Loaal Bureau of Standards

Library, N.W. Bldg

FEB 6 1964

Reference book not to be

taken from the library.

^ecUnic&l *^lot& 233

SENSITIVITY INDICES FOR HALL GENERATORS

SHERWIN RUBIN AND GEORGE J. ROGERS

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director

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NATIONAL BUREAU OF STANDARDS

^ecltnicai <Vlot& 233

ISSUED FEBRUARY 6, 1964

SENSITIVITY INDICES FOR HALL GENERATORS

Sherwin Rubin and George J. Rogers

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C., 20402 - Price 20 cents
CONTENTS

Page

1. Introduction 1

2. The Hall Effect 3

3. Sensitivity Indices 4

4. Conclusions 16

Appendix I. Proposed Specifications 17

Appendix II. Hall Voltage Derivation 19

li
SENSITIVITY INDICES FOR HALL GENERATORS 1

Sherwin Rubin and George J. Rogers

The Hall generator is usually characterized

as a product-sensitive device, but product

sensitivity (the ratio of the Hall voltage to the

product of control current and magnetic induction)

is unsatisfactory as a figure of merit for many

applications. For this reason, two additional

sensitivity indices, one a measure of the effect

of magnetic induction on output voltage and the

other a measure of the effect of control current

on output voltage, are proposed, defined, and

illustrated by an example.

1. INTRODUCTION

The sensitivity of any device may be defined as the

ratio of output response to a specified change in the input

variable. Since the output of Hall generators is

proportional to the product of two inputs, magnetic induction

and control current, the product sensitivity (the ratio of

Hall voltage to the product of control current and magnetic

induction) has long been used to characterize the sensitivity

of these devices. Product sensitivity is useful when the

application depends upon the product relationship between

the two inputs, for example when the Hall generator is used

to measure power or for analog multiplication and division.

Hall generators are also used to measure magnetic induction,

however, and in other applications in which one of the two

This work was supported by the Navy Department, Bureau of

Naval Weapons, and is part of a project to develop standard

terminology and measuring methods for galvanomagnetic devices.
input variables is held constant. In these cases indices

are needed which express the sensitivity of the Hall

generator output to either control current or magnetic

induction independently of the other.

When the Hall generator is used as a magnetic induction

meter, for example, the control current is held constant,

while the magnetic induction varies. Product sensitivity

can be very misleading when used to predict performance in

such applications. What is really needed is the magnetic

sensitivity, or the ratio of the output voltage to magnetic

induction at the specified control current. In other

applications, the magnetic induction is held constant and

the control current is the input variable, in which case

the control-current sensitivity is needed, where controlcurrent sensitivity is defined as the ratio of output voltage

to control current at the given magnetic induction. Even

when the application is one in which the control-current

magnetic-induction product is the input variable, a knowledge

of the control-current and magnetic sensitivities may be

needed to determine the output requirements for the driving

source.

Product sensitivity is defined only in the linear case,

i.e., when it is assumed that the Hall generator characteristic can be closely approximated by a straight line through

the origin. Small-signal applications do not depend on the

linear characteristics of the Hall generator however, and

are not confined to the linear region of operation. For these

2
applications, the definitions for control-current and

magnetic sensitivity must be extended to include the smallsignal case, especially since a small-signal product

sensitivity is not now available in a useful form.

2. THE HALL EFFECT

When a charged particle moves, it generates a magnetic

field. When a charged particle moves in an external magnetic

field, the interaction between the magnetic field of the

particle and the external magnetic field in which it moves

causes the particle to be deflected from its initial path in

a direction perpendicular to the direction of the magnetic

induction. The charge distribution resulting from the

deflection of charged particles moving through a conductor

or semiconductor causes a difference of potential to be set

up across the conductor or semiconductor in a direction

mutually perpendicular to both the direction of current flow

and the direction of magnetic induction. This phenomenon

is called the Hall effect, and the voltage is called the

Hall voltage.

A slab or plate designed to generate a Hall voltage when

it carries an electric current (control current) in an

external magnetic field is called a Hall plate. The Hall

plate, together with its leads or pins, encapsulation, heat

sink, and other appurtenances attached during the manufacturing process, is called a Hall generator. For a Hall

3
generator with a Hall plate (fig. 1) of thickness d, and with

a control current I perpendicular to a magnetic induction,

2

B, the Hall voltage is:

V

H

= (R

H /d)I

c B, (1)

where V„ is the open circuit Hall voltage in volts,

I is the control current in amperes,

2

B is the magnetic induction in teslas (webers/m ),

d is the thickness in meters, and

Ru is the Hall coefficient in meters /coulomb

(volt-meter/ampere-tesla)

.

The Hall coefficient, R„, is a characteristic of the material

3 of which the Hall plate is made and the thickness, d, is

determined in manufacture. For a given Hall generator,

therefore, the output voltage is proportional to the product

of the control current and the magnetic induction.

3. SENSITIVITY INDICES

The product sensitivity index, Vm/I-B, is identified in

the literature by the symbol K„, K , or S. In this paper it

will be designated C„ to avoid the use of the symbol preferred

by any particular person or group. Although it would appear

from equation (1) that product sensitivity is equal to R^/d,

in a practical device C„ may be less than Ru/d because of

2 See Appendix II for the development of this equation.

See Appendix II, equation II. 4.
the finite area of the Hall electrodes and because of the end

effects caused by the control current electrodes. The

product sensitivity index is usually determined empirically,

therefore, rather than by correcting Ru/d to account for the

physical configuration of the Hall plate.

The performance of a Hall generator may be characterized

by families of curves which define Hall voltage as a function

of magnetic induction or control current for particular

values of load and source impedance, as illustrated in

figure 2. In figure 2(a), the curves are obtained by setting

the control current to a constant value and measuring the

Hall output voltage as the magnetic induction is varied, and

in figure 2(b) by setting the magnetic induction to a constant

value and varying the control current. Although these curves

have been idealized by drawing them as straight lines passing

through the origin, they are very close approximations to the

output characteristics of Hall generators within the normal

operating region.

The slope of any of the curves of Hall voltage versus

magnetic induction in figure 2(a) is the magnetic sensitivity,

4

If the length-to-width ratio is greater than 3, no

correction is ordinarily required; for a length-to-width

ratio of 1, C„ is about 701 of Ru/d. In this context,

length always refers to the dimension which forms the path

of the control current, and width to the dimension across

which the Hall voltage is developed, so that the length-towidth ratio may be less than unity.
Yg, at the specified value of control current. Since the

magnetic sensitivity of each curve is different, the value of

the control current for which the sensitivity was measured

must be stated. Similarly, the slope of any curve of

figure 2(b) is the control-current sensitivity, y t

» at the

corresponding magnetic induction.

The magnetic and control-current sensitivities could be

given by plotting the magnetic sensitivity against control

current and the control-current sensitivity against magnetic

induction, as shown in figure 3. If the Hall voltage is a linear function of both magnetic induction and control current

each of these curves will also be a straight line passing

through the origin. It will be noted that both these curves

have slopes equal to V.,/1 B, so that the slope of either

curve is the product sensitivity index, C„, previously

defined.

Data for the curves of figures 2 and 3 can be generated

from equation (1) if only one of the factors on the right

side of the equation is a variable. The actual characterThe symbol y is used to denote sensitivity, with the

subscript differentiating between magnetic and control

current sensitivities. It has been used without a subscript in the Russian literature to denote magnetic

sensitivity. For example, see Yu. A. Vasilevskii, "The

Use of Hall Pickups in Magnetic Heads," Avtomatika i

Telemekhanika , Vol. 21, No. 9, pp. 402-408, March 1960,

or the hnglish translation, Automation and Remote Control , November 1960, pp. 279-282.
istic of a Hall generator is not an absolutely straight line

because RH varies slightly as control current and magnetic

induction change. For the present discussion, however, this

departure from linearity is assumed to be negligibly small,

and, for the particular Hall generator, R„ is considered a constant so that C„, which is a function of Ru/d" is also a constant. Therefore, equation (1) can be written

V

H

- C

H

I CB. (2)

If the product of control current and magnetic induction

is taken as the independent variable, a plot of equation (2)

would have the same slope as the curves of figure 3. This

slope is the previously defined product sensitivity, since

V

H /IC

B = C

H

. (3)

On the other hand, if the control current is assumed to

be constant and magnetic induction the only variable, control

current can be included in the constant term and equation 1 may be written

V

H

= (CH

I C

)B.

which is the equation of the curves of figure 2(a). The

slope has been defined as magnetic sensitivity, y b

, so that

r B

= v

h /b = cH

i

c

, (4)

where I is the magnitude of the control current included in

the constant term. Similarly, if B is assumed to be constant

and I the only variable, the equation is the one plotted in
figure 2(b), whose slope is control-current sensitivity, y T

,

*I " VT c

= C

H

B > CS)

where B is the constant value of magnetic induction.

The product sensitivity index, C„, has the same value

for any magnitude of control current and magnetic induction

within the linear region, that is, at any point within the

normal operating region of the Hall generator. The magnetic

sensitivity, y

b is a constant only for the magnitude of

control current at which the measurement was made, however,

and the control-current sensitivity, y t

» is valid only for

the magnetic induction at which it was measured. For linear

conditions, the sensitivity at any other value of control

current or magnetic induction can be obtained by taking a simple proportion.

The magnetic sensitivity carried on the data sheet will

be specified at the nominal control current unless otherwise

noted. Since nominal control current is defined as the

maximum control current at which the specified linearity can

be obtained over the specified range of magnetic induction,

this magnetic sensitivity is the highest obtainable from the

Hall generator with the degree of linearity specified. At

smaller control currents, the linearity is better, but the

magnetic sensitivity is reduced by the ratio of the desired

control current to the nominal control current. At control

currents higher than nominal, the magnetic sensitivity will

increase but the linearity will be degraded. If poorer

8
linearity can be tolerated, operation is permissible without

reducing the magnetic induction provided the control current

does not exceed the specified maximum safe operating control

current.

Control-current sensitivity will be specified at one

decitesla (one kilogauss) unless otherwise noted. Controlcurrent sensitivity at magnetic inductions other than one

decitesla can be obtained by multiplying the specified

sensitivity index by the ratio of the desired induction to

one decitesla. (This is the same as multiplying the controlcurrent sensitivity index by the desired induction expressed

in decitesla, or kilogauss.) The specified degree of

linearity will be obtained over the range of magnetic

induction specified for the particular Hall generator, and

operation is permissible at values of magnetic induction

higher than the maximum specified in this range if the control

current is reduced to keep the dissipation within safe

operating limits. (At constant control current, dissipation

increases with magnetic induction because of the magnetoresistive effect.)

The maximum safe operating control current is the highest

control current at which the Hall generator can operate

without irreversible changes in the characteristics.
The magnetic and control-current sensitivities are

specified in different ways to make the information available

in the form in which it is most likely to be needed. When

magnetic induction is the input variable, Hall generators

are usually picked on the basis of the maximum sensitivity

that can be obtained with the specified degree of linearity,

since it is relatively easy to obtain whatever magnitude of

control current is required. When control current is the

input variable, on the other hand, it is likely that the level

of magnetic induction will be determined by the sources

available, and a comparison of control-current sensitivities

at the same induction level is the most meaningful. One

decitesla (one kilogauss) was chosen as the induction level

at which to make the comparison because it is within the

normal operating range of most Hall generators. Under these

conditions the control-current sensitivity can also be used

as a relative measure of product sensitivity, since product

sensitivity and control-current sensitivity are proportional

when the Hall generators are compared at the same magnetic

induction.

The following definitions of the proposed indices are

consistent with the practice used in specifying vacuum

tube and transistor characteristics. For a given magnitude

of control current, the static magnetic sensitivity, Yg» is

the ratio of Hall voltage to the magnetic induction which

10
produced it,

v

H

Y D = — , I = constant. (6)

B

B c Similarly, for a given magnitude of magnetic induction, the

static control-current sensitivity, y t

» is the ratio of the

Hall voltage to the control current which produced it,

y

V

T H

T = -2- , B = constant. (7)

1

I c The upper case subscripts indicate that these are static

values, which are compatible with the presently used product

sensitivity index.

If the instantaneous values are needed to specify the

slope at a point on the curve when the linear approximation

will not suffice, a similar pair of indices can be defined

in terms of small changes in the independent variables. If,

in equation (1), both control current and magnetic induction

vary simultaneously, the total differential is

6 V 5 V

dVH

= —- dB + —- dl (8) " 6B 61 C

c 6V

H in which is easily identified as the small-signal

6B

5V

H magnetic sensitivity and as the small-signal controlcurrent sensitivity. Therefore,

dV

H

= Y b

dB + Yi dl

c

(9)

11
where the lower-case subscripts indicate the small-signal

magnetic and control-current sensitivities.

Equation (9) is valid even if a simple equation relating

Hall voltage, control current, and magnetic induction cannot

be written, since the small-signal sensitivities can be

determined empirically for any value of control current and

magnetic induction. The small-signal control current

sensitivity, y- t is the ratio of the change in Hall voltage

to the corresponding small change in control current about

the desired point, and the small-signal magnetic sensitivity,

Yl, is the ratio of the change in Hall voltage to the corresponding small change in magnetic induction.

Equation (9) can be used as a basis for defining the

small signal sensitivities. If the control current is held

constant, dl is zero, and: ' c

dV

H

= Y b dB, so that

dVH Y b = —- t i = constant. (10)

dB c

If the magnetic sensitivity is held constant, dB is zero, and

dV

Y H

i = , B = constant. (11)

dI

c Since the product sensitivity index is usually the only

one specified in present Hall generator data sheets, it will

often be necessary to derive the magnetic and control current

sensitivities from the product sensitivity. This is possible

12
only in the static case. The necessary relationships are

given in equations (4) and (5). If it is desired to find

the static magnetic sensitivity at a particular value of

control current, I',

Y

B (I c

- !•)" V'< ^

Similarly, the static control-current sensitivity at a magnetic induction B = B* is:

Y

= CH B». (13)

I (B = B') H

The advantage of specifying magnetic and control-current

sensitivities can be illustrated by an example. Table I

lists the pertinent data as it appears in the manufacturer's

specification sheets for two Hall generators. The values for

the product sensitivity index listed in Table I make it

appear that there is only a slight (201) difference in the

sensitivity of the two Hall generators. For many Hall

generator applications this would be a reasonable conclusion,

but for induction-sensitive applications this first

impression would be misleading.

nt

13
TABLE I

Typical Hall Generator Specifications

Characteristic Type A Type B

7

I nominal 150 mA 400 mA

7

B nominal 1 Tesla (10 kilogauss) 1 Tesla

(10 kilogauss)

V

H for B and I c 95 mV 300 mV

nominal

CH 0.063 mV/mA-decitesla 0.075 mV/mA - decitesla

Linearity error 1% 1 percent of maximum

"0

For practical purposes the nominal values of control

current and magnetic induction are the highest at which

the Hall generator will operate with the degree of

linearity specified. To some extent one of these limits

can be reduced to compensate for increases in the other,

but both cannot be exceeded simultaneously without

degrading the linearity.

14
When the magnetic and control-current sensitivities

are needed, it is necessary to calculate them from the

product sensitivity, using the nominal value of control

current and a magnetic induction of one decitesla. For

generator A,

Yr C H

I

f

. nan = 0.063 (150) B H c(nominalj v

'

= 9.45 mV/decitesla, I = 150 mA,

and Yl = C

H

B

(1 decitesla) » 0.063 (1)

= 0.063 mV/mA, B = 1 decitesla.

For generator B,

Y D

= C„I r „ • ,.= 0.075 (400)

B H c(nominal) v '

= 30 mV/decitesla, I = 400 mA,

and Y T = C„B ri Ao „^acV^ = 0.075 (1)

I H (. 1 deciteslaj v *

= 0.075 mV/mA, B = 1 decitesla.

Thus it is seen that while there is no significant

difference between the product sensitivity indices of the

two Hall generators, their static magnetic sensitivities

differ by a factor of three to one. As expected, the

control-current sensitivities are proportional to the product

sensitivities

.

15
In this example the magnetic and control-current

sensitivities were easily calculated because the value of

the nominal control current was available from the specifications. It is common practice in the literature, however,

to use product sensitivity to specify the sensitivity of

magnetic induction meters, often without stating the control

current at which the meter operates. Without information

about control current, it is impossible to compare two such

devices. This situation would not occur if the magnetic

sensitivity of the device was given, rather than the product

sensitivity.

4. CONCLUSIONS

The use of product sensitivity alone to specify the

sensitivity of Hall generators is inadequate for two reasons.

First, it is not specific enough when the application is one

in which either magnetic induction or control current is the

input variable, and second, it cannot furnish small-signal

information for applications in which the static sensitivity

is not a satisfactory index. The two proposed indices,

magnetic and control-current sensitivity, have been defined

for both the static and small-signal case to provide the

necessary information. In addition, the control-current

sensitivity is specified in such a way that it can be used

as a substitute for product sensitivity.

16
APPENDIX I

Proposed specifications for the magnetic and control current

sensitivities of Hall generators

Control current sensitivity : The control current sensitivityis a measure of the effect of control current on Hall voltage

for a given magnitude of magnetic induction under given

temperature conditions. The static control-current sensitivity, Y I, is the ratio of Hall voltage to control current

at any point on the curve of Hall voltage versus control

current for a constant magnetic induction. The small-signal

control current sensitivity, Y i, is the ratio of a change

o in Hall voltage to the corresponding small change in control

current about the desired point on the curve.

Magnetic sensitivity : Magnetic sensitivity is a measure of

the effect of magnetic induction on Hall voltage for a given

magnitude of control current under given temperature conditions. The static magnetic sensitivity, Y B, is the ratio of

Hall voltage to magnetic induction at any point on the curve

of Hall voltage versus magnetic induction for a constant

control current. The small-signal magnetic sensitivity, Y b,

is the ratio of a change in Hall voltage to the corresponding

o small change in magnetic induction about the desired point on

the curve.

g The change is considered small if doubling its magnitude

does not produce a change in the sensitivity that is greater

than the specified accuracy of the measurement.

17
Conditions of measurement : Unless otherwise stated, these

sensitivities shall be measured at nominal control current

and at a magnetic induction of one decitesla (one kilogauss).

The Hall generator shall be suspended by standard size leads

in still air at 25°C when the measurements are made. Both

the decrease in sensitivity for optimum linear loading, and

the limiting range of magnetic induction to insure a given

maximum percentage non-linearity shall be stated.

For a specified range of magnetic induction, nominal control

current is that magnitude of control current which, if

exceeded, would cause the linearity error of the device to

exceed the rated magnitude under optimum drive and/or load

conditions.

18
APPENDIX II

HALL VOLTAGE DERIVATION

A standard derivation for the equations relating Hall

voltage to control current and magnetic induction follows:

If a Hall plate, oriented along cartesian coordinates

x, y, and z, as shown in figure 1 has a current flowing in

the x direction and is acted on by a magnetic induction in

the y direction, there will be an electric field set up in

the z direction. This field is caused by the action of the

magnetic field on the moving charges, which action concentrates the charges along the xy edge of the plate.

There are two forces acting in the z direction. The

Lorentz force due to the magnetic induction B acting on the

charges, e, which move in the x direction with a velocity,

V is

F = ev

x

B

y

. (II. 1)

The force due to the electric field, E , caused by the

concentration of charges along the edge of the plate is

F = eE

z

. (II. 2)

At equilibrium these two forces are equal, and

eE

z = ev

x

B

y

,

so that e„ = v B . (II. 3) z x y

19
Since the charge velocity v = pE , where u, the charge

A A. mobility, is equal to the drift velocity per unit field, and

field strength E = J /a, where a is the conductivity of the

material of the Hall plate in mhos per unit length and J

A

is the current density,

v

x

= y(J

x /a ),

and E

z = m.(J x /o ) B

y

. (II. 4)

Charge mobility and conductivity are properties of the

Hall plate material, and their ratio, \i/a is a constant of

the material called the Hall coefficient, RH

, so that

equation II. 4 can be written

E

z = VxB y' (II,5)

EquationH.5 shows that the field strength set up by

the Hall effect is a function of the Hall coefficient of the

plate material, the control current density, and the magnetic

induction, and is independent of the dimensions of the Hall

plate. In a practical device, however, it is preferable to

replace field strength and current density by their

equivalents, volts per unit distance and current per unit

cross section area. It is conventional in describing Hall

effect devices to define length, 1, as the dimension which

forms the path of the control current, width, w, as the

dimension across which the Hall voltage is developed, and

thickness, d, as the dimension parallel to the magnetic

induction. Using this convention, E = V /w, and

20
J = I /wd. Furthermore, V is the Hall voltage, V,,, and I A A «# i 1 A

is the control current, I ; the magnetic induction is usually

written without a subscript. Making these substitutions,

V

H

= R

H

J

x (w)B

= (RH /d)I

c

B

(11. 6)

(11. 7)

R,. and d are grouped together because they are a character'

istic of the Hall plate, while I and B are independent

variables.

x

4

e ©

L-ft

1

V.

Figure 1. A Hall plate carrying control current, I , in a A

magnetic induction, B , producing a Hall voltage, V .

The plate is assumed to be of n-type material with conduction predominantly by electrons. If p-type material were used,

conduction would be primarily by holes, and the polarity of the

Hall voltage would be reversed.

21
cd

1

300

o>

1

1 200

«=E m

100

>

E

CD

2 4 6 8 10 MAGNETIC INDUCTION (B)-DECITESLA

(a) Hall voltage versus

magnetic induction.

CONTROL CURRENT (Ir )-m c (b) Hall voltage versi & control current

Figure 2. Typical Hall generator performance curves

o

> 60

E

i CD 50 - /

\

I

>

40 - /

i

> 30

Lo2

LU 20

CO

CD

1— 10

U_l ^^CD f

«t / I l l 1 1 1 2 4 6 8 10 CONTROL CURRENT (I

c )-mA

12

(a) Magnetic sensitivity

versus control current

.

2 4 6 8 10 MAGNETIC INDUCTION (B)-OECITE

(b) Current sensitivity

versus magnetic induction

Figure 3. Hall generator sensitivity.

The slope of either curve is 5 millivolts per decitesla.

22

U.S. GOVERNMENT PRINTING OFFICE : 1964 O —718-5-

bS;
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hysical Chemistry. Thermochemistry. Surface Chemistry. Organic Chemistry. Molecular Spectroscopy. Eleentary Processes. Mass Spectrometry. Photochemistry and Radiation Chemistry.

ffice of Weights and Measures.

-DER, COLO.

jenic Engineering Laboratory. Cryogenic Processes. Cryogenic Properties of Solids. Cryogenic Technical

ices. Properties of Cryogenic Fluids. CENTRAL RADIO PROPAGATION LABORATORY

_jhere Research and Propagation. Low Frequency and Very Low Frequency Research. Ionosphere Re-

:h. Prediction Services. Sun- Earth Relationships. Field Engineering. Radio Warning Services. Vertical ings Research.

)Sphere and Space Telecommunications. Data Reduction Instrumentation. Radio Noise. Tropospheric

urements. Tropospheric Analysis. Spectrum Utilization Research. Radio-Meteorology. Lower Atmosphere

ics.Systems. Applied Electromagnetic Theory. High Frequency and Very High Frequency Research. Fre-

;y Utilization. Modulation Research. Antenna Research. Radiodetermination.

Atmosphere and Space Physics. Upper Atmosphere and Plasma Physics. High Latitude Ionosphere

sics. Ionosphere and Exosphere Scatter. Airglow and Aurora. Ionospheric Radio Astronomy.

RADIO STANDARDS LABORATORY

Standards Physics. Frequency and Time Disseminations. Radio and Microwave Materials. Atomic Fre-

:y and Time-Interval Standards. Radio Plasma. Microwave Physics.

Standards Engineering. High Frequency Electrical Standards. High Frequency Calibration Services. High

lency Impedance Standards. Microwave Calibration Services. Microwave Circuit Standards. Low Frequency

bration Services. NBS LABORATORY ASTROPHYSICS GROUP

(Joint Institute for Laboratory Astrophysics at Univ. of Colo.)