Frequency Response Characteristics of a Rate

Gyroscope Using a Microsyn

JERZY GRYGLASZEWSKI

Summary-The method of obtaining a frequency response of arate gyroscope, as described in this article, is applicable to anygyroscope which is equipped with a variable reluctance transformer(microsyn). The damping ratio of the gimbal system can be obtainedunder any environmental condition without mechanical rotation ofthe gyroscope and without breaking the electrical connections to it.

The microsyn is electrically excited to exert upon the gimbal asinusoidally varying-in-time torque of constant amplitude and ofcontrollable frequency, and at the same time is used to indicate theresulting displacement of the gimbal.

INTRODUCTIONI N A RATE gyroscope, the damping ratio and the

natural frequency of the gimbal system can be ob-tained by observing gimbal movement in response

to an alternating torque of constant amplitude appliedto it. Presently used methods for the evaluation ofdamping utilize the gyroscopic torque produced by rota-tion of the device about its sensitive axis. The short-comings of these methods are self-evident particularlywhen testing a gyroscope under varying environmentalconditions.

In this method the microsyn is used to exert a torqueupon the gimbal system and at the same time to indi-cate the displacement amplitude of the gimbal. Whentwo sinusoidal voltages of different frequencies are im-pressed across the primary and secondary windings ofthe microsyn, the microsyn produces a useful torquewhose frequency is equal to the difference of excitingfrequencies.The amplitude of this torque is constant for con-

stant excitation currents. If, in addition to the excitingvoltages, a carrier signal is injected into one of the wind-ings, it will appear across the terminals of the otherwinding in a modulated form. The amplitude of themodulated carrier wave and the frequency of modula-tion are proportional to gimbal displacement and itsfrequency of oscillation, respectively. Thus in order toestablish a frequency response characteristic of the de-vice, we need to vary the frequency of excitation to oneof the windings and read the amplitude and frequencyof the modulated carrier wave appearing across theother winding.

ALTERNATING TORQUE PRODIJCED BY MICROSYN

The instantaneous torque produced by a microsyn isproportional to the product of the instantaneous values

Manuscript received April 9, 1962.The author is with Teledyne Systems Corporation, Los Angeles,

Calif.

of its primary and secondary magnetizing currents andis given by

T = Cipmism gm cm (1)

where

C = 1.597 X 1t-o gm cm/a'.g

(2)

If the voltages impressed across the windings have dif-ferent frequencies, then the magnetizing currents havethe form

ipm = 2Ipm sin (pt -p)

ism = V\2 Ism sin (wct - O8 + a). (4)

Substituting for i, and i8m into (1) we obtain

T = 2CIpmIsm sin (wot - 4p) sin (w1t- 4) + a)

= 2CIpmIsm{ cos [(Wy - w8)t + 48- 4)p - a]

-Cos [(c, + cW,,)t - Pp -4 + a]} gm cm. (5)

Eq. (5) shows that the torque produced consists oftwo components of equal amplitude, one at a frequency(fp+fs) cps and the other at (f,-fe) cps. If eitherfp or

f, is large compared to the natural undamped frequencyof oscillation of the gimbal (microsyn rotor is an integralpart of the gimbal), then the response of the gimbal totorque at (fp+fs) cps will be negligibly small and canbe disregarded. Assuming, then, that this is the case,the torque able to produce the displacement of gimbal is

T = 2CIpm[sm cos [(cop- W')i - p, + 0)8 - a] gm cm. (6)

Neglecting phase angles, the expression for torquemay be written

T = Tma,x cos wt (7)

where

Tmax= 2CIpmIsm gm cm

w = p- ws radians/sec.

(8)(9)

Displacement of Gimbal as a Function of Applied TorqueA general equation governing gimbal response to

torque Tmax cos wt with no gyroscopic or frictional

100

Cryglaszewsk: Frequency Response of Rate Gyroscope

torques present is

d'6 dOJ-+ B-+ KO = TmaxCOCs t. (10)

di' di

Steady-state solution to (10) has the form

0 = A, sin cot + A2 coswt. (11)

Upon evaluation of constants A1 and A2, (11) becomes

TraT =F) sin(t +4) radians, (12)K

where

1

V/ Cd-\-4 (d 2

1 - -Cn 2@} (t - 222)

= frequency function

V = sin-' F(w) [1 - radians

/K)n = / radians/sec

B

2VJK

The EMF Transformation Ratio of Microsyn in Terms ofFrequency Function F(w)The EMF transformation ratio of a microsyn is de-

fined as the ratio of voltage induced in the primary tothat induced in the secondary by the air-gap flux and isgiven by

N'-= 2C-WrC8

or

CpDWrC8

Therefore, if, in addition to the exciting voltages thatproduce torque, a carrier signal of frequency f, cps isinjected into one of the windings, it will appear acrossthe terminals of the other winding as a wave that is 100per cent modulated at a frequency fr-f, cps. The ampli-tude of the modulatcd wave is proportional to the fre-quency function F(co), or effectively to the amplituderesponse of the gimbal system. The above are generalproportionality statements, but in order to realize thesein practice the following conditions have to be fulfilled:

1) The amplitude of torque exerted upon the gimbalsystem must be constant and independent of fre-quency.

2) The amplitude of the modulated wave must beara fixed relationship to the displacement amplitudeof the gimbal system.

An analysis of an equivalent circuit of a microsynwith both windings excited simultaneously indicates away in which, for all practical purposes, the conditions1) and 2) can be fulfilled.

ANALYSIS OF AN EQUIVALENTCIRCUIT OF A MICROSYN

In order to simplify the mathematical treatment of anequivalent circuit, the following assumptions are made:

1) Leakage reactances of the windings are neglectedsince they are small as compared with the resist-ances of the respective windings.

2) The current components supplying iron losses are

also neglected bccause of very small flux densitiesinvolved.

3) Magnetizing reactances of the microsyn are inde-(17) pendent of currents fed into its windings (i.e. mag-

netic circuit of the microsyn is not saturated). Asa result of this assumption the equivalent circuitconsidered is a linear one and hence allows the use

(18) of superposition principle.

(15)

(16)

and

2C.1V =-x. (19)

Substituting for 6 from (12) we obtain the EMF trans-formation ratio as a function of the frequency functionF(w). Thus,

CpD TmaxN' = W F(w) sin (cot + Vb). (20)

Wr,f K

Eq. (20) indicates that the EMF transformation ratiovaries sinusoidally in time with an angular frequencyo=Cp--cv8 radians/sec. Also, its amplitude is propor-tional to the frequency function F(w), provided theamplitude of torque exerted by the microsyn is constant.

Fig. 1 shows an equivalent circuit of a microsyn witha constant voltage source applied across the primarywinding and the secondary terminated by the outputimpedance of the secondary voltage source.

Fig. 2 shows a similar circuit but with the secondaryexcited with a constant voltage and the primary termi-nated by the output impedance of the primary voltagesource.

Each circuit is solved independently and the resultantcurrents in the windings may be obtained by addingthe instantaneous currents calculated from circuits ofFigs. 1 and 2.Let the voltage applied across the primary in circuit

in Fig. 1 be given by

VP - V/2 Vp sin w,t. (21)

F(c) =

101

102 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT December

LrpV I

Fig. 1-Equivalent circuit of a microsyn excitedfrom the primary side.

The circuit equations pertaining to Fig. 1 are

Ldp. + (Rp + Rpg)ip = vpdt

dimpNLp d - (R + Rsg)isp = 0

dt

1pm - Nisp = p.

where

RsaLF3

(22)

(23)

(24)

After algebraic transformations (21)-(24) yield

dipm Rp + Rpg . = V2V2Vp .- + tpn = sin (ptdt K Kp

-[1-+N2(Rp + Rpg)]Rs + Rsg

Solution to (25) has the form

ipm = A sin co I + B cos wpt.

(25)

(26)

(27)

Upon evaluation of constants A and B the primarymagnetizing current is

/2 Vvipm= x-2V sin (cet - 4P). (28)

AlzpIAlso,

=\/VpVip = 1 sin (wpt -p)

V/2 VpXpN2+ (R 8)IzICos (cop'l - )

(R, + Rsg)IZ\/2 VpXpN

P - ( + R cos (()Pt -p)R.s + Rpg) Zp

Fig. 2 Equivalent circuit of a microsyn excitedfrom the secondary side.

By similarity with circuit in Fig. 1 we obtain

-sm sin (ot-4s + a)

-\/2 V,i8s -- sin (c - , + a)

V2 VgXS(N')2

± (Rp + Rpg)I ZSjs

cos(t - Os

± a)

V/2 VXs8N'(RI R5) Cos (w0st - + a),(Rp + Rpg) |Zs

where

(N')2(R, + RSg)]Ks = LsF1 +IRp + Rpg

Zs I = 8\(R, + Rs)' + W, KsXs = wsLs

WsK,58-= sin-'

(34)

(35)

(36)

(37)

(38)

(39)

(40)

Equating the expressions for magnetizing currents asgiven by (3), (4) and (28), (34), respectively, we obtain

VPIpm=

IzpiVe

im= I~ .

sl

(41)

(42)

(29) Combining (41), (42), (8), and (20), the EMF trans-formation ratio becomes

(30)

where

(31)(32)

Zp -V\I(Rp + Rpg)2 + Wp2K 2

-1 cpKp

N'= 2CDCpVBVpWrC8=

2 pV,Vp K F(wo) sin (wt +±0. (43)

If a carrier signal voltage V, is injected in series with thesecondary excitation source, then by analogy with (36)the current circulated through the primary winding dueto this voltage is

i-/2 VcX8cN'

(Rp + Rpg) ZscI Cos ot,(44)

Let the voltage applied across the secondary in circuitin Fig. 2 be

Zsc = V\(Rs + Rsg)2 + CO02K82. (46)

where

8sc - cs (45)VI, -, \/2 V, sin (w,t + ce). (33)

Gryglaszewski: Frequency Response of Rate Gyroscope

1AIPL/P/EE CAeei

VAt/A2LCt

4eL2a/D SappevrOt E4a/PAiWr

-Block diagram of a circuit used in the measurement offrequency response of a rate gyroscope.

Substituting for N' into (44) from (20)

2V/2 VcVsVPCCpDX8c 1

(Rp + Rpg)WrCsK zsc Zp |Z8*F(,,) sin (wt + i) cos wct, (47)

and the carrier voltage appearing across the microsynprimary is

2A/2 VCVSVPCCPDXSCRpg 1

(Rp + Rpg)WrC,K Z ZP|C ZSF(,) sin (wt + t) Cos wct. (48)

It is noted from the above equation that the primarycomponent of voltage due to carrier signal, consists ofa wave that is 100 per cent modulated at a frequencyat which the gimbal oscillates. Furthermore, providedthe term

IZC I I ZP I IZ81

gyroscope is shown in Fig. 3. This technique does notrequire breaking of the circuit in which the microsynis connected. Also, provided there is no rate input aboutthe sensitive axis of the device, the measurements arenot affected by the spin motor speed. This is because theaxis of the microsyn rotor is orthogonal to the sensitiveaxis of the gyroscope.The amplitude response for the required frequency

spectrum is obtained by variation of frequency of supplyto the secondary of the microsyn. All remaining param-eters must be kept constant. For the damping ratios be-low 0.7 a definite peaking is observed in the curve relat-ing amplitude and the frequency of gimbal oscillation.Position of this peak in the amplitude ratio and fre-quency coordinate system uniquely defines the dampingratio and the undamped natural frequency of the gimbalsystem. Eqs. (49) and (50) state the damping ratio andthe undamped natural frequency in terms of the fre-quencyfp at which the amplitude ratio reaches its peakand has a value A.

is constant, the amplitude of the modulated voltagewave is directly proportional to the frequency functionF(@,w.As shown in Appendix I, the term

1

Z IZPI 1IZ.,is subject to a very small variation, provided the am-plitude of oscillation of the gimbal structure is small. Iflarge amplitude of oscillation and high degree of accu-racy are required, it is necessary to increase the primaryand secondary source impedances.

METHOD OF MEASUREMENT ANDINTERPRETATION OF RESULTS

A typical circuit arrangement used in the measure-ment of frequency response characteristics of a rate

t= - o0.5 -- (Ap+ l)(Ap- 1)v Ap

fn= p-

,V1 -2~2

(49)

(50)

Eqs. (49) and (50) are derived from the frequencyfunction, (13). Knowledge of the natural frequency ofoscillation is required in the evaluation of dampingratios which are above 0.7. (Value forfn is usually sup-plied by the manufacturer of the gyroscope and is notsubject to variation with environmental conditions.)Calculation of the damping ratios for t>0.7 is carriedout in the conventional manner, i.e. by plotting theamplitude ratio vs frequency ratio and comparing theresulting curve with standard curves. A family of suchstandard curves is shown in Fig. 4.

1031965

104 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT December

- - - - - - - - - - AIMPZ/ITUaD PESPONSE

| SH+ ' f h N \ + + oc A RA7!F' <FYe0 7P0 ,4A

{E,/0 a.04

o 0...2 04 6 0.8 /1.0 1.2 /1./6 a.8 2.0 2.2 z.4 2s 2.7

Fig. 4-Amplitude response of a rate gyro to a sinusoidal input.

APPENDIX I

DEPENDENCE OF THE TERM Z I ZVI I Z UPON 0

Combining (46), (37)1/Z8c becomes

and (18), the admittance term

1

(R + Rg)2 + cL[1 + ( CD + Rpg 021

We obtain in a similar manner

1

ZPI

1

1

/(Rp + Rpg) 2 +pp I +( C,D 2Rp + RpgL kWrCp/ R8 + Rag ]

1

~i(R8 + Rsg)2 + woL,[i+(Q)+ +R0021t/ r Rp ~~~~~+Rpg

(51)

(52)

(53)

Gryglaszewski: Frequency Response of Rate Gyroscope 105

The product of (51), (52) and (53) constitutes theonly term in (48) that is a function of 6. Inspection ofthis product makes clear that the less the dependence of

1

Z.c" IZPI IZSIupon the gimbal deflection angle 0, the smaller the valueof 0. However, even for large values of 6 this dependencecan be made insignificant when large resistances Rpg andRsg are used. This is best illustrated by an example.

In a microsyn with D=0.686 inch, W=0.062 inch,C8=876, Cp=292, Rp= 126 ohms, R8= 1100 ohms, cpLp= 78 ohms, W8L, = 707 ohms, r = 1.44 and f, = 10,000 cps,the percentage variation in the amplitude of (48) isshown in Fig. 5 for two cases where Rpg =R8s=0 andRpg = 2000 ohms, Rsg = 4000 ohms. Note that in the casewhere Rpg =Rsg=0 a large error is introduced at highervalues of 0, while in the case where R1,g= 2000 ohms andR,g = 4000 ohms the error is considerably reduced.

VI4

0

t44

APPENDIX I I

LIST OF SYNIBOLSB =gimbal damping constant (gm cm/radians

/sec)Cp= number of primary conductors per slotCs=number of secondary conductors per slotD = microsyn air-gap diameter (cm)g =length of air-gap (cm)

tpm = primary magnetizing current instantane-ous value (amperes)

-,=secondary magnetizing current-instanta-neous value (amperes)

'pm= primary magnetizing current rms value(amperes)

'Sm=secondary magnetizing current rms value(amperes)

J= moment of inertia of the gimbal system(gm cm2)

K = elastic restraint constant (gm cm/radian)/=net axial length of iron in microsyn (cm)

Lp= magnetizing inductance of primary winding(henry)

L= magnetizing inductance of secondary wind-ing (henry)

N= ratio of induced voltages-secondary toprimary

N'=ratio of induced voltages-primary to sec-

ondaryr = flux leakage coefficient

Rp =resistance of primary (ohms)Rpg= primary source resistance (ohms)RS = resistance of secondary (ohms)Rsg = secondary source resistance (ohms)

s = number of slots on wound members of themicrosyn

t= time (sec)vp = primary voltage-instantaneous value

(volts)v,= secondary voltage instantaneous value

(volts)V, =applied carrier (volts)Vp = primary voltage rms value (volts)V.= secondary voltage rms value (volts)W=width of tooth on wound member of micro-

syn (cm)x = (D/2)0 (cm)= damping ratio= displacement of microsyn rotor from its nullposition (radians)

4, a, = phase angles (radians)W,n = undamped natural angular frequency of

oscillation (radians/sec)wp=angular frequency of primary supply (radi-

ans/sec)w,= angular frequency of secondary supply

(radians/sec).

/00R- 4 I I&000A, Aps=2ayXC

_~~~~~~~s __ 0-80

70

60

0 /.0 2.0 9.0

A4ARMPOra6 a'-'/Ma4L af4

Fig. 5-The effect of R., and Rp, upon the accuracy.

1965