The servo system is a special type of feedback amplifier, usually including a
mixture of electrical, mechanical, thermal, or hydraulic circuits. With suitable

21 Ele

design, the behavior of these various circuits can be described in the universal
language of linear systems.
Further, if the servo system is treated in terms of
circuit response to sinusoidally varying signals, it then becomes possible to

draw upon the wealth of linear feedback amplifier design based on frequency
analysis.

This paper discusses a typical analogy between electrical and mechanical
systems and describes, in frequency response language, the behavior of such
common servo components as motors, synchro circuits, potentiometers, and tach-

ometers. The elementary concepts of frequency analysis are reviewed briefly,
and the familiar Nyquist stability criterion is applied to a typical motor-drive
servo system. The factors to be considered in choosing stability margins are
listed—system

variability,

noise

enhancement,

and

transient

response.

The

basic gain-phase interrelations shown by Bode are summarized, and some of
their design implications discussed. In addition to the classical methods,
simple approximate methods for calculating dynamic response of servo systems
are presented and illustrated.
Noise in the input signal is discussed as a compelling factor in the choice of

servo loop characteristics. The need for tailoring the servo loop to match the
input signal is pointed out, and a performance comparison given for two simple
servos designed to track an airplane over a straight line course.
The use of
subsidiary or local feedback to linearize motor-drive systems, and predistortion

of the input signal to reduce overall dynamic error are described.
1. INTRODUCTION

SIMPLE servo system is one which controls an output quantity
according to some required function of an input quantity. This
control is of the ‘‘report back” type. That is, some property of the output
is monitored and compared against the input quantity, producing a net
input or “error” signal which is then amplified to form the output. The
first statement defines the servo as a transmission system; the second, as a
feedback loop. The problem of servo design is then to fashion the desired
transmission properties while meeting the stability requirements of the
feedback loop. This is the familiar design problem of the feedback amplifier.
2. THe

SERVO

CIRCUIT

The design of linear feedback amplifiers has been developed to a high
degree in terms of frequency response; that is, in terms of circuit response
to sinusoidal signals.'. The servo system is a special type of feedback
amplifier, and usually can be made fairly linear. Thus, it is logical to
analyze and design the servo circuit on a frequency response basis. Also,
1See “Network Analysis and Feedback Amplifier Design,” by H. W. Bode, D. Van
Nostrand Co., 1945.
616

Cont

which
rotatio

Fig. 1:
pliane

angule
analog

dimen
the ra

resist
torqu
Sor
chani
convé
d6/di
betw

actin
circu
load

LINEAR

SERVO

THEORY

617

servo systems usually are combinations of electrical, triechanical, thermal,
or hydraulic circuits. In order to describe the behavior of these various
circuits in homogeneous terms, it is desirable to recognize the analogous
relationships established by similarity of the underlying differential equations. Before proceeding to a discussion of frequency analysis, a typical
analogy between electrical and mechanical systems will be described.

2.1 Electrical-Mechanical Analogy
Confining the discussion to rotating mechanical systems, the analogy
which will be chosen here puts voltage equivalent to torque, and current to
rotational speed. This choice leads to the array of equivalents shown in
Fig. 1; inductance, capacity, and resistance corresponding to inertia, compliance, and mechanical resistance, respectively. Charge is equivalent to
angular displacement, and both kinetic and potential energy are selfanalogous. The ratio of voltage to current, or torque to speed has the
dimensions of resistance.
In an interconnected electro-mechanical system,
the ratio of voltage to speed or torque to current may be called a transfer
resistance. Similarly, the ratio of voltage to angular displacement, or of
torque to charge, is a transfer stiffness (reciprocal of capacity or compliance).
Some commonly used devices for coupling between electrical and mechanical circuits are shown in Figs. 2 and 3. The motor, Fig. 2a, is used to
convert an electrical current 7 into a mechanical speed or “current” @ (=
d6/dt). The electrical control current 7 is produced by the voltage difference
between an applied emf e and a counter-rotational emf (not indicated),
acting upon the total electrical mesh resistance R,.* In the mechanical
circuit, a torque proportional to i forces a “‘current” 6 through the mechanical
ond Ky, J.
An equivalent mechanical mesh directly relating shaft speed to the applied
emf is shown in Fig. 2b. A fictitious generated torque u,e acts upon the
mechanical load through an apparent mechanical resistance Rm. uy
is a transfer constant determined both by the motor properties and the
electrical mesh resistance R,. Ry is similarly governed and is inversely
proportional to R, .
The motor may be compared to a vacuum tube having an amplification
factor yw, and a plate resistance R.,.. However, the motor is usually much
more a bilateral coupling element than the vacuum tube, due to the effect
of the counter emf upon the electrical mesh.
The potentiometer, tachometer, and synchro circuit shown in Fig. 3 are
all means for converting a mechanical quantity to an electrical one. All
three are substantially unilateral coupling elements. The potentiometer
* R, includes both the source resistance and the motor winding resistance.

618

BELL SYSTEM

TECHNICAL

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delivers an output voltage e proportional to its shaft angle
6. Thus, the
ratio of € to @ is a transfer stiffness constant S,. The synchro
circuit
con- |
sists of a synchro generator connected to a control transformer,
and delivers
an output voltage e proportional to6; — 6, the angular
difference between

ELECTRICAL

MECHANICAL

INDUC TANCE
L

J

O00 wo
t

@

>

6

—»

COMPLIANCE

c

egg

,

c

}

8
T=

Cc.

,gyy-Y
Fee

Brig.

g

SPRING

SS

ie

le

a.
e=c

Fe
/idt

T= ae? /edt

RESISTANCE

RANE A,

RESISTANCE

Pee

R

om\/\\—m0

ar

om \V\\—o
|

——

SERIES L-R-C

SERIES J-R-C

B.
<<
VV bao

a——

@€ —__»

si

J
om]

L

e=Lddepist

te /igt
/t0
Fig.

—=

DASHPOT
H

R

-

¢
{|

we
6

———_—_

2

cf
q

T —

T=RO

Jt
Om OO

8

6

@=Ri

at

6

pean ee

.

~

\)

FLYWHEEL

de
dt

CAPACITY

a

4

|

Tey

e=_——

etal.

~

.
\
San)

ee

di
dt

tran

:

TO 0

-

tion

-

INERTIA

_—

<——

TI
to el
chro
pro}
is tk

6

J

(7
{ /

| Ts wn
YY

,

Cc

RL

fT—__

=) “Yap
98 (Ore
sp641 /odt
/j

2.

1—Electrical-mechanical analogy.

the two shafts. Thus the action of the synchro pair is that of a combined
transfer stiffness and differential. The tachometer is a
generator which
produces an output voltage e proportional to 6, the angular
speed of its
shaft. The ratio of e to 6 is a transfer resistance constant
R, .

|

It
ir
q
is
s

LINEAR

SERVO

THEORY

619

There are many other specific devices used to convert from mechanical
to electrical quantities.
Most are equivalent to the potentiometer or synchro circuit, one such being a lobing radar antenna, which delivers a voltage
proportional to an angular difference. A different, less widely used device
is the accelerometer, a generator which delivers an output voltage proportional to the angular acceleration of its shaft.
Its characteristic is that of a
transfer inductance or inertia.
'

Rm

A A

WN

ee.
=

<Rm

TapeOy | © 4
|

=

2,
Oo

J ef+(Rmm+Rin)O
=pye
(a)

‘b)

Fig. 2—Motor asa transfer device.

(a) Motorand load.

(b) Equivalent mechanical mesh.

POTENTIOMETER

a

wea

—

e=S;,8e

e

(

<<
SYNCHRO

CIRCUIT

—————

ae

|

=

=

$,+(0)-6
eee

"e

TACHOMETER
Rea
eee

:
e

.
e=R,6

—————

Fig.

3—Mechanical-electrical transfer devices.

2.2 Frequency Analysis
A brief review of the basic concepts of periodic analysis will be presented.
It is assumed that the driving force applied to a network may be analyzed
in terms of a series of sinusoidal components of various amplitudes, frequencies, and phases. The network response to each sinusoidal component
is then evaluated, and the over-all result obtained by a summation of all
such elementary responses.
This is the formal procedure.
Actually

620

BELL SYSTEM

TECHNICAL

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it is often unnecessary to perform these precise operations in order to obtain
a broad picture of the network behavior.
The method for determining the network response to a sinusoidal input is
developed as follows.
It is assumed that the circuit parameters are constant, independent of signal amplitude. Then, as indicated in Fig. 1, a
single R-L-C or R-J-C mesh may be represented by a constant-coefficient
linear differential equation. Choosing the electrical mesh for illustration,

(Lp + R+ 1/Cp)i) = etd),
where p” = d”/dt”, 1/p = [a and e(#), i(¢) are the mesh voltage and current respectively.

This may be further abbreviated as

Z(p)i(t) =e(?),

(1)

where Z(p) = Lp + R+ 1/Cp. For purposes of frequency analysis we
are interested only in the forced or steady-state solution of (1), where e(?)
is a sinusoidal voltage E sin wt. This steady-state solution is

i(t) —.

|Z(jw)

sin (wt + 9),

where jw has replaced p in the function Z(p), and the phase shift ¢ is the
negative of the phase angle of the complex number Z(jw).? This result is
conventionally abbreviated as

E
Z (jw)

fm ge,

(2)

where J is a complex number whose magnitude equals the peak amplitude
of the current, and whose phase angle gives the associated phase shift.
The function Z(jw) is called the impedance of the mesh.
The relationship between torque, angular speed, and mechanical impedance is of course the same as expressed by (2). That is,

=

‘g

_—.,

Z(jw)

(2.1)

where 6 is the complex peak amplitude of the sinusoidally varying speed, T
is the peak amplitude of the applied sinusoidal torque, and Z(jw) is the
mechanical impedance obtained by substituting jw for p in the operator
Z(p) = Jp+R+1/Cp.
Since the angular displacement is the time integral
2

is used to represent frequency in radians/sec, or 2 times frequency in cycles/sec.

LINEAR

SERVO

THEORY

621

of the speed, the expression for @ may be obtained by dividing both sides of

equation (2.1) by p or, for the periodic case, byjw.

Thus

mi pie
jal (jw) ”

(2.2)

The function jwZ(jw) is the complex stiffness of the mechanical mesh.
The phase shift of 6 relative to 6 is —90 degrees, as seen from a comparison
of (2.1) and (2.2).
For an electro-mechanical network consisting of a number of interconnected meshes, a set of simultaneous differential equations of the type of
(1) may be written. If jw is substituted for p in these equations, there
results a set of simultaneous algebraic equations which lead directly to the
steady-state periodic solution.
If a sinusoidal voltage or torque is applied
at some mesh of the network, the resulting sinusoidal current or speed
response in some other mesh is given by, using the notation of (2),
(Response)

= (Cause)
Z.(jw)

where Z,(jw) for the chosen pair of meshes is obtained from the solution
of the algebraic equations. 7Z,(jw) is called a transfer impedance, and may
express the ratio of a voltage to current or speed, or of a torque to current
or speed. The above relation also may be written as
(Response)

=

Y,(jw)- (Cause),

where Y,(jw) = 1/Z;(jw) is called the transfer admittance between the two
chosen parts of the network.
In this form the response amplitude is obtained by multiplying the forcing sinusoid by | Y;(jw) |, while the phase
shift is given directly by the angle of Y,(jw). The transfer ratio between
like or analogous quantities at two parts of the network is similarly a
complex function of frequency, having the dimensions of a pure numeric.
Servo systems usually consist largely of elementary networks isolated
by unilateral coupling devices (vacuum tubes, potentiometers, .etc.).
Thus, over-all transfer ratios often may be evaluated by taking the product
of a number of simple ‘‘stage’’ transfer ratios, rather than by solving a large
array of simultaneous equations. If the absolute magnitudes of the transfer
ratios or ‘“‘transmissions” are expressed in decibels* of logarithmic gain,
both the over-all gain and phase shift of a number of tandem stages may be
obtained by simple addition of the individual stage gain and phase values.
The transfer ratios of the conversion devices shown in Figs. 2 and 3
3 The gain in decibels for a given transfer ratio is taken to be 20 logio of the absolute
value of the ratio.

622

BELL SYSTEM

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may be written by inspection.
Referring to Fig. 2b, the transfer admittance
of a motor with resistance and inertia load may be written as

of the
assem
equal

6 ss

E

the fo

Mt

jw + Rn + Ri,’
Mee

1

J jo + wm’

where
equal
partic
servo

where

o, = Km t Rm
m

Bi

ad

wm is the reciprocal of the time-constant of the motor and control mesh,
and is 2% times the ‘‘corner” frequency at which the inertial impedance just
equals the apparent mechanical resistance.
Writing the transfer characteristic in terms of shaft position, rather than speed,

a

EJ

jw(jw + wm)

(3.1)

For values of w small compared with w, , 6/E is proportional to 1/jw.
This factor has a phase shift of —90 degrees and approaches infinity as w
approaches zero. This is merely a statement in frequency analysis language
that the motor shaft angle is the time integral of the applied voltage, for
slowly changing voltage.
or more rapidly varying voltage, such that w
is large compared to w, ,0/E is proportional to 1/(jw)? or — 1/w", the angular
variation being shifted — 180 degrees with respect to the voltage variation.
The transfer ratio of the potentiometer, Fig. 3a, may be written as
E
--9
while

for the tachometer,

=et

(4.1)
(4.1)

joR: °

(4.3)

Fig. 3c,

E

6

=

Usually at some point in the system a compensating or “equalizing”
network will be included to modify the transfer ratio of the basic components to the desired over-all transmission characteristic.
Frequently

this equalizer is incorporated in the electrical section of the servo because

23 S$
Th
ing ¢

LINEAR

SERVO

THEORY

623

of the comparative ease with which electrical circuit components may be
assembled in desired combinations.
The transfer characteristic of the
equalizer may be simple or complicated, but in general may be written in
the form,
—, Go + o1)(jw + ws)

Geo + 0a) (joo+ wa) -->”

ee

where the constants «1 , w2 , etc. may be real or complex.
The synthesis of
equalizing networks is a well known art and will not be discussed here,
particularly since most of the equalization characteristics used in present
servo’ systems can be realized with simple networks.
2.3 Simple Servo System (Single Loop)
The simple servo system may be divided into two basic parts, an amplifying circuit and a monitoring or comparison circuit. Such a division is
f

,(t)

F, +BFo

q

BFo

Fo

Fo(t

,

Fo

BS

LOOP

TRANSMISSION= U6

TRANSFER

CHARACTERISTIC

e

ae
gs

1-UB

Fy

Fig. 4—Simple servo system.

indicated in Fig. 4, where uw and @ are the transfer characteristics of the amplifying and monitoring parts, respectively.
/; and Fy, represent typical
sinusoidal components of the total input and output quantities /,(¢) and
fo(t),* while » and 8 are complex-valued functions of jw as described in the
previous section.
The return signal 8F2 from the monitoring circuit is added
input F; to form a net u circuit input /; + 8F2.

to the servo

The servo transfer char-

acteristic is found by setting

PF; = w(Fi + Bo),
from which

ae
2
ee

1

i

(6)

4 That is, F; and F; are complex quantities employed in the same fashion as J in equation (2).

624

BELL SYSTEM

TECHNICAL

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The closed system formed by the two basic circuits in tandem is of course
feedback loop, the loop transmission characteristic being given by yf.
Any desired form of servo transfer ratio may be obtained by an unlimited
number

of uw and @ circuit combinations.>

However,

the 8 characteristic,

which is usually determined by a passive network or an inherent propert)
of a monitoring device, tends to be more stable with time and varying signa
amplitude

than

that of the uw circuit, which

may

include vacuum

tubes,

motors, and other variable components.
Consequently, it is desirable to
have the servo transfer characteristic largely dependent upon the 8 circuit
properties alone. This may be accomplished by making the loop trans
(a)

8

GEN

et.

GIN
SouT~=B =6iN

Fig.

5—Synchro follow-up system.

mission w8 very large compared to unity over the essential frequency range
of the servo input signal f;(¢).. Under this condition, equation (6) becomes,
F,

a

up |

(6.1)

Thus the external transfer characteristic is set by 8.* If, for instance, F;
and fF» are similar or analogous quantities and it is desired to have the servo
output a replica of the input, 8 may be chosen as —1, yielding F2 ~ F,.
It is not always easy to determine the basic parts » and 8 of a servo by
inspection of a schematic diagram of the system. An example is furnished
by the synchro follow-up system shown in Fig. 5a. As previously discussed,
the characteristic of the synchro comparison circuit is that of a differential
® Feedback stability requirements place certain restrictions on the permissible forms of
u8.

This will be discussed

in the next section.

* The error arising from the approximate
section as one type of “servo error.”

nature of (6.1) will be discussed in the next

transf
given

Fig. 5
in the

relati

LINEAR

SERVO

THEORY

625

transfer stiffness S, , the voltage output of the control transformer being
given by S;(@: — 62). However, as seen from the modified diagram of
Fig. 5b, the 8 characteristic is simply —1, the transfer constant S, appearing
in the w circuit. Thus if the loop transmission is kept large, the essential
relation demanded between 6; and 4 does not depend upon the value of S;

y

Fig. 6—Potentiometer loop.
TACHOMETER.
——
6
+

4

+

:

Ep ~ -R,e

oa

Fig.

E

7—Tachometer loop.

(as is obvious from physical considerations).
This result also applies to a
radar angle-tracking loop, where the received deviation or error signal is
proportional to the difference between the angular coordinate of the target
and that of the antenna system.
Figures 6 and 7 represent two servo systems where the input is electrical
and the output mechanical.
In Fig. 6 a potentiometer is used as a monitor-

626

BELL SYSTEM

TECHNICAL

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ing device, the transfer stiffness in this case appearing in the 8 circ)
If Ais regarded as the output, then 8 = —S;, ,* and the transfer characteris
is, for high loop gain,

As us
forman
restrict

to be t

A~

=

—£

=

compat

where C, = 1/S;. Thus the over-all characteristic between input voltage
Ip
and angular displacement is simply a transfer compliance constant.
Fig. 7 a tachometer monitor is used.
Regarding angular speed 6 as the
output,

then

8 =

—R,,

the transfer resistance of the tachometer.

The

transfer equation is thus
6

aie

oe. me

=

g. E,

where g;(= 1/R;) is a transfer conductance.

is neitk
the loo
such as
distrib

The ef
cause |
It will
the lo

freque
reduce

tribut
3. DESIGN OF SIMPLE LINEAR SERVO SYSTEMS

The majority of servo systems in use, while often greatly extended in

space and frequently including highly diversified transmission elements,
may be represented by one essential feedback loop. However, a well
designed servo often will incorporate numerous subsidiary or local feedback
loops around stages of the system, in order to obtain a desired degree of
linearity or performance with easily obtainable circuit components.
Common examples of such local feedback loops are electrical feedback around
vacuum
tube amplifiers, and tachometer (velocity) feedback around motordrive systems. These subsidiary feedback loops are almost always designed
so that they are individually stable when the over-all feedback loop is
opened (assuming the method employed for opening the over-all loop does
not disturb the impedance terminations of the local feedback stages).
If it is thus assumed that any subsidiary loops are individually stable, then
the primary servo loop design may be treated simply as that of a single loop,
whose over-all loop transmission is obtained by taking the product of the
external transfer ratios of the various stages.
The design of a single loop servo may be divided into the design of the
loop transmission 48, and one of the remaining parts, » or 8. As previously
described, it is usually desirable to fix 8 according to the required basic
input-output relationship of the servo, thus leaving uf as a single characteristic to be chosen.
*It is assumed here that the transmission of the yu circuit is basically positive. The
negative signs associated with S;, of Fig. 6 and R; of Fig. 7 are then introduced (by poling)
to make the loop transmission yf essentially negative. This stipulation ensures what is
commonly called “negative feedback,” when the loop delay is zero.

The

signal
Input
overlo

are re
teristi
by (6
by rec
On
tainir
trans

nitud
upon
Th
noise
tuati
strict
SA.

LINEAR

SERVO

THEORY

As usual, specification of the form of u8(jw)

is beset

627

by a series of per-

formance objectives on the one hand and a set of physical limitations and
restrictions on the other. Assuming the relationship expressed by (6.1)
to be the required one, it would seem desirable to make u8(jw) very large
compared to unity at all frequencies.
However, there are reasons why this
is neither possible nor actually desirable.
As the value of w is increased,
the loop transmission is eventually controlled by paras: ic circuit elements
such as distributed capacity and inductance in the electrical circuits, and
distributed inertia, compliance, and backlash in the mechanical circuits.
The effect of these parasitic elements at the higher signal frequencies is to
cause |u@ | to decrease as a very high order of 1/w with increasing frequency.
It will be shown, however, that feedback stability considerations require
the loop transmission to be decreasing comparatively slowly through the
frequency region where | u@ | is of the order of unity. Thus u& must be
reduced below unity at a frequency sufficiently low to avoid excessive contribution from the parasitics.
The presence of “noise” or undesired disturbances in the servo input
signal is another compelling factor in the design of the loop characteristic.
Input noise is harmful both in causing spurious output fluctuations and in
overloading the power stages of the servo system.
Both of these effects
are reduced by narrowing the frequency band of the servo transfer charac-

teristic.

Referring to the expression for the transfer characteristic given

by (6), it may be seen that a restricted transfer bandwidth may be obta‘ned
by reducing u and uf well below unity at a small value of signal frequency.°
On the other side of the picture is the requirement of fidelity in maintaining the desired input-output relationship. Undue narrowing of the
transfer bandwidth of the servo results in large dynamic error, the magnitude of which depends both upon the character of the input signal and
upon the chosen transfer characteristic.
The optimum design of a servo system, for a specified input signal and
noise, thus is a compromise between dynamic error and output noise fluctuations, with stability considerations and parasitic circuit elements restricting the possible choice of loop transmission characteristics.
3.1 Stability of Single Loop Systems
The word stable as applied to a servo system is used here to imply a system whose transient response decreases with increasing time. It is possible
6 When the 8 characteristic is under suitable design control, another method is available:
Thus if 8 is made to rise in the frequency region of the desired transfer cut-off, and if 48
is maintained large beyond this region, (6.1) shows that the desired restriction is effected

For a given transfer characteristic, this cut-off method requires a wider frequency range
for uf and is thus more vulnerable to the effects of parasitic circuit elements.

However,

shaping of both the uw and @ circuits permits a more rapid cut-off of the servo transfer
characteristic than is possible with « circuit shaping alone.

628

BELL SYSTEM

TECHNICAL

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to determine the stability of a completed servo design by obtaining
+e
transient solution of its differential equation. Though often very tediv:is,
this is fairly straightforward.
However, this method of procedure often js
of little help either in guiding the initial design or in predicting the necessary
changes, should the trial design be found unstable. The addition of even
one circuit element to a design will generally create an entirely new differential equation whose solution must be found.
An alternative method for determining servo system stability, based on
frequency analysis, furnishes the necessary information in a form which
greatly facilitates the design procedure. This method is relatively simple
to apply, even when the system has a large number of meshes and a high
order differential equation, and the additive effects of minor circuit modifications are easily evaluated.
The stability of a single loop servo system—or of a primary loop, when
the subsidiary loops are individually stable—may be investigated by plotting the negative of the loop transmission, —u8(jw), on a complex plane
for real values of w ranging from minus infinity to plus infinity. (The
negative signis introduced because the loop transmission yf is generally
arranged to have an implicit negative sign, apart from network phase shifts.
Thus —uyf is a positive real number when the network phase shift is zero.)
Then the necessary and sufficient criterion for system stability is that the resulting closed curve must not encircle or intersect the point —/,0.* This type of
plot is commonly called a Nyquist diagram, and is widely used in the design
of electrical feedback amplifiers. An added stipulation is necessary if uB(jw)
becomes infinite at a real value of w, say w’. In this case an infinitesimal
positive real quantity « must be added
to jw; that is, the function
to be
plotted is 48(jw + €). This has no effect upon the plot except in the neighborhood of the singularity, where u@(jw + e€) is caused to traverse an arc of

infinite radius as w is varied through the value w’.
As seen from (3.1), inclusion of a motor in a servo loop of the type shown
in Figs. 5 and 6 will cause an infinite loop transmission at w = 0, assuming
there is transmission around the remainder of the loop at zero frequency.
The motor is the only commonly encountered circuit element capable of
producing an infinite loop transmission at' real frequencies.
In order to illustrate the use of the Nyquist diagram, a motor servo system of the type shown in Fig. 6 will be chosen. Again referring to equation
(3.1), it may be seen that the transfer ratio of the motor and potentiometer
1S,

* This criterion is due to H. Nyquist, “Regeneration Theory,”’. B. S. T. J., January 1932.
Detailed descriptions of stability criteria for single and multiple loop systems are given by
Bode, loc. cit., and by L. A. MacColl, “Fundamental Theory of Servomechanisms,” D.
Van Nostrand Co., 1945.

It is as
the ove

where .
given 1

where

The t
unity
given
to pl

in th
tity,
ingl}
ima
axis

equ

LINEAR

SERVO

THEORY

629

ig ed
pm

J

= jw(jw + wn)

It is assumed that the amplifier includes an equalizing network such that
the over-all amplifier characteristic is

4 (ete\
1

jo + we)

y

’

where A, w, ,and w, are positive real constants.
‘The loop transmission is
given by the product of these two transfer factors and thus may be written
as

3)

uB(Gw)

= —20( om

2 (e—\

fe tom) 4)
w

J\jwo
+

7

ar)’

@)

where w is a positive real constant given by
jin

on
a

AS: ut

va

a

—

Om J

AS
=

ui

:

.

Rn + Rm

The three factors in parenthesis have been so grouped that they all approach
unity for small values of w. Thus the low-frequency behavior of —y is
given by wo/jw. This quantity has a pole at w = 0, so that it is necessary
to plot the function

—uB(jo + «-)

~—“-,

(7.1)

jute

in the neighborhood of w = 0. Aswis, in succession, a small negative quantity, zero, and a small positive quantity; the expression of (7.1) is correspondingly a large positive imaginary, a large positive real, and a large negative
imaginary. Thus (7.1) traverses an infinite arc from the positive imaginary
axis to the negative imaginary axis as w increases through the value zero.
Assuming the numerical values
wo = 200 sec
jimh
wm

=

10

“*

w

=

200

“,

equation (7) may be rewritten as

er

eee
uB( Jo)

ee

jo +10\/

jw (;,+. et

200
Ys + 200)

)
°

es
ia

The phase angle of —y@ in degrees is, from (7.2),

B = —90— an tan” w+ tan
ta +=-10 —

3tan” 00’
-

(7.3)

630

BELL SYSTEM

TECHNICAL

while the absolute magnitude is given by
eer eee
r

JOURNAL

the syst
to knov

eS Lae a Nr

co ffom
1a) = 2w V (K
EE
P.
1+ w
10?
200? + w*?/ °
general

Pe

eT

e

pee
a
lhe Nyquist diagram of (7.2)
is shown in Fig. 8. To emphasize the jim.
portant features, radial magnitudes have been plotted on a logarithm)

tions, ¢
sign dt
unallov
instabi
and pe
signal.
Stal
betwee
two n

INFINITE
NEGATIVE

will val

w

These
Gay 2
short

gain I
increa
crease
facto!
a
negat

abou
were
coinc
Th
A
POSITIVE

:

w

marg
serve

the «
rewr
Fig. 8—Nyquist diagram of —y.

scale.’ The arrows indicate the direction of traversal as w is varied from
—* to-+«,.
The infinite arc traversed as w varies through zero is indicated symbolically by the dotted semicircle in the right half plane.’ As
is the case for any physical system, the plot for negative values of w is
simply the mirror image of the positive frequency plot.
he

.

.

a
Since the polar plot does not encircle or intersect the ‘critical’
point —1,0,

7 Except in the immediate neighborhood of the origin, where a linear scale must be employed to plot the value ug = 0
* The exact shape of this arc is of no consequence.

The
hoes

2

circl

tip «

LINEAR

SERVO

THEORY

631

the system is seen to be stable.” From a practical standpoint it is necessary
to know not only that a design is stable, but that it has sufficient margin
against instability. The need for proper stability margin arises from two
general considerations.
First, the loop transmission of the physical system
will vary with time due to aging, temperature changes, line voltage fluctuations, etc. Also the physical embodiment will depart from the paper design due to errors of adjustment and measurement, and to the effects of
unallowed-for parasitic elements.
Second, a design which is too near
instability will have an undesirable transient response—large overshoots
and persistent oscillations—and will unduly enhance noise in the input
signal.
Stability margin is measured in a sense by the minimum displacement
between the polar plot and the point —1,0.
In feedback amplifier design,
two numbers often are taken as a measure of margin against instability.
These are called the gain margin and the phase margin.
The gain margin,
G» , Measures

the amount,

in decibels, by which

the magnitude of y@ fails

short of unity, at a phase angle of +180 degrees. The numerical value of
gain margin for the system of Fig. 8 is about 18 db, which is the required
increase in amplifier gain to make the servo unstable.” That is, this increase in amplifier gain would multiply the curve of Fig. 8 by a constant
factor such that it would intersect the point —1,0. The phase margin,
B», , is equal to the absolute magnitude of the angle between —y8 and the
negative real axis, at |u8| = 1. Figure 8 illustrates a phase margin of
about

50 degrees.

That

is, if the points on the curve

where

ywB

=

1.0

were swung toward the negative real axis by about 50 degrees they would
coincide with the point —1,0, and the servo would be unstable.
The type of transient response obtained with reasonable gain and phase
margins is indicated in Fig. 9, which shows the response of the illustrative
servo system to an input step. The initial overshoot is about 25°), and
the oscillation damps out very quickly. For the general case, (6) may be
rewritten

in the form
os
F, =

The

relation

F, =

F,/—8

may

|

1—

F
HB

wp}

|

be regarded

: ‘

(8)

—B

as the desired

bracketed factor acting as the inevitable modifier.

one,

with

the

Then if the quantity

With more complicated systems it may not be obvious whether or not the plot en
circles —1,0. A simple test employs a vector with its origin at the —1, 0 point and its
tip on the curve. If the vector undergoes zero net rotation as it traces along the curve
from w = Oto w = ~, the curve does not encircle the critical point.
10 In some servo systems a decrease in amplifier gain also may cause instability.
Such
systems are still covered by the polar plot criterion of stability, and are commonly called

‘Nyquist stable,” or “conditionally stable.”

632

BELL SYSTEM

TECHNICAL

JOURNAL

—yf exhibits gain and phase margins of the order of 10 db and 50 deg:
respectively, the transient response of the modifying factor to a step func:
will be well-damped and generally not overshoot more than about 25
If the gain margin is sufficient, the phase margin usually will be the dominant
factor in determining the size of the initial overshoot.
The required phase
margin for critical damping depends upon the exact shape of uB(jw), but
in general is about 60 degrees. The gain margin needed in a particular
design will depend upon the expected variability of the loop transmission
Radar tracking loops should usually have gain margins of the order of 15

RESPONSE
SERVO

0.1

0.2

0.3
TIME

IN

0.4
0.5
SECONDS ->

0.6

0.7

0.8

Fig. 9—Transient response of illustrative servo system.

db or more because of the large number of factors which may cause the Joop
gain to vary.
While the polar diagram gives a clear picture of stability considerations,
it is usually more convenient for design purposes to plot the gain and phase
of —yf as separate curves on a logarithmic frequency scale, for positive
values of w. This is illustrated in Fig. 10, for the sample servo system.
Under two commonly met conditions, the requirement for single loop"
stability on this type of plot is simply that the absolute value of phase shift
be less than 180 degrees at zero db gain (|u8| = 1). The conditions are
that the connective polarity be such as to make —yf positive when the
" Again, multiple loop systems may be included if all subsidiary loops are individually
stable.

network
one fre
An ac
forms 0
and thr

be seer

2000,
are dra

LINEAR

SERVO

THEORY

633

network phase shifts vanish, and that the gain curve cross zero db at only
one frequency.”
An advantage of this logarithmic diagram is that commonly encountered
forms of |u8 |vary as w*" for intermediate or asymptotic frequency regions,
and thus plot as corresponding straight line segments.
From (7.4) it may
be seen that the illustrative form of |u@| behaves, in turn, as 200w,
200w~*, 20w~!, and 1.6 X 10%w~4, as w is increased. These asymptotic lines
are drawn in lightly in Fig. 10, the actual gain describing smooth transi-

- 100

-120
”

a
Ww

©

-140

Ww

te
«
C
o

o

z

z

_

z

-160

<

4
<q

0

a

8

-180

a

-200

1000
wo

Fig. 10—Loop characteristic of illustrative servo system.

tions between adjacent asymptotes.
Since the logarithmic slope of w*" is
+6k db/octave," the successive asymptotic slopes in Fig. 10 are —6, —12,
—6, and —24 db/octave. The junctures of adiacent asymptotes occur at
values of w of 1, 10, and 200. These juncture frequencies are called ‘‘corner’’ frequencies, and may be seen from (7.2) to coincide with the real constants or “roots” added to jw in each of the three factors in parentheses.
The corner associated with the factor 200/jw is of course atw = 0. The
corner of the last, or cubed factor is a multiple one, joining two asymptotes
differing in slope by 18 db/octave.
From a knowledge of such corner
12 This discussion assumes that uf has a low-pass configuration; that is, only the high
frequency cut-off is considered. If u8 has a ow-frequency cut-off also, then a corresponding requirement identical with the above must be added.
13 An octave is taken to be a 2:1 span in frequency, and 6 db is of course very closely
equivalent to a 2:1 increase in |uA |.

634

BELL

SYSTEM

TECHNICAL

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frequencies, and the fact that the actual gain curve lies 3 db from an
lated simple asymptotic corner, the gain curve can usually be drawn jn
witheut further computation.'* The phase curve also is easily constructed
by adding up the elementary phase curves associated with the various corners. As may be seen from (7.3), these component phase curves all w)\!
have the same shape on a logarithmic frequency plot, merely being shifte
along the frequency scale. The phase contributed by each simple corner
will be + 45 degrees at the corner frequency, the sign depending upon
whether the associated root appears in the numerator or in the denominator.
It is an extremely important fact that the very requirement of stability
imposes an unambiguous interrelationship between the gain and the phase
shift of most types of transfer characteristic!
By the general mathematical methods leading to the previously discussed stability criteria, Bode!’
has shown that this is true for the broad class of network structures com
monly used in feedback loop design. ‘That is, if either the transfer gain
or phase shift is specified at all frequencies, the attendant phase or gain can
be computed without further information.
This class of networks is called
minimum phase. Any stable structure composed of lumped circuit ele
ments will have a transfer characteristic of the minimum phase type, pro
vided it does not include an all-pass section.!® All-pass characteristics are
seldom used in the design of feedback loops, since their inclusion in the loop
always reduces the stability margins achievable with a given high-frequency)
cut-off. Thus the unique interrelationship between phase and gain may
be assumed for the loop characteristic —y@ in single-loop feedback systems.
The nature of this relationship is discussed in detail by Bode.
Briefly, the
phase shift at any frequency w, is proportional to a weighted average of
the gain slope in db/octave, over the entire logarithmic frequency scale.
The weighting factor sharply emphasizes gain slopes in the immediate vicinity of w, , while the contributions of gain slopes at remote frequencies are
reduced in proportion to the logarithmic frequency span from the particular frequency w,.!? For transfer characteristics of the form w’,
having a constant gain slope of +6k db/octave,'* the associated phase shift
is also constant and equal to +90 degrees. For transfer functions which
behave approximately as w*” over a finite frequency span, the phase shift
‘* The corner frequency concept is less useful if the roots are complex. However a
great many servo systems are so constructed that uf has only real roots.
* Loc. cit. Also see “Relations between attenuation and phase in feedback amplifier
design,” by H. W. Bode, B. S. T. J., July 1940, p. 421.
6 An all-pass section is one which has constant gain but varying phase shift versus

of +9
is incl
Thi
Fig. 1
appro

—10.

—90
shift |
appre
Fo

the s
able
desig
terist

zero
shift:

ing 1
that
char
able

$.2 .
A
tion
sub:

que
whi
able
of t
que
Th
noi

inf

frequency, and is usually composed of a lattice, bridged T, or other bridge type circuit.

1 About 60% of the area under this weighting function lies between w = 0.5 w, and
w = 2w., 80% between 0.25 w, and 4 a, .
16 That is, for transfer characteristics whose

absolute magnitude is given by wt*---.
sit!

LINEAR

SERVO

THEORY

635

of +90k degrees is approached more and more closely as the length of span

is increased,
This may be observed qualitatively from the transfer characteristic of
Fig. 10. For w «1, the gain slope is —6 db/octave, and the phase shift

approaches —90 degrees.

For 1 < w < 10, the average gain slope is about

—10 db/octave, and the phase shift near w = 3 is —148 degrees (instead of
—90 X 10/6 = —150 degrees). As w increases toward 200, the phase
shift increases rapidly due to the asymptotic slope of — 24 db/octave, finally
approaching — 360 degrees (—90 X 24/6) for w> 200.
Foreknowledge of the inevitable gain-phase relationship is of great value to
the servo designer, in making clear the comparatively small class of realizable gain-phase combinations and thus averting attempts at non-physical
designs. For example the design use of too-rapidly falling loop gain characteristics in the region of the high-frequency gain cross-over (that is, near
zero db loop gain) is not permissible because of the large negative phase
shifts which must accompany the steep gain slopes. Another way of stating the advantage of an early realization of the gain-phase laws is to say
that the designer is assured in advance that any paired gain and phase
characteristics which he chooses within the basic restrictions will be achievable with stable physical networks."
3.2 Dynamic

Error

A servo system is usually designed to transmit some class of ing.ut functions with a required degree of fidelity. Thisiclass of functions may reduce
substantially to one specific input signal whose time variation or whose frequency spectrum is known, or it may include a great variety of signals
which have certain properties in common.
In the latter case it is conceivable that definite limits may be placed upon the allowable amplitude ranges
of the input signal and its various time derivatives, or certain limiting fre-

quency spectrum characteristics may be specified for the input function.
Servomechanisms are subject to several types of transmission error.
The systematic error, or dynamic error, is predictable from knowledge of the
noise-free input signal and of the transfer frequency characteristic of the
servo system.
For simplicity, the discussion of error will be limited to the
case where the output signal is desired to be a replica of the input, and
where 8 = —1. Thus the loop transmission 48 becomes simply —u.
The
input-output relationship as given by (6) is therefore

P= 1+u_*_f,
.

d

x

(9)

19 With some necessary reservations as to practicable dissipation constants and parasitic circuit constants.

.

636

BELL SYSTEM

TECHNICAL

JOURNAL

where F; and F2 are again typical sinusoidal components of the input an
output respectively. Thus the corresponding sinusoidal error componeni
may be written as

(9.1),
mand
signa
comp

s=h-

hot.

(9.1)

The methods which may be used to determine the actual dynamic error
A(t) from (9.1) depend both upon the nature off;(#) and the type of information available about f(#). If the input signal is a known periodic function,
A(t) may be found by applying (9.1) for each sinusoidal component of the
input and summing the resulting terms. If the input is non-periodic in
character, then the error may be calculated from the Fourier integral expression
A(t) =

Ms

” Fi(w)
F
Be
ae e** du,

(10)

where F;(w) represents the continuous seadaned spectrum of fi(#), as obtained from
F,(w) = | file** dt.

(10.1)

The problems of calculating /;(w) from fi(¢) and A(é) from F;(w) often may
be avoided by consulting well-known tabular lists of paired time and frequency functions.”
Equation (9.1) may be used as a broad guide in selecting the type of u
characteristic best suited to a particular input signal. It has been mentioned previously that because of input noise and parasitic circuit elements,
the servo transfer bandwidth usually should be kept as narrow as possible,
consistent with dynamic error requirements.
The transfer characteristic
u/(1 + yu) will be closely equal to unity while |x| > 1, will rise slightly”
in the region where |» | = 1, and fall off as u when u is small compared
with unity. The “‘cross-over” frequency, for which |» |= 1, may be taken
as a rough measure of the transfer bandwidth.
Thus, the requirement of
minimizing the bandwidth may be restated as that of minimizing the crossover frequency, while holding the dynamic error within specified limits.
Reasoning in a general way, this requirement may be met by designing
so that the amplitudes of the sinusoidal error components, as given by
20 An excellent list is given by G. Aj Campbell and R. M. Foster in a Bell System monograph ‘‘Fourier Integrals for Practical Application,” ag ig 1931. “A table of Laplace
Transforms, which also may be used, is given by
F. Gardner and J. L. Barnes in
“Transients in Linear Systems,” John Wiley por /
Ared Inc., 1942.
21 Assuming a phase margin of the order of 60 degrees.

over

signa
even
reduc
ina |
3.21
Fr

type
ing f
of le
ener}
versi
inpu

|w |
redu
cont

lecte
serie

whe
B
crea

few
tota

whe

pro]
of t
spec
mis
For
large
sma

)-

o

LINEAR

SERVO

THEORY

637

(9.1), are roughly constant with frequency over the servo band.
This demands that 4 have somewhat the same frequency distribution as the inpul
signal spectrum (for |u| >> 1). Because of stability requirements and
complexity of the necessary apparatus, this rule can usually be followed
over only a part of the servo frequency band, especially when the input
signal spectrum falls off very rapidly with increasing frequency.
However,
even a rough adherence to this desired relation is usually of real worth in
reducing the noise errors of the servo.
An illustration of this will be given
in a later section.
3.21 Approximate Calculation of Dynamic Error
Frequently the servo requirement is to transmit, with great accuracy, a
type of signal whose frequency spectrum falls off very rapidly with increasing frequency. As may be seen from (9.1) this demands very large values
of loop transmission » at the lower frequencies where the input signal
energy is concentrated, but permits a rapidly dropping loop transmission
versus frequency commensurate with the falling amplitude spectrum of the
input signal. Such a rapid reduction in loop gain is practicable while
|u| > 1. However. stability considerations force a more gradual gain
reduction as the region of gain cross-over is approached.
As a result,
contributions to the servo error from this frequency region may be neglected compared with those from the lower frequencies.
This suggests a
series expansion of (9.1) in the form,
A =

[ao a ay( jw) +

@2(jw)? a a3( jw)? 4+ -- “| F, P

(11)

where do , a; , etc. are real constants.
Because of the assumed rapid drop in component amplitude F; with increasing frequency it is often unnecessary to take account of mote than a
few terms of the expansion.”
It is,¢asy to show that (11) may be rewritten on a time basis to give the

total #foamic error as
A(t) = aofi(t) + ay fi(t) + aa} 1(t) 4 see,

(12)

d( )/dt. Thus the coefficient ao gives the error component
‘nal to input displacement.
Similarly, a; and a are the coefficients
ror components due to input velocity and input acceleration, reiy. For a great many motor-drive servo systems the loop transu approaches infinity as 1/jw when w approaches zero. This enseries may be said to converge rapidly in a practical sense, for the following reason:
i values of w the higher order terms are negligible.

For values of w sufficiently

large ts (: the high order terms may no longer be neglected the coefficient F; has become so
small w to make the contribution of the entire series negligible.

638

BELL SYSTEM

TECHNICAL

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sures that a and thus the displacement error will be zero, leaving pri
cipally the velocity and acceleration errors to be considered.
The coefficients ao , a; , a2 , etc. may be calculated easily for any partir
larcase.
For illustration, the three common forms ofu characteristic sho,
in Fig. 11 will beexamined.
(As previously discussed, the designated forn
of uw need hold only for | u | > 1.)

Agair
a cele

Type
Th
servo

and t

and «
com]

3.3 1

Tt
signa
|u|
DECIBELS
IN

whe!
noise
calet

ever
stan
ated
nois

Wy

LOG

W—e

easil
freq

Fig. 11—Elementary yu forms.
Type

1.

—6

db/octave,

a=

wo/Jw

The error expansion becomes,

1 :
A(t) = —f,

L's
—-sfA@+---.

3)

Wo

(12.1)

For the combination of high accuracy and rapidly converging input spec
trum, the first term is the only one of importance.
Thus this type of system has essentially a velocity error.

Type 2. —12 db/octave, up = (wo/jw)?
Here the error is
is

pe

A(t)
f(t)“ — wi hil + Sad
(¢) = we!

LINEAR

SERVO

THEORY

639

Again for the rapidly converging case, this system will have principally an
acceleration error.
Type 3. —6, —12 db/octave, wp= wows/Jw(Jw + w1)
This is perhaps the most commonly encountered characteristic in simple
servos. The corresponding error expansion is
-

oe

A(é)

=

—

f(t)

a
-f-

Wo

,
fil

—

,

2

Wo W1

Ail)

—.

eee

(wo

>>

@)).

(12.3)

Wy W1

and the principal error for this type system thus is a combination of velocily
and acceleration components.
Either the velocity or the acceleration error
component may be predominant, depending upon the various parameters.
3.3 Noise

Errors

The typical sinusoidal component of servo error due to noise (unwanted
signals or irregularities) in the input signal may be written as*

4, =

——— N,

(13)

1+u

where N represents the corresponding sinusoidal component
noise.

of the input

If the noise signal #(/) is known, the total noise error A,(/) may be

calculated from

(13) in the ways described

ever, the noise input is seldom

known

for the dynamic

in this sense, although

error.

How-

certain out-

standing components sometimes may he estimated and their effects evaluated. On the other hand the average disturbance due to random input
noise, of the kind described as ‘‘thermal noise”’ in electrical circuits, may
easily be calculated.
This type of noise has constant amplitude versus
frequency, and the total power in the output noise error may be found from
@

P.
40

|

mv

rotons
1 of uM

2

1-day,

os

(14)

.

where K is a constant dependent upon the input noise power.
Input noise also causes overloading of the power amplifier and overheating of the motor.
These effects are aggravated by the falling transfer
characteristic versus frequency of the motor, as seen from the following discussion. The servo transfer characteristic is maintained approximately
at unity out to the cross-over frequency.
However the transfer ratio of
the motor, equation (3.1), will be falling at least at 6 db/octave, usually
at 12 db/octave, at frequencies below this point. Thus the transfer
* Again assuming 8B=

—1.

23 Assuming that the mechanical load impedance is a series combination of resistance
and inertia.

640

BELL

SYSTEM

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characteristic (loop closed) from the servo input up to the motor and pow
amplifier must rise correspondingly with frequency, out to the cross-o.
point. Again assuming input noise of the uniform amplitude versus fr
quency type, the total noise power at the motor input is therefore,

and tk
mecha
that tl

more |
3.4 Ce

| om

nee

K,

|

_

i,

(w

+

wn)w

dw.

(15

In ¢

Again, w» is the reciprocal time-constant of the motor and K, is a propor
tionality constant.
If @» is less than about half the cross-over frequency,
then the noise power at the motor input increases as the fifth power of the
bandwidth of the servo transfer characteristic.* Thus, if the input signal
noise ratio is small, this effect may be an important design consideration.
Sull other servo errors may result from local extraneous signals or from
coulomb and static frictional effects. These error sources are in a some
what different class from those discussed previously, in that they are more
nearly under the designer’s control. That is, such extraneous signals
and friction may be kept small by proper design and the residual friction
effects further reduced by the use of local feedback.
In the absence of
local feedback, the servo error resulting from frictional or other torque disturbances at the output shaft readily is found to be
a.
T

Lee,

S(jo)

(16)

1+ 4

S(jw) is the actual! stiffness (loop opened) of the output mesh, and T is the
disturbing torque.
7 conceivably may represent static or coulomb friction,
load-torque irregularities due to fluctuating running-friction, or wind torque
Again assuming the mechanical impedance to be resistance and inertia in
series, the mechanical stiffness is, from (2.2), S(jw) = jw(R + jwJ).

Thus

the error is
Ar

T

1

~ jo(R + joJ) 1+ p’

(16.1)

and the apparent output stiffness (loop closed) is
S’ = jwo(R + go) (1 + py).

(16.2)

If T is taken as the static load torque, the resulting static error is found
by setting

w =

O in (16.1).

Assuming

that » behaves

as wo/jw when w

approaches zero, the static error is
i
Ar

=

w) R’

(16.3)

* This assumes a constant functional form for the transfer characteristic. However,
the statement holds approximately, even with considerable variation in this form.

for a]
an

au

linear
6, bei

angle

LINEAR

SERVO

THEORY

641

and the apparent low-frequency stiffness is woR, being the ratio of the
mechanical resistance to the velocity error coefficient.
It may be noted
that the static error will vanish if the loop transmission approaches infinity
more rapidly than 1/w as w approaches zero.
3.4 Comparison ofu Characteristics for a Particular Input Signal
In order to illustrate the advantages of shaping the loop characteristic
for a particular input signal, a brief discussion will be given of the design of
an automatic radar loop to track an airplane in azimuth over a constant
linear-velocity course.
The servo configuration is that given by Fig. 5b,
6, being the azimuth

angle.

angle of the target and @ the corresponding antenna

The lobing radar antenna has been assumed to take the place of the
T
2

“4

-6

-4

2

6

en

TIME IN SECONDS —=

Fig. 12—Azimuth

8

angle for constant linear velocity course.

synchro pair. Thus 8 = —1, and an error signal proportional to 6; — 4 is
25
developed.
Assuming a constant linear-velocity course having a maximum azimuth
rate of 30 degrees/sec, the target azimuth angle is given by *
A(t) = tan

.5244¢,

(17)

which is plotted in Fig. 12.
This course will develop a maximum azimuth acceleration 6 of +10.3
degrees/sec? and a maximum @, of —16.4 degrees/sec*. The continuous
frequency spectrum of 6;(¢) may be found from (10.1) to be
Fie)
25Assuming

a low elevation

course.

=

¢

—1.9]a]
——-..

jw

(18)

% The azimuth angle has been so taken that zero azimuth is obtained at the point of
nearest approach.

642

BELL SYSTEM

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A logarithmic plot of |F,(w) | is shown in Fig. 13.*

It may be seen that tie

input signal spectrum falls at 6 db/octave for w K 0.5, at 12 db/octa
forw = 0.524, and 30 db/octave at w = 2.1.
Assuming that the permissible dynamic error is 0.3 degree, a comparison
will be made between the type 1 and type 3 loop characteristics of the pr
vious section.
For the type 1 system, which will have essentially a pur
velocity error, (12.1) shows the required value of wo to be 30/0.3 or 100
20

Baal

-12 DB/OCTAVE SLO
PE

lus
DE
in

-20
-30 0B/OCTAVE

ROR

-40

s

(W)|
|DECIBELS
IN
Fy

+

+
|

|
|

+

-80}-———-

~——-——+

|
|

-10

‘

085)

02

0.4

06

08
10
Ww

4

20

40

60

Fig. 13—Target frequency spectrum for constant velocity course.

Thus » = 100/jw. Figure 14 shows a logarithmic plot of the corresponding
|u|. This characteristic departs rapidly from the shape of input signal
spectrum given by Fig. 13, as w is increased abeve 0.1.
The type 3 characteristic permits a considerably better match.
Choosing
a compromise value for w of 0.1, (12.3) may be used to calculate the necessary value of wo as 415. Thus the loop transmission becomes php =
41.5/jw(jw + 0.1). Figure 14 shows a plot of the corresponding | x |,
modified near the gain cross-over to satisfy .the stability requirements.
This curve is a considerably better average match for the target frequencyspectrum up tow = 1. The resulting type 3 system has a predominant
acceleration error as judged from the maximum velocity and acceleration
errors of .072 degree and 0.25 degree respectively.
The total dynamic error curves for the constant-velocity course are given
*| Fi(w) |= x has been taken as the zero db level.

644

BELL

SYSTEM

TECHNICAL

JOURNAL

in Fig. 15. The velocity error of the type 1 system is always a lagging err:
and is maximum at the point of nearest approach. The type 3 composit.
of velocity and acceleration errors is lagging over about the first half of th:
course and leading for the second half, having lead and lag maxima at point
closely grouped about the point of nearest approach.
Although the two loop characteristics develop the same maximum dy
namic error on the specified target course, their transient responses to an
input step differ widely, as may be seen from Fig. 16. The rise time for the
type 1 loop is about .03 second compared with an initial rise in 0.17 second

the ne
to be
wante

RS
Th
servo
tube
ties,

1.4

Ww
”
z
9

—+-—

-

|
4+ ——_____+--___J

+>}

”)
ud

«

°
>

+

$$$

—_—_——_+—_—___+___

a
ad

”

+————_}+————-++
|

t

a

|

|

|

0.3

i
0.4

|
0.5

0.6

07

0.8

TIME IN SECONDS —>

Fig. 16—Transient response of tracking servos.

for the type 3 system. Also, because of the overshoot the type 3 system
requires about 0.7 second to settle within 5% of the equilibrium value.
For a final comparison of the two systems the corresponding transfer
characteristics, u/(1 + qm), are plotted in Fig. 17 on arithmetic amplitude
and frequency scales. It may be seen that the type 1 system is vulnerable
to noise and interfering signals over a far wider frequency range than the
type 3. Again assuming uniform input noise versus frequency, (14) may
be used to show that the ratio of output noise power for the two systems is
about 7.5:1.
Thus the luxury of crisp transient response as obtained with the type 1
system may demand a heavy penalty in terms of output fluctuations due to
noise and other unwanted signal variations. This is a clear illustration of

|

sup
frec
“6

sta

tac

fee

ing
tac

ant

le

LINEAR

SERVO

THEORY

645

the necessity for designing the servo loop to match the type of input signal
to be transmitted, particularly for radar tracking systems where the ‘‘unwanted variations” are ever present.
3.5 Use of Local Feedback
There are many examples of the use of local or subsidiary feedback in
servo systems. The more common of these include feedback around vacuum
tube power amplifiers to obtain improved linearity and impedance properties, and over-all feedback around amplifier and motor-drive systems to

RATIO
TRANSFER

te)

40

80

120

160

200

240

280

320

W->

Fig. 17—Frequency response of tracking servos.

suppress frictional effects, increase output stiffness, and modify the inherent
frequency characteristics of the basic components.”
The tendency toward
“8 circuit dependency” as previously discussed also produces greater constancy of the stage transfer characteristics with time, temperature, etc.
Perhaps the simplest and most useful kind of local feedback is negative
tachometer (velocity) feedback around motor-drive systems. This type of
feedback widens the transfer frequency band of the drive system by reducing its time-constant, and increases the linear speed range of the motor.
This may be illustrated by referring back to Fig. 7, which shows a typical
tachometer loop. Assuming the transfer ratio of the amplifier to be a con27 In a slightly different class are the servo systems used to provide automatic frequency

and gain control in radio systems.

646

BELL SYSTEM

TECHNICAL

JOURNAL

stant A, the transfer ratio of the motor and amplifier without feedback js.
from (3.1),
tr

=

=

6

— (loop

(loop

open)

=

open)

Mo

1

"— ———___—.

(19

J jw(jw + wm)’

where the constant wo = Apw;. (To avoid confusion with primary loo;
quantities, the tachometer loop will be represented by the symbols ur and
8,,rather than and.)
The quantity w,, was defined as the ratio (R,, 4
R;,)/J (see Fig. 2b), and is the reciprocal of the motor time constant.
Replacing (R,, + Ry) by R for convenience, (19) may be rewritten as
@

.

Mo

. = — (loop open) = ————;. .
aie loop of
jwo(R + juJ)

.

Equ

where
The
show!

witho
such
a
Wm/

Wy

low-fi

19.1)

The transfer ratio of the tachometer is
Br

Es
6

:

=—=

—jwR, ’

and thus the loop transmission characteristic is
2
Mr Br =

po Ry

—-_-—_,
R+ joJ

(20)

For values of w small compared with w,, this loop transmission is constant
and closely given by wrBr(0) = —poR:/R.
When w > wm, urBr approaches the form —yoR;/jwJ, and thus falls off at 6 db/octave. Consequently the maximum phase shift of the factor —u787 is — 90 degrees, and
no stability problem arises for the local tachometer loop.”®
From (19.1) and (20), the over-all transfer ratio with feedback is
H (loop closed) = yen
(21)
—_

Ho

Sees

~ jo(R + woR: + jot)”
Comparing (21) with (19.1), it may be seen that the sole effect of the tachometer feedback upon the over-all transfer ratio has been to add an apparent
“ohmic”

friction

or mechanical

resistance

oR;

to the original

value

R.

(It will be shown that this increase in apparent mechanical resistance also
is effective in increasing the mechanical output impedance, although no
power is dissipated in the added component yoR;:.)
8 Actually, the effects of parasitic elements always modify this situation somewhat,
especially if unusually high loop transmission is sought. However tachometer loops often
require little or no stabilizing equalization.

ste

LINEAR SERVO

THEORY

Equation (21) also may be written as
: (loop closed)
where w,, =

=

Lo

1

I jae
+oh)’

a

(R + woR,)/J is the new corner frequency.

The change in over-all transfer ratio due to the tachometer feedback is
shown in Fig. 18. The solid line diagrams A and B are the transfer gains
without feedback and with feedback, respectively.2® At low frequencies
such that w < w,,, the feedback reduces the transfer ratio by the factor
wm/W@m, the ratio of the two corner frequencies.” In order to restore this
low-frequency loss in transmission, it is necessary to provide an added
a)

al
w

@

Uv

wW

{a}

z
°
é

<
c
x

WwW
u
”
Zz
qt
c

|

|
|
|
|
1

a

Wm
LOG

W

Fig. 18—Effect of tachometer feedback on motor characteristic.

amplification wm/wm.

If this is accomplished by increasing yo and decreas-

ing R, so that the product yok; remains constant,"

the resulting transfer

ratio will be that shown by the dotted lines C in Fig. 18. Comparing A and
C, it may be seen that the net result of applying tachometer feedback and
increasing the amplifier gain is to widen the transfer bandwidth by the
factor wm/wm.

The required increase in amplification is the cost of widening

the transfer bandwidth either by tachometer feedback or by non-feedback
means, such as the use of a ‘“‘forward-acting’ ’ equalizer in the amplifier.
(However, such forward acting equalization fails to provide the increased
over-all linearity and mechanical impedance obtained by the feedback
method.)
At frequencies sufficiently high that w >> w,, the change in
transfer ratio due to the feedback disappears, the mechanical inertia becoming the controlling element.
29 The straight line asymptotes have been drawn instead of the actual gain curves.
380 This is also the factor by which the feedback reduces the output speed obtained for a
steady input voltage, neglecting circuit non-linearites and coulomb

friction.

31 This ensures a fixed loop transmission, and thus an unchanging value for ws .

648

BELL

SYSTEM

TECHNICAL

JOURNAL

been in

For w small compared with w,, , (21) becomes
.

.

,

,

local fe
6

Mo

z - |(loopop closed)
closed)

a: jo(R
——————
+ wR)’

If the tachometer feedback is substantial (wm >

approximated

(w
K

wm), this may

,

w,).

be further

as

0

(loop closed)

~

ene

1

-—~

JoR,

w Kw»

,

a

Wm >> Wm

P

21.2)

ah

and the corner frequency becomes

the tw
eductic
torque

may b
3.6 Er
Ins
may b
systen

iR

Win ~ —

:

(cm > Wm).

Thus for reasonably high feedback, the over-all transfer ratio (21.2)
depends only upon the tachometer characteristic, being substantially independent of changes in the original mechanical resistance R or the amplifiermotor factor uo. The corner frequency wm is similarly independent of
changes in R, although still a direct function of uo. Thus the principal
non-linearity of two-phase induction motors, namely variation in electrical
damping with speed, is effectively suppressed by this type of local feedback, and systems employing such motors up to 80°% of their synchronous
speed may be designed on a linear basis.
The increase in mechanical impedance due to the feedback may be shown
by assuming a torque disturbance 7 applied at the output shaft. Without
feedback, the resulting speed disturbance is
6 (loop

open)

=

F: =

RE

°

Tha
inpu
to r

With feedback, the corresponding shaft speed disturbance becomes

6 (loop

closed) =

( tipi

T

1

3- - ————.

Zan

i-

,

MrBr

T

R + wR: + jw] ”
Thus the apparent mechanical resistance, and therefore the protection against frictional
torques, has been multiplied by a factor
(1+ ywoR:/R) = wm/wm.
If the motor-drive system with tachometer feedback is employed in a simple follow-up system of the type of Fig. 5,
equation (16.3) shows that the resulting low-frequency output-shaft stiffness
will be wo(R + pwoR:) or (wm /tom oooR22 Therefore the output stiffness has
*2 The low-frequency loop transmission of the follow-up loop is again taken to be wo/jw.

acceé
sim)
intc
syn
sys!

the

Hal’
sho

LINEAR

SERVO

THEOR}

649

been increased by the factor w,/w, over that obtained without the use of
local feedback, assuming identical follow-up loop characteristics (u8) for
the two cases.
The ratio w,/w, thus directly measures the feedback
eduction of static and low-speed errors of the follow-up system due to
torque disturbances.
In practice the resulting increase in static accuracy
may be of the order of 10 to 100 times.
3.6 Error Reduction by Non-Feedback

Means

In situations where the noise associated with the input signal is small, it
may be desirable to reduce the dynamic errors obtained with a given servo
system by the use of forward-acting equalization external to the loop.
ADDED

‘cn
hr)

|

|

THoF

fy (t | Fy

Fy-Fo|

|

1

>

Fo

Ho

fo(t

SERVO
LOOP

Fo

.

ena

FOR

A

ZERO

DYNAMIC

Cnr

1

Meri

Fig. 19—Forward-acting error compensation.

That is, the dynamic error characteristic may be computed, and the servo
input or output modified by supplementary networks in such a fashion as
to reduce the over-all error.
An illustrative arrangement, which is suitable when the input member is
accessible,® is shown in Fig. 19.

For convenience the servo is taken to be a

simple follow-up system having 8 = —1. The yu circuit is shown divided
into two parts, uw: and we. Typically, u: may be the transfer stiffness of a
synchro pair (Fig. 3b), and we the transfer characteristic of a motor-drive
system. The normal dynamic error component for such a loop, omitting
the dotted line, has been shown to be F;/(1 + uw).

If an additional signal

ual’; is obtained from the input member and injected into the system as
shown by the dotted line, then

St

oe

—

ten

i

HF

Home

oe

i
7

l+u
38 This is not the case for a radar tracking loop, for instance.

650

BELL

SYSTEM

TECHNICAL

JOURNAL

us is gi

Thus the over-all error becomes

for sub
F,

ai

F,

=

(1
a

bb +

pot) Fy,

pared ’

M

closely

at the

or
Fi

1

ee De: ee
2

from t

1 — pape My
»
eee
1
Lu
1

(22

ratios
maint:
The

If the added transmission path is so designed that

loop c
stabili
when
becau
FORWARD-ACTING

FEEDBACK

TACHOMETER

TACHOMETER

—

aR
—

INPUT
8;

——

? kL
ee
_SYNCHRO
PAIR
G Eger ES
L—/
_

bean

|

OUTPUT

|

85

!

|
|

4

M

Ccctemaninaied

4

\——~<% b

es

F

-~jJWR,O5

y
(FOLLOW-UP
LOOP

!
|

;

_
|
|
|

|
|
|
|
|

FOR ZERO DYNAMIC ERROR, Rt Ry
Fig. 20—Forward-acting tachometer system,

then fF; = F;, and the dynamic error vanishes. Thus the desired form
of the added transmission depends only upon the ye portion of the loop
characteristic.
It will not be possible to satisfy the condition given by (23)
exactly, especially at the higher frequencies where noise enhancement and
parasitic effects will become increasingly important.
However, it is often
possible to obtain the proper form for wa over the range of frequencies responsible for the bulk of the dynamic error. If ue has the proper frequency
characteristic but is too large by 10%, fcr instance, it may be seen from (22)
that there still remains a 10/1 increase .n dynamic accuracy.
The foregoing method is especially applicable when ye represents the transfer characteristic of a motor-drive system employing tachometer feedback,
as shown in Fig. 20. Here the basic input-output comparison is obtained
by means of the synchro pair, while a tachometer coupled to the input shaft
provides the error-reducing signal. Thus the transmission ya is equal to
jwRt, where R; is the tachometer transfer resistance. The expression for

LINEAR

SERVO

THEORY

651

us is given approximately by (21.2) as 1/jwR,. Thus, by (23), R, = R,
for substantial cancellation of the dynamic error (at frequencies small compared with Wm).

That is, the output voltages of the two tachometers must

closely annul each other when the input and output shafts are travelling
at the same speed. Since the tachometers may be closely alike and excited
from the same supply line, it is comparatively easy to keep their transfer
ratios closely matched.
In practice an error reduction ef 20/1 is readily
maintained by this method.
The error compensation scheme described above does not change the
loop characteristic 48 of the basic servo loop, and thus does not create

new

stability problems.
Its use to obtain high servo accuracy is desirable
when the input noise is small and when a high loop gain is difficult to obtain
because of parasitic elements or equipment complexities.

HCl-sa
with k

which <

Abstracts of Technical Articles by Bell System Authors
Quai
Computation of Interfacial Angles, Interzonal Angles, and Clinographic Pro
jection by Matrix Methods.' W.L. Bonp. A way of setting up the general
crystallographic axes a, b, c on unit orthogonal axes x, y, z is used to afford a
matrix method of computing interfacial angles and zonal angles. It also
affords a method of making clinographic projections.
A Current Distribution for Broadside Arrays which Optimizes the Relationship between Beam Width and Side-Lobe Level? C. L. Doupu. A oneparameter family of current distributions is derived for symmetric broadside
arrays of equally spaced point sources energized in phase. For each value
of the parameter, the corresponding current distribution gives rise to a
pattern in which (1) all the side lobes are at the same level; and (2) the beam
width to the first null is a minimum for all patterns arising from symmetric
distributions of in-phase currents none of whose side lobes exceeds that level.
Design curves relating the value of the parameter to side-lobe level as well
as the relative current values expressed as a function of side-lobe level are
given for the cases of 8-, 12-, 16-, 20-, and 24-element linear arrays.

compe
crysta’

ence 0
quartz
of the

develc
imme
our 1
and r
cles c
Tech

Th
quar!
speci
cryst

varic
othe

the |

Paper Capacitors Containing Chlorinated Impregnants—Mechanism of Stabilization L. EGerton and D. A. McLean.
The stabilization of paper
capacitors containing chlorinated aromatic impregnants with small quantities of organic additives is well established commercially.
Although for
practical reasons anthraquinone was chosen for initial commercial application, other quinones are also effective, as are the nitroaromatics, maleic
anhydride, and sulfur. Evidence is given that the mechanism of stabiliza-

mos:
men
crys
mer

tion consists of the formation of barrier films on the electrodes.

tail

These bar-

rier films, which in certain cases may cover only the active points on the
electrode surface, reduce the catalytic decomposition of the chlorinated impregnant by the electrode metal, prevent attack of the electrodes by liberated
hydrogen chloride, and hinder electrolytic action. It appears likely that the
film-forming properties of the stabilizers are dependent upon their oxidizing
power. A secondary effect of stabilizers may be the formation of complexes

with aluminum chloride to diminish the activity of the latter or change the
nature of the reactions which it induces.
1

Conductivity

American Mineralogist, Vol. 31, pp. 31-42 (1946).

2 Proc. I.R.E. and Waves and Electrons, June 1946.
8 Indus. and Engg. Chemistry, May 1946.
652

measurements in

the

an
for
chi
fou
arl
tol

cu
pr

ABSTRACTS

OF

TECHNICAL

ARTICLES

653

HCl-saturated chlorinated dipheny! containing soluble additives are in line
with known hydrogen-bonding tendencies of the additives. Compounds
which are strong organic bases do not stabilize capacitors.

ral
da
lso

Quartz Crystals for Electrical Circuits* R.A. Hetstnc.
This book is a
compendium of information, both theoretical and practical, on quartz
crystal plates, their design and manufacture.
It embodies the vast experience of the Bell Telephone Laboratories in research and in manufacture of
quartz crystals. It originated from a series of lectures given by the members
of the Laboratories technical staff who had carried out the early studies and
developments in this field. By this means, engineers were trained for the
immense expansion in crystal manufacture required to meet the demand of
our military forces during the War. These lectures have been reorganized
and rewritten, and are published together in this comprehensive book. Articles covering some of the various chapters have appeared in the Bell System
Technical Journal.
The treatment covers in full the theory and practice of the preparation of
quartz crystals, the instruments used, including new types developed for
special purposes, the problems encountered in the various uses of quartz
crystals, and the full details of the methods devised for their solution. The
various processing chapters, dealing with cutting and grinding, plating and
other topics of equal importance, include much information that appears for
the first time in any book. The account of the manufacturing process is
most complete. There are discussions of new practical methods of adjustment to frequency, of the new performance indicator, of a new type of
crystal cut that operates at very low frequencies, and many new developments that represent notable advances in crystal technology.
Geometrical Characterizations of Some Families of Dynamical Trajectories.
L. A. MacCott.
The chief problem considered in this paper is that of obtaining a set of geometrical properties which shall completely characterize
the five-parameter family of trajectories of an electrified particle moving in
an arbitrary static magnetic field. A solution of the problem is found in the
form of a set of four principal and four subsidiary properties. A geometrical
characterization, in the form of a set of two properties, is also given of the
four-parameter family of trajectories of an electrified particle moving in an
arbitrary static magnetic field with an arbitrarily prescribed value of the
total energy. Various other properties of the families of curves are discussed, and the paper closes with a brief consideration of some analogous
problems in which the particle moves in a fixed plane.
‘Published by D. Van Nostrand Company, Inc., New York, N. Y., 1946.
5 Amer. Math. Soc. Trans., July 1946.

654

BELI.

SYSTEM

TECHNICAL

JOURNAL

Comparison of Natural and Synthetic Hard Rubbers® G. G. Wrisprax.
GR-S, nitrile, and natur.|
D. B. HERRMANN, F.S. MAtm, and A. R. Kemp.
hard rubbers are compared as regards compounding, processing, vulcanizs
tion, and physical and dielectric properties. Natural rubber and GR-S
compounds intermediate in sulfur content between hard and soft rubber also
are compared.
GR-S and nitrile rubber compositions suitable for commercial ebonite fabrication are described.
Extensive breakdown of the basi:
copolymers has little effect on the physical properties of synthetic ebonites.
The time required for the beginning of exothermic reaction in vulcanization
is longer for GR-S than for natural rubber ebonites. Rockwell hardness is
Some GR-S ebonites are penetrated to the same depth
greater for GR-S.
as natural ebonites, with a greater tendency toward instantaneous recovery,
The two are similar in impact strength, but the ability to withstand a sharp
bend is characteristic of natural ebonites alone. The latter are superior to
GR-S ebonites in heat deformation below 60° C., but above this temperature

the reverse is true and nitrile ebonites are superior to both.

WALL

1936; \
been pr
FREL

versity.

DSc. (
Electri
Labore
Labor

GR-S ebonites

fundar

are more stable and nitrile ebonites less stable chemically than natural
ebonites. GR-S ebonite dust as a filler increases brittleness. A diatomaceous earth improves the processing properties of GR-S hard rubbers. The
adverse effect of ultraviolet light on surface resistivity is reduced when a
GR-S hard rubber is filled with whiting. Natural and GR-S hard rubbers
are alike in dielectric behavior.

switch
device
Rol

joinec
has s
work

Signal and Noise Levels in Magnetic Tape Recording." D. E. WooLpRipce.
The primary object of the work described here was to determine what properties of the tape and associated magnetic elements are responsible for the
noise and signal output levels of magnetic recordings and, if possible, to display in specific equations the pertinent relationships connecting noise and
signal levels with the physical properties of the tape and polepieces. In the
course of the study, methods appeared for decreasing the noise and increasing the useful signal reproduced from magnetic tape. These methods and
some of the use that Bell Telephone Laboratories and Western Electric have
made of them are mentioned in the discussion. While some of the work
described in this paper has implications for more than one type of magnetic
recording process, perpendicular recording on tape is the actual subject matter dealt with. In every case discussed, the record medium was 0.050 inch
wide and 0.0022 inch thick. Except where otherwise noted, a chrome-steel
tape was used at a speed of 16 inches per second.
SIndus. and Engg. Chemistry, July 1946,
7 Elec. Engg., Trans. Sec., June 1946.

phon
in co
used

1917

Eng
Cor
par

wo!

~
Te
net
19:
ing

656

BELL

SYSTEM

TECHNICAL

JOURNAL

E. J. Ryper, E.E., Polytechnic InstituteofBrooklyn, 1935. Engineering
Department, Western Electric Co., 1922-25; Bell Telephone Laboratori:
1925-.
In the Physical Research Department engaged in contact studies
and during the war the development of switching devices used in radar.

The Bell System Technical Journal 1946-10: Vol 25 Iss 4


[75] Inventors: Larry A. Coldren, Holmdel, N.J.;
Karl J. Ebeling, Gleichen, Fed. Rep.

of Germany

[73] Assignee: AT&T Bell Laboratories, Murray

Hill, N.J.

[21] Appl. No.: 572,682
[22] Filed: Jan. 20, 1984
S51] Unt. C14 ec ccccecsteeeeennenreeseerensesens HOIS 3/10
[52] ULS. CL, woe eeecenceseensreeeennereneees 372/32; 372/92;

| 372/97
[58] Field of Search ............:::see 372/50, 29, 32, 92,

372/97

[56] References Cited

PUBLICATIONS

W. Bludau et al, “Characterization of Laser-to-Fiber
Coupling Techniques by their Optical Feedback”, Ap-
plied Optics, vol. 21, No. 11, Jun. 1982, pp. 1933-1939.
K. R. Preston, “Simple Spectral Control Technique for
External Cavity Laser Transmitters”, Electronics Let-
ters, vol. 18, No. 25, (1982), pp. 1092-1094.

W. Tsang, N. A. Olsson and R. A. Logan, “High-Speed

Direct Single-Frequency Modulation with Large Tun-
ing Rate and Frequency Excursion in Cleaved—Cou-
pled-Cavity Semiconductor Lasers”, Applied Physics
Letters, vol. 42, No. 8, (1983), pp. 650-652.

H. Kogelnik and C. V. Shank, “Stimulated Emission in
a Periodic Structure”, Applied Physics Letters, vol. 18,
No. 4, (1971), pp. 152-154.

K. J. Ebeling, L. A. Coldren, B. I. Miller and J. A.
Rentschler; ‘““Generation of Single-Longitudinal-Mode
Subnanosecond Light Pulses by High-Speed Current
Modulation of Monolithic Two-Section Semiconduc-
tor Lasers”, Electronics Letters, vol. 18, No. 21, (1982),

pp. 901-902.
Primary Examiner—James W. Davie

[57] ABSTRACT

A control circuit for stabilizing the emission wave-
length of the coupled cavity semiconductor laser 1s
described which uses measurements of the voltage
across the laser cavity to maintain stable single longitu-

dinal mode output.

12 Claims, 3 Drawing Figures

33 35
| REFERENCE | | DC BLOCKING
OSCILLATOR [oo *SY~CO*«é RGU
f | | kHz
PHASE jf
SENSITIVE dc
DETECTOR a a—(+)—0 If
» REF

U.S. Patent

ARB.
UNITS

-10

FIG. 3

Nov. 11, 1986 4,622,672
“OVe, Ie.
5, Pete
“3
wr
— . f
g
mA
| 33) 399
REFERENCE | OC BLOCKING
OSCILLATOR [7 *«)~«3=s CIRCUIT
3) } [kHz
PHASE Sf
SENSITIVE dc
DETECTOR (+)e——0 Ip
» REF

4,622,672

I
SELF-STABILIZED SEMICONDUCTOR LASERS

TECHNICAL FIELD

This invention relates generally to semiconductor
lasers and particularly to such lasers that operate stably
with a single longitudinal mode output.

BACKGROUND OF THE INVENTION

Optical communications systems as presently con-
templated and constructed use a light source, such as a
semiconductor laser or light emitting diode, which 1s
optically coupled to a photodetector through a glass
transmission line. The transmission line is commonly
referred to as an “optical fiber.” Either amplitude or
frequency modulation may be used to convey informa-
tion. If amplitude modulation is used, the information ts
transmitted as a bit stream of 1’s and 0’s with the bits
being both transmitted and detected within predeter-
mined time intervals.

Such systems presently operate at wavelengths be-
tween approximately 0.8 wm and approximately 1.6 um
with the longer wavelengths, that is, the wavelengths
greater than appoximately 1.2 jum, presently being con-
sidered most desirable for many systems applications
because optical fibers presently used exhibit minimum
loss and dispersion within this wavelength range. These
characteristics facilitate design of optical communica-
tions systems having desirable properties such as high
transmission rates, long distances between repeaters.
etc.

However, to most efficiently utilize the minimum loss
and dispersion properties of the optical fibers, the light
source should not only emit radiation within this wave-
length range but should also have its output concen-
trated in a narrow spectral range. In practical terms, this
means that the light source should be a semiconductor
laser emitting output of a single longitudinal mode. A
single longitudinal mode is narrow enough in spectral
width that it may be thought of as being a single fre-
quency. The output contains unwanted secondary
modes but these are greatly suppressed in intensity with
respect to the wanted primary mode. The requirement
that the output be a single longitudinal mode is easily
understood by considering that a finite width pulse will
spread, i.e., broaden, because of the dispersion proper-
ties of the fiber. If the spread becomes large enough,
adjacent pulses will overlap and recovery of the trans-
mitted information by the receiver will be impossible;
i.e., the bits will not arrive at the photodetector within
the proper time interval.

Accordingly, a variety of approaches has been taken
in attempts to develop single longitudinal mode lasers.
For example, distributed feedback (DFB) lasers have
been developed. See, for example, Applied Physics Let-
ters, 18, pp. 152-154, Feb. 15, 1971. Semiconductor
lasers operating with an external cavity to produce
single mode operation have been reported. See, for
example, Electronics Letters, 18, pp. 1092-1094, Dec. 9,
1982. Additionally, coupled cavity lasers have been
developed and have been shown to produce single lon-
gitudinal mode output even under high speed modula-
tion. See for example, Electronics Letters, 18, pp.
901-902, Oct. 14, 1982, and Applied Physics Letters, 42,
pp. 650-652, Apr. 15, 1983.

The operation of a coupled cavity laser is described in
detail in the preceding references and may be briefly
summarized as follows. A coupled cavity laser has two

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cavities which are optically coupled to each other.
Each cavity has separate electrical contacts, i.e., the
laser is a three-terminal device. One cavity operates as
an oscillator while the other cavity acts as an etalon and
provides mode selection. That the laser produces a
single mode output is understood, for the case of gap
widths approximately nA/2, where n is an integer and A
is the wavelength of the radiation, by considering that
the mode of the coupled cavity laser must satisfy Fabry-
Perot mode conditions in both cavities. More generally,
modes are enhanced at local loss minima because of the
constructive interference of optical energy reflected
from the gap with that fed back from the other cavity,
i.e., the etalon. This may occur near resonance or anti-
resonance of the etalon as well as intermediate points.
This is possible only for a limited number of discrete
frequencies.

Variation of the etalon current permits tuning of the
wavelength across the gain profile of the laser. How-
ever, a large number of mode hops may occur as the
etalon current is increased from zero to values above
threshold. Although stable output is observed between
the mode hops, the mode hop boundaries, as a function
of etalon current, vary with temperature and also per-
haps with device aging. Thus, simply maintaining the
etalon current at a constant value will not guarantee
stable single mode operation. Mode hops are undesir-
able because they may cause information to be lost
because, for example, bits are not transmitted as desired
or they are lost because they are not received within the
proper time interval. Thus, techniques to maintain sin-
gle mode operation at a desired wavelength are desir-
abie.

Wavelength stabilization techniques have been previ-
ously developed for both coupled cavity lasers as well
as various other types of lasers such as semiconductor
lasers operating with an external cavity. A common
element of these prior art schemes is the requirement of
an external optical element, such as either a photodetec-
tor or a spectrometer, to monitor the light output from,
e.g., the laser. Feedback loops are then used to adjust,
for example, the laser current or a property of the exter-
nal cavity such as its length if it is a passive cavity.

SUMMARY OF THE INVENTION

We have found that wavelength stabilization of a
semiconductor laser having two cavities optically cou-
pled to each other, such as a coupled cavity laser, may
be obtained by using the lasing cavity as a photodetec-
tor to provide a discrimination signal for a feedback
control circuit. The feedback control circuit comprises
means: for measuring the voltage variation across the
laser cavity, means for comparing the measured voltage
to a reference characteristic, and means for varying a
property of a second cavity by an amount determined
by said comparing step. In one embodiment, the laser is
a coupled cavity laser and the circuit comprises a dc
blocking circuit, a phase-sensitive detector which is
electrically connected to the laser cavity, and a refer-
ence oscillator electrically connected to both the block-
ing circuit and the phase-sensitive detector. The output
from the phase-sensitive detector goes to an amplifier
where it is compared to a reference value. The outputs
from the amplifier, which is a function of the difference
between the measured and references voltages, and
blocking circuit are superimposed on the etalon current.

4,622,072

3
BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 1s a schematic representation of a coupled
cavity semiconductor laser;
FIG. 2 plots a typical differential voltage (laser)/cur-

rent (etalon) characteristic, 1.e., dVz/dIz, vertically
versus the etalon current horizontally in units of mA;
and

FIG. 3 is a schematic representation of a wavelength
stabilized coupled cavity semiconductor laser accord-
ing to our invention.

DETAILED DESCRIPTION

10

Our invention will be specifically described by refer- |

ence to the coupled cavity laser schematically depicted
in FIG. 1. For reasons of clarity, the elements of the
device are not drawn to scale. The laser comprises laser
cavity 1, etalon 3, electrode 5, laser electrode 7, and
etalon electrode 9. Electrode 5 is a common electrode
and contacts both the laser cavity and the etalon. Both
cavities comprise a narrow active stripe 11 where elec-
tron hole recombination takes place when the laser
cavity and etalon are forward biased. The laser and
etalon cavities are spaced from each other but are opti-
cally coupled to each other. The space between the
cavities may be formed by, for example, etching or
cleaving a unitary structure. The details of exemplary
laser and etalon cavities, as well as their fabrication, are
well known to those skilled in the art and need not be
further described in detail.

As shown, currents I, and I¢ flow through the laser
and etalon cavities, respectively. The voltages across
the two cavities are Vr, and Vg for faser and etalon
cavities, respectively.

The operation of the control circuit will be better
understood if the operation and light output characteris-
tics of the coupled cavity laser are first explained by
assuming that a constant current, Iz, which is above
threshold, flows through the lasing section. If the mag-
nitude of the etalon current is now varied, the changes,
diz, of the etalon current cause variations of the light
distribution within the coupled cavity system. These
variations also lead to variations, dV ,, of the lasing
junction voltage as a consequence of the internal photo-
effect. These variations are typically extremely small.

FIG. 2 shows a typical differential signal, dV7/dlIz,
plotted vertically in arbitrary units as a function of the
etalon current plotted horizontally in units of mA. The
laser was an InGaAsP/InP buried crescent coupled
cavity laser emitting at 1.3 um. The lengths of the two
cavities were 222.0 and 191.0 um and the space between
them was less than 1.0 um. The current through the
lasing section was 20 mA above threshold. The laser
produces stable single mode output when the etalon
current is above 3 mA.

As the etalon current increases, mode hops occur to
neighboring modes as well as across the entire gain
profile of the semiconductor laser. The arrows indicate
mode hops across the entire gain profile. It was found
that a mode hop results in a sudden decrease in the
differential signal with an abrupt mode hop producing a
sharp negative peak while a more gradual mode transi-
tion produces a comparatively smooth step. Between
successive mode transitions, the differential signal usu-
ally increases as the etalon current increases. Thus, the
behavior of the junction voltage, Vz, is a function of the
etalon current, Iz, and may be used to detect mode
hopping. The points of maximum suppression, with

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respect to the primary mode, of the unwanted second-
ary modes are indicated by the solid circles. These cir-
cles generally lie approximately in the center of each

single mode regime. The dV;/dlz slope may therefore
be used as a discriminating signal. The wavelength sta-

bilization circuit of this invention tracks the etalon cur-
rent to the point of optimum mode suppression and
maintains the current at that point by adding the ampli-
fied discriminating signal to the preset etalon bias cur-
rent. If the etalon current is preset before the feedback

loop is turned on, the desired mode can be maintained.

FIG. 3 1s a schematic representation of our wave-
length stabilization circuit. In addition to the laser ele-
ments described with respect to FIG. I, the stabilization
circuit further comprises a phase-sensitive detector 3I, a
reference oscillator 33, a dc blocking circuit 35, and an
amplifier 37. The output from the reference oscillator

goes to both the de blocking circuit and the phase-sensi-

tive detector. The phase-sensitive detector is connected
to the laser cavity and to an amplifier which also has a
reference input, A;yey The amplifier output is a function
of the difference between the signal from the phase-sen-
sitive detector and pret. The output of the amplifier, dc
blocking circuit, and the etalon current, Iz, are com-
bined and sent to the etalon cavity. Each of these ele-
ments is well known to those skilled in the art and need
not be described in further detail.

The laser cavity is driven above threshold and a small
dither signal from the dc blocking circuit is superim-
posed on the etalon current. A typical amplitude is 10
uA and a typical frequency 1s 1 kHz. The correspond-
ing ac voltage change at the lasing cavity is detected
with the phase-sensitive detector. If the measured ac
signal deviates from a reference value, indicated as Ayey,
required for optimum suppression of the unwanted sec-
ondary modes, a de correction current is fed from the
amplifier to the etalon thereby forcing the overall eta-
lon current to the optimum value.

The laser section current, for the laser previously
described, was set at I, =20 mA‘above threshold and
the etalon current was set at 11.1 mA to select the de-
sired mode. The temperature was then varied and the
output spectra recorded. With the wavelength control
circuit on, the emission stayed in the same mode over
the temperature range between 14 degrees C. and 26
degrees C. while several mode hops occurred with the
circuit off. )

It will be readily appreciated that our circuit may be
used with other types of lasers. For example, the feed-
back circuit may be used to adjust the length of the
external cavity used with a semiconductor laser.

What is claimed 1s:

1. A wavelength-stabilized semiconductor laser com-
prising a laser cavity, 4 second cavity, said laser cavity
and said second cavity being optically coupled to each
other, means for measuring a voltage variation across
the laser cavity, means for comparing the measured
voltage to a reference characteristic, and means for
varying a property of said second cavity by an amount
determined by said comparing step.

2. A laser as recited in claim 1 in which said laser
comprises a coupled cavity laser and said second cavity
comprises an etalon.

3. A laser as recited in claim 2 in which said means for
varying changes the current to the etalon.

4. A laser as recited in claim 1 in which said second
cavity comprises an external cavity optically coupled to
said laser cavity.

4,622,672

.

5. A laser as recited in claim 4in which said means for
varying changes the optical length of said external cav-
ity.

6. A laser as recited in claim 1 in which said means for
measuring comprises a reference oscillator and a phase-
sensitive detector electrically connected to said laser

and to said oscillator.
7 A laser as recited in claim 6 in which said means for

comparing comprises an amplifier having an input from
said detector and a reference input, said amplifier hav-

ing an output.

.)

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8. A laser as recited in claim 7 in which said means for
varying comprises the output of said amplifier.

9. A laser as recited in claim 8 in which said laser
comprises a coupled cavity laser and said second cavity

comprises an etalon.
10. A laser as recited in claim 9 in which said means

for varying changes the current to the etalon. |
11. A laser as recited in claim 8 in which said second

cavity comprises an external cavity optically coupled to

said laser cavity.
12. A jaser as recited in claim 1] in which said means

for varying changes the optical length of said external

cavity.
* * * ae *

Self-stabilized semiconductor lasers