,-* ,•*"-

Rtyj Hu H V

1 ^W

W

'"„r*
>f

\

NBS TECHNICAL NOTE 692

U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards

NBS

Pub!

cations

Servo Techniques in Oscillators

and Measurement Systems

QC

100

.U5753

nO.692

1976
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act of Congress March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau consists of the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology,

the Institute for Computer Sciences and Technology, and the Office for Information Programs.

THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United

States of a complete and consistent system of physical measurement; coordinates that system

with measurement systems of other nations; and furnishes essential services leading to accurate and uniform physical measurements throughout the Nation's scientific community, industry,

and commerce. The Institute consists of the Office of Measurement Services, the Office of Radiation Measurement and the following Center and divisions:

Applied Mathematics — Electricity — Mechanics — Heat — Optical Physics — Center

for Radiation Research: Nuclear Sciences; Applied Radiation — Laboratory Astrophysics" — Cryogenics 2 — Electromagnetics- — Time and Frequency 1'. THE INSTITUTE FOR MATERIALS RESEARCH conducts materials research leading to improved methods of measurement, standards, and data on the properties of well-characterized

materials needed by industry, commerce, educational institutions, and Government; provides

advisory and research services to other Government agencies; and develops, produces, and

distributes standard reference materials. The Institute consists of the Office of Standard

Reference Materials, the Office of Air and Water Measurement, and the following divisions:

Analytical Chemistry — Polymers — Metallurgy — Inorganic Materials — Reactor

Radiation — Physical Chemistry.

THE INSTITUTE FOR APPLIED TECHNOLOGY provides technical services to promote

the use of available technology and to facilitate technological innovation in industry and Government; cooperates with public and private organizations leading to the development of

technological standards (including mandatory safety standards), codes and methods of test; and provides technical advice and services to Government agencies upon request. The Insti- tute consists of the following divisions and Centers: Standards Application and Analysis — Electronic Technology — Center for Consumer

Product Technology: Product Systems Analysis; Product Engineering — Center for Building

Technology: Structures, Materials, and Life Safety: Building Environment; Technical Evaluation and Application — Center for Fire Research: Fire Science; Fire Safety Engineering.

THE INSTITUTE FOR COMPUTER SCIENCES AND TECHNOLOGY conducts research and provides technical services designed to aid Government agencies in improving cost effec- tiveness in the conduct of their programs through the selection, acquisition, and effective

utilization of automatic data processing equipment; and serves as the principal focus within

the executive branch for the development of Federal standards for automatic data processing

equipment, techniques, and computer languages. The Institute consists of the following

divisions: Computer Services — Systems and Software — Computer Systems Engineering — Information Technology.

THE OFFICE FOR INFORMATION PROGRAMS promotes optimum dissemination and

accessibility of scientific information generated within NBS and other agencies of the Federal Government; promotes the development of the National Standard Reference Data System and

a system of information analysis centers dealing with the broader aspects of the National Measurement System; provides appropriate services to ensure that the NBS staff has optimum

accessibility to the scientific information of the world. The Office consists of the following

organizational units: Office of Standard Reference Data — Office of Information Activities — Office of Technical

Publications — Library — Office of International Relations — Office of International Standards.

1 Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington, DC. 20234.

- Located at Boulder. Colorado 80302.
lOO ' 3

Servo Techniques in Oscillators

and Measurement Systems

<

~zLe-c_.K *v MD^^5 CVd.

Fred L. Walls

S.R. Stein

Time and Frequency Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

*** 0F cc

o Mi, ( *

U.S. DEPARTMENT OF COMMERCE, Elliot L. Richardson, Secretary

Edward 0. Vetter, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

^.C.NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued December 1976
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 692

Nat. Bur. Stand. (U.S.), Tech Note 692, 23 pages (December 1976)

CODEN: NBTNAE

US. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1976

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402

(Order by SD Catalog No. C13. 46:692) Price 55 cents (Add 25 percent additional for other than U.S. mailmf
SERVO TECHNIQUES IN OSCILLATORS

AND MEASUREMENT SYSTEMS

Fred L. Walls & S.R. Stein

Nearly every precision oscillator includes a frequency or phase servo system.

In the case of cesium standards, a crystal oscillator is frequency locked to a particular resonance line in atomic cesium; in the case of an oscillating hydrogen

maser, a crystal oscillator is phase locked to the very weak signal coming from the

microwave cavity. The first section of this note treats the errors and offsets of

frequency-lock loops which result from background, noise, and other effects. Third

harmonic lock systems and square wave frequency modulation are analyzed as possible

solutions to some of these problems. The second section is a general treatment of servo system response which is applicable to both frequency-and phase-lock loops.

The effects of noise on such servo systems are discussed in detail, and an example

is given of how to obtain optimum performance from a pair of phase-locked oscillators.

A simple circuit is suggested for phase-locking high quality oscillators, which has

many advantages over previous circuits.

Key words: Atomic frequency standard; frequency discrimination; frequency domain

stability; frequency-lock loop; phase-lock loop; phase noise; servo techniques;

time domain stability.

I

.

Basic Requirements

In order to lock our local oscillator, a number of requirements must be

met. We must be able to detect a reference signal at the frequency of

interest, or we must be able to detect the beat frequency between the

reference signal and the local oscillator. The reference signal could be

the n-th harmonic or subharmonic of a stable oscillator (including the

fundamental n = 0) or it might be the absorption line of methane, CH. , at

14 0.88 x 10 Hz, etc.

Next, the local oscillator must be tunable. Usually the local oscillator

is voltage tunable at least over the frequency range of interest and we will simpy call it a VCO, for voltage-controlled oscillator.

Next, some means for determining if the VCO is higher or lower in frequency

than the reference, i.e., frequency discrimination, must be employed.

This usually means frequency modulating the VCO or using some kind of phase

detector.

The type of frequency discrimination chosen determines to a large extent

the details of the amplifier and feedback-servo-system which actually

controls the VCO. The servo system amplifies the difference signal from

the frequency or phase discrimiator and processes it to create the

appropriate correction voltage for the VCO in order to steer it close to

the desired reference frequency. The precision and speed to which the

VCO tracks the reference is determined to a large extent by the magnitude

and frequency dependence of the servo gain. [1,2]

II

.

Types of Frequency Discrimination Commonly Used

A. Sine Wave Modulation

The most common way to tell if the frequency of a local oscillator
is above or below a reference resonance line is to frequency

modulate (FM) the local oscillator and detect the phase of the

resulting amplitude modulated signal.*

1 .o r

2 0.4

0.2

_L

0.2

0.0

>

H>

a

wQ

A % B

VC0 FREQUENCY -

FIGURE 1 Reference line (a) and its first derivitive (b)

For the moment, let's assume that the width of the VCO output is

totally symmetric and very narrow compared to the width of the

reference line shown in figure 1, curve a. If the center of the VCO

is at the point A, then the output signal obviously increases as the

frequency of the VCO is increased, and at point B, the signal decreases

as the VCO frequency increases. If the frequency of the VCO is

swept back and forth, then the signal has both a dc and an ac component. If the deviation of the FM is small compared to the linewidth

2W, then the ac component of the signal, referenced to the phase of

the FM as in a lock- in detector, fairly accurately reproduces the

derivative of curve a as is illustrated by curve b in figure 1. Note that in curve b, the point of zero signal which also

* Passive frequency discriminators yielding directly an output vs. frequency

similar to curve b in figure 1 exist for RF frequencies and are used in FM radio

reception and other applications. Unfortunately, they usually do not have the

high stabilities typically required of a frequency reference.

It is also possible to inject a low power reference signal into an oscillator

and have it lock to the reference. The difficulty with this approach is that the

parameters cannot be easily controlled or measured, and the stability of injection

locked oscillators are typically much less than the reference. Unwanted injection

locking can be a serious problem when comparing two oscillators if there is not

sufficient isolation between them.
has the steepest slope nominally occurs at the center of the resonance line. This curve is referred to as a frequency discriminator

curve. This type of signal can then be used to steer the VCO because we now

have a dc signal proportional to the frequency error and of opposite

sign depending whether the VCO is higher or lower than the center of

the resonance. Now let's examine this process in a little more detail. Assume more generally that we have a Lorentzian* line being observed on a sloped

and curved background. Then:

Signal Amp = 4/ -5 *- 7 + k (u, - u ) + k, (to - o> n ) 2 V v' + (to - uj„r 10 2

Now let id - a), = B cos fit. Then

Signal Amp = - / . \l + -j [(u>, - a) ) + B cos fit] Y

+ k-jj ( to 1

- ojq) + B cos fit]

+ k

2

[(to 1 - 10 ) + B cos fit] 2

where y = 2ttW relates to the linewidth, co is the instantaneous angular frequency

of the VCO,^ is the average frequency

of the VCO, and B is the amplitude of the

FM which occurs at an angular frequency

fi. For simplicity, let the offset u - to be small compared to the sweep amplitude

B, and let B be small compared to y . Then using:

1 6 +

3

K 2A. - X -

2 +8

6 + yJlT&

*By this we mean a resonance which can be accurately described by the first

term in the following equation.
Signal Amp i 1 r 2 1 ~ —J

luj l ~ ^0 + cos nt ^

+ k, ( l

(co l ~ w + B cos fit ^

j.+

3 r 4 —J lt0j_ - w + B cos fit]

8r

+ k,[2 1^2. ~ w + B cos ^^'

Expanding the brackets and grouping according to the terms Q, 2fi,

etc. , yields

A= 1 --i.

2Y'

(C0 1

-03 ) + ^((O^Wq) + k

2

( Ul -

U(J )

(to 1

-o) ) ^7~ 0" coefficient

(o) 1

-cj )B

2 - 2y k. (w-i-wg) 4 ^ k

2

2 (oj 1

-(jj )

3B(u£u ) ~T~2"

2y cos fit: 1st order coefficient

B

2 .1 3 ^!-^ )

2" 2 Y k, . . .] cos 2fit : 2nd order

coefficient

3B (u) -

u)q)

2y

T cos 3fit : 3rd order coefficient

Note that a non-flat background has sizable contributions to the

1st order coefficient. This means the VCO locks to the wrong frequency. The corresponding frequency shift is found from setting to zero the first order coefficient, using only the terms linear

in frequency for a suitable approximation,

2 -

2 ^ k

l

A

2 - 4y kk_ =

or

(u>.

2 u

Y k -

1 i -a k

2 which is essentially the ratio of the slope of the background to the

slope of the derivative. This effect can be avoided to a high
degree by using the coefficient of the 3ft term, which discriminates

against the sloping background through the quadratic term, k_.

At this point, it should be mentioned that harmonic distortion will

shift the average frequency of the VCO also. This distortion can be

caused by either a harmonic content in the modulation itself or by

asymmetries in the resonance line. In both cases, the magnitude of

the shift can be explicitly found by using the formalism we developed

above. For example, if the frequency modulation were:

B cos fit + k_. cos 2ftt,

then there would be an additional term in both the 1st and 3rd order co2 efficients. For the first order term, it is -Bk-./2y . The frequency

shift in a 1st order lock, assuming no baseline problems, is

co.. - cu n = -k /2. The shift is equal to half the amplitude of the

second harmonic modulation. This is an extremely severe condition.

Take the primary cesium beam standard, for example: to achieve an accuracy of 1 part in 10 , one has to keep the second harmonic of

the FM, k-. (due to harmonic content in the modulation and/or line

asymmetry), below 0.002 Hz while sweeping across the line width of

30 Hz.

There is another lesson here as well. The VCO has a finite amount

of noise at frequencies equal to to.. + ft, to.. ± 2ft,......., which will

mimic FM at ft, 2ft,.... While this won't yield an offset if. they are symmetric about the line center, the terms ft and 2ft will add to the

noise in the detected signal of frequency ft and hence lead to perturbation in the locked frequency of the VCO. These effects are treated to first order in appendix A.

Squarewave Frequency Modulation

If baseline problems are not present, squarewave modulation provides

an attractive alternative to sine wave modulation. For example, the

second harmonic problems do not exist. The experimental problems are

now: (1) how to switch the frequency without any transients and

(2) keeping the power constant. Both would, of course, shift the

lock point of the VCO away from line center. The optimum way to

process the signal now is through an integrator which is switched

synchronously with the FM. If the reference level of the integrator

is the average of the two signal levels, then this system is even independent of small differences in the time spent at the two frequencies. In fact, the line can be split to higher precision with

square wave than with sine wave modulation. This is because the
2 2 12

average signal power from sine wave modulation is A cos Qt = yA

~2 2 while the average from square wave is A = A . This factor of two

increase in power translates to a 50% increase in the resolution for

picking line center; i.e., into a corresponding increase in lock

accuracy and frequency stability.

Direct Phase Comparisons

The technique of direct phase comparison is most commonly used to

lock one oscillator to a reference signal derived from another osc- illator although it can be used whenever- one can directly detect the

beat frequency between the reference signal and the VCO. Figure 2 shows a typical setup:

REF SIG MI XER VCO

OUTPUT

-180 -90 90 180

PHASE DIFFERENCE —

FIGURE 2 Top: A double balanced mixer used as a phase detector.

Bottom: Output voltage of phase detector vs. phase difference

between the reference and the VCO.

III.

The beat frequency is caused by the accumulation of phase A cot associated with the frequency difference Aco between the two signals.

If the mixer output is applied to the control of the VCO, it will cause the VCO to lock at +90 or - 90 phase difference from the

reference signal. If the phase difference is maintained constant,

then the frequencies must be the same. The stability of the lock is

determined by the noise at the mixer and the gain of the servo loop.

This will be discussed at some length in section IV.

Part of the observed noise at the mixer is due to the mixer itself;

i.e., changes in observed phase between the two signals due to

thermal effects of the diode or diodes used. Amplitude changes in

the signals change the observed phase difference also. For this

reason, amplitude limiting is often used before the mixer.

Servo Systems

A. Basic Concepts

6
B.

The function of the servo system is to process the error signal

derived from the frequency discriminator and use it to control the

frequency of the VCO. For simplicity, it will be assumed that the servo is a proportional continuous one, although digital and sampled

servos are sometimes used. As long as the error signal is constantly

monitored and processed, there is no theoretical reason that one kind is superior to another.

The function G (co) which specifies the gain around the loop as a servo £-a f

function of angular frequency and n(cj), the servo noise as a function

of frequency, fully specify the servo system. In order to determine

the optimim c values of G (u>) , one has to know the details of the servo phase shift around the loop from dc up to the desired unity gain

point c or 1/t , where x is the fastest correction time desired. We

v v

call it the attack time. The minimum gain at dc is set by the maximum phase error permitted and the size of the open loop phase shifts

expected.

Example of a Phase Locked Loop

Problem: Design a servo system to lock a 5 MHz VCO to a reference

oscillator with an attack time of 1 second (this is equivalent to a unity gain frequency of 1/2tt Hz = .16 Hz). Also calculate the maximum

open loop frequency difference that can be tolerated between the two

oscillators so that the residual phase error at the mixer doesn't

exceed 10°. The voltage control of the VCO causes a frequency

change of 5 x 10

_3 Hz/volt, and the phase sensitivity of the mixer

is approximately .003 v per degree phase deviation from + 90 . The

solution is shown in figure 3a.

FIRST ORDER PHASE-LOCK LOOP

5 MHz

VCO

3 7Kfl 150«

A/ NTDOUBLE

BALANCED

MIXER

50fl

5 MHz

REF OSC

FIGURE 3a Example of a 5 MHz VCO phase-locked to a reference oscillator with

a first order loop.
Ideally, the phase locked loop forces the phase difference between

two oscillators to

variable time T is

two oscillators to approach zero. The servo gain Gservo ^^ at

in degrees volts/degree

G rvo

(^) = (360) (5 x 10 3 Hz/volt) t (3 x 10

3 /°) G (u)

where the frequency dependence of the amplifier is contained in

G (co) . Note that the phase detection yields a G (co) which

amp c 2 servo

falls off as 6 dB/octave even if the amplifier gain is independent of

frequency. This means that the amplifier gain should fall off less than

6 dB/octave or the servo will be unstable. In fact, for the best

stability,J G (co) should fall between and 3 dB/octave for

amp ' angular frequencies near 1/t . For simplicity, let G (co) be a constant. Then G (co) = 1 at t = 1 second leads to an servo amplifier gain of

G u) =

1 ^ = 185

amp

(360) (5) x 10 J

3 x 10

The 50 SI resistor and the 0.1 yf capacitor terminate the 5 MHz

signals at the X-port. The additional 150 Q resistor terminates the

dc signal with 200 fi. To achieve a gain of 185 requires a feedback

resistor of 37 kf2. The capacitance from input to output is typically

of the order of 20 pf so the RC time of the amplifier is

RC = 3.7 x 10

4 (20 x 10~ 12

) = 74 x 10~ l! seconds. Since this is

smaller than 1 second, it won't change the gain slope at frequencies

less than = .16 Hz. Note that the equivalent frequency

2ttt error gain at dc is automatically very large due to the phase detection of frequency differences. The gain at 1 hour is 3.6 x 10 . The dashed line in figure 3 3 b illustrates the function G „(co). servo

The optional variable capacitor at the mixer outnut is adjusted to

parallel resonate at the sum frequency. This typically increases the

phase sensitivity near the zero crossings by a factor of 2.*

'suggested by Charles Stone, Austron, Inc.
12 dB/OCTAVE

.0001 .001 .01 .1 1 FREQUENCY a

FIGURE 3b Servo gain, G servo

(uj) , as a function ot frequency for a first

order (dashed line) and second order loop (solid line

If the amplifier input or double-balanced mixer drifts 1 uV, the servo will force the VCO to drift 10~ 6 /0.003 = 0.00033 degrees in phase

relative to the reference oscillator. The maximum frequency offset that

can be tolerated to keep the phase error at the mixer output less

than 10 is

10° (3 x 10 3 V/ ) (185) (5 x 10 3 Hz/volt) = frequency offset

Af

-2

f = 2.5 x 10 ' Hz or only 5 parts in 10 at 5 MHz.

The frequency difference over which the phase lock loop in this example

will stay locked is so small as to be very troublesome in many

applications. In a phase measurement system, the phase error must be

kept small (0 <10°) so that the phase sensitivity of the mixer is

constant.

The following circuit increases the long term gain of the phase lock

loop by several orders of magnitude, enabling one to reduce the phase

error to nearly zero.
SECOND ORDER PHASE-LOCK LOOP

DOUBLE

BALANCED

MIXER

150ft 50ft —*—vwi—•— X-^ F ZL7

5 MHz

REF OSC

-t

FIGURE 3c A 5 MHz VCO phase locked to a reference oscillator with a second

order loop. Loop attack time is Is.

The first stage is identical to figure 3a.

The second amplifier has unity gain for frequencies which are larger

than 1 If R-,C, is set larger than the attack time of the loop

R

1 C1 then the loop is unconditionally stable. The best step response is

obtained for R,C, between 1 and 5 times the attack time t . The loop 11 v

is critically damped for R.C, = 4 t . The inherent gain in long term

is increased by the factor R(_ over that of figure 3a where RL is the

leakage resistance across C. . The solid curve of figure 3b shows

10
G (<jj) for Ri_ = 10 . This increase in gain holds the phase error

servo Hr. 3 r R

l very close to zero and increases the frequency difference between the

two oscillators that can be accommodated up to the limit imposed by the

voltage swing of the second amplifier and/or the maximum tuning available in the oscillator. The solid line in figure 3b shows the total servo gain 3 G (co) for figure 3c. The dashed curve of figure 3d servo ^ 3 shows the measured step response of the first order phase lock loop

of figure 3a while the solid curve of figure 3d shows the measured

step response of a second order loop similar to figure 3c with

R,C, = t . 11 V

FIGURE 3d Response of circuit 3a to a step phase change (dashed line) and

circuit 3c (solid line) .

Second order loops such as illustrated by figure 3c can also be used

to optimize frequency lock loops as well, and are very easily accomplished

with present day operational amplifiers. The second order loop filter

of figure 3c can be realized with a single operational amplifier- However,

for clarity it was shown with two stages.

iy. Overall Estimates of Performance

We can now estimate the overall performance of the locked VCO. The frequency stability is calculated using the following formula for the spectral

densitv:

11
S(f) =

G (1/2TTT) servo

1 + G (1/2ttt: servo

S (f , ref ) + S (f , n) S(f, VCO) servo (1/2ttt)

G (1/2^) is generally

servo ^ J

where G (f = 1/2 m) is the measured servo gain, S(f, ref) is the servo 3 ' spectral density of phase fluctuations S (f) for a phase lock loop or

the spectral density of frequency fluctuations S (f) for a frequency

lock loop of the reference signal, S(f, n) is the corresponding spectral

density that would be required in the VCO to produce the servo system

noise, and S(f, VCO) is the corresponding spectral density of phase or frequency fluctuations of the unlocked VCO.

a monotonically increasing function of T . This makes it possible to

draw some general conclusions about the output spectrum of the servoed

oscillator. The noise in the reference oscillator and in the control

loop is low pass filtered, while the noise in the VCO is high pass

filtered. This leads to the most common situation - the output spectrum

is dominated by the reference oscillator at low Fourier frequencies and

by the VCO at high Fourier frequencies. However, in the event that the

stability of the VCO, S(f, VCO), is much worse than the stability of the servo or reference, we see that S(f) is ~[S(f, VC0]/[1 + 0(1/2^ ] 2

. 1 4 If G (^ ) has a maximum value of 10 , for example, then the stability

of theservolocked

2-m; VCO can never be better than S(f, VCO)

^2

/ ( 10 , 001 ) , even if

the servo and reference are much more stable at the sample time t .

This indicates that we want G (,.,) as large as possible.

servo w a jt However, the maximum G

sev. Tr^w) is fixed bv the rolloff slope and the minimum value of T which can be tolerated either by

the short-term stability of the reference signal or by the additional

phase shift in a first order servo loop at high frequencies which would

cause the gain rolloff to exceed 9 dB/octave. If the rolloff exceeds

12 dB/octave at the unity gain point, the servo will oscillate.

The required ^ shapec of G (f) to reduce S(f, VCO) below S(f, ref) for servo Fourier frequencies below 1/2 7TT can be determined from the above

--2, equation. For example, to transform random walk of phase (S^ccf ) to

white phase noise (S « f ) requires a single integration (i.e.

1 <j) ... IG (f)|oc An analogous result can be derived for deterministic

l servo I f

r processes which can not be described in terms of spectral densities.

For example, the unlocked VCO may exhibit frequency offset [ <J> (t)«t ],

2 frequency drift [<j,(t)<xt ], or even frequency acceleration. If the parameter to be controlled has an open loop behavior proportional to t then

the requirement for the closed loop system to have zero dc error is

12
lim fpG (f) = °°. Thus if the oscillator in a first order phase lock loop has

_ servo

f->-o a frequency offset from the reference, then the closed loop system has a phase error. This problem is corrected by adding a second integration. Insufficient servo gain is even a greater problem in frequency

lock systems since they lack the intrinsic integrating property of

the phase-lock loop. The dc gain from a single stage servo is quite

often insufficient to reduce the frequency drifts in the VCO to an acceptable level. The solution is then to use a second order loop

to achieve the desired dc gain, being careful to keep the combined

rolloff curves between 6 and 9 dB/octave near the unity gain point.

Figure 4 shows a typical result of insufficient dc gain in a frequency

lock system.

Figure 4a shows the spectral density of phase fluctuations S,(f) of

an oscillator typified by curve "VCO", which is locked to an oscillator

typified by "ref " The servo gain is assumed to have a maximum value

4 of 10 falling 6dB/octave and reaching a value of 1 at the indicated

attack times.

Figure 4b shows the time domain stability resulting from the various

attack times. Note that in this example the frequency stability for

sample times less than 100 s is almost independent of the attack

time. The noise bandwidth is assumed to be >> 1 MHz.

13
a:

i.

FOURIER FREQUENCY f (Hz)

FIGURE 4a Spectral density of phase for a VCO locked to a reference with a first order loop as function of loop attack time. G ( w ) assumed

10

10

10

10"

D -10

10

u

to be limited to 10 . -2

10

10

•12

14

10 -16

J L I I I I l l ' I l I

I L 10"° 10

-6

lO"

4 10' 2 10 10° ~\Q

1 10" SAMPLE TIME t(s)

FIGURE 4b Corresponding time domain stability for the reference and VCO

signals of Fig. 4a as a function of loop attack time. Noise

bandwidth assumed to be > 1 MHz.

14
Figure 5a shows another example where the reference (curve b) has

better stability than the VCO (curve a) only at low Fourier frequencies

Curve c shows the resulting phase spectral density when the VCO is

locked to the reference with a second order loop with the gain

specified by figure 5b. Figure 5c shows the resulting time domain

stabilities. The optimum attack time can be found by minimizing the

equation for S(f) as a function of x . For this example ,. lms <x <lms

v v produce nearly identical results. If x is decreased below . Iras then

the higher frequency phase noise in the reference begins to cause a deterioration in a (x>x ) and for x >lms the low frequency components

of the VCO are not reduced enough so that again a (x>]ms) is not as low

as it could be.

There are two limitations as to how much stability improvement may be

obtained. In the first place, if the noise of the control loop is

larger than the reference noise, then the VCO spectrum will be

degraded compared to the ultimate performance. For example, excessive

white noise in the detector of a frequency lock loop will result in

-1/2

o decreasing only as x . In the second place, it is not possible " -1

for a to roll off faster than x because it is the integral of

the phase fluctuations up to some high frequency cut-off.

Thus, the zero crossings of the signal will always have some rms fluctuation, not less than the integral of the high frequency phase

noise. Since the frequency is phase divided by measurement time, the

uncertainty in the frequency decreases as t _1

. The exact performance

for a given phase spectrum may be computed from the formula:

o\(x) = ±y ±5. Kr J

h

dfS,(f) Sin * (lTfT) 2 2 1 f h 4

X TT~ V y 2 2 _2

As a result of this characteristic, the long term stability may be degraded if the

attack time of the servo is too slow.

15
10 -4

10

10

-6

CVI 10

S-

•10

£ io12

10

•14

10

10

-16

•li

V \

i i v \ V \ a. FREE RUNNING ( DSC. V \ \ \ \ \ \ \ b. REFERENCE "

\ \ C \ c. SYSTEM LOCKED WITH

\ \ \ SERVO GAIN OF \

^

\ b\ N \

a

FIG. 5B

- \

\\DRIFT-LIKE

A \ f

" 5

-

- / \

f

" 3\ b / i\ \

-

^FLICKER-FREQUENCY

f"

3

\\

\ \

\ \

\ \c „

\

x f

\ \ \

i i

\

\ a>c

• " \

10 -4

10 -2 ,0 10' 10

10 10" 10 10" 10' FOURIER FREQUENCY f (Hz)

FIGURE 5a The dashed line shows the S (f) which results from locking a free

running oscillator typified by curve a to a reference typified by curve b, using the servo gain shown in Figure 5b.

60

40

20

—

20

SUGGESTED SERVO GAIN

FOR FIG. 5A

-40

-60

-6dB/0CTAVE

-2

10

-1 10" 10' 10° 10" 10'

FOURIER FREQUENCY f (Hz)

FIGURE 5b Suggested servo gain for Fig. 5a. This form of servo gain provides

near optimal time domain performance for the system. See Fig. • 5c.

16
10 -5

10

10

-7

10

10

10

11

13

15

-

I

. b

1

/a

i 1 1

a. FREE RUNNING OSC.

-

a,c\

b. REFERENCE

c. SYSTEM LOCKED WITH

\ s

SERVO GAIN OF FIG. 5B

b,c

j i i 10" 10 -2

10' i(r 10* lo

1

SAMPLE TIME t(s)

10

L 10

10 10

12

FIGURE 5c Time domain stability calculated from S (f) shown in Fig. 5a. Note

that the time domain stability of the system is much better than

4 either component between approximately 0.5 and 10 seconds. Noise

bandwidth f, = °°.

17
APPENDIX A: The Effect of Frequency Noise at ft and 2ft Away from the VCO Carrier on Stability

A. Noise at Fourier Frequency ft When the VCO is modulated at frequency

ft, then the detected 1st derivative signal

is to first order given by:

(w, - co Q

) B

Sig Amp = * Y The RMS amplitude of the frequency

excursion of the VCO at Fourier frequency

ft is equal to

) 1 V S (n/2-tr) = b ,

which we will set equal to b. S (f) is the

spectral density of frequency fluctuations.

The effective FM amplitude is now given by:

B- = J(B +-^) 2 + (-^) 2 - B(l + -i- | >

where B is the initially applied FM amplitude

and b/B is assumed to be less than 1. The signal is now

Sig Amp = ^j Uu> 1 - u) Q

) + <S(i) 1( T v

)) ^ + ^r B

;

where cu, - co is the average frequency

offset, and 6co is the absolute frequency

fluctuation at the servo attack time T . We see that as long as

b = u^ Js (ft/2-rr]

is less than the amplitude of the applied

FM that the precision in picking line center

is not seriously impaired.

18
B. Noise at Fourier Frequency 2^

Neglecting everything but the extra

contribution due to noise at a Fourier

frequency 2q removed from the carrier,

the detected 1st derivative signal is given

by:

Sig Amp = 7[2B( W;L - u) Q

) + Bk

3

] 2 Y

_B

2 Y

( UW - Wq + 60)n + k,) ,

where k

3 = Ul J S (2^/2-n)

We see that the requirement

here is that k, must be small against

6u)-i = a)-.a (x ) in order not to introduce

any additional noise. To make this comparison

requires a knowledge of the dependence of

S (f) on f; that is, the character of the

noise. If S (f) is proportional to f or

2 y

f ; i.e. , if flicker of phase or white

phase noise dominates at times of order

t to l/2fi, then the noise at Fourier frequencies

2Q probably is the major term.

-2 -1 If S (f) is proportional to f , f , ^

or f , then the noise at 2fi is probably

a minor term. The exact values of course

can be found by substituting in the measured

values for S (ft/Tr) and a (x ) . Y y v

19
REFERENCES

[l] Gardner, F.M. , Phaselock Techniques (John Wiley, New York, 1966).

[2] Kesner, D. , Take the Guesswork out of Phase Locked Loop Design, EDN, 5,

pp. 54-60, January 1973.

20
NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBS-TN-692

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Servo Techniques in Oscillators and Measurement Systems

5. Publication Date December 1976

6. Performing Organization Code

277.04

7. AUTHOR(S)

Fred L. Walls and S.R. Si-pin

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2774148

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

same as Item #9

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

Nearly every precision oscillator includes a frequency or phase servo system.

In the case of cesium standards, a crystal oscillator is frequency locked to a particular resonance line in atomic cesium; in the case of an oscillating hydrogen

maser, a crystal oscillator is phase locked to the very weak signal coming from the

microwave cavity. The first section of this note treats the errors and offsets of

frequency- lock loops which result from background, noise and other effects. Third

harmonic lock systems and square wave frequency modulation are analyzed as possible

solutions to some of these problems. The second section is a general treatment of servo system response which is applicable to both frequency and phase-lock loops.

The effects of noise on such servo systems are discussed in detail, and an example

is given of how to obtain optimum performance from a pair of phase-locked oscillators

A simple circuit is suggested for phase-locking high quality oscillators, which has

many advantages over previous circuits.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Atomic frequency standard; frequency discrimination; frequency domain stability;

frequency-lock loop; phase-lock loop; phase noise; servo techniques; time domain

Si ahil j t-y.

18. AVAILABILITY [^ Unlimited

j For Official Distribution. Do Not Release to NTIS

| X

1 Order From Sup. of Doc, U.S. Government Printing Office Washington, D.C. 20402, SD Cat. No. CIS 4 6 l bV 2 J Order From National Technical Information Service (NTIS)

Springfield, Virginia 22151

19. SECURITY CLASS

(THIS REPORT)

UNCLASSIFIED

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

21. NO. OF PAGES

23

22. Price

$0.55

fru S GOVERNMENT PRINTING OFFICE: 1977—777-045/1224 REGION NO 8 USCOMM-DC 29042-P74

NBS TECHNICAL PUBLICATIONS

PERIODICALS JOURNAL OF RESEARCH reports National Bureau

of Standards research and development in physics,

mathematics, and chemistry. It is published in two

sections, available separately:

• Physics and Chemistry (Section A)

Papers of interest primarily to scientists working in these fields. This section covers a broad range of physical and chemical research, with major emphasis on

standards of physical measurement, fundamental constants, and properties of matter. Issued six times a year. Annual subscription: Domestic, $17.00; Foreign, $21.25. • Mathematical Sciences (Section B)

Studies and compilations designed mainly for the mathematician and theoretical physicist. Topics in mathematical statistics, theory of experiment design, numerical

analysis, theoretical physics and chemistry, logical de- sign and programming of computers and computer sys- tems. Short numerical tables. Issued quarterly. Annual

subscription: Domestic, $9.00; Foreign, $11.25.

BJMENSIONS/NBS (formerly Technical News Bulletin)- -This monthly magazine is published to inform

scientists, engineers, businessmen, industry, teachers,

students, and consumers of the latest advances in science and technology, with primary emphasis on the work at NBS. The magazine highlights and reviews

such issues as energy research, fire protection, building

technology, metric conversion, pollution abatement,

health and safety, and consumer product performance.

In addition, it reports the results of Bureau programs

in measurement standards and techniques, properties of matter and materials, engineering standards and services, instrumentation, and automatic data processing.

Annual subscription: Domestic, $9.45; Foreign, $11.85.

NONPERIODICALS

Monographs—Major contributions to the technical liter- ature on various subjects related to the Bureau's scientific and technical activities. Handbooks—Recommended codes of engineering and

industrial practice (including safety codes) developed

in cooperation with interested industries, professional

organizations, and regulatory bodies. Special Publications—Include proceedings of conferences sponsored by NBS, NBS annual reports, and other

special publications appropriate to this grouping such

as wall charts, pocket cards, and bibliographies.

Applied Mathematics Series—Mathematical tables, manuals, and studies of special interest to physicists, engineers, chemists, biologists, mathematicians, computer programmers, and others engaged in scientific and technical work.

National Standard Reference Data Series—Provides

quantitative data on the physical and chemical properties of materials, compiled from the world's literature and critically evaluated. Developed under a world-wide

program coordinated by NBS. Program under authority

of National Standard Data Act (Public Law 90-396).

NOTE: At present the principal publication outlet for these data is the Journal of Physical and Chemical

Reference Data (JPCRD) published quarterly for NBS

by the American Chemical Society (ACS) and the American Institute of Physics (AIP). Subscriptions, reprints,

and supplements available from ACS, 1155 Sixteenth

St. N.W., Wash. D. C. 20056. Building Science Series—Disseminates technical infor- mation developed at the Bureau on building materials,

components, systems, and whole structures. The series presents research results, test methods, and performance criteria related to the structural and environmental

functions and the durability and safety characteristics

of building elements and systems.

Technical Notes-^Studies or reports which are complete

in themselves but restrictive in their treatment of a

subject. Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject area. Often serve as a vehicle for final reports of work performed at NBS under the sponsorship of other government agencies.

Voluntary Product Standards—Developed under proce- dures published by the Department of Commerce in Part

10, Title 15, of the Code of Federal Regulations. The

purpose of the standards is to establish nationally rec- ognized requirements for products, and to provide all concerned interests with a basis for common understanding of the characteristics of the products. NBS

administers this program as a supplement to the activities of the private sector standardizing organizations.

Consumer Information Series—Practical information,

based on NBS research and experience, covering areas

of interest to the consumer. Easily understandable language and illustrations provide useful background knowledge for shopping in today's technological marketplace.

Order above NBS publications from: Superintendent

of Documents, Government Printing Office, Washington,

D.C. 20402. Order following NBS publications—NBSIR's and FIPS

from the National Technical Information Services,

Springfield, Va. 22161. Federal Information Processing Standards Publications (FIPS PUBS)—Publications in this series collectively constitute the Federal Information Processing Standards Register. Register serves as the official source of information in the Federal Government regarding stand- ards issued by NBS pursuant to the Federal Property

and Administrative Services Act of 1949 as amended,

Public Law 89-306 (79 Stat. 1127), and as implemented

by Executive Order 11717 (38 FR 12315, dated May 11, 1973) and Part 6 of Title 15 CFR (Code of Federal

Regulations).

NBS Interagency Reports (NBSIR)—A special series of interim or final reports on work performed by NBS for outside sponsors (both government and non-government). In general, initial distribution is handled by the sponsor; public distribution is by the National Technical Information Services (Springfield, Va. 22161) in paper copy or microfiche form'. BIBLIOGRAPHIC SUBSCRIPTION SERVICES

The following current-awareness and literature-survey

bibliographies are issued periodically by the Bureau:

Cryogenic Data Center Current Awareness Service. A

literature survey issued biweekly. Annual subscription: Domestic, $20.00; Foreign, $25.00.

Liquified Natural Gas. A literature survey issued quarterly. Annual subscription: $20.00.

Superconducting Devices and Materials. A literature survey issued quarterly. Annual subscription: $20.00.

Send subscription orders and remittances for the pre- ceding bibliographic services to National Bureau of Standards, Cryogenic Data Center (275.02) Boulder,

Colorado 80302.
U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Washington. DC. 20234

OFFICIAL BUSINESS

Penalty for Private Use. $300

POSTAGE AND FEES PAID

U.S. DEPARTMENT OF COMMERCE COM-215

Fourth Class Mail