to the requirement that the instrument natural frequencies be separated

j by at least 0.396 octave, the same restriction as is imposed by quadrant ' point frequency considerations.

i

I 'Analytical examination of the expressions for X , X and X ! o^ g m ' shows that they are all real and positive for any bandwidth greater than

| 0.396 octave.

Incidently <t> •= if3 is the frequency at which the system gain is

a maximum, independent of bandwidth. Also the ratio of instrument criti-i

cal damping resistances is given by ^

i (6.3.24)

a

I

II

Ao) - Bo)

g o

Bcju - AcjO 8 o
87

6.3.3 Direct coupled system, "constant acceleration" response

, The technique exemplified by Eqs. 6.3.18 through 6.3.24 suggests an

i elegant procedure for developing a "constant acceleration" system.

Let A and B have such values that the third and fourth terms on

the right hand side of Eq. 6.3.19 are zero. Then, since I y/oo^ I =

' m

'

(|)^

I y an equation analogous to Eq. 6.3.22 may be written,

4 -JB W I 1

4 K ' 0

OJm

- i (6.3.25)

where « + (X^ = 2 8 o

(XTm

(6.3.26)

and 2AB = 2 + f

03m

as before (6.3.27

Again explicit solutions for and are messy.

03^, the quadrant frequenqy> is given by

03^ 2o3^ + (03^ + 03^)'

_v « -J2 g o

CD^ 4o3^ (03^ + 03^)

m m g o

(6.3.28)

Comparing Eqs. 6.3.28 and 6.3.23 it will be noted that the normalized

quadrant frequencies are reciprocals.

It can be shown that the system is physically realizable provided

that 03 /o3 or 03 /o3 is greater than \T3, i.e. the two instrument

g/ o o' g * ^ ' natural frequencies are separated by more than 0.396 octave.

Considering the two lines of attack on possible "constant amplitude"

midband response characteristics given in paragraph 6.3.2 the "constant

velocity" case considered in paragraph 6.3.1, and the "constant acceleration" case above, it appears that one may design a system showing a midband response which is either rising or falling with increasing frequency,
88

by a proper choice of the ratio of the two critical damping resistances.

Further, having set k /\ to provide a desired overall response then

the proper choice of series loop resistance for the electrical circuit

result in a mean ratio to critical damping, any desired

amount of "saddle” or "droop" in the midband response. One may therefore,

within limits, adjust the system response for optimum detection capability

for seismic disturbances in terms of the spectral distribution of local

background noise by proper selection of these parameters. Additional

flexibility in the tailoring of response- frequency characteristics is

available as shown in Section 7.

While the flat velocity, direct coupled response characteristic

developed in Section 6.3.1 produces the highest possible gain for a given

ratio of seismic mass to galvanometer moment of inertia, it surfers from

the disadvantage that no conveniently adjustable gain control may be placed

between the two instruirients. That is, its full capabilities require that

each instrument pair be designed for its intended location with a prior

knowledge of the noise background obtaining at that location. While this

requirement is, in principle, not severe, in practice certain difficulties

are encountered. The noise background is often not known in advance and

further, the hard facts of economics suggest that a certain degree of

instrument standardization is desirable. Hence the remainder of this sec- tion is devoted to the following hypothetical but practical problem.

A seismometer and galvanometer are available which will have the

desired amplitude-frequency response when used as a flat velocity fourth

order direct coupled system. The sensitivity with a certain recording j system is adequate to the quietest proposed location. The dynamic range

^ of the galvanometer-recording system is 40 db between noise level and i excessive non-linearity. For any given set-up the operating dynamic

range (smallest to largest useful excursion) is 20 db. The seismometer

and galvanometer can be replicated within adequate tolerance. The ques- | tion is, what is the minimum alteration of frequency response which can

be obtained with the insertion of fixed T pads having attenuations which:

are multiples of, say, 20 db, in order to use replicates at more noisy

locations?

For a T pad with ~ 20 db attenuation, C = 0.1 and 0^ « 1 so that Eq. 6.2.1, with X = 0 and d/dt *= jco, may be written

6.4 Further Remarks on the T Pad Problem

(6.4.1) r
89

Recalling the remark following Eq. 6.2.6, let the values of the

resistors in the T pad be chosen so that X = X » X and using

o g °

<t> ® form

4^

^

00

M

K

m

0

(4> + (4> -i)p(

00o V) 00m

- 2

'OO - 00 V

00m

- &X‘

/OO - 00 » O ^ 00m

4X i (6.4.2)

So that if the attenuator be designed to have X such that

i.e.

/OO + 00 \ /OO “ 00 \ 4X^ \ - 2 - 8X^ = 0

OOm I vr 00m

2X‘

fCO -:. . 00 ^ 2 2

00 + CO R O

(note the similarity to Eq. 6.3.6)

(6.4.3)
90 -

Then Eq. 6. A. 2 may be written

go-+u>^\ 'j

M

K

Z |2.

9 '

/ ^l-^

i\

- O) V o ^ 0)m I/

+ 2 2 2 i CO + Oi i g o ;

(6.4.4)

This response curve may^ for convenience^ be called the "flat velocity fourth order loose coupled case." On comparison v;ich the flat velocity fourth order direct coupled case. Section 6.3.1, it can be seen that

the frequency function is the same but the constants are different (Eq.

6.3.9 compared to Eq. 6.4.4) and the resulting frequency response curve

will, as a result, be altered by more than a multiplicative constant.

First, in the matter of \, several- points, may be brought out. In

the direct coupled case it is necessary that, the critical damping resist - ances of the two instruments be the same, while for che T pad case it

is necessary that the resistors of the T pad be adjusted so that the

instruments are operated at the same fraction of critical damning. In so far as T pads are more easily fabricated, adjusted or altered than the

magnet structures of galvanometers or seismometers, the T pad linked

system is easier to build.

The optimum value of X depends only upon the octave separation of

the natural frequencies of th-e two instruments. For both the direct

coupled and the T pad arrangements X 0 as CO - CO and

as 00 00-00 , while for intermediate values of

g o

'

^ ^

I the optimum value of X will be slightly higher for the T

pad system than for the direct coupled system.

CO CO

For a given value of earth velocity, calling the galvanometer deflection for the direct coupled system 9^ and that for the T pad

system one may write the ratio of the midband gains as
91

fod + O) \ ^

i_-S 9)

00^ + 00^ g o

(6.4.5)

by dividing Eq, 6.4,4 by Eq. 6.3.9b,

Equation 6.4.5 shows that, unless the octave separation of the

natural frequencies is large, the midband attenuation is not given by

a. With regard to the hypothetical question with which this section was

'introduced it follows that for prescribed midband attenuations to be

obtained by the insertion of T pads, Eq. 6.4.5 must be used to obtain

the appropriate values of o: to define the resistor values required to

fabricate the T pads.

Second, as regards the 3 db points, it has been shown by Eqs.

6.3.7 ar^ 6.3.8 that the 3 db points for the direct coupled case are at

CO and cjo . The 3 db points for the T pad arrangement are given by

"

<t>^3 db

CD - CD CD = (- CD

-) (1 + 2

m

m CD*" 4- CD^ (6.4.6)

While algebraic solution of Eq. 6.4.6 is messy, its numerical evaluation

for specific values of cd and cd o g

the direct coupled system response, a

a wider passband. Table 6.1 gives some typical values showing that the

fractional increase in bandwidth for the T pad arrangement is greater

for narrow band (oD^/a) -*• 1) systems.

is relatively easy. As compared to

T pad containing system will have
92

Table 6.1

Computed 3 db Points for Direct and T Pad

Linked Fourth Order Flat Velocity Systems

Natural frequencies of

instruments and 3 db Octave 3 db points when

points direct coupled separation T pad coupled

U) Cb 0) CD 0

0)m

g

CD

ra

CDm

CDm

0.707 1.414 1 0.632 1.58

0.500 2.000 2 0.443 2.26

0.250 4.000 4 0.237 4.21

0. 125 8.000 6 0.123 8.12
93

Section 7

SPECIAL RESPONSE-FREQUENCY CHARACTERISTICS

7.1 General

In the course of this work several systems have been studied which

lead to differential equations of orders higher than the fifth. Of these

only two have yielded results of sufficient interest to justify their

inclusion in this report. The first, which we have chosen to call the

"Notch Filter", appears to be useful for suppressing a narrow band of high

amplitude noise within, or adjacent to, an otherwise interesting spectral

region. Depending on the number of such notches included, the resulting

equations may become of very high order. The utility and operation of a

notch filter are discussed chiefly from elementary considerations. The

equations of motion of such a system are also pertinent to the question of

using several galvanometers with a single seismometer, to record simultaneously two separate spectral regions which may or may not overlap. The

second system of interest is characterized by a sixth order differential

equation leading to overall response curves which we have chosen to call

"the flat velocity sixth", "the flat amplitude sixth" and "the flat

acceleration sixth". They appear to be useful in working with a seismometer

system having very high gain, and, as will be shown (subsection 7.4) are

well adapted to use with telemetering techniques.

7.2 The Notch Filter

Suppose one wishes to set up a broadband, high gain, seismic system

to record information in the frequency range of say 0.03 to 3 cps. For

the system to have high gain and yet keep the total weight of the seismometer

manageable one may elect to use one of the direct coupled arrangements of

Section 6. Unfortunately, one knows a priori, or quickly learns by

experiment, that the desired instrument location has a noise background

consisting largely of microseisms of, say, 6 second period whose amplitude

is such that they completely dominate the record. One might consider sensing

the galvanometer coil position electronically and include within the electronic

portion of the system a suitable filter. While this would eliminate the 6

second energy from the record it would still be present in the earlier portions

of the system and could be great enough severely to affect the dynamic range

of the overall system. It would therefore be desirable to eliminate this

energy as early in the system as possible. This leads to the thought that

a conventional LC trap might be inserted in the electrical loop connecting

the seismometer and galvanometer.
94 -

Considerations of resonant frequency. Impedance level, circuit

resistance, and desired 0 suggest that the inductor and its associated i condenser bank required to produce a suitable conventional LC trap would ! be difficult to obtain, costly and bulky. Referring to Fig. 4.1 it would

appear that, considered as a two terminal network, a galvanometer having ' negligible mechanical damping and a period of 6 seconds might well serve ' as the desired LC trap. Since, as the mechanical damping of a galvanometer

is decreased, R increases, it follows from equation 4.3.6 that the

d I attenuation at the natural frequency of the added galvanometer may, in

principle, be made as great as we please. That is, the amplitude response

curve of the system would have a deep notch centered on the natural frequency * of the added galvanometer.

As to the width of the notch, a detailed mathematical analysis is messy,

j however approximations suitable for preliminary calculations may be developed

as follows:

wherein the total circuit resistance is taken -as R; the portion to the ! right of the dotted line being the electrical equivalent of the added galvanometer.

Calling the natural frequency of the added galvanometer and its

critical damping resistance^,,, we see that when o) and o) are far

“removed from co^, we may approximate the system by the circuit of Fig. 7.1

.1

4 Figure 7.1 1
95

From equation 4.3.5 we may write

e

i - R + Rg

- CO

1 + j Hr

IL

2 2,2 %

' “

j Hr J

(7.2.1)

or « + (2R + Rq) ( 7 . 2 . 2 )

For values of oi well removed from" equation 7.2.2 approaches

[e/i[-*R. The 3 db points at the sides of the notch will occur at those

values of u) for which |e/ip » 2R^. Combining this with equation 7.2.2

gives

+

/.‘4 ~ “3 db \

\2ayii3

J

2

(7.2.3)

In many air damped galvanometers of period shorter than one second, the

mechanical damping may be neglected. Then ()2,j^«Rg»R, and equation 7.2.3

simplifies to

H- ‘”3 db

j “3 db “n

j

or “3 db

(7.2.4)
9b

Table 7.1 shows the notch width, separation of 3 db points, for several

values of such as might be obtained by using a galvanometer with a wide

range of adjustment for(]^^, such as a galvanometer with adjustable flux.

Table 7.1

Notch Width as a Function of Filter Damping

(For galvanometer having negligible internal losses)

3 db points as Approx octave separation

of 3 db points

1/4- ^ 0.781 1.281 0.8

v^2/4 0.707 1.414 1

1/2 0.618 1.618 1.3

'41 jl 0.518 1.932 1.9

1 0.414 2.414 2.9

'J'l 0.318 3.146 5

2 0.236 4.236 9

Having designed a system consisting of a seismometer and galvanometer,

direct coupled, according to a desired amplitude-frequency response curve ; as outlined in Section 6 and having computed a notch filter as indicated

above, it is of interest to insert these parameters into the equations of i motion for a direct- coupled system which includes a notch filter in the |, electrical loop. Such a system is diagrammed in Figure 7.2. f

Figure 7.2
97

Since the electrical circuit is series connected, we may lump the

various coil resistances and any added circuit resistance as the single

resistor R. Using the subscript N to denote the notch galvanometer and

W to denote the working galvanometer, the basic equations of motion, with

mechanical damping neglected, may be written

X-M^=M^ + Sx- GI

dt^ dt'^

^ (7.2.5)

0 =

d^0. —f + VwV

(7.2.6)

dt

dt'^

(7.2.7)

E = G,

^ +

de.. de _N + RI

1 dt Ww J N Jdt

(7.2.8)

These four equations may be used to eliminate x, I and 0^. For

X = 0 (no force applied directly to the bob), E = 0 (no external EMF

applied to the circuit), d/dt = jot (sinusoidal excitation)^ remembering

that 00 ^^ = n/U^^/K^^ and defining Xq as(^^^/R where

the resulting equation may be written
I

98

2 2 2 2

(CD^ - 03 ) (g^ - (X> )

+ 2j(o - (X^ + (7.2.9)

2 2 2 2 ^Tn

+ 2j03 (u) - U) ) (03^ - 03 ) 2

(03^

- 03 )

It Is of interest to note that the parameters of the notch filter

appear only in the final term and also that algebraic manipulation of

equation 6.3.1 will put it into precisely the form of equation 7.2.9

except, of course, for the absence of this final term.

Returning to the supposition with which this subsection was opened, the curve marked with circles of Fig. 7.3 is the response curve for a flat velocit;

fourth order direct-coupled system according to equation 6.3.9 with

03go® 0.06tc and 03 = 6jt. From equation 7.2.4 a notch filter centered on 6

seconds with the 3 db points of the notch approxim.ating 3 and 12 seconds

corresponds to “ 0.333jt and. = 0.750. Using ~ ” 0.693

from equation 6.3.6 together with the above values for the other parameters,

,

equation 7.2.9 yields the curve marked with triangles on Fig. 7.3. The

differences represent the errors which result from first approximation design.

In any real case the errors introduced by ignoring the mechanical losses,

especially in a low frequency galvanometer, may represent a greater deviation

of the resulting amplitude- frequency response curve than suggested by Fig. 7.3

and any final design should take account of the mechanical losses by their

inclusion in an equation appropriate to the special case at hand.

Reexamining equation 7.2.9 it can be seen that as 03

^^ approaches

closely to either U3 or 03 the value of the last term oscillates violently

o g

for small changes of 03 in this region. In particular for o3 and

for 03^<ji3<h3j^ the real term on the right side of equation 7.2.9 may be very

small and the first imaginary term will be positive while the second is

negative. There is one value of o3 for which the two imaginary terms cancel

so that the system gain becomes quite large - possibly much greater than at

any other valve of co. A similar situation arises when the notch filter

frequency is adjacent to, but outside the frequency region bounded by 03^ and 03 .

o

g
99

On elementary grounds the existence of these maxima may be explained

as follows. From the electrical equivalent circuit any of these instruments

I shows capacitative reactance above its natural frequency and inductive

reactance below its natural frequency so that^ when two or more are series

connected, there is always at least one frequency at which series resonance

.occurs. For a single galvanometer and seismometer this has been shown to be

(i) . From this line of reasoning it follows that maxima of this type may be

suppressed at the expense of lessened attenuation in the notch by shunting

the notch galvanometer with a resistor. In so far as the coil resistance of

the notch galvanometer is small compared to the total circuit loop resistance

such an added resistor is operationally indistinguishable from R for the

) d

notch galvanometer.

This line of reasoning shows that the use of two galvanometers in the

same electrical loop to broaden the rejection band is impractical since at

some intermediate value of frequency the system gain will tend to rise to

'substantially the same value as it would have without the added instruments,

Such behavior, in a real system would, generally speaking, be undesirable,

jlhis instability of equation 7.2.9 indicates that in this region the response

of a real system would be dominated by factors ignored in equation 7.2.9,

namely the mechanical losses in the two instruments having proximate natural

frequencies.

An alternative solution to the problem of spurious resonance between

instruments having proximate natural frequencies is to tolerate the increased

mass required.' and use a T pad connection between the seismometer and working

j galvanometer .

' A -notch filter may then be safely placed in that arm of the

T not containing the instrument of proximate natural frequency, provided

„a«l.

7.3 Multiple Frequency Response Systems

Having inserted a second galvanometer into a direct-coupled system to

.reject energy which would normally appear in the working galvanometer it is

' reasonable to ask whether it is practical to record the motions of the notch

galvanometer coil as a means of monitoring the rejected frequency band. On

elementary grounds it would appear that the frequency response curve of the

seismometer- notch galvanometer combination would be the difference between

the "circle” and "triangle” curves of Fig. 7.3. This would be true if, 1)

I the addition of the notch galvanometer did not alter, at any frequency, the

impedance seen by the seismometer, 2) there were no interaction between

galvanometers and 3) the seismometer pier may be considered as a truly

infinite impedance source. The third assumption seems reasonable; in a

direct-coupled system the first two are not.

Equation 7.2.9 takes account of the interactions among the three

instruments. As far as a record of the excurs-ions of the notch galvanometer
100 -

coil is concerned the working galvanometer may be considered as a notch

^ filter on the notch galvanometer. Therefore equation 7.2.9 may be used

; to compute the amplitude frequency response of the seismometer-notch - galvanometer by interchanging the parameters assigned to the working and

notch galvanometers. This has been done for the particular system

discussed in Section 7.2 and plotted as the "square" curve of Fig. 7.3. J

I Since a notch galvanometer will generally be added to reject relatively:

high level signals there is no requirement for a small moment of inertia

in the notch galvanometer. In fact the reverse may be true, as a -^relativelyi

large moment of inertia may be required, since throughout this discussion

it has been an essential though implicit assumption that the notch galvan- , ometer perform as a linear element.

Extension of the above discussion to a system consisting of, say, a one second seismometer series-connected to a 60 second galvanometer and

a 5 cps galvanometer simultaneously is fairly obvious.

The possibilities associated v/ith a system of two galvanometers each ! having added series resistance, parallel connected to a single seismometer

^ appear interesting. The appropriate equations have not been attempted;

however their development, at least for special cases, appears straightforward in the light of the material presented in Section 6.

7.4 The Telemetering Problem

The operation of a seismometer at high gain may require that it be

located at a distance from cultural noise. If the position of the galvanometer coil be sensed electrically then- it becomes feasible to use conven- tional telemetering techniques to transmit the intelligence over great

distances. Conventional telemetering techniques frequently include filters

to restrict the passband of the circuit and thereby reduce the effect of

noise generated in the circuit. Conventionally these filters are designed

to provide a passband only broad enough for the purpose at hand since the

effective noise of the circuit is reduced by limiting the passband. Since

the presence of this filter does alter the system frequency response the

question may be asked as to whether one can take advantage of this fact

and so modify the response of the seismometer-galvanometer combination that,

when combined with filters inserted just ahead of the final recorder, the

overall system has a desired response-frequency characteristic. To show

a general line of attack we will set up the development of the sixth order

flat velocity direct-coupled system in paragraph 7.4.1 and in paragraph

7.4.2 show the alterations of circuit necessary to produce a "flat

amplitude sixth" and a "flat acceleration sixth".
lOi

7.4.1 A velocity sixth" order response

The block diagram for a potentially useful seismometer-galvanometer

system is shown in Fig. 7.4.

transmission

link

• Equipment at the seismometer vault

required

Recorder.

Equipment at the recording site

Figure 7.4

The following assumptions will be made with respect to the block

diagram of Fig. 7.4.

The seismometer has negligible inductance and mechanical losses^

and the symbols previously assigned will be used. R is the total

resistance of the electrical circuit connecting tne seismometer and

galvanometer. It includes the coil resistance of the two instruments,

connecting cables and any additional resistance which may oe aeliberately

t
102

added. Since the galvanometer Is presumed to have negligible inductance

^ and mechanical loss, the symbols assigned and used in Sections 4, 5 and

6 will be used here. Electrical sensing of the galvanometer coil position^

is here preauned to be accomplished by a photoelectric amplifier which

produces a voltage strictly proportional to coil position and independent

of intelligence frequency. It will be presumed to follow the relation

e - ke

•

^ 2 gain at all

be assigned a

The coder-decoder combination will be presumed to have unity.

Intelligence frequencies. The high pass element will

time constant T, so chat the 3 db point for this element

is at The low pass element RjCj will be assigned a time constant

giving a 3 db point for this element at 0O2

. It is presumed that

some type of unity gain isolation is provided between and ^

2^2

'

It is further presumed that the recorder is not frequency selective and

represents no load on the low-pass filter. The equation relating the

output voltage to the galvanometer coil position is

1 i d + W d + 0).N E I/O (7.4.1)

which, for sinusoidal excitation, may be written

Q + 1 + j (

CO,

00 ) ! (7.4.2)

I!

If the time constants of the high and low-pass filters be sec so that

2

cjo,u 12m 3» = CO then the signal attenuation due to the electronic portion of |

, the system will be the same at co and gd . With this restriction, the

^ o ^

' overall system, will have the same gain on § velocicy basis at to and u)

-n ^ ^ when ,scv.o'-'gogm ' or X = X = X . As a matter of convenience the subscript

^For practical instruments of this type k may be 500 to 1000 volts per

radian.

2“Available gair.s may ran^^e higher; the authors have used syscem.s with an overall gain of five. Low level voltage controlled oscillators have been

Duilt which would give gaxns as high as 1000 with practical decoders.

I
i03

of \ will be suppressed in this subsection. Therefore the over-all

system response will be given by using equation 7.4.2 to eliminate 0

from equation 6.3.5 to give

0),

0)m

V K E

» c

1

CO, + r

I (<!> -

COm L

If4> ' - (

CO - CO 2

£

COm J

+ (®^j. o

CD

) (» - J-) + j (< - 7-)-!

m

r

! 12\

r

! (<t> L —+ CO 1

°)/

CO. + C0_ CO (-

CD ^ CO ( m m r J

CO

CO

CO 27

m

(7.4.3)

For any given values of co and co equation 7.4.3 represents a

double infinity of curves depending on the octave separation of CO,

and CO2 and the value of the circuit resistance in the loop connecting

the seismometer and galvanometer^ i.e. X. We wish to select that value

of X and those values of co,12and co- ((Consistent with co,co„ 12m= co^)

yielding the flattest midband response. This is equivalent, to finding

the curve having the highest order tangent to a horizontal straight line

at CO = CO . This is equivalent to finding those values for X and

m

(co, + co_) /co which will make the first n derivatives of the amplitude1 2 ' m

frequency response curve zero, where n is as large as possible.

2 .22

The algebraic symbolism will be simplified by letting c =;k co 2My /co^co^KE

2 2 2

* b = (co, + co„) /co . and a = (co - co ) /co in writing the expression for

1 2 ' m g o m

the absolute value of the gain

4X

2 c *» (<t» - —

1^ l‘^f2 2 22 ) “ ^ ) ! ^ “2a + 4X (a +4)

l^f22 2 99'> 7 42

+ (<D - ja (a - 2b^) + 4X b^ (a*" + 4)/ 4- a^o

L

(7.4.4)
104

1.2

The first and second derivatives of c with respect to (<J> - -)

are both zero at 0) *= 1 when

(a^ “ 2b^) + 4\ c (a + 4) - 0 (7.4.5)

and

2 2 2 2

b"^ - 2a"^ + 4X (a^ + 4) - 0 (7.4.6)

respectively.

Solving equations 7.4.5 and 7.4.6 simultaneously for X and b I* gives

2o3 , = |o3 - Ci I + - CJO )^ -

1 / g O] - g o m

(7.4.7)

I I ! 2 2

2oio = - 0)J± i(oD - oo) - 4co 2 j g oj y

' g o m

(7.4.8) •

and

.00 - 00 -S 2

00 +00

g o (7.4.9)

When the loop resistance R and the filters C^R^) are adjusted according to the upper sign in equations 7.4.7, 7.4.8 and 7.4.9

then equation 7.4.3 takes the forms

, : 00 - 00 1 1 00^ rr . k I g o I _£_ j M y_

00 i 00 + 00 A‘00, .'I K E

' g ojl 1 ^ <

00 - 00 _g o m

00m

/

00 - oo 2 ! O.

00m

0 > J

+ j (0 - -r ) {

L

0-)

00

- 2 (

- 00 2 ]

00 (7.4.10)

m
105

and

,2 oi - CO 2 co-M

k o. _2__

(jo^ 'oo + CO _ CO, K

m g

iL_,

£

o

(. - 6 CO +

CO

-CO 6

2)

m

(7.4.11)

Examination of equations 7.4.7 and 7.4.6 shows that; 1) The

3 db points of the individual RC filters are within the pass-band defined

by'og CO and co for all values of coog-"*^‘ and co and they aporoach coogand co as the pass-band is widened. 2) The use of the lower sign in each

equation represents the "crossed filter" setting of these two filters.

The over-all system gain is reduced without any increase in system dynamic

range^ except for the trivial case where the system dynamic range is

limited by the recorder. 3) There is a minimum separation of co and co o g

for real values of This separation is co /co = 3 r 27 2 or

2.542 octaves

co^ and co^. For narrower bandwidths the values of co^ and co^ given

by equations 7.4.7 and 7.4.8 are imaginary, 1. e. the two RC filters must

be replaced by a single LC filter of prescribed Q.

From equation 7.4.9 it can be seen that, as in the fourth order

direct-coupled flat velocity case (eq 6.3.6), as co /CO X-> 0 while

the limiting value of X is l/2 critical as the pass-band is oroadened.

This limit is lower than for the corresponding fourth order case where X

may approach 0.707 critical.

The system midband gain is (from eq 7.4.10 or 7.4.11 w

1 Eo ("m" k

2

1 “m ) ^‘^r. \

I '«

1 y

CO

^ K co^ 1 2 2

1 CO - CO

' g 0 -

i ico - CO /

^ vs 7

<? = 1 )

(7.4.12)

m

while at co = co^ and = co^ the system gain is

1 nr k /

"COm

!co , CO “ r

j o’ g ^2 V K CO, 2 _ . 1 I '"g

- “o ' ' *

\f COm

CO - CO 7

I E

o y

(7.4. i3)
106

Therefore the sixth order flat velocity direct-coupled system resembles

the fourth order case in that the 3 db points for the overall system f gain are at cjo and oo . i 0 8

^ Having determined the requirements for the sixth order flat velocity }j system it is of interest to note the gain, as observed at the galvanometer*

mirror, at cn^, midband, and Inserting the value of a. given by

equation 7. A. 9 back into equation 6.3.5 one finds that

• >

(7.4.14)

He is led to conclude that a devious method has been used to find those | values for the time constants of a CR-RC filter which will make it 3 db | down at cd and o) vis-a-vis co (equations 7.4.7 and 7.4.5) and that

o g m

^ value of \, or. circuit loop resistance which yi-elds a midband droop of

3 db vis-a-vis the fourth order direct coupled flat velocity system, cf.

equation 6.3.7 vis-a-vis equation 7.4.14.

As a matter of interest one may readily compute the quadrant points

of the phase- frequency response curve for the sixth order direct-coupled i flat velocity system. Taking '\if as the angle by which, for sinusoidal

excitation, the voltage at the recorder leads the earth velocity we may

j write, from equation 7.4.10,

(«> -hrN / ' tan y

-

4)'' ~

03 - CD 1 2 7

2 ( -V °

) -(-hi

CD - CD2_

! CD

/ ^

CD - CDh]

(7.4.15)

m J
107

where o) - o) in the denominator is always to be taken positive. The

g o results are given in Table 7.2 from which it can be seen that there is

a total phase shift of the output of three quadrants across the pass-band,

as compared with the two quadrant shift characteristic of the fourth

order case.

Tab 1 e 7.2

Phase Lead of the Recorder Voltage with Respect

to Earth Velocity for the Sixth Order

Direct-Coupled Constant Velocity System

CO radians/sec electrical degrees

0 270

180

135

o

"2 2

) + 4o

m

90

0

m

90

135

g

g o . i g o 180

00 270
108

7.4.2 A "Flat Amplitude" and a "Flat Acceleration Sixth" order response

Consider a pair of decoupled RC filters. These may be arranged in

three ways giving three slightly different differential equations

'=1

if03, R,

it

032

d^e

dt

(7.4.16)

S

03,

^2

VWV

03«

^^2 d d

03„ 7- = (j- +03.) (j- +03.) E,

2 dt ^dt 1' Mt 2' o

(7.4.17)

AAAA-|

03i —

2 WW03.

(7.4.16)
- :g9 -

In section 7.4.1 use was made of equation 7.4.17 with the relation

€2 “ If in Fig. 7.4 one interchanges the position of and

then equation 7.4.18 applies. A derivation paralleling, in detail, that

leading to equation 7.4.3 will yield equation 7.4.20. Alternatively the

reader will note that the quantity de

2

in equation 7.4.17 is replaced

dt

by equation 7.4.18. If one replaces y in equation 7.4.3 by

y one obtains

1m “i^

03 V’ 03, V K

m 1

^ = k I- =

V K E right hand side of eq. 7.4.3

(7.4.19)

Then under the restrictions of equations 7.4.7, 7.4.8 and 7.4.9 on

03i, 03^ and X, equations 7.4.10 and 7.4.11 become

03 03

03 +03

g o l^y.

K E

03 “ “of

/'

03

tn 11

I !c

2(b - r>

2^ 03 - 03 )

03

in (7.4.20)

+ j

(i> )^ ($ - - 2

03 (_a.

03

03 m

and

(^ “ V, / ^

o. ^ ^

03go+ 03 K I ^1

/ ol

03 - 03 = (<t> - r-) + (

03m

(7.4.21)
no

The reader will note that the frequency variable function (right

side) is the same in equation 7.4.10 and 7.4.20. The gain variable in

equation 7.4.10 is earth velocity while in equation 7.4.20 the gain

variable is earth amplitude. That is, we now have a "flat amplitude

sixth order response" (eq. 7.4.20). The phase-frequency relation for

equation 7.4.10 (Table 7.1) is changed for equation 7.4.20 only in that

the reference phase is earth displacement rather than earth velocity.

The designer will note that the Interchange of and in Fig. 7.4 removes from the system all high pass filters other than that associated

with the mechanical portions of the seismometer and galvanometer. In

particular the displacements on the record will include drift of the

galvanoiTieter rest-point and of the electronic equipment as well' as d-c

offsets, if any. He may, fortunately, insert high pass filters as required provided their time constants are so long cotripared to the

longest period of interest that they have negligible effect on the

amplitude and phase response in the pass-band.

Turning our attention to an interchange of and C

2 in Fig. 7.4

rather than R^ and as was done above then we must use equation

7.4.16 rather than 7.4.17 or 7.4.18. Doing this, equation 7.4.3 becomes

I" " ^r e" " 7.4.3, (7.4.22)

o m o

R [m

. CJD, * K

2 m 1

The equations corresponding ’to 7.4.10 and 7.4.11 may be written

, 1 oi - GO : r~ .. k \ _g 01 I ^ ^ 2 1 ! T OO I GO + GO ' V K E m I g o, ‘ o

2(0

00 - 00

r g O

GOm

2

'

I

and

+ j (0

GO - GO 2 (• GOm

-) rj

(7.4.23)

GO

o> -r GO g

Lj = (0

K E / o

'

6

GO - CO + -“)

GOm

(7.4.24)
Ill

The reader will note that the frequency variable function (right

side) is the same in equation 7.4.10 and 7.4.23. The gain variable in

equation 7.4.10 is earth velocity while in equation 7.4.22 the gain

variable is earth acceleration. That is, we now have a "flat acceleration sixth order response" (equation 7.4.24). The phase- frequency

relation for equation 7.4.10 (Table 7.1) is changed for equation 7.4.23

only in that the reference phase is earth acceleration rather than

earth velocity.

The designer will note that the interchange of and in Fig.

7.4 removes from the system all low pass filters other than that associated

with the mechanical portions of the seismometer and galvanometer. In

particular the displacements on the record will include the telemetry

carrier, line pick-up, hum, electronic equipment noise, etc. He may,

fortunately, insert low pass filters as required provided their 3 db

points are at ' frequencies so high compared to the highest frequency of

interest that they have negligible effect on the amplitude and phase

response in the pass-band.
112

Section 8

SENSITIVITY LIMITS IN PRINCIPLE

AND IN PRACTICE

8.1 General

I

In this section we shall discuss some effects introduced by the

extraneous variables to which any real (as distinguished from ideal)

seismic system is subjected in the course of "normal” use. Throughout

this section it will be presumed that one is attempting to detect the i smallest possible motions of the seismometer pier under the most

difficult of field conditions. It will also be presumed that electroni( means are used to sense the angular displacement of the galvanometer

mirror and to relay this information to a recording location. The

limitations in principle are considered with respect to idealized instruments^ particularly in that each is presumed to have one and only

one (iegree of mechanical freedom.

8.2 Brownian Motion

Brownian motion^ or the thermal agi-tation of the movable portions

of the mechanical system is not ordinarily thought of as being great

enough to represent a sensitivity limitation on real systems. It does

however represent a noise source which cannot be reduced by refinements

of design or construction and therefore represents a limitation in

principle on the maximum useful magnification of a system.

It will be presumed that the seismometer, the galvanometer, and

the interconnecting cables are all at some constant temperature, T,

degrees Kelvin. Any mechanical system has an average kinetic energy

associated with it amounting to ^l/2 for each degree of freedom,

where ^is Boltzmann's constant. This is the inherent thermal energy

of the device and actually defines its temperature. The spring-mass

combination of the seismometer or galvanometer has one degree of mechanical freedom^ with which is associated an average kinetic energy of

1t/2, an equal average potential energy and therefore an average total

energy of The potential energy considered here is that increment

of energy stored in the spring due to the displacements of the mqss | from its long- time average position. The kinetic energy is that energy - stored as velocity of the bob with respect to the frame in its excursions

in the neighborhood of its long time average position. Any persistent ' change in these energies may properly be said to be a change in the

temperature of the spring-mass system.

We are here explicitly ignoring modes of motion generally classed as spurious resonances.
113

A point which has sometimes been overlooked in the literature but

which seems quite clear from the above remarks is that the coupling of

any other system, which is at the same temperature, to a spring-mass

combination cannot alter the long time average of either the potential

or kinetic energy of the spring-mass system. If there were any change

in the long time average of these energies it would represent a permanent

displacement of thermal energy between two systems initially at the same temperature, in clear violation of the second law of thermo-dynamics.

Specifically the Johnson noise in the electrical circuit arises

from the therrrial agitation of the electrons in the conductors. When

the resistors are at the same temperature as the mechanical components,

there can be no net flow of energy between the electrical and mechanical

degrees of freedom. Therefore the mean energy of thermal agitation of

the bob of the seismometer and of the armature of the galvanometer is

neither increased nor decreased oy connection to the electrical system.

> We do not mean to imply that the spectral distribution of the noise

amplitude of a spring-mass system is independent of its coupling to other

systems - it almost certainly is not - but the rms value of the Brownian

motion cannot be altered by coupling between systems which are at the same temperature.

Proceeding on this basis it becomes a very simple matter to compute,

quantitatively,- as a function of frequency, that nris value of sinusoidal

earth motion which will produce galvanometer excursions whose rms value

is equal to the rms value of the Brownian motion of the galvanometer

coil.

Each of the amplitude- frequency response curves relating seismometer

excitation and galvanometer response which has been developed may be put

in the form

where F is a dimensionless function of frequency and the constants of

the system.

For a sys-tem in which the final record gives equal weight to equal

galvanometer velocities then eq. 8.2.1 may be rewritten as

( 8 . 2 . 1 ) CD

mIyI

( 8 . 2 . 2 )
114 -

The left side of equation 8.2.2 is in the fonh of the ratio of

two kinetic energies. The denominator is the kinetic energy of the

galvanometer coil resulting from a sinusoidal earth velocity, y, at

a frequency co . If, for the denominator, we substitude the mean

kinetic energy of Brownian motion we get

2

My (8.2.3)

00^

from which, assuming a value for the absolute temperature T, one may

calculate, at various frequencies, that sinusoidal earth velocity

which will produce the same mean kinetic energy in the galvanometer

as the average value due to Brownian motion.

On the other hand, for a recording system which gives equal

weight to equal galvanometer displacements then equation 8.2.1 may

be written in the form

I M y

4 U 0^

= —%

F

00g

(8.2.4)

so that

My2 =|j (8.2.5)

is the appropriate form for computing that value of sinusoidal earth

velocity which will produce the same mean potential energy in the

galvanometer as the average value due the Brownian motion.

The difference between equation 8.2.3 and 8.2.5 is related to

t he frequency distribution of Brow-nian mo tion energy. _

It is of interest to note that the various amplitude- frequency

response curves which have been developed have been quite independent,

both as to form and as to system gain, of whether the seismometer

or the galvanometer had the higher natural frequency Equation 8 2.5

shows that as regards sensitivity limited by Brownian motion one can do

better, for a given inertial mass, by using a high frequency galvanometer and low frequency seismometer rather than vice versa.
- 115

We leave it to the reader to decide what ratio of signal to

Brownian motion "noise" represents a "just detectable signal" as well as how to arrange the recording system so as not to inject

noise of effective amplitude greater than that due to the Brownian

motion of the galvanometer coil.

An interesting corollary of equation 8.2.3, obvious as soon as

it is pointed out, is that, given a recording mechanism adequate to

record the Brownian motion of any galvanometer, the minimum detectable

earth motion is independent of the moment of inertia of the galvanometer

used. The availability of such a recording mechanism would ease the

problems of the galvanometer designer considerably.

Another point which is samewhat obvious is that, while equations

8.2.3 and 8.2.5 have been developed assuraing that the useful sensitivity of a proposed system is Brownian-motion limited, corresponding

equations can be written for any less sensitive recording system by inserting the appropriate value of galvanometer rotation. We also see

that by doubling the useful sensitivity of the recording system one

may reduce by a factor of four the mass required to record a given

earth motion. While this may be trivial in situations requiring only

a few grams of inertial mass the same is not true when tens of kilograms are .required.

...Returning to the relative merits of T pad and direct coupling

between the two instruments consider two systems based on the same galvanometer and recording mechanism, one containing a T pad and the

other direct coupled. Assetming that the over-all response curves of

the two systems are approximately the same then, for a factor of ten

attenuation of the trace amplitude due to the T pad, an increase. in

the inertial mass by a factor of 100 is required to obtain earth motion

sensitivity with the T pad system comparable to that of the direct

coupled system. For high gain systems where the m1 nxiTiUTTi mass otctateci

by Brownian motion considerations is on the order of a kilogram this

factor weighs heavily in favor of direct coupling.

S.3 Quantum Effects

Seismic systems can be devised with such calculated gains as to

give reasonable trace amplitudes for pier displacements of atomic

dimensions. The question may be raised as to whether quantum con- siderations with their concept of finite energy increment represent a limitation, in principle, of system performance.

2 On the latter point it should be mentioned that shere are commercially

available galvanometer-photoelec trie assemblies whose equivalent input noise

computes rather closely to the expeccea brownian motion
116

Three such possibilities come to mind: (a) The application of the

Heisenberg uncertainty principle to the position of the galvanometer,

seismometer, or pier. This sets a limit to the product of the un- certainty in the coordinate and the uncertainty in the momentum. As

applied to a galvanometer this may be written

(A6) (A Ku>0) — h/27t-^).0 joule sec,

-4 ^ -11 2

If K 10 gr cm or 10 kg m

and the highest frequency of interest be cu = 10 /sec

then lO"® (A0)^ 10

-34

-13

A0 10 radians, a value much smaller than can be

observed in the present state of the art. For a seismometer of 1 kg

• 18

mass, one obtains Ax~' 3x10 meters, again far beyond present

measurement techniques. For the pier the mass is larger and the cor- responding displacement smaller. (b) One might be concerned as to

whether energy is transmitted from the earth" to the seismometer and

galvanometer in quantum packets of observable size. (c) There is a possibility that the "noise” background is quantized in observable

steps. In either of these cases the quantum step, of the order of

h\3, must be significant in comparison with the average instrumental

background which corresponds to a total energy in the galvanometer of

at least^T.

The importance of these considerations may be estimated in terms

of the quantity h\3/|f^T which appears prominently in the Dirac theory of

specific heats, the Planck radiation theory, and in fact throughout the

quantum statistics. Since Planck's constant h is of the order of

-34 -21

6.6 X 10 joule seconds and At is approximately 4 x 10 joules

at room temperature, h\D/j^T has significant magnitude for frequencies of

12

10 cps or more. This is above the frequencies with which this study is

concerned. In view of these considerations it appears that a classical

approach to the problem will be adequate in the present state of the art.
117 -

8.4 Spurious Voltages in the Electrical Circuit

In the construction and installation of a high sensitivity

seismometer- galvanometer system one naturally attempts to select materials

for the electrical conductors and to arrange the conductors so as to

minimize thermo-electric voltages and stray electrical pickup in the

conducting loop. In principle these spurious .voltages may be reduced to zero. In practice one may be more or less successful in reducing these

disturbances to insignificant levels. In this paragraph we shall show

that one may readily compute the quantitative effect ox such voltages

as a guide to indicate their relative importance in any given system.

Let us take as an example a direct coupled system having negligible

inductance, as shown in Fig. o.l. Here represents the combined effects

of thermo-electric junctions under the influence of fluctuating temperature

and theftemf due to magnetic induction, e^. is taken as the voltage generator

O

Figure 8.

simulating electro-static induction in series with c, the. capacity between

conductor's in the inter- instrument cabling. It has been assumed, wicn re- spect to e that all the resistance is at the seismometer. This is the

worst case^with respect to electrostatic induction.
118 -

The pertinent equations nvay be written

(M^ + S) X - G,I

0 + 0) 0 - G .(I + c 3

em *’1 dt * G de + RI

8.4.1 ^

)

I

\ 8.4.2 I

f

I 8.4.3 ^

i

Eliminating x and I, letting d/dt = joi, remembering that 2X O) =

o o G,^/RM and 2X a) = G ^RK one may write

1 ' g g 3

>/2\ 0) o o ^

) ®ni +

- 0)^ + 2j ^ vr

(-

o o

00 ) joicv R Cg ='

00 CO 00 00 00 to 00 00

.) (_S ) + 2j (-S ) + 2j (- 1

00 00 00 00.g

00 00g‘ g 00 GO u

O ! \/ 2X GO J g g

8/4.4 i

1

I The reader will note that each of the four terms in equation 8.4.4

has the dimensions of the square root of power. More important is the

fact that a given earth velocity raultipled by ^he coefficient of y and I divided by the coefficient of either em or 6g will give the magnitude of j that foreign voltage which, at that frequency, >7ill produce a galvanometer ' deflection equal in magnitude and phase to that produced by the given

earth velocity. The galvanometer parameters do not enter the ratio of 1 earth velocity to either foreign voltage for a given galvanometer excursion
- 119 -

In the equations for the various direct coupled cases, the reader

may have noticed that only the ratios of circuit resistance to the critical

damping resistances entered the final equations. No preference between

high and low impedances could be inferred from the equations . Here we see

that the effect of thermoelectric or electroraagnetically induced voltages

is reduced by raising the instrument critical damping resistances as high

as engineering limitations will permit so as to have the highest possible

circuit resistance. The improvement will be proportional to the square

root of the increase of resistance. On the other hand, the effect of

electrostatic induction is reduced by lowering the critical damping resistances and circuit resistance. A practical compromise would minimize the

combined effect.

rt is probable that the electrostatic effects can be reduced to

negligible proportions by reasonable shielding of the instruments and

cable as well as the use of twisted pair cable. If any added resistor or notch filter galvanometer is used, it should be placed at the working galvanometer rather than at the seismometer. Certainly all foreign voltages

will be reduced by keeping the cable lengths between instruments as short

as possible and enclosing the instruments in conducting containers.

It may be very difficult to eliminate the effects of electromagnetic

induction in the coils of the- instruments and thermoelectric emfs at cable

junctions, binding posts, etc.

As an aid in visualizing the frequency dependence of the effect of

thermoelectric voltage, and to show the effect of interchanging natural

frequencies, the values in Table 8,1 have been computed. We have chosen

a fourth-order-flat-velocity direct-coupled system, with natural frequencies

of 0.8 cps and 5.0 cps and a series loop resistance of 144 ohms. From

equation 6.3.6, is 0.512. The seismic mass is one kilogram. The values

of are those equivalent to earth displacements of 1 millimicron

(10"^ meters) so that a number having larger absolute magnitude represents

a more desirable situation.
120 -

Equation 8 4.4 shows that at the natural frequency of the seismometer

a thermoelectric voltage has no effect on the record and Table 81 shows

a very sharp maximum in equivalent thermoelectric voltage at this frequency

In the development of specific amplitude response equations it has

been pointed out that overall system gain is quantitatively independent

of whether the seismameter or galvanometer has the lower natural frequency.

With regard to the effect of thermoelectric voltage it can be seen that

the system in which the seismometer has the lower natural frequency is

slightly more tolerant in the pass-band, much more tolerant at frequencies

below the pass-band of Interest and slightly less tolerant at frequencies

above the pass-band, to foreign voltage in the electrical circuit.

8 . 5 Air Buoyancy

The vertical accelerations of the earth's surface and the pier as a result of changes of atmospheric pressure are not considered as limitations

on system sensitivity since purely seismic equipment cannot distinguish

among various causes of pier motion.

The buoyant effects of the atmosphere acting directly on a seismic

system will depend in large measure on the details of construction of

individual instruments. In horizontal instruments bouyant effects, since

they are perpendicular to the axis of motion, probaoly may be neglected.

Vertical component instruments, on the other hand, are sensitive to

variations of atmospheric pressure. Diurnal variations oj. atmospheric

pressure amounting to several percent are the rule in the temperate zones

and, insofar as they are outside the pass-band of the equipment, the

effect is chiefly to vary the rest-point of the inertial mass. This presents no problem for an instrument with an adequate range over which the

rest-point may be varied without significant change in operating parameters.

The effects of barometric pressure fluctuations within and adjacent

to the instrument pass-band are more severe. For any particular seismic

system they may be estimated according to the following line of reasoning.
- IZl -

Table 8.1

Thermal Voltage Equivalent to

One Millimicron Earth Displacement

Fourth Order, flat velocity, direct coupled system, one kilogram

seismic mass, natural frequencies 0.8 and 5.0 cps, resistance 144 ohms

Frequency Earth Velocity Equiv. Voltage - Microvolts

cps millimicrons/second Seis at 0.8 cps Seis at 5 cps

0.10 0.628 + 2.7 X O

1 + 1.7 X O

1

0.20 1.26 + 2.3 X 10-3 + 1.4 X 10-^

0.50 3.14 + 5.5 X 10-3 + 2.2 X o

1

0.70 4.40 + 3.9 X

10-3 + 6.0 X 10-3

0.80 5.02 • + OC + 9.0 X 10-3

1.0 6.28 - 4.7 X

1 o + 1.8 X 10-3

2.0 12.6 - 4.1 X

10-3 + 1.6 X 10-3

5.0 31.4 - 8.8 X

10-3 + OC

10 62.8 - 1.7 - 5.7

20 126 3.4 - 9.

50 314 _ 8.6 -21.6
122 -

The equations of motion of a seismometer all involve a term

(X - Md^y/dt^), c.f. equation 2.6.10, from which frequency response

equations have been developed by suitable combinations with other

relations appropriate to the system under consideration. In obtaining

equations for response to earth motion, y, we have assumed in the course of the algebraic manipulation that there were no forces acting

directly on the inertial mass, i.e. X was taken as zero. Here we are concerned with the system response to forces applied directly to the

inertial mass when earth motions are presumed zero. Had we made the

alternative assumption that y = 0 and performed precisely the same algebraic manipulations with X as the independent variable we would

have the previously obtained frequency response equations with y

replaced by X/cd^M or y replaced by -jX/aM. (Since we are concerned

with absolute amplitude relations the coefficient - j may be ignored)

Therefore, the galvanometer response to sinusoidal earth motion and

to bob force will be the same when

y or X/cM y 8.5.1

Let Ap^ be the change in density of air and be the effective

density of the bob. Then the variations of buoyant force X = gMA-p^/p^^,

and equations 8.5.1 become

gAP

§ /^y -or

') g

oa

y

PM

8.5.2

The effect of air buoyancy changes is, therefore, independent of

the mass of the bob and becomes important as the frequency is lowered.

Also, obviously, the effective density of the bob should be high.^

As seismometers are usually used, the variations of air density

with humidity and temperature at constant pressure occur so slowly as to be lumped with diurnal pressure changes. Variations of air density

3

This might lead to a preference for brass over aluminum for

the bob, but would not justify the use of lead.
123 -

with pressure are given by AP/P - with sufficient accuracy for

our purposes. Equations 8.5.2 may, therefore, be written

gp AP gP^AP _a_ = y or = y

PmPo. PjjPm

8.5.3

There seems to be very little data in the literature on variations

of atmospheric pressure to be expected at periods on the order of one to

thirty seconds. To get some idea of the order of magnitude of the effects

to be considered one may assume ah inertial mass having an effective

density of 8.6 g. fee (brass), the density of air to be 1.2 mg/cc and

compute the magnitude of that pressure change which is equivalent to one millimicron (10” ^ meter) of earth displacement at a period of 6 seconds

(o3 = 1) . The computed pressure change is 0.5 microns of mercury or 0.7

dyne/cm^. This is the change in buoyancy which would result, in a quiet

atmosphere, from raising the instrument 8 mm!

From the above it seems obvious that any vertical seismometer

intended for use in a high gain system should be sealed in a rigl^ case.

8.6 Air Convection

The possibility that convection currents around the inertial mass

result in spurious deflections of the galvanometer snoald be considered.

This again represents a possible limit in practice rather than in principle.

Some data bearing on this question was presented by Messrs. H. A. Bowman,

L. B. Macurdy and H, E. Aimer at the meeting of the American Association

for the Advancement of Science held in Washington, D. C. in Decemoer 1958.

They found that with the air at the top of a balance case 0.1 °C

colder than that at the bottom, convection currents were set up which

produced erratic motions of the balance pointer equivalent to as much

as several tenths mg unbalance of the pans (peak to peak). The pans

of the balance used each had an area of 84 square cm. If one assumes

that convection currents of comparable magnitude exist around a seismic

mass’ of comparable projected area then Equation 8.5.1 may be applied.

Assuming 0.1 mg unbalance at a period of 6 seconds (cu = 1) and a mass

of one kg the effect would be equivalent to an earth motion of 0.98

microns or 0.98 microns/sec. Turbulence due to vertical thermal gradients will, of course, be absent when the gradient is less than about

4

To be published in NBS Jour, of Research.
124 -

0.01“C per meter (the adiabatic lap&e rate for dry air) so that in

practice there is a naturally occurring stabilizing effect against

turbulence due to vertical gradients.

Horizontal gradients produce similar effects without, however, a stabilizing effect analogous to the adiabatic lapse rate. Fortunately

when the vertical gradient is less than the adiabatic lapse rate it serves to suppress the effect of horizontal gradients, especially when

the latter are small.

These gradients are, however, clearly factors to be reckoned with.

Three courses appear open to the designer. 1) The sealed, rigid case suggested in Section 8.4 may be made of a good thermal conductor and it

may be supplemented by thermal insulation., 2) He can provide a close

fitting shield around the inertial mass and the moving portions of its

support. By close, we mean clearance on. vertical surfaces of the order

of 3 mm and on horizontal surfaces on the order of 6 to 8 mm. 3) He can arrange to keep the top of the seismometer case slightly warmer

than the bottom by dissipating a watt or so of power at the top of the case. This is the most reliable way of avoiding convection currents.

Convection currents around a galvanometer mirror would, at first

glance, appear 'to be more troublesome than those around the inertial

mass. Fortunately several factors generally existing in a seismic

system tend to suppress this effect. First the system response to earth

motion is effectively increased in magnitude at this point and second,

the galvanometer case is generally small. This decrease in size makes

it inherently more nearly isothermal and also the suspension is automatically moderately well baffled. The use of a light beam does imply

the presence of a local "hot spot" wherever light is absorbed in either

the suspension or the case, so that care should be taken to be sure that

the beam falls only on reflecting portions of the mirror and is of as low a total intensity as can be used (this will also tend to reduce

thermo-electric effects in the galvanometer)

8.7 Light Pressure and the "Radiometer Effect"

The possibility that photon pressures could result in spurious

signals which are large enough to concern a seismologist seems utterly

ridiculous until one realizes that there is no a priori reason to assume that their effects are either greater or less than Brownian

motion effects

A seismometer in a sealed opaque case, all of whose parts are at

the same temperature, is in a region of uniform radiation density and

the inertial mass is tnerefore subjected to no unbalanced forces due to
125

light pressure. On the other hand a galvanometer with an optical sensing system operates under quite different conditions of photon balance.

A general solution for the forces involved is not so graphic as a numerical example.

Assume that the total photon emission of the galvanometer lamp is one watt and that the collimating lens is 2 cm in diameter at a distance

of 10 cm from the filament. The lens subtends- a solid angle at the filament of 0.01 steradian or 2.5 x 10"^ part of the sphere. Assuming 607o

losses in the optical system the energy striking the mirror is one milliwatt or 10^ erg/sec. A.ssuming that the center of the light spot is 1 mm

off the axis of rotation of the galvanometer and that the light is totally

reflected then the torque due to light pressure is

2 X TO^ X —rrjr X 10 ~ X 10 ^ dyne cm.

3 X

10^^

P

If the stiffness of the galvanometer suspension is 0.12 dyne cm/radian

the resulting deflection is 5.56 x 10 ^ radian.

The energy stored in the galvanometer suspension is

1 ( U0 ) 0 = I

X IX 10'^ X 5.56 X 10"^ = 1.85 x lO'^'^ ergs.

This may be compared with the Brownian motion energy for one degree

of freedom which, at room temperature, is about 4 x 10"^^ erg (averaging

1/2 kinetic and 1/2 potential). That is, for the assumed conditions, the

steady state deflection due to photon pressure represents about the rms

value of the deflection due to Brownian motion, and therefore photon pressure per se need give us no concern.^

The radiometer effect on a galvanometer coil is not capable of so clean cut an evaluation as photon pressure. Since early attempts to

detect photon pressure were completely obscured by the radiometer effect,

there is a strong suspicion that it is several orders of magnitude larger

than photon pressure. Certainly, in any case, a light beam should be

well centered on the galvanometer mirror and so masked that it cannot

strike less reflecting parts of the galvanometer.

For instance, a 10% change of light intensity would represent one- tenth

the effect of Brownian motion. it would be difficult to balance a phototube amplifier so that it would tolerate so large a change in lamp intensity.

Therefore, the effect of variation of light intensity on the phototube would

swamp the variation of light pressure on the galvanometer mirror.
126 -

8.3 Magnetic Noise

The operation of any transducer whose output depends upon the inter-,

action of a magnetic field and a conductor will be adversely affected y anv effects which cause the field to change in an unpredictable manner. , One such effect, known as the "Barkhausen effect" or "Barkhausen ^jo^se

results in abrupt changes or steps in the magnetic field associated with

| ferromagnetic materials. Its origin is presumed to be abrupt upsets of

, Individual magnetic domains. Equipment capable of demonstrating Barkhaus

noise is easily assembled and it is readily observed when the magnetic

induction of a ferromagnetic material is sufticiently great that increased

induction is not readily obtained by the orderly

oriented domains at the expense of their neighbors. This form or noise

, is only observed as the magnetic induction is changed and xs not detect- | able at low values of the induction,

|

In moving coil transducers the magnetic induction of the ferromagnetic materials is perturbed only by the currents generated in the

electrical circuits. The perturbations may safely be considerea as too

small to generate Barkhausen noise.

In transducers of the variable reluctance type there may be rela-^

tively large changes in magnetic induction of the iron witnin tne coixs

as the mass moves relative to the frame, so that it may in fact oe possible for the Barkhausen effect to be present and geL.erate spurious

^ signals. In instruments of the variable reluctance type, as^ presently

constructed, Barkhausen noise . is avoided by keeping the total magnetic

induction of the iron sufficiently low.

Another source of magnetic field fluctuation must exist in principle although to the authors' knowledge it has neither been named

nor observed. Consider a magnetic structure composed of a permanent

magnet, soft iron and an air gap. Since the atoms of both the magne.

A 100 kg horizontal instrument of the Benioff variable reluctance type was examined for Barkhausen noise by Mr. Sanford of the

Magnetic Measurements Section. The eifect was not detected.
127 -

and the iron are in motion due to thermal agitation, there must be random

fluctuations in both the magnetomotive force and the circuit reluctance.

These fluctuations result in random fluctuations of the strength of the

field in the air gap. Since the magnet is presumed to be in thermal equili

brium with its surroundings and with any electrical circuit to which it

may be coupled, again we have a circumstance wherein the act of coupling

may change the spectral distributions of the energy fluctuations although

it cannot alter the time average of the thermal energy in either the

electrical or magnetic circuit (cf. subsection 8.2). We have therefore

another instance of a noise source whose effect on the system is not to

reduce its theoretical ultimate sensitivity.

By way of recapitulation we should like to remark that thermal agitation as manifest by fluctuations of a magnetic field is nameless. When

it is observed as the motion of a particle it is called "Brownian motion".

When observed as a voltage fluctuation, the phenomenon is called "Johnson

noise". The use of different names for different manifestations of the same basic cause does not warrant the assumption that by coupling between

the phenomena one can cause thermal energy to flow persistently from one system to another at the same temperature, in violation of the second law

of thermodynamics

8.9 External Magnetic Fields

The possibility that a high gain seismometer using an electromagnetic

transducer may be influenced by externally created weak magnetic fields

must be considered. The magnitudes of these effects will be markedly a function of the detailed construction of individual instruments so that we

can only indicate some of the coupling meclianisms involved, leaving calculations or measurement of the expected magnitudes to the designer or user. A magnetic field, whether uniform or having a gradient in space, which is

constant in time will affect -- at most -- the rest-point of the instrument

When a magnetic field varies in time, a number of possible effects must be

considered.
128

8.9.1 Field Unifonn in Space, but Varying In Time

Although it is possible to have a pier and shelter in which the magnetic field is quite uniform before installation of an instrument the

presence of magnetic materials in the instrument will introduce gradients

of the field so that the uniform field condition is a somewhat idealized

situation. For a unifonr. field the effect of time variations is chiefly

to Introduce spurious voltages in the coils of the instrument (cf. Section 8.4). One may assume as a worst case that tha iron surrounding a moving coil instrument presents no shielding or as a best case that the

iron contained in the coils of a variable reluctance pick-up has a negligible perturbing effect.

It is obvious, though not generally realized, that a magnetic field

which would be uniform in space in the absence of a magnetic material

can exert only torques on the material. Such a field cannot exert a translational force even though the magnetic object has fringing fields

or definite poles of its own. In considering a specific instrument it

must be recognized that the structural linkages may couple this torque

into the degree of freedom being monitored. These points become of

special interest when one considers springing a magnetic structure as its own seismic mass in an attempt to build an. instrument having a high

ratio of sprung mass to total mass.

8.9.2 Field Non-Uniform in Both Space and Time

When a magnetic field does have a space gradient then translational

forces are exerted on magnetic material and the time variation of these

forces must be considered. These effects are in addition to the spurious

voltage mentioned above. They could well be quite significant during

geomagnetic storms especially on a sprung mass having fringing fields.

For any given instrument these effects can be calculated (at least

in principle) and the computed forces inserted in equations 8.5.1;

however, the user will certainly prefer the result of tests on assembled

instruments.
129

Section 9

CALIBRATION

9 . 1 General

In considering the calibration of a seismic system it will be

convenient to discuss the subject under three headings: 1) Over-all

system calibration by use of a shake table, 2) application of known

forces to the inertial mass or known emfs in the electrical circuit

(semi-direct calibration) , 3) measurement of appropriate parameters

of individual components or sub-assemblies, (indirect calibration).

This choice is admittedly somewhat arbitrary.

9.2 Shake Table

To the user who is interested in knowing the performance characteristics of his equipment with an accuracy of say 1 to 5T{>, a shake

table which simulates pier motions represents the most direct way of

obtaining over-all system calibrations. Unfortunately a good shake

table, aside from its cost, is a piece of laboratory equipment, and

laboratories are usually noisy locations. Thus calibrations must be

run against the relatively high noise level of the laboratory. This

requires that the seismometer have a large d3mamic range, so that the

parameters at high and low levels are the same. Also, there is always

the possibility that after calibration, the instrument parameters may

change during shipment and installation.

Many workers have attempted to devise relatively cheap and simple

shake-tables for permanent installation between the pier and the instrument. As such a device seems not to have^been generally adopted,, we

may presume that the attempts have not been too successful.

9.3

Semi-direct Calibration

In the course of developing equations relating a recordable output

to pier motions, relations were noted between the pier motion and forces

applied directly to the inertial mass which will produce equal recorded

output (cf. equations 8.5.1), Also, a relation has been shown between

spurious voltage in the electrical circuit and earth motion for equal

recorded output (cf. equation 8.4.4). For want of a better term we will call methods based on these relations semi-direct to imply that

they represent calibration of an over-all, operational system, but are not so unambiguous as the shake table, since one must presume that a relation such as equation 8.5.1 is valid.
130

In this section we will discuss possible ways of calibrating an operational system based on these techniques.

9.3.1 Weight Lift •

The weight lift measurement has become a well recognized way of

estimating over-all system performance and '‘magnification'*. The test

is easily performed, at least on unsealed instruments. The recorded

output is that resulting from a step- function in the fcrce applied to

the bob. If the mass ot the bob is known, this disturbance is equivalent to a step in the acceleration of the pier. If the effective

. stiffness of the spring is known, the disturbance is equivalent to the

release of the bob from a non- equilibrium position. From the detailed

shape of the record the user can, in principle, compute a relative

amplitude- frequency response curve. In practice one can only infer that

the system performance has or has not changed by comparison with previous .re»cords . The peak amplitude on the -record, resulting from the

weight lift, is related to the sinusoidal midband gain of the system;

however, in the present state of the art one should use an arbitrary

constant determined as the result of shake- table studies. The use of

such an arbitrary constant presupposes that the frequency response of

the system remains unchanged.

With all its limitations the weight lift has the advantages of

simplicity and unambiguous polarity indication. Unfortunately the

engineering required to arrange a weight lift calibrator within a sealed unit takes it out of the "simple” class.

9.3.2 Pneumatic Calibrator

In subsection 8.5, it was pointed out that the vertical component

seismometer is quite sensitive to small variations of atmospheric pressure so that a high gain instrument should be enclosed in a rigid,

sealed container.

If we are committed to a sealed, rigid container for the instrument it is logical to examine the possibility of using buoyancy forces

to calibrate the system.

Referring to subsection 8 . 5,

a

calibrating signal equivalent to

50 millimicrons peak to peak at 6 sec period requires a density change

on the order of 35 x 10"^, if the inertial mass has an effective

density of 8.6 gm/cc and the case is filled with gas to a density of

.0012 gm/cc (air at 1 atmosphere pressure and room temperature).
131

Suppose the case has a volume of 10 liters and is connected by fine

tubing to a 1 cc hypodermic syringe. The required volume change of

0.35 cc peak to peak may be produced by a motor driven eccentric with

a stroke of the order of 2 cm. With a variable speed motor one may

cover a range of frequencies and remote operation is quite feasible.

Using equation 8.5.3 one may obtain the operating amplitude- frequency

response curve of the over- all system.

This idea has at least two disadvantages, 1) It cannot be

directly applied to horizontal instruments. 2) As outlined above,

inadvertent polarity reversals within the system are* not easily

detected. (A second syringe with solenoid withdrawal and spring

return will indicate polarity)

9.3.3 Electromechanical Calibrator

Equations 8.5.1 suggest that any mechanism which will apply

known sinusoidal forces to the inertial mass may be used to obtain

an amplitude- frequency response curve for the over-all system.

Such a device might consist of a motor-driven eccentric coupled

to the mass by a light spring. From the measured spring stiffness

(a watch or clock hairspring would probably be about right) and the

eccentricity of the drive, the applied force is easily obtained.

Again a variable speed motor is indicated. 'Among the engineering

difficulties to be considered are gear noise and electrical cross-talk

between the motor and the main coils of the seismometer since the

whole mechanism would certainly be mounted within the case.

This device, like the remotely operated sinusoidal pneumatic calibrator, cannot check a system for polarity.

9.3.4 Electromagnetic Calibrator

Another method tor subjecting the inertial mass to known sinusoidal forces .is to provide the seismometer with a second coil working

in an auxiliary gap preferably arranged to have minimum electromagnetic coupling to the main coil and its circuit.

Calculation of the motor constant required to have one milliampere

equivalent to 50 millimicrons Indicates that a few meters of wire working in a field on the order of 10 to 100 gauss is all that is required

for a few kilograms at 2 cps
132 -

Such an arrangement l 3 as readily adapted to remote operation as a

| pneumatic or electromechanical calibrator and has several advantages over ; either, provided line losses can be measured and presumed constant. The , amplitude and frequency of the input signal is readily altered over a

^ wide range. The device is not even restricted to sinusoidal inputs, a , step function of calibrator current accurately simulates the weight lift.

One may determine the system response to any transient pier motion for

which, using equation 8.5.1, a suitable time varying current can be

generated. Obviously the calibrator must be driven from a constant cur- . rent generator, i.e. it must see a source impedance high enough so that (i the emf * s generated in the calibrator coil by bob motion produce negli- i

gible modulation of th? calibrator current. This presents no special

engineering difficulties.

f Of '^he many possible methods for determining the motor constant of i; the calibrator coil several will be discussed in subsection 9.4. One f method is so direct and specifically applicable to an operational seismic ^ system as to be properly included here. It consists of matching the recordj

produced from a weight lift by the output due to a step- function of current. The weight lifted should produce a record comfortably above the

local background level but not so great as to approach non-linearities

in the system. Here the system is used only to compare two forces and i is only required to be stable throughout the course of the measurements. i Having obtained the motor constant of the calibrator one may then ener- v gize the calibrator sinusoidally to obtain the amplitude- frequency

response curve for the over-all system.

Over-all system polarity may be checked by verifying the polarity

|

of the circuits used to energize the calibrator. This is generally simpiei.|

than checking the polarities of the several instruments and connections r in the signal circuits of the system.

j

if 9.3.5 Electrodynamic Calibrator r

1 : The two engineering problems which exist when using an electromagnetic calibrator: namely, "cross-talk" between the calibrator and *

recording circuits, and the possible ambiguity of polarity, have led ^1 Mr. Ben S. Melton to point out that an electrodynamic calibrator may be ; devised which replaces these problems with others which may be more . tolerable. i
133

If one arranges two coils so that their magnetic fields interact and

>asses currents through them then a force will be developed between the

:oils which is proportional to the product of the currents through them.

Ln particular, for two coils series connected, the instantaneous force

Is proportional to the square of the current. Thus, the polarity of the

force is independent of the polarity of the current and, for sinusoidal

excitation the resultant force also varies sinusoidally, but at double

frequency.

Attaching one coil to the instrument frame, the second to the mass,

and connecting them in series gives a calibrator whose polarity is deternined by the construction. Further, the output from the seismic system

should be separable on the record from system "cross-talk" since the former

is at double the driving frequency.

This technique reduces the problems of polarity ambiguity and "crosstalk" inherent in electromagnetic calibrators at the expense of being able

to calibrate with signals displacing the mass to only one side of the

operating rest point. In order to inject signals of arbitrary amplitude

and wave- form, one must generate a current in which the instantaneous

amplitude is the square root of the desired instantaneous pier acceleration.

The same calibration arrangement can be. excited by an amplitude

modulated carrier of frequency comfortably above the pass band of the

system. The attenuation of a transmission line may be larger and more

variable for a carrier signal than for one of the modulation frequencies.

Many users would prefer the limitations of an electrodynamic calibrator to those of an electromagnetic device’.

9.3.6 Injected Voltage

9. 3. 6.1 Amplitude of Series Injected Voltage

The effect of spurious voltages in the electrical circuit between

the seismometer and galvanometer has been discussed in subsection 8.4.

It should be possible to turn this sensitivity to advantage and use it

as a method of calibration. To cover a range of frequencies (cf. Table

8.1) it appears desirable to include several four-lead precision resistors in series as a portion of the total loop resistance. Known currents
134 -

through one of these resistors can be related to equivalent earth motion

by an equation such as equation 8.4.4.

This method appears to have two advantages over those considered so

far. First, one may make an overall system calibration without the need

for modification of the seismometer to include an electro-mechanical or electromagnetic calibrator. Second, it appears serviceable as a means

of calibrating a system which consists of several matched seismometers,

separated in space, series or parallel connected to a single galvanometer.

The problem of polarity determination could be quite severe, especially if "the seismometer" is a group of interconnected "matched"

instruments

9. 3. 6.

Bridge Methods

The authors have not attempted to develop methods for calibrating

assembled seismic systems utilising bridge measurements. One such possibility is that of Willmore which appears to be useful for calibrating

installed systems having instruments w’ith readily lockable masses.

It is of interest to consider phase measurements using the circuit

of Fig 4.2 or Fig 4,3, For example, one may open the electrical circuit

of a direct coupled system and consider it as a two terminal network.

Neglecting mechanical losses one may write the im.pedance from equation

4.3,4 as

9. 3. 6.

3

Phase Measurements

Z = R . 1 + 2 ju) ( CO - CO o

X CO o o

2 2

CO - CO g

X co_ 9.3.1

By inspection it can be seen that the imaginary part approaches

infinity as co approaches co^ or co , The imaginary part is zero when
135 -

X oj (oi^ - oi^) + X 00 (o)^ - 00^) =0 9.3.2

o o ^ g g g

' o

Equation 9.3.2 with oo = oo^ is equivalent to the defining equation

of 00^

(cf. equation 6.2.4) so that measuring these three phase points

give the natural frequencies of the seismometer and galvanometer and also

the ratio of their critical damping resistances.

In what follows it will be convenient to rearrange equation 6.2.4

or 9.3.2 slightly and define a quantity B as

X 00 o o X 00 g g

2

00o

2

00V

2

00V

2

00g

00 2 o o Z B 9.3.3

Using equation 9.3.3 to eliminate X^ -and X^

we may write the general fourth order direct coupled

from equation 6,3.1

response as

00 /oo_S

oo \00 • +

00o

00g

0000m

(B+l )

- J_

2>un

+

00m

oo + 00o

00m

9.3.4

If we can determine X then the phase- frequency response curve and

m

the relative amplitude- frequency response curve are known. A knowledge

of the effective mass of the seismometer and the moment of inertia of the

galvanometer serve to make the amplitude- frequency response curve quantitative.

1
136 -

Either of the folloving two measurements of phase, together with a measurement of the d-c resistance of the seismometer-galvanometer circuit,

serve to determine X . m

Given a suitable inductor with resistance and inductance L con- nected in series with the seismometer and galvanometer, the impedance of

the circuit (again neglecting mechanical losses) may be written

Z=IL + R il+1cori +

X oi

^ o o

0) - CjD

X 0) —a. S

o:

-)i': 9.3.5

1 1

O) The imaginary part of equation 9.3.5 approaches infinity atto and so = 00 as before. It is zero when

o g

X 00 m m

2 2. , 2 2,

(OO^ - 00 ) (oOg - 00 ) 2 2 1

(oo^ - oop -(B + ^) . V D

9.3.6

Each of the two frequencies satisfying Eq. 9.3.6 will be slightly

above the natural frequency of one of the two instrumex'.ts.

Alternatively one may connect a suitable condenser of capacity

in parallel with the network terminals. The reactive component of impedance will be zero at two frequencies each of which is slightly below

the natural frequency of one of the instruments. The imaginary part of

the impedance of the parallel combination vanishes when

X CO m m ^

- 00^) (0)2 - co^)

4oo^RC

e

(oo^

V

- oi^) (B + h

B

(1 ± ^1 - Aoo^iCC*^e 9.3. 7

9.4 Indirect Methods

Under indirect methods we include all raethods of calibration which

involve determination of the pararaecers of the seismometer, galvanometer,

T pad if any, and recording mechanisms as separate components since these

methods require that the qver-ali system be presumed to respond according

to a known equation sucn as those developed in earlier sections.
137 -

9.4.1 Recapitulation

The performance of an inertial type seismometer having negligible

Internal losses using a variable reluctance transducer is completely

defined by a knowledge of M, R and L for the instrument. All

these parameters except M can be measured by electrical means as indicated

in paragraph 3. 2. 1.3. M may be measured directly if the instrument is disassembled. Alternatively it can be obtained from a measurement involving

one mechanical quantity, e.g. a motor constant.

The performance of a galvanometer having negligible internal losses

and inductance is comjiletely defined by K, co , R and . Measurement of

these parameters has been discussed in subsection 4.4.^

• > A moving coil seismometer may also be defined by the methods discussed

in subsection 4,4. Equation 4.4.1 will require either a micrometer microscope, an electrical contact mounted on a micrometer screw or a similar

method of determining mass position,

9.4.2 Special Tests

In connection with checking the "linear range" of a moving coil

instrument several tests have been used for measuring the motor constant

directly, for both the main coil and an auxiliary electromagnetic calibrator.

Adjustment of the "rest point" of an instrument is readily made, for

test purposes, by injecting a constant d-c current from a high impedance

source into the main coil of the instrument.

*A specific mass position may be sensed by means of an electrical

contact (it must be very light and on a very compliant spring) which can

be precisely adjusted in the line of mass motion.

The motor constant of an instrument can be obtained by a force balance using the electrical contact to determine the balance point. For

instance (testing a vertical instrument) one may bring the mass to the

desired restpolnt, adjust the electrical contact to the "just make"

position, add a known mass to the inertial mass and then by changing

the d-c current in the coil return the mass to its initial position as indicated by the electrical contact. The force applied by the known mass,

I

i

1
138 -

, r- r'liT-TPnt ttives directly the motor constant

divided by the change of d-c current, gives ui-i. / at the initial rest-point.

Uslna a balllsUc galvanometer calibrated In volt seconds ^"d providing the Inertial mass with stops allowing only a small, precisely

trLe! one may easily determine an average generator constant

f rhf> galvanometer deflections obtained on moving the inertial mass

f^ror^cop to the other. Parenthetically it may be noted

thif the generator constant is usually thought of as volts per (cm per

sef). aifia volt sec per cm. the quantity measured directly by this

test

The following method is somewhat tedious in the taking of data and

laborious in its rectification but, knowing the effective mass, it is

capable of approaching 17. accuracy in the

the motof/generator constant of both tno mam and caiibr.tor co*is - an

instrument having negligible inductance.

Consider the circuit of Fig 9.1. Here we may assume

Fig 9.1

and in a well designed instrument the electra

the coils will be negligible so that 0
- 139 -

Equations 2.8.6 and 2.8.8 may be written

/ d ^

_x d__ + 0) ) X

o

'=1^1

M M

9.4.1

° = S

9.4.2

Eliminating x and substituting gives

° =f4 + +

L at

2 d . 2i , ^2 — ^ ) — h .60 1 ( M (R^ t dt oj^

G,

,1 (R, + R )

1 d E,

+ — — (~is)

M dt R.in

9.4.3

Letting d/dt = joo, remembering (from Fig 4.1) that R^ may be considered as

2 a resistor internal to the instrxjment with 'a numerical value equal to

and that = - ^2^0

E. G_ R. R + R„ + R 03^-03^

in ^ _2 in / o 2 x\ _

.i /_o \ / E., G, R R 03 R G^

2 1 X o o 2

R + R- R. M

o 2\ in 9.4.4

Multiplying by the complex conjugate gives

E (2

‘ „ 2 _ „ 2

in

E,

R + R,

_ (_£ £

R

R, Mo3 “ f

) (-i£_£) (_° - i_)

G,G, f f 12 o

2 2 2

G- R, R 4- R- + R —5^

1 o X

9.4.5
140 -

Equation 9.4.

me

is linear in the variable ^ ~ ^ • ^7

asuring | E. /E

j at a number of frequencies in the neighborhood of f^

^

and plotting j ^ ordinate with {tjf - f /f^)^ as abscissa

j ^ one may obtain the slope m, and intercept on the ordinate, b. Then

R + R

v/m G,G- = R. Moj (_2_^—

12 in o R

) ana

/b!l

G„

R. _ in

R

R -i- R., + R

o 2 X

R

X

from which it follows that

G, =

R.

(-iB)

R

2

K/JC)

V mb

<R -1- R- + R

o 2 X' 9.4.6

X

and =

/b/B ym (R + R.,) (. R

2 \l -jx o R + R„ + R

o 2 X

9.4.7

A few remarks on instrumenting Fig. 9.1 may be of interest. We have i used a low frequency (not d-c) amplifier and recorder with a high input impedance as a voltmeter. R^^ was made up of a resistor in parallel with a j, precision potentiometer. At each frequency the araplifier was connected across' R and a section of record was made. The amplifier v/as then connected across '

the potentiometer output and the potentiometer adjusted to give a x’ecord of | the same amplitude. This avoided questions of the ampl itude- frequency charac-j'

teristic of the recording amplifier and its calibration as a voltmeter. A ' range of frequencies of ± 3^4 octave centered on f was adecuate for piottit o *

, purposes. A value of appropriate to l/4 critical damping was satisfactory;}

higher values of result in excessive magnification by the seismometer of' any harmonic distortion in the signal generator or power amplifier as the testj

frequency approaches f /2. Lower values of R will require either more input

o' o ‘

I current or a higher gain in the voltmeter. A d-c constant current supply

| across served to shift the operating rest point of the inertial mass for

^ • r r. studying the change of G. and 0, with mass position. K i
141

) A slight error in selection of for plotting the data results in a

very slender parabola, frequencies above f lying on one half and below f

. ® o

lying on the other. The slope and intercept of its axis will give the

’desired yalues of m and b. .

The circuit of Fig 9.2 gives an excellent check of the ratio of the

average value of to G^. Differences in their non-linearity over the

range of mass motions are easily seen.

)

The main coil is connected to- a potentiometer having a total resistance

large compared to either the coil resistance or critic-al damping ' res istance

of the seismometer. At the proper adjustment of the potentiometer the galvanometer is remarkably insensitive to violent . motions of the mass The

ratio of G /G is R / (R + R, ) . As we have used it, the galvanometer was

L Z. II p jw heavily overdamped but the point seems immaterial as long as it is a sensitive instrument
142 -

Section 10 fi

Nomenclature and Pertinent References ’

10.1 General 'i,

This section is in two parts. The first contains a complete list U

of the symbols used in this report. Each is given with a brief descrip-,

tion of its significance and the paragraph in which it is first intro- «

duced. In several cases a single symbol has been assigned different -O

meanings in different sections and these are noted by multiple listings. J|

In these cases the first meaning is generally not used after the second

has been introduced. ]

The arrangement of listings is strictly alphabetical, English pre- M

ceding Greek. For each letter the arrangement is print capital, script

I'j

capital, print lower case followed by script lower case. Primed symbols

follow immediately after unprimed symbols. ^'There a. symibol is used with

a number of subscripts the arrangement of subscripts is numerical followed

by alphabetical as indicated above.

The second part of this section contains a list, alphabetical by

authors, of some of the pertinent literature. Many of the symbols used

by others correspond identically to symbols or relations among symbols

used in this report. Accordingly, tables have been prepared indicating '

J some of these identities. Throughout most of this report we have presumed a seismom.eter in vfnich the principal inertial mass is constrained

to linear miotion so that the instrument natural frequency is definable f in terms of a stiffness and a mass while many authors have considered

specific instruments in which the inertial m.ass is pivoted on an arm S

so that it is constrained to rotate. In this case the natural frequencyM

is defined by a moment of inertia and a restoring torque. The similarityj

of concept between mass and moment of inertia and betv^een stiffness and j restoring torque has led us to include listing, of the corresponding ?

symbols v?ith the notation that they are analogous rather than identical.

The remark "analogous" has also been used for other cases of similar but , not identical symbols. I
- 143

10. 2 Nomenclature

A

A

m

a

B

(6.3.2)

( 2 . 1 . 1 )

(2.3)

(2.4.1)

(6.3.1)

(6.3.2)

(9. 3. 5. 3)

a constant introduced for simplification of algebraic

manipulation. An = 2 (^. + \ co ) , hence a measure

m og g o'

of damping.

first coefficient of a coordinate of a constraint,

its magnitude is determined by the linkages among

the constraints.

area of the magnet in the instrument.

a weighting factor relating the axial motion of an

element of mass linked to the bob and the coordinate

of bob motion.

a constant introduced to simplify algebraic notation,

a measure of the octave separation of the instrument

natural frequencies, a = jcOg - co^[/co^

a constant introduced for simplification of algebraic

manipulation. ^

B<jd = 2(\oo + X.cjo) m ' o o g

g^ a constant introduced to simplify algebraic manipulation

B

m

(3.2.2)

(2.3)

0) g

three arbitrary constants in the transient solution

for a seismometer connected to a resistive load.

the magnetic induction in an element of a permanent magnet.

(g (2.1.2) the magnetic induction

b (2.4.1) a weighting factor relating the transverse motion of an element of mass linked to the bob and the coordinate

of bob motion. Only used in this sense in this section.
- 14A -

b (7.4.1) a constant introduced to simplify algebraic manipulation

b = (o), -ku^)/co . a measure of the octave separation of

1 .L m

the final K.C filters.

(9.4.2) intercept on the ordinate, of a straight line obtained

by rectification of data taken during measurements on

a seismometer.

(7.4.1) condenser of the first of two final RC filters. It is

associated with to give a time constant and a

3 db point at

CM O (7.4.1) condenser of the second of two final RC filters. It

is associated with to give a time constant and

a 3 do point of

c

e (4.3.2) a capacitor added to a galvanometer, mea s_uremer.es on

the combination permit determination ofOi-^.

O

C

X

(3.2.4) numerically M/G?. The eie.ctrical capacitv ecuivolent

to .the inertia o^: the bob in the electrical analog of

the seismometer.

<=0 (4.3.2) numerically K/G?. The electrical capacity equivalent

to the inertia or tne armature in cne electrical ana.tOg

of the galvanometer. ^

c (7.4.1) a s;;-mr.bol introduced to simplify algeoraic manipulation.

1,2 ,,,2/2 -.,-.2

c = j K ct^My /CO CO, AJi

1 2 m 1 o j

(8.3) the capacity between the conductors connecting a seismometer and galvanometer.

D (3.2) defined as d/dr = d/ dco t. D is the derivative operator in normalized time.

J? (2.1.2) the electric displacement of an electrostatic field.
145

dm (2.4) an infinitesimal element of mass of the instrument.

£ (3.2) voltage across a seismometer and external resistor in

series with it.

E

0

(2.3) a supplementary emf necessary to preserve constant

current in the coil simulating the instrument magnet.

(4.4.1) output of a device for producing a signal proportional

to galvanometer deflection.

(7.4.1) output voltage of a system consisting of seismometer,

galvanometer, telemetering .device and final filter.

=1 (2.2.3) the generalised force of the electrical coordinate, the

electromotive force. Also used as the external emf

applied to a coil of the seismometer.

^2 (2.8.3) emf applied to a second real coil. A prime is used to

distinguish the numerical value from that obtained under

different test conditions.

(4.2) the potential difference at the terminals of the galvanometer circuit.

E.in (9.4.2) emf across to monitor the driving current into a calibrator coil.

P (2.1.2) Electric field strength of an electrostatic field.

e (3.2.4) defined as Gdx/dt,

(7.2) voltage across a circuit, or portion of a circuit.

e0 (4.4.1) output voltage of a device for converting galvanometer

coil rotation into voltage during special tests.

(7.4.1) the output of a device for producing a voltage proportional to the deflection of the galvanometer mirror.

"2 (7.4.1) output of a data transmission system. Input to the

final pair of RC filters.

®3 (4.4.1) external emf applied to a galvanometer coil. (special

test)
(3.3) an extraneous voltage generated in the electrical

circuit between a seismometer and galvanometer as by

electromagnetic induction or thermoelectric junctions.

(8.3) an extraneous voltage generated in the electrical

circuit between a seismometer and galvanometer as by

electrostatic induction.

(8.2) some function of the frequency variable o>.

(3.2) normalized frequency defined as co/ao^.

(5.2)

> natural frequency of the galvanometer when the natural

frequency of the seismometer is taken as unity.

• P

f = 0) /oa . g

g' o

(3.2. 1.1) defined as t^fy 2. q. f \)q considered as the

normalized apparent natural frequency of the instrument with a superconducting coil short-circuited at

its output. It is also the series resonant frequency

(normalized) of the electrical inductance and the

reflected impedance of the spring-mass system.

(2.6) The generator/motor constant of an instrument^ i.e.

emf per unit velocity or mechanical force per unit

current.

(2.8.2) the generatcr/motor constant associated with one real

coil of the seismometer.

(2.8.2) the generator/raotor constant associated with a second

real coil of the seismometer.

(4.2) the motor/generator constant of the galvanometer.

(7.2) when a seismometer drives two galvanometers simultaneously, refers to the generator/motor constant of

that galvanometer whose output is not being recorded.
147

(7.2) when a seismometer drives two galvanometers simultaneously refers to the generator/motor constant of

that galvanometer whose output is being recorded.

g (2.4.3) the steady upward acceleration ’to which an instrument

must be presumed to be subjected in order that its

parts shall exert downward forces equivalent to those

due to gravitational attraction.

H (2.1.2) the intensity of the magnetic field.

H (2.3) n Magnetizing force in the magnet.

h (8.^)

-34

Planck's constant. 6.6x10 . joule sec.

I (3.2) electrical current flowing into the seismometer from

an external circuit.

(2.2.4) the phantom current in a simulated • coil replacing the

permanent magnet in the instrument.

(2.2.3) electrical current--the generalized velocity of the

electrical coordinate. Also used as the current flowing into the coil of the instrument.

•(2.8.2): . current in the first real coil.

(2.8.3) A prime is used to distinguish the numerical value from

that obtained under different test conditions.

I2 (2.8.2) current in a second real coil of the instrument.

I3 (4.2) current in the galvanometer coil.

h (7.2) when a seismometer drives two galvanometers simultaneously it is necessary to distinguish between them.

refers to the current through the galvanometer whose

output is not being recorded.

Iexp

(3. 2. 2. 2) an electrical current decreasing exponentially with time.
1A8

I (3. 2. 2. 2) osc an electrical current varying in time as a damped

sinusoid.

CM M • when a seismometer drives two galvanometers simultaneous ly^ refers to the current through that galvanometer whose output is being recorded.

i (7.2) current through a. circuit, or portion of a circuit.

(3. 2. 2, 2) current at time t^^.

(3. 2. 2. 2) current at time t^.

i3 (4.4.1) a prescribed dc current into a galvanometer used in

establishing the performance of a device for converting

coil angular position into voltage.

(4.3.2) current through an electrical inductor simulating the

suspension stiffness of a galvanometer. This current

is proportional to the instantaneous angular displacement of the coil.

j (2.8)

K (4.2) moment of inertia of the galvanometer coil. The angular

equivalent of M^.

^

(7.2) when a seismometer drives two galvanometers simultaneously, refers to the moment of inertia of that

galvanometer whose output is not being recorded.

K (7.2) w

when a seismometer drives two galvanometers simultaneously K refers to the moment of inertia of that

galvanometer whose output is being recorded.

k (2.1.1) the designator of a push rod into a Lagrangian black

box.

(4.4.1) the proportionality constant, relating output voltage

to mirror deflection, of a device for producing a voltage proportional to galvanometer deflection.
(8.2)

•23

Boltzmann's constant, approximately 1.37 x 10 joule/

deg.

(2.6) The self inductance of the instrument coil as measured

with the mass blocked.

(2.8.2) the self inductance of one coil of the instrument as measured with the mass blocked.

(2.8.2) the self inductance of a second coil of the instrument

as measured with the mass blocked.

(4.2) the electrical inductance of the galvanometer coil plus

series inductance (if any) in the galvanometer circuit.

(3.2.4) numerically G^/S. The electrical inductance equivalent

to the effective stiffness of the suspension in the

electrical analog of a seismometer.

(4.3.2) numerically G^/U. The electrical inductance equivalent

to the stiffness of. the suspension of the galvanometer

in the electrical analog.

(2.1.1) the Lagrangian function. The difference between the

kinetic and potential energies associated with a Lagran

gian black box. T-V.

(2.3) length of the magnet in the seismometer.

(3.1) the mass of the bob. In this and subsequent chapters

the assumption is made that M = M . To preserve the

X xy

distinction would be more precise but tend to obscure

the relations deemed to be important.

(2.8.2)

(2.4.1)

the mutual inductance between two real coils of the

instrument.

o inertia (apparent mass) associated with x , independent

of y, for computing mechanical kinetic energy. The

linear equivalent of K.

(2.4.1)
150

M (2.4.1)

xy

inertia (apparent mass) associated with the product

xy for computing mechanical kinetic energy.

M (2.4.1)

y

* ^2 inertia (mass) associated with y , independent of k,

for computing mechanical kinetic energy.

m (3. 2. 2.1) the mass used in a weight-liff calibration.

(9.4.2) slope of a straight line obtained by rectification of

data taken during measurements on a seismometer.

n (2.2.3) number of turns in a coil of the instrument.

P (8.4) ambient atmospheric pressure around the inertial mass.

Q; (.2^.1) one of the forces on a pus.h rod associated with a con- straint within a Lagrangian black box.

\ (2.1.1) external force applied to the k^^ push rod into a Lagrangian black box.

(2.2.2) The generalized force associated with the coordinate

of bob motion.

(2.1.1) first coordinate of a constraint within a Lagrangian

black box. A constraint may be considered as a push

rod not accessible from the outside.

(2.1.1)

ttl generalized coordinate^ the position of the k push

rod into a Lagrangian black box.

\ (2.1.1) velocity of the k^^ push rod into a Lagrangian black

box.

R (6.3) the series loop resistance of a direct coupled seismometer-galvanometer combination.

\ (9.4.2) Load resistor across the main coil of a seismometer.

(2.8.2) electrical resistance of one real coil of the instrument

(5.2) when a seismometer and galvanometer are connected by a resistive T pad the resistance of the input arm of the

T pad plus the resistance of the seismometer coil is

indicated by R^.
151

R (7.4.1) resistor of the first of two final RC filters. It is

associated with to give a time constant and

a 3 db point

R (2.8.3) electrical resistance of the second real coil of the

instrument.

(7.4.1) resistor of the second of two final RC filters. It

is associated with to give a time constant T

2

and a 3 db point od^.

the electrical resistance of the galvanometer coil

plus series resistance (if any).

when a seismometer and galvanometer are connected by a resistive T pad the resistance of the output arm of

the T pad plus the resistance of the galvanometer

coil is taken as R^.

resistance of the coil of the seismometer.

electrical resistance of the galvanometer coil.

resistor in series with a calibrator coil, to monitor

the, driving current.. •

the electrical resistance of an inductor used with a galvanometer; measurements on the combination permit

determination of ^

g.

R ( 5 . 2 ) the resistance of the shunt (common) arm of a resistive

T pad connecting a seismometer to a galvanometer.

R (3.2.4) numerically G^/r . The electrical re.sistance which is

X X

equivalent to the internal losses in the instrument in an electrical analog of the seismometer.

Rg (4.3.2 numerically G_/r^ . The electrical resistance equivalent to the internal losses of the galvanometer in the

electrical analog.

R

3 (4.2)

(5.2)

R (2.2.3)

c

Rg (4.3.2)

Rin (9.4.2)

Rl (4.3.3)

I
152

CB„ (6.2)

o

(4.2)

(% (7.2)

Ce„ (7.2)

(4.4.2)

r (2.2.2) X

r, (4.2)

S (2.6)

JC^ = G^/2a)^M. In a seismometer having negligible

internal losses and negligible inductance this is the

critical damping resistance (CDRX + coil resistance)

G^/2oo^K. This is the critical damping resistance

(CDRX plus coil resistance), of a galvanometer having

no internal losses and negligible inductance.

when a single seisnometer drives two galvanometers it

is necessary to distinguish between them, refers

to the critical damping resistance of the galvanometer

whose output is not being recorded.

when a seismometer drives two galvanometers simultaneously, refers to the critical damping resistance

of that galvanometer whose output is being recorded.

resistors in a bridge network for determining instrument

parameters.

losses within the instrument tending to reduce the

velocity of the inertial mass such as residual eddy

currents, air damping, etc. Presumed proportional

to velocity. The linear equivalent of r . U

mechanical damping "constant” of the galvanometer.

Usually chiefly air damping but would include losses

due to eddy currents in a conducting coil form. The

rotational equivalent of r^.

the stiffness of an ideal mechanical spring equivalent

to the rate of change with displacement of the sum of

the various restoring forces on the inertial mass when

the seismometer is subjected to no external forces.
153

s (4.4.1) galvanometer sensitivity in amperes per radian of coil

motion.

T (2.1.1) the kinetic energy associated with a Lagrangian black

box.

T (8.2) absolute temperature, degrees Kelvin.

T

mag

(2.1.2) magnetic kinetic energy.

Tmech (2.1.2) mechanical kinetic energy.

t (2.1.1) time.

particular values of time.

U (4.2) stiffness of the suspension of the galvanometer coil,

the angular equivalent of S.

Ujj (7.2) when a seismometer drives two galvanometers simultaneously, refers to the restoring torque of that galvanometer whose output is not being recorded.

U (7.2) when a seismometer drives two galvanometers simultaneously refers to the restoring torque of- that

galvanometer whose output is being recorded.

V (2.1.1) the' potential energy associated with a Lagrangian black

box.

Velectrostatic (2.1.2) electrostatic potential energy associated with

a Lagrangian black box.

Vgr^v (2.2.1) gravitational potential energy.

V (2.1.2) mechanical potential energy associated with a Lagrangian

^ black box.

V

s potential energy stored as elastic distortion of the

primary and other springs.

(2.4.4)
154 -

W (3.2.2) defined as (p - + !)• W is introduced

as a matter of convenience to explain the construction

of Fig. 3.1.

(5.2) total power dissipated by a dc emf introduced in one branch of a resistive T pad.

W (2.4.2)

oo electrical inertia associated with the current I in

o a hypothetical coil equivalent to the instrument magnet.

In (2.7) it is shown to be the self- inductance of the

simulated coil.

(2.4.2) electrical inertia associated with the product of the

current in the simulating coil and the current in the

real coil. In (2.7) it is shown to be the mutual

inductance between the simulating coil and real coil.

W^2 (2.8.2) electrical inertia associate4 with the product of the

current in the coil simulating the instrument magnet

and the current in a second real coil of the instrimient.

The mutual inductance between the simulated coil and a second coil on the instrument.

(2.4.2) electrical inertia associated with the current in the

real coil. In (2.7) it is shown to be the self inductance of the real coil of the • ins trument.

W^2 (2.8.2) electrical inertia associated with the product of the

currents in two real coils of the instrument. Identified as the mutual inductance between two real coils

of the instrument.

W22 (2.8.2) electrical inertia associated with the current in a second I'eal coil. Identified as the self inductance of

a second real coil.

W

3 (5.2) power dissipated by currents in one branch of a resistive

T pad due to a dc emf introduced into the other branch.
155

X (2.2.2) forces applied directly to the inertial mass parallel

to the motional axiS;, as by an electromagnetic calibrator,

or in a vertical instrument, air bouyancy. The linear

equivalent of 6.

X (2.2.2) linear displacement of the center of gravity of the

principal inertial mass, parallel to the motional axis

of the instrument, with respect to the frame of the

instrument. For vertical instrument taken as positive

upward. The linear equivalent of 0.

X (2.2.2) dx/dt time derivative of x. In a vertical instrument

taken' as positive for upward velocity of the bob with

respect to the frame.

X ( 2! 6) d^x , the acceleration of the bob with respect to the

dt^

frame. In a vertical instrument it is taken positive for

increase of upward velocity.

X (2.6)

0

the equilibrium position of the inertial mass with

respect to the instrument frame when the instrument is

subjected to no external forces, i.e. the "rest point"

of the inertial mass.

Y (2.2.1) force exerted on the seismometer frame by the pier.

Taken as positive if it tends to produce a positive

displacement y.

(4.4.2) vertical deflection of an oscilloscope trace, also a switch position, during the measurement of instrument

parameters.

Y' (4.4.2) vertical deflection of an oscilloscope trace, also a switch position, during the measurement of instrument

parameters.

y (2.2.1) earth displacement parallel to the motional axis of the

instrument. For vertical instrument y is taken

positive upward. The linear equivalent of q.

y (2.2.1) dy/dt, earth velocity parallel to the motional axis or the

instrument. For a vertical instrument d\'/dt is taken as positive upward.
156 -

sc

a

a

(2^ 6 )^ d^Y acceleratior. of the frame of the instrument with

dt^^

respect to a "stationary" point.

(9. 3. 5. 3) by opening the electrical loop of a fourth or sixth

order system one may consider It as a two terminal

network whose electrical impedance is Z.

(4.3.2) the electrical impedance of a galvanometer considered

as a two terminal network.

(4.3.2) the electrical impedance of a galvanometer in series

with a' capacitor.

(4.p.2) the electrical impedance equivalent to the mechanical

portions of a galvanometer when it is considered as a two terminal electrical network.

(3.2.3) the fraction by which a variable may be attenuated, as

a voltage by a potentiometer Used- in this sense only

in this section.

(5.2) the mean coupling coefficient, a measure of the attenuation provided by a resistive T pad,

^4

(3.2.2.)

(3.2.2)

a . (Rl 4- RP (R3 + RP

a root of the auxiliary equation for the transient

solution of the differential equation for a seismometer

connected to a resistive load.

the three roots of the auxiliary equation for the

transient solution of the differencial equation for a seismometer connected to a resistive load.

defined as G/4 LS = G/cu LM.

/ o generator/raotor constanc.

7 (3.2) 7 is a normalized
(4.2)

(4.2)

(4.2)

(6.4)

(7.2)

(6.4)

(7.2)

(6.3)

angular position of the galvanometer case, with respect

to a "fixed” direction in space. The angular equivalent

of y.

mechanical torque applied directly to the galvanometer

coil. The rotational equivalent of X.

angular position of a galvanometer coil with respect to

the case. The rotational equivalent of x.

galvanometer deflection for a direct coupled system for

comparison with the output of a T pad coupled system

subjected to the same excitation.

when a seismometer drives two galvanometers simultaneously, 0^ refers to the angular position of that

galvanometer coil which is not being recorded.

galvanometer deflection for a T pad coupled system,

for comparison with the output of a direct coupled

system subjected to the same excitation.

when a seismometer drives two galvanometers simultaneously, refers to the angular position of that

galvanometer coil which is being recorded,

angular deflection of the galvanometer mirror at o>^.

(6.3) angular deflection of the galvanometer mirror at a'* .

(6.3.1) ratio to critical damping, fraction of critical damping.

In the special cases where X = X = X no point is

o g m

served by preserving the subscript.
158

X (6.2)

o

\

(^- 2 )

fraction of critical damping, or ratio to critical

damping, of the seismometer due to the circuit

resistance as seen by the seismometer (includes coil

resistance of the seismometer).

fraction of critical damping, or ratio to critical

damping, of the galvanometer due to the circuit resistance as seen by the galvanometer (includes coil

resistance)

\ ( 6 . 2 ) m

Nl (7.2)

\ < 7 - 2)

Mo (2-3)

mean ratio to critical damping, the geometric mean of

the fraction of critical damping of seismometer and

galvanometer. X = '/x X . ^ m o g

when a seismometer drives two galvanometers simultaneously it is necessary tc distinguish between them.

Xj^ refers to the ratio to critical damping of the galvanometer whose output is not being recorded.

when a seismometer drives two galvanometers simultaneously X^ refers to the fraction of critical damping

which the circuit resistance presents to chat galvanometer whose output is being recorded.

permeability of the core upon which is wound the coil

simulating the instrument's magnet.

(2.3) the "reversible" permeability of the magnet of the

instrument, as used.

V

V

P

(3. 2. 2.1) when two of the three values of p are complex they

may be written as cr + jv and a - jv.

(8.3) The vibration frequency which when multiplied by Planck's

constant gives the permissible energy increments of a system.

defined as R/co L. p is the ratio of resistive to 2 Q

inductive impedance at the open circuit frequency of

the seismometer.

(3.2)
159

• 00

a. density of air surrounding the seismic mass.

P„ (8.4) m effective density of the inertial mass. \

(3.2) defined as r /co M. p is the normalized mechanical damping coe^ficen^.

a (3. 2. 2.1)

«*

when two of the three values of p are complex they

may be written as a + jv and a - jv.

T (3.2) defined as co t. t is a normalized time variable,

o When measured in this unit any seismometer has an open

circuit period of 2:t.

(7.4.1) time constant of the first of two final RC filters.

^2 (7.4.1) time constant of the second of two final RC filters.

^ (2.2.3) magnetic fliix linked with a coil of the instrument.

(6.3.1) a normalized angular frequency- defined as 0 =

0 (2.2.4)

0

the magnetic flux linked with the simulated coil

replacing the permanent magnet in the instrument.

(2.8.2) the magnetic flux linked with the first real coil of

the instrument".

^2 (2.8.2) the magnetic flux linked with the second real coil of

the instrument.

f (7.4.1) The phase angle by which the voltage at the recorder

leads earth velocity.

f (3. 2. 1.2)

X

The phase angle by which the system output leads force

applied directly to the seismic mass.

^ (3. 2. 1.1)

y

The phase angle by which the output current leads earth

displacement.

I
160

0)

0)

O),

CO,

O)

CO,

00m

(2.8.3)

(2.8.3)

(7.4.1)

(7.4.1)

• >

(4.2)

(7.2)

(6.3.1)

(6.3.1)

( 6 . 2 )

2jt times frequency in cps, angular frequency, frequency

of a sinusoidal variable in radians per second.

defined ayv/S/'M , called the natural angular frequency

of the instrument. It is the frequency, in radians per

second, at vhich the inertial mass would oscillate after

an impulse if it had no internal damping, when disconnected

from an electrical circuit. i

3 db point of the first of two final RC filters, frequency

at which the capacitative and resistive impedances of the

filter, elements are equal.

3 db point of the second of two final RC filters, frequency

at vjhich the capacitative and resistive impedances of the

filter elements are equal.

natural angular frequency of a galvanometer, defined as

nTu/K.

when a single seismometer drives tw^o galvanometers it is

necessary to distinguish between them. oils refers to

the natural angular frequency of the galvanometer whose

output is not being recorded.

angular frequency at which the high frequency asymptote

to the amplitude- frequency response curve intersects the

midband tangent.

angular frequency at which the low frequency asymptote

to the amplitude- frequency response curve intersects

tne midband tangent.

the geometric mean of the two instrimient natural angular

irequencies.

(jd vT

m

CD CD . o S

h
161

0)pc

(4.3.2) that angular frequency at which the parallel combination

of a capacitor and galvanometer shows' no reactive component of impedance.

CO (4.3.2)

PL

that angular frequency at which the parallel combination

of an inductor and galvanometer shows no reactive com- ponent of impedance.

CO (3. 2. 1.1)

s defined as 2:rf^. The "short-circuit" angular frequency

of the seismometer.

that angular frequency at which a capacitor and galvanometer in series show no reactive component of impedance.

CO (6.2) V

' ' that angular frequency of earth motion at which the

galvanometer displacement is in phase with earth

velocity.

CO (7.2) w when a seismometer drives two galvanometers simultaneously, co^ refers to the natural angular frequency of

that galvanometer whose output is being recorded.
162

10.3 Reference.s

Chakrabarty, S. K.

Response Characteris flcs of Electro^iagnetic Seismographs and Their

Dependence on the Instrumental Constants.

Bui. Seism. Soc. Amer. 205-218, (1949).

J & M Rem.arks

D rX

identical

II

g s

Possiole ambiguity of

K

identical

M M identical

R

.

II

r

'‘a

II

S R

s

II

t t

II

U

II

X X

II

y y

II

a a

M

<? m o

II

Oi COo

II

cneCO generally identical

0)g

a identical
163

C. J & M Remarks

approximately equal

1 ^ 1

+ 3r^)

S tn o aZ - r^)

J & M found no need to separate the

effective stiffness into its components

S^/ (R + S) (r + S) Why C. should state that "For efficiency and sensicivity" CC^ "is

always less than about 0.05" is

obscure^ cf. p. 211 line 6.

Como, Neil M.

Private Communication dated June 1955

C. J & M Remarks

3

% ^o P - identical

^ o^o^'p M

a 0) p S o

II

CD CD f

s os Como's CD^ approaches J & M's

CD f as the circuit resistance

o s approaches zero.
164 -

Como, Neil M.

Theory and Analysis of the Benioff Variable Reluctance SeismometerGalvanometer System.

Master's Thesis, Rensselaer Polytechnic Institute, Troy, New York

June 1955. Library No. C-73.

C_^ J.& M Remarks

a aPp identical

o o

a. identical

1 o s

^2 V

II

D r

II X

‘0

tl

G

If

g

II

I I

II

K s

11

L L

II t M M

K

II

n n

M

R

1 c R

c ^1 Probably identical

R R- It II g 3

R

s R

s identical
165 -

c_ J & M Remarks

r R

S

identical

"i

U

X X

•X

•X

ff

X X

y y

II

y y tl

»

• y y

II

e 0

\ X

s s

4) similar

m m

00 00 identical

00 00 identical^ except in Como’s Figure

S g XXVII-b.

00 00 identical

n o

00 00 f Como's 00 approaches J & M's oo f

s o s , S . . os,

as the circuit resistance approacn<

zdro.
166

Cosson^ J.

Equivalent Electrique d un Galvanometre a Cadre Mobile

Mesures, 143 , 127- 130, ("April 1949)

The following are identical.

J &: M

A

"e

B G /U

> C u

• P

i h

K K

L

t t

T

U ^3

y

7 c.

d

C7 c,

p R

g

4>

. °3

r
167

Coulomb, J.

Accroissment de la periode d'un galvanometre par emploi d'un

condensateur.

Mesures, 179 , 221-224 (1952)

£i

c

f

J & M Remarks

identical

0)

K

R

U

e

GO

identical, except that J & M may use i

where no ambiguity can exist.

identical
168

Coulomb, J. and Grenec, G.

Nouveaux Principesde Construction des Seismographes Electromagnetique.

Annales de Physique, 11, 321-369 (1935)

C & G J 6c K Remarks

D rX

analogous

d

3

identical

G h

analogous

g =3 identical

K analogous

k K identical

R identical when a T pad is used.

1

‘ R identical when instruments are direct coupledi

S identical

U S analogous

u U identical

a identical when internal losses of galvao nometer may be neglected.

P identical when internal losses of seismometer,

0

may be neglected.

0 0 identical

Z 2\ CD identical when internal losses of instrum m

ments may be neglected.

X analogous !

I n CD

1 identical | o O 1

1 CD' CDm

identical J

CD CD identical i O g
169 -

Dennison, A. T.

The Design of Electromagnetic Geophones

Geophysical Prospecting, _1, 1-28 (1953)

J & M believe that Fig. 2 (b) is in error, that the intended input is as shown in Fig. 6 (b), c.f. D's text p. 7.

D J & M

C

h

K

L

C

X

L

L' L

M M

X

R R

R' R

c r

r'

S

w Oi o o

W- 00 CD 1 so

X X or y

Remarks

identical

identical when internal losses are negligible

identical

identical

identical

identical

identical, i.e. J 6c M's total circuit resistance

in section 3

identical

identical

identical

identical

identical

identical

D. uses X with both meanings
170 -

Eaton^ J. P.

Theory of the Electromagnecic Seismograph

Bull. Seism. See. Amer. 37-75 (1957)

_E.

D

d

G

g

K

k

R

r

S

tu

u

X

O)o S-/ (S- + Q“)

J & M

rX

G.

X G.

j

M

K

R.

t

Su

y

§

c

0

X

O)

00m

00a

Remarks

analogous

identical

analogous

identical

analogous

identical

II

M

II

analogous

identical

identical

identical when E '

s

a

o is negligible

identical when E ' s ^O is negligible

identical

analogous

identical

identical

identical

identical
170a

Fix, James E.

Theoretical and Observed Noise in a High-Sensitivity Long-Period

Seismograph

Bull. Seism. Soc. Amer. 1979-1998 (1973)

Of the sixty three~symbols used in this reference, the following

thirty five are identical to those used in this report:

f

G,

J

K

M

R

01.

X

T

t

U

0

n

6

0)

(X)

(X)m

c

X +X Q

g

9

X +x

o X

2ti/(x)

(X)

The following six identities should also be noted

F Remarks

identical

2Tr/to
-• .V,. i' -

.

’"' boli*'J*saoJ v3J:vi3lan»2-i<»l» • •«to« bwaadO boa

:

*"

‘1< rfqa-raoj

.£i \r>o^ .««i»a ^

-<S SH.«aa»rVi

aiitJ at ba«w aiod«t»-*»-»^^

:3To<j« jfli bae« »ao4J^<^‘^bij^t»bt oia wil

fh'> ,.j| idandlcAl ^ ^

#rtal<>gou# 'X ^iiiaoeicai®

J^lQgoua

*>%

j tdaottcal M ; »

'4."i a

% . •- *

. uif:

1

*' 4ba3oir ftd astB blt«Mlaf*^b4^tf5W^bt xta ^at«*Ollol «

identical M

"f" KdL

idnnUeaX 1

iitnficA}
171

Maxwell, J. C.

i Treatise on Electricity and Magnetism

Article 831; Part IV, Chapter V

3rd Ed. Clarendon Press^ Oxford^ 1904.

Pomeroy, Paul W. and Sutton, George H.

fhe Use of Galvanometers as Band-Rejection Filters in Electromagnetic

Seismographs.

Bull, of the Seism. See. Amer. 50, 1, 135-151 Jan. 1960

P 6c S

CC

fl

fl

J & M

-M /M

xy^ X

-M

xy

N

M

x

*S)

K

w

JSu V

Remarks

identical

identical

identical except for possible

ambiguity of sign

identical except for possible

ambiguity of sign

identical except for possible

ambiguity of sign

used identically

identical

identical

identical except for poasible

ambiguity of sign

identical

used identically

'identical
J & M

Remarks

“n

identical when P & S are restricted

to a single galvanometer as a filter

LR

identical when T pad is used

identical for direct coupled system

^3

identical

R

s identical

X identical

y

identical

\ )“(

)

identical

identical

identical

“( ) identical

4X \ oi CO o g o g

identical
jlybner, Jo r gen

ijlnvesfigations on the Theory of the Galitzin Seismograph

[jSerland s Beitrage zur Geophysik, 21^ 259-281, (l93]^.

I R. J & M Remarks

! D rX analogous

' d

CD identical

E E J & M use various subscripts for

special situations.

G G,

X analogous

g s

identical

I h

identical

i I3 identical

K M analogous

k K identical

n (JD0

identical

COg

identical

n

P d^Vdt^ identical

R

> i

-c identical when a T pad is used

R when direct coupled J & M lump

resistors together as R.

r sometimes identical

; t>

1 "3 identical when a T pad is use

1

R Vhen direct coupled J £c M lump th

resistors together as R.

S R

s identical

t t identical

u S anal ogous

u u identical
X

X

174 -

R. J & M Remarks

e

c

6

\ O) g S

identical when internal losses of the

seismometer may be neglected

identical when internal losses of the

galvanometer may be neglected.

identical

T

(1 - a^)/a^

analogous

identical

identical

• A

Rybner^ Jorgen

The Determination of the Instrumental Constants of the Galitzin Seismograph in the Presence of Reaction.

Gerland's Beitrage zur Geophysik, 375-401^ (l937>

(correction of a numerical error^ ibid.^ 55>~ 303-313, (^1939)

In addition to the notation of the previous paper the following

symbols are introduced.

R. J & M Remarks

T

1

a

2rt/co^ identical

X analogous

6 identical

1 - identical when internal losses in the

seismometer are neglected

CC identical when internal losses of both

the seismometer and galvanometer are neglected
175 -

Scherbatskoy, S. A. and Neufeld, J.

Fundamental Relations in Seismometry

Geophysics 2j 188-212 (1937)

S & N

C

mg

e

i

i o

i

g

K

g

K

gm

L

o M

m

m

R

R

g

rm

rmg

sT

T

e T

m

V

V

m

J & M

G

3

G

1 W

oo

M

X

R

c R

g

rX

Remarks

identical

identical

identical

identical

identical

identical

identical

identical

identical

identical

identical^ J & M believe the definition

on p. 201 is a misprint,

identical

identical ^

identical

identical

X identical

T identical

(mag

identical

mech identical

V identical

V identical

s

6 9 identical

g
176 -

Silverman, Daniel

The Frequency Response of Electromagnetically Damped Dynamic and Reluctance

Type Seismometers

Geophysics 4, 53-68 (1939)

Q J & M

h

M

S

X

y

y

O)

S

X

y

y

05

Remarks

analogous. S. includes internal damping

which J & M generally presume zero,

identical

It

It

II

II

II

Sparks, Neil R. and Hawley, Paul F.

Maximum Electromagnetic Damping of a Reluctance Seismometer

Geophysics 4, 1-7, (Jan. 1939)

S & H J & M Remarks

D

identical

m p

II

05o

05o

• 1

^1 05S

II

a + jp a + jv

II
177

Sutton, George and Oliver, Jack

Seismographs of High Magnification at Long Periods

Ann. de G^ophysique, J_5, 423-433, (l959>

S & 0 J & M Remarks

K M /M

xy' X analogous

s R

3

identical

Y \7J identical

e e identical

X X identical

00 00o identical

00 00 Identical except that J . e not restrict the running

to earth motion

00g . 00g

identical

R + R

0

Identical

r + rg

^3 identical
?r

ccVjq

178

Wenner, Frank

A New Selsmoe,eter Equipped for Electromagnetic Damping and Electro

magnetic and Optical Magnification.

Natl. Bur. Stand. J. Research 2 , 963-999, (l929) RP66

0

J & M

d

r

e

G G

1

S

sH

j

M

K

M

Remarks

analogous

identical

analogous

identical

identical

identical

identical

analogous

identical

identical

S

T

o

t

+ Rl R3 + R3R,

R

s

2jt/oi

identical

identical

identical

identical

U

uX

0

U

y

0

analogous

identical

identical

identical when W. uses 0

to refer to the galvanometer

armature

analogous

identical although W. restricts

his use to earth motion.
179 -

Wenner, F. and McComb, H. E.

The Galitzin Seismoaieter. Discrepancies Between the Galitzin Theory and

The Performance of a Wilip-Galitzin Seismometer.

Bull. Seism. Soc, Amer. 2^, 317-‘322 (1936).

W & M

D

d

G

j

K

k

r

t

U

u

X

&

0

Oi

O)o

ML

J & M

?1

M

X

K

R

t

S

U

y

e

X

CD

CD & CD O g

M

xy

Remarks

analogous

identical

analogous

identical

identical

analogous

identical

identical

identical

analogous

identical

identical

identical

analogous

identical

W & M are concerned only with systems for which

CD = CD . o g

analogous
180 -

Whittaker, E. T.

A Treatise on the Analytical Dynamics of Particles. Section 27, Chapters

II and VIII. 4th‘ Ed. Dover Puolications, 1944.

Willmore, P. L.

The Theory and Design of Two Types of Portable Seismograph

Mon.. fJot. Royal Astro. Soc., Geophys. Suppi. 6, 2, 129-13S, (^1950-

W

A

I

M

s

e

CO

CO

J & M

y

K

M

9

CO

CO

Remarks

identical

W. uses I as the moment of inertia of a galvanometer mirror. It therefore^

represents a lower lirait ror J o. K. » K when optical sensing is used,

identical

identical

identical

identical

CO CO identical
131

Willmore, P. L.

The Application of the Kaxwell Impedance Bridge to the Calibration of

Electromagnetic Seismographs

Bull. Seism. Soc. Pn\er. 49 , 1, 99-114, ,^an, 1959^;

The following symbols appear to be identical.

D

J & H

X

2

K

M

S

X

e

CO

- G,

M

S

y

X

d

COO)

Wolf, Alfred

The Limiting Sensitivity of Seismic Detectors

Geophys; 2, 2, 115-122, .(Apr, 1942;

W. J & M P-emarks

A

k

identical for sinusoidal earth motion

identical

M

T

X

COCOo

H identical

T identical where J 6; to absolute temy-cr;

X identi cal

CO idenr ica

0) identical

o

G. identical
182

Worrell, Francis T.

On the Calibration of the Wenner Seismograph.

Bui. Seism. Soc. Amer., 32, 31-43, /194^.

W J & M Remarks

D rX analogous

d identical

G analogous

g =3 identical

K M analogous

k K identical

R %o identical when is negligible

R

c 'X - R

o c identical when r is negligible

r R

g

identical

r identical when r^ is negligible

r

c g 8

identical when r, is negligible

Si S/G^ identical

s identical

T

o

CN identical

t

to

CM identical

u. S analogous

03o identical

0) COg

identical

US-COIM-XBS-DC
Fig.

3.1a

Graph

for

Obtaining

Reol

Roots

of

an

Algebroic

Cubic
and V

against

fj

and p !
IT) ro CsJ o

Choracter

of

the

Transient

Response
(3) fs =2 , p= 2.8 (4) f 5

= 2 , ^

= 6

°eriod or Sei;>mometer on open circuit

Fig. 3.0 /^inc'log CoT'.putsr Resporsse tc Simiilctec Weight Lif^

J'MIiHltl
Fig. 3

(6) f 5

= 3 , /D = 4.28

Period of Seismometer on open circuit

.4 Analog Computer Response to Simulated Weight Lift
Fig. 3

(6) fs = 3 , p=6.3

l< H

Perrod or 3e;snicrr!.3Ter on open circuit

5 Anoio.j Computer Response to Smuiated Weignr Lift
Fig. 3.

(9)fs = 3,/>=I0

Period of Seismometer on open circuit

6 Analog Compufer Response to Simulated Weight Lift
(10) f s

= 4 , p-- 1.5

H- H

Period of Seismorriefer on* open circuit

Analog Computer Response to Simulated Weight Lift Fig. 3.7

(14) f s

= 4 , p= 20

(15) f s

= , yo = 50

h H

Period of Seismometer on open circuit

Fig. 3.9 Analog Computer Response to Simulated Weight Lift
OJ/cJm

FIG. 6.1 FOURTH ORDER FLAT VELOCITY
Fig. 6.3 A Speciai Case, Amplitude- Frequency response

Curve, Direct Coupled Instruments

'
o

o

o

o

rO

r!7» 3

o

3

Fig.

6.4

Damping

Required

For

Flat

Ampiituca

With

Direct

Coupled

Instruments
E

•'i i

TM.rf

j ^

w

- o . — «rv

_l! t»j

*''' »«

' '•'" o ^

^: A*—* 2j?.. ^iM<a] ^ *' T> . p- O

I ,^_ ._,,i ^

K<»' pr

n , ai -3-'

nu Ci>^

o'- ,

V

-

"

j K- "‘

r:«Mit- icw#3fo:\

w

‘•itn ' i i ^

SDfe''

'' tv^ s m o5i>^

jfl?

o

.•Jc’& JHB I

i 0/> ,?e

,

' " '''wIjS »• ^

'4«| AmplitJae- Frequency

Coupled Instruments ' is , ;’

..ik.

ir' * »
RADIAN

SECONDS

Fig. 7.3 A Possible Broad -Band Direct-Coupled Sysiem

Using A Galvanometer As A Microseism Filter
t\0 so 1.0 ^f-Oi so.

«SH ,Ya«aupst«

m9le\(8 balqucO-fosiia bno8-bDOt§ aldieaoq A

i9lli^ maisaoioiM A sA isfsmolovloO A pniaU

Jr iT>, T i.

^ -A. :1

1) * . ^

. . * 1

. . .
IREV. 7-73) ^T. OF COMM.

pOGRAPHIC DATA

' SHEET

[iTANO SUin iTU1. RJhI.ICATION OK Ki;iK)KT NO.

NBSIR 76-1089/

NBS Report 7454

2. (iov’t Accfs.sion No.

AN ANALYSIS OF INERTIAL SEISMOMETER-GALVANOMETER

COMBINATIONS

3. Kfi- ipicnl 's Ai'cc.s.sion No.

5. Publication Date July 1976

6 . Performing Organization (a

213.00

HOR(S) Daniel P. Johnson and Harry Matheson

8 . Performing Organ. Report No.

pRMING ORGANIZATION NAME AND ADDRESS

j NATIONAL BUREAU OF STANDARDS

I DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 0601- 20-06405

11 . Contract/Grant No.

poring Organization Name and Complete Address (Street, City, State, ZIP)

! Same as No. 9.

13. Type of Report & Period

Covered

Final

14. Sponsoring Agency Code

p=LEMENTARY NOTES

i This NBSIR is a reissue of the NBS Report 7454, originally printed

j March 7, 1962, and is the fourth printing

ITRACT (A 200-word or less factual summary of most si^ificant information. If document includes a significant

Mraphy or literature survey, mention it here.)

This report develops the fundamental equations for an inertial seismometer with

electromechanical transducer. If the inductance as well as the resistance of the

jtem are included, the equations of motion are of the third order. These are

bssed in some detail. Response curves are developed for several seismometerivanometer combinations, some of which are suitable for telemetry. Techniques are

Icussed for calibration, both in the laboratory and in the field. Practical and

loretical limitations are treated for circumstances in which thermal agitation is

|ajor source of instrumental noise. The bibliography includes lists of equivalent

jations

.

I ,

• K 1 nORDS (six to twelve entries; alphabetical order; capitalize only the. first letter of the first key word unless a proper

y, separated by semicolons)

lelerometer; calibration; electromechanical transducer; galvanometer; galvanometer

Ilifier; seismometer; seismic system

AII..AH1L1TY Uni imited

, lot Offici.il Distribution. Do Not Release to NTIS

,^'tiler !• rom Sup. of Doc., U.S. C'lovernmenf Printing Office

"'slungton, D.'O. 20 lOJ, SD ( .it. No. ( M

I'tJcr 1- tom N.ttion.il lecbnic.il Inform.ition .''ervice (N l l.'^l ''rtiiigf ield, Virgini.i 221^1

19. SECURITY CLASS

(THIS REPORT)

UNCI. ASSIFIED

20. SE( URll V ( LASS

(THIS PAGE)

UNCLA.SSlElliD

21. NO. OE PAGES

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