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FUNDAMENTALS

where the constants K

, K

are ideally equal, but

differ slightly in practice, and represent the rotor-sta-

tor transfer functions;

, α

, and α

, are ideally zero (or small and equal),

but are appreciable in practice, and represent the

rotor-stator time-phase shift at the carrier frequency;

and ω= 2πf, where f is the carrier (or reference) fre-

quency used to excite the system.

With these variables accounted for the basic synchro

signal relationship is a shown in Figure 1.11a.

Similarly, a 4-wire set of resolver signals (see Figure

1.10d) measured at terminals S1 and S3 (V

), S2

and S4 (V

), and corresponding to the

spacial

phase

angle θ, would be represented mathematically by:

where K

and K

are ideally equal transfer-function

constants like K

, K

, and K

;

and α

are ideally zero time-phase shifts, like α

, and α

;

and ω = 2πf where f is the same as in the synchro

equations.

V = K

sin θ sin (ωt + α )

V = K

cos θ sin (ωt + α )

150

210

270

330

360

θ (DEGREES)

CCW

S4-S2 = V COS(θ)

MAX

S3-S1 = V SIN(θ)

MAX

- V

MAX

+ V

MAX

In Phase

with RH-RL

Out of Phase

with RH-RL

Figure 1.11b.ResolverSignal Relationships.

“repeaters”), or perform other low-energy mechani-

cal work. Table 1-1 relates torque components to the

analogous control components described above.

The most important fact about all synchro and

resolver signals is that they present information

about the angular position of a shaft in the form of

relative amplitudes of a carrier wave.

All

signals, rotor

and stator, input and output, are at the same fre-

quency, and (except for imperfections in the compo-

nents) in perfect time-phase synchronization.

Although it is common to speak of the “phase angle”

of the shaft, and of the winding as a “3-phase”(syn-

chro) or “2-phase” (resolver), all carrier signals are

sine waves, in phase with all others in the system...

except for the imperfections and undesired side-

effects to be discussed later. To differentiate

between time-phase angle and shaft-position angles

(or their electrical equivalents), we refer to the latter

as “spatial-phase”angles.

A 3-wire set of synchro signals (see Figure 1.10a),

measured between pairs of terminals, S1, S2 and

S3, corresponding to the spacial phase angle, would

be represented mathematically as:

V = K

sinθ sin (ωt + α )

= K

sin (θ + 120˚) sin (ωt + α )

= K sin (θ - 120˚) sin (ωt + α )

2-3

3-1

1-2

150

210

270

330

360

θ (DEGREES)

CCW

S3-S1 = V SIN(θ)

MAX

S2-S3 = V SIN(θ + 120°)

MAX

S1-S2 = V SIN(θ + 240˚)

MAX

- V

MAX

+ V

MAX

Out of Phase

with RH-RL

In Phase

with RH-RL

Figure 1.11a.Synchro SignalRelationships.

Standard Synchro Control Transmitter (CX)Outputs as a Function of

CCWRotation From Electrical Zero (EZ).

Standard Resolver Control Transmitter (RX)Outputs as a Function of

CCWRotation From Electrical Zero (EZ).

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FUNDAMENTALS

That is, regardless of dθ/dt, the rotational velocity of

the shaft, or even of d

θ/dt

, the angular acceleration,

the value of θis always given by:

However, there is an undesired effect in synchros and

resolvers called “speed voltage,” which can cause

appreciable deviations from the ideal relationship stat-

ed above.Speed voltageis discussed in Section VII.

Speed voltage is only one of several undesirable

effects that cause synchro and resolver behavior to

depart from the ideal relationships discussed above.

Among the others are: harmonic distortion; quadra-

turecomponents (of output voltage);loading (of or by

the synchro or resolver);time-phaseshift in the syn-

chro or resolver (the parameter α, mentioned above);

and nonlinearities in the synchro or resolver - i.e.,

departures, due to mechanical imperfections in the

magnetic or windings that cause departure from the

ideal transfer functions described above. All of these

are discussed in some detail in Section VII.

As a final item in our discussion of useful mathematical

relationships, let us examine the nature of digital sig-

nals. A digital signal consists of a set of voltage levels

(or current or resistance levels), at a set of terminals,

each of which has been reassigned a certain “weight”

(i.e., a certain relative

importance

), in accordance with

a certain “code”. The most common code is the so

called “natural binary” code, in which the weight of

each successive voltage-level terminal, from the small-

est to the largest, is given by the simple relationship:

where n is the number of terminals, which varies

from 1 to N (the maximum number of terminals). In

a digital signal, the voltage level at each terminal

may have either one of only two values, nominally:

The ONE state (for example, +4 Volts)

The ZERO state (for example, 0 Volts)

weight = 2

n-1

θ = tan

-1

With these variables accounted for the basic resolver

signal relationship is a shown in Figure 1.11b.

Thus, for any static spatial angle θ, the outputs of a syn-

chro or a resolver are

constant-amplitude

sine waves at

the carrier frequency. In resolver format, ignoring

imperfections, the ratio of the amplitudes would be:

This ratio is independent of the frequency and

amplitude of the reference excitation (carrier fre-

quency and amplitude).

Similarly, for the same static spatial angle θ, the

ratios of the amplitudes of the synchro-format sig-

nals, ignoring imperfections, would be:

These ratios are independent of the frequency

and amplitude of the reference excitations, as in

the resolver-format case, above.

As we shall see, all data-conversion techniques in

current use operate on

resolver-format

(sin/cos) sig-

nals, for convenience. This is done by converting

from synchro to resolver-format before digitizing or

operating on the signals.

So far, we have spoken only of

static

shaft angles. In

practical systems, of course, θ changes, sometimes

rapidly, sometimes slowly and sometimes intermit-

tently. At all times, however, the

instantaneous

value

of θcorresponds to the

instantaneous

ratios of V

V sin (θ)

V sin (θ + 120˚)

2-3

3-1

V sin (θ + 120˚)

V sin (θ - 120˚)

1-2

2-3

V sin (θ - 120˚)

V sin (θ)

3-1

1-2

Note that this

set of ratios is a

single position, θ,

and is the same

information as is

contained in Vx/Vy

above.

V sin θ

V cos θ

= = tan θ

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FUNDAMENTALS

The examples given are completely arbitrary, the

ONE and ZEROstates may be any two easily distin-

guished values. All that matters is that they be dif-

ferent enough so that noise, drift, and other circuit or

signal imperfections are not able to create any doubt

as to which state exists at a particular terminal.

Each terminal in a set is said to present a “bit” of

information, while the entire set, in the pre-assigned

sequence, is called a “word.” The 10-bit digital word:

1101000101

would represent voltage values of +4, +4, 0, +4, 0, 0,

0, +4, 0, +4 Volts and the conventional way of writing

the word would indicate that the extreme-right-hand

bit (a one, in this example), would be the

least-signif-

icant bit

(or LSB) - i.e., its weight would be:

with respect to the extreme-left-hand bit (also a ONE in this

example), which is, by convention, the

most-significant-bit

(or MSB), the weight of which (relative to the LSB) is:

In other words, a ONE level in the MSB position (i.e.,

a +4 V ONE state on the MSB

terminal

) has 512

times as much weight as a ONE in the LSB position.

The actual value of the digital word 1101000101 is

found by adding the ONE bits, in proper weight, and

to count the ZERO bits as zero:

(MSB) 1 = 2 = 512

1 = 2 = 256

0 = 0 = 000

1 = 2 = 064

0 = 0 = 000

1 = 2 = 004

(LSB) 1 = 2 = 001

837

MSB = 2 = 2 = 512

n-1

LSB = 2 = 1

Note that the

maximum value

that can be represent-

ed by a 10-bit word is (when all bits are in the ONE

state) is (2

-1) =1,023, or one less than 2

. The

resolution to which the digital word can be varied

then is ±LSB, which is:

But the digital word need

not

be interpreted in terms

of an LSB=1. In the case of angle data, for example,

we might set the maximum value of an N-bit word

equal to 360°, so that

but this would, for most values of N, give us rather

inconvenient values for each bit. A more appropriate

scheme is to set the MSB=180°. Then the bits would

have values in descending order, of:

In this scheme, then, a 10-bit digital word in natural

binary code would have a resolution of:

Note also that the

first two bits

of any natural-binary-

coded digital word scaled in the manner described

above determine the

quadrant

in which the angle lies

(see pages 47 and 48 for a detailed explanation).

± 1LSB = ≅ 0.35˚

360˚

1024

(MSB) 2 = 180˚

N - 1

2 = 90˚

N - 2

2 = 45˚

N - 3

(LSB) 2 =

(

)˚

180˚

N - 1

...

n = N

n = 1

= (2 + 1) = 360˚

N - 1

Resolution =

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FUNDAMENTALS

Typical Synchro/Resolver System Architecture

Synchro and resolver components can be intercon-

nected and combined (mechanically and electrically)

with other devices and circuits in hundreds - perhaps

thousands - of useful configurations. A number of

these combinations are shown in Section IX of this

handbook, but, before considering them it would be

best to review a “pre-digital” use of synchros in the

all-analog configurations in which these components

were originally used.

Single-Speed System

Figure 1.12a shows what might be called the “clas-

sic” combination of synchro components into an

electro-mechanical follow-up mechanism. The input

angle (θ

) is established by the position command -

a setting of the shaft position of the control transmit-

ter, CX, by hand or by some director mechanism -

and the position of the mechanical load (e.g., a valve,

a turntable, a tool-bit feed screw) is made to conform

to this command, rapidly and accurately. The

sequence of events is as follows:

(1) The CX puts out a 3-wire representation of θ

the position command.

(2) The CT transforms θ

into a 2-wire signal pro-

portional to the sine of (θ

- θ

(3) The output of the CT is amplified and used to

drive the servo motor.

(4) The servo motor positions the load, through the

gear train, which simultaneously drives the shaft of

the CT.

(5) When the load has reached the correct (com-

manded) position, θ

= θ

, the output of the CT is

then zero, and the motor stops.

The above will be recognized as a vastly oversimpli-

fied description, but it should serve to identify the

function

of each element in the follow-up servo.

With the advent of digital electronics, it became pos-

sible to replace one or more of the synchro (or

resolver) components in such a system by an accu-

rate, small, reliable, and economical electronic cir-

cuit. In the system of Figure 1.12a, for example, the

control transmitter could be replaced by a digital-to-

synchro converter,as shown in Figure 1.12b, which

would accept

digital

commands, from aprogramming

device - such as a computer, or a punched paper

tape, or a “Read-Only Memory”(ROM)- and produce

the input to the CT. Such a combination is called a

“hybrid” synchro system, because it combines

electromechanical and electronic devices for angular

data conversion.

In all the systems shown so far, only one or two

angles are involved in the data manipulation and

control. In many applications, as we shall see,

many

angles may be measured, monitored, or controlled;

therefore, the systems shown are relatively simple

(although very important) examples.

CONTROL

TRANSFORMER

CONTROL

TRANSMITTER

SERVO

MOTOR

MECH LOAD

GEAR TRAIN

Figure 1.12a.Typical Electromechanical Follow-up

Servo.

SOLID-STATE

CONTROL

TRANSMITTER

CONTROL

TRANSFORMER

AMPL

SERVO

MOTOR

DIGITAL

MECH LOAD

D/S

Figure 1.12b.Follow-up Servo with D/S converter

replacing CX of Figure 1.12a.

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FUNDAMENTALS

In all of the synchro systems shown and discussed

so far, the relationship between θ(the shaft angle to

be monitored, measured, or controlled) and the

angular setting of the synchro or resolver transducer

has been a constant ...usually shown as unity (direct

coupling), but possibly geared up or down. In all

such “single-speed”systems, the product of dynam-

ic fidelity and resolution is the figure of merit - i.e.,

how fast one may track, measure, convert, and react

to, variations in θ, to an accuracy consistent with how

many bits of resolution.

See Figures 1.13 through 1.16 for examples of other

types of synchro systems.

Multispeed System

There is a type of synchro/resolver system (shown in

Figure 1.17) in which much higher accuracies and

resolution may be achieved. This is called a multi-

speed system.

A multispeed synchro system consists of two or more

synchros or resolvers geared together, usually with a

gear ratio of some whole number.The most common

are two-speed systems with ratios of 1:8, 1:16, 1:32,

1:36 but other ratios can also be found. To under-

stand how these systems achieve high accuracy let

us examine a typical two-speed system.

In the single-speed case (see Figure 1.12a) the sys-

tem will, essentially, have an accuracy dependent

upon the accuracy of the CT (we will assume a per-

fect transducer) and the positional resolution of the

servo loop. Assume we can position the CT with a

certain accuracy, say within ±0.1°. If we now gear this

CT to another with a gear ratio of 1:n (called a

“coarse”CT) where the CT we are positioning rotates

n turns for each single turn of the other, then 0.1° of

rotation of the fast CT (called the “fine”CT) will turn

the slow CT (called the “coarse”CT) 0.1÷n, so that an

inaccuracy in the fine CT position is effectively divid-

ed by the gear ratio (often called the speed ratio).

CX (ANGLE SENSOR)

DIGITAL

DISPLAY

183.9˚

SOLID-STATE

SYNCHRO/DIGITAL

CONVERTER

(

OUT

)

CONTROL

DIFFERENTIAL

TRANSMITTER

CDX

MECH LOAD

D/S

DIGITAL

SERVO MOTOR

β = θ − α

= β AT NULL

Figure 1.13.Angle Position Readout.

Figure 1.15.HybridFollow-up Servo System with

D/S Converter and CDX used to produce output

equal to difference between two angular inputs, one

digitaland one analog

MSB

LSB

D/S

DIGITAL

INPUT

ECDX

SOLID-STATE

ELECTRONIC

CDX

MECH LOAD

DIGITAL

INPUT

= θ

− θ

AT NULL

Figure 1.14.HybridFollow-upServo System with

ECDXto add two angular inputs in digital format.

MECHANICAL

LOAD

REFERENCE

EXCITATION

INPUT

SOLID-STATE

CONTROL

TRANSFORMER

(SSCT)

DIGITAL

COMPUTER

Figure 1.16.HybridFollow-up Servo System Used

to Interface Computer with Motor-Driven Load.

FUNDAMENTALS

we use it to drive a servo to position we can theoret-

ically increase our positioning resolution by a factor

of n.

Let us examine why. If we have a servo loop which

can position θ to a point where the null is less than

some value, say 2mV, this will represent some posi-

tional resolution, say 0.2° (this figure is determined

by the loop gain of the system). Assume now that

our servo system input is the 1xCT error voltage and

we have driven to a 2 mV null. This means θand φ

are still 0.2° apart. If we now switch the servo input

from the 1xCT output to the nxCT output, the servo

will see a 2n mV voltage and position θ so that the

nxCT output is a 2 mV null and:

To implement such a system it is apparent that we

must have a means of determining when to switch

from the coarse (1x) output to the fine (nx) output.

φ = θ within 0.2˚ ÷ n

Perhaps the easiest way to explain their operation is

to examine a basic system as shown in Figure 1.18.

Assume θ= φ;that is, assume the control transform-

ers (CT) are at the same angle as the transmitters

(CX). For convenience let θ = φ = 0 so all synchros

are at 0°. Under these conditions the output of both

CT’s will beat null, there will be no error signal from

either the one speed (1x) or n speed (nx) CT.

Now assume we change the input shaft position by

some small amount, say 2°, so θ is not equal to φ.

The output of the 1xCT is now some value, E

1xCT

and since the nxCT is geared to it by 1:n the nxCT

output will be n times E

1xCT

. Since φ still equals 0°

the output of the 1xCT can be expressed as:

where A=F.S., and the nxCT output can be

expressed as:

Note: 1x signifies that the coarse CX is connected

directly to the shaft being monitored. It has a 1:1

relationship with it.

It can be seen then that as an additional bonus the

error gradient at nullout of the n speed CT is n times

the one speed CT gradient (see Figure 1.19) and if

E = A sin (θ - φ ) = A sin (2n˚ - 0˚)

1xct

E = A sin 2n˚

1xct

E = A sin (θ - φ) = A sin (2˚ - 0˚)

1xct

E = A sin 2˚

1xct

"COARSE"

SYNCHRO

"FINE"

SYNCHRO

ANY ARBITRARY

RATIO*

SHAFT

COUPLING

TRANSDUCERS

TO INPUT

DATA

SPEED RATIO=GEAR RATIO=S

(FOR IDENTICAL NUMBER OF POLE PAIRS

ON THE TWO SYNCHROS)

*USUALLY 1:1

Figure 1.17.Two-Speed Synchro Transducer

Configuration.

TRANSDUCER

1XCX

nXCX

1XCT

nXCT

ref:

1X OUTPUT

nX OUTPUT

for changing θ

error

180

3600

let n = 5

Figure 1.18.Two-Speed Servo Loop.

Figure 1.19.Relative RMS Magnitudes of Coarse

and FineOutputs.

FUNDAMENTALS

This can be accomplished with a sensor to monitor

the coarseoutput level and control which error signal

is used. The crossover level is set within 90° or less

of the finespeed null(approximately 3° in a 1:32 sys-

tem) to prevent hang ups at false nulls on the fine

output. Since the fine CT turns n times for one turn

of the coarse CT there will be n points where its out-

put will be at null, therefore, the fine error signal is

only used when it is within 90° of the true null. Or

more correctly when the coarse null is within (90°/n).

Stick-off Voltage

If Figure 1.20 is examined it can be seen that for both

even and odd ratios a fine and coarse stable null

exists together only at 0° (or 360°). A stable null is

defined as the point where the error signals pass

through zero in the positive direction. If a servo is set

up to drive towards 0° misalignment it will drive away

from 180° because the phaseof the error signal for a

displacement will be opposite than that at 0°. An

unstable null will exist at 180° but the slightest dis-

turbance will cause it to drive to the correct null.

If the coarse/fine ratio is even, the unstable coarse

nullat 180° will be accompanied by a stable finenull.

If the coarse/fineratio is odd, the coarse and fine sig-

nals each have unstable nulls at 180°, and there is

no danger of the system accidentally remaining

aligned at 180°, but if an even ratio system happened

to be at 180° from the true stable null (as can happen

at power turn on or some forms of switched systems)

then the servo would sense the coarse null (even

though an unstable one) and switch to the fine error

signal. Since the fine error signal is at a stable null

the system would “lock in” at 180° from where it

should.

To enable even-ratio systems to function without the

possibility of nulling at 180°, a stick-off voltage is

added to the coarse control signal.

Figure 1.21 shows a simplified circuit for this pur-

pose. When a fixed AC voltage of the correct phase

is added to the coarse signal, and the stator of the

coarse control transformer (or transmitter) is suitably

repositioned, the coarse system can be made to null

at 180° of fine rotation, which is unstable fine null.

The null at 0° is unaffected.

In detail, the coarse control transformer signal/mis-

alignment curve is shifted obliquely so that it still

passes throughout the same nullat 0° but a different

null at 180°. The stick-off voltage shifts the coarse

null horizontally by 90° of fine rotation, and the

coarse synchro offset shifts the error signal by a fur-

ther 90° of fine rotation. At 0° the two 90° changes

cancel; at 180° they add up to prevent a fine stable

null at 180°.

“Stick-off” voltage can be added to any even-ratio,

two-speed system, regardless of the method of

adding the coarseand fine signals. The voltage must

have the same phase as the normal coarse signal to

avoid introducing quadrature components at null. A

two-speed S/D converter is described in detail on

pages 33 through 36. Three-speed, four-speed, and

even more highly articulated synchros are practical,

although they are rarely used. In all multispeed ser-

vos, the accuracy of the gearing must be high

enough to support the added resolution provided by

the fine synchro.

Where mechanical gearing size and/or errors cannot

be tolerated, electrical two-speed synchros can be

used. Electrical two-speed synchros are devices

with two rotor/stator winding sets. The two-speed

EVEN RATIO n = 8

ODD RATIO n = 5

360˚/0˚

180˚

1x (coarse)

nX (fine) n=EVEN

1x (coarse)

nX (fine) n=ODD

Figure 1.20.CT Voltage/Misalignment Curves for

Even and Odd Ratios.

FUNDAMENTALS

COARSE CT

STICKOFF

VOLTAGE

LEVEL

SENSOR

CROSS OVER

SWITCH

SWITCHED ERROR

VOLTAGE, Es

FINE CT

WITHOUT STICKOFF

WITH STICKOFF & OFFSET COARSE CT

FALSE STABLE NULL

STABLE NULL

UNSTABLE

NULL

STABLE

NULL

OFFSET DUE TO

COARSE CT

OFFSET

180

360/0

OFFSET DUE TO

STICKOFF VOLTAGE

Figure 1.21.Unstable False NullShifted by Adding ‘Stick-Off’Voltage to CoarseSignal and Displacing

Coarse Control Transformer Output.

FUNDAMENTALS

ratio (N:1) is achieved by having Nx as many in the

“fine” rotor/stator set than there are in the “coarse”

rotor/stator set. Since the individual rotor/stator sets

on an

electrical

two-speed synchro are brought out

separately to appropriate sets of terminal, they may

be treated (i.e., connected) in the same manner as

separate

mechanical

two-speed transducers.

The advantages of electrical two-speed synchros

(resolvers are also available in this configuration)

are:no inaccuracies due to gear train wear or back-

lash, increased reliability due to fewer moving parts,

lower driving torque required, and smaller size. With

all these things considered, the cost differential

between a mechanical two-speed arrangement (two

synchros and a gear train) and a single electrical

two-speed unit is marginal. Electrical two-speed syn-

chros are generally available in binary ratios, i.e., 8,

16, and 32 to 1. Electrically there is no difference

between electrical and mechanical units.

Digital Data Conversion Techniques for

Synchro/Resolver Systems

This handbook does not attempt to study every cir-

cuit technique ever used for synchro/resolver data

manipulation.It does not even attempt to present an

encyclopedic survey of every kind of circuit in current

use; instead, it reflects a selective concentration on

what authors deem to be the most important and

effective modern techniques. To justify our selec-

tions, we offer the following brief review of older con-

version techniques.

INPUT

Vref

GATE ON

GATE OFF

ZERO

CROSSING

DETECTORS

CLOCK

PULSE IN

DIGITAL VALUE OF θ

COUNTER

RESOLVER

(OR SYNCHRO

PLUS SCOTT "T")

Vref

INPUT

2θ

Figure 1.22.Single-RC Phase-Shift Synchro-Digital Converter.

Figure 1.23.Two-RCPhase-Shift S/D Converter (uses same zero-crossing detectors, gate, counter, and

clock-pulses as that of Figure 1.22).

FUNDAMENTALS

1.Single-RC phase-shift approaches (Figure 1.22).

2.Double-RC phase-shift approaches (Figure 1.23).

3.Real-time-function-generator

approaches

(Figure 1.24).

4.Ratio-bounded harmonic oscillator approaches

(Figure 1.25).

5.Demodulation of sine and cosine with µP and

A/Ds (Figure 1.26).

Let us briefly consider each of them in turn, recog-

nizing that the examples of each type given here are

subject to wide variation.

The single-RC phase-shift synchro-to-digital

converter shown in Figure 1.22 operates by com-

paring the zero-crossing times of the reference wave

and the phase-shifted sine-to-cosine (resolver-for-

mat) wave, V

at point A in the diagram. It can be

shown that, if ωRC=1 (where ω = 2πtimes the refer-

ence carrier frequency) the phase shift between the

voltage from point A to ground and the reference

FUNCTION

GENERATOR

PROGRAMMING

CIRCUITRY

ANALOG

COMPUTING

CIRCUITRY

FUNCTION

GENERATOR

FUNCTION

GENERATOR

A/D

CONVERTER

DIGITAL SIGNALS (θ - φ)

ANALOG VOLTAGE

PROPORTIONAL TO (θ - φ)

DIGITAL

OUTPUT

= θ

WHEN

(θ - φ) IS

AT NULL

Vy f (φ)

Vx f (φ)

SIN

DC SIN

COS

DC COS

DEMODULATOR

A/D

DIGITAL

ANGLE

OUTPUT

DIG SIN

DIG COS

SCOTT

REF

µP

BOUND

INTEGRATOR NO.1

INTEGRATOR NO.2

BOUND

UNITY GAIN

INVERTER

Figure 1.24.Real-Time Trigonometric — Function-Generator S/D Converter.

Figure 1.25.Harmonic OscillatorS/D Converter.

Figure 1.26.Demodulation, A/D, and µP approach to S/D.

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