Basics of Gyroscope

8/10/2019 Basics of Gyroscope

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The effective path lengths are

Where the transit times and are

For light waves travelling in the clockwise and counter clockwise directions, respectively. The

free-space speed of light is denoted by C, and R is the radius of the ring. The transit time

difference t between the counter propagation waves, in the case of n loops that enclose an area,

(or Sagnac area — SA), can be expressed as

t = -

=

The assumption is made that >>

. The resultant optical path length Difference , L, is

C*∆t

L =

In the case of the analog or Interferometric fiber optic gyroscope (IFOG), the Sagnac phase shift

caused by a rotation can be expressed in terms of L as

L

Where is the wavelength of the free-space optical energy. Multiple wraps of fiber can be

wound to significantly increase L, thus improving the sensitivity. However, the optical

attenuation tends to limit the length of fiber to several kilometres.

In the case of the digital or resonant fiber optic gyroscope (RFOG), the energy in the

counter propagating beams is coupled into the fiber loop at two different frequencies in the

presence of a rotation. The relative frequency difference ƒ between the counter propagating

waves can be written in terms L of as



where ƒ = c/ and L = 2 RN is the total distance traversed. The fundamental RFOG equation that

relates ƒ to Ω is given by

Basic Configuration of FOG:

A basic FOG configuration is shown in figure. Light from a broadband source, such as a

super luminescent diode (SLD), is projected into a 3-dB fiber optic coupler that splits the light

into two waves. After traversing the coupler, the two light waves propagate equally in opposite

directions around the fiber optic coil. The light waves interfere upon return to coupler and project

a fringe pattern onto a photodetector.

Figure 2:Basic FOG configuration

In accordance with any two-wave interferometer, the intensity on the photo detector,

which represents a mixture of the two light waves, varies as cosine of Sagnac phase with its

maximum value at zero as shown in Figure This intensity is expressed as

where Io is the mean value of the intensity. The detected intensity is used to calculate the rotation

rate. In the case of no rotation, = 0, the light waves will combine in phase, which results in

maximum intensity.



Figure 3:Optical intensity versus phase difference between interfering waves

In the presence of a rotation, the light waves travel different path lengths and mix slightlyout of phase. The intensity is reduced due to the degree of destructive interference. The cosine

function, which is symmetrical about zero, has its minimum slope there. For small rotation rates,

it is impossible to determine the direction of rotation (CW or CCW) from the symmetrical aspect

of above figure, where the slope is near zero. Furthermore, the gyroscope operating in this mode

has minimum sensitivity near zero. Incorporating a dithering phase modulator with drive

modulation capability a symmetrically in the loop (near one end of the coil) provides a means to

introduce a nonreciprocal phase shift to bias the gyroscope to its maximum sensitivity point. This

corrective measure solves both the low-sensitivity problem and the issue of ambiguous direction

of rotation at low rotation rates.

Figure 4:Fiber optic gyroscope



When a phase modulator is used, the expression for the intensity on the photo detector is

where S is the Sagnac phase shift .

Where =wavelength of the source.

c=velocity of light.

L=length of the fiber coil.

D=diameter of the fiber coil.

Ω=rotation rate.

Why phase modulation?

A major problem of the basic configurations presented in Figure1 is the output

nonlinearity for small S = 0, which hinders high sensitivity measurements of small rotation

angles without sign ambiguity. This limitation is overcome by transforming the baseband cosine-

dependence into a sinusoidal function, for example, by translating the output signal from

baseband to a carrier at angular frequency w. Although different solutions have been proposed

and demonstrated, the optical phase modulation technique is nowadays commonly used. A phase

modulator is inserted in the fiber coil, close to a coupler output, so that a different phase delay is

cumulated by the counterpropagating waves.

The open loop configuration with phase modulation:

The all-fiberversion phase modulator is constructed by winding and cementing a few fiber turns

on a short, hollow piezoceramic tube (PZT). By applying to the PZT a modulating voltage, a

radial elastic stress and a consequent optical pathlength variation due to the elasto-optic effect

are generated.



Figure 5: All-fiber gyroscope with phase modulator.

As a result, the CCW and the CW propagating waves experience a phase delay Φ(t) and Φ(t +

τ), espectively, where τ=L/v is the radiation transit time in the fiber of overall length L.

The relative phase difference on the detector is then

ΦCCW - Φ CW = S + Φ(t) - Φ(t + τ)

which can also be written as

ΦCCW - Φ CW = S + Φ(t - τ/2) - Φ(t + τ/2)

Applying a phase modulation at angular frequency ωm

Φ(t) = Φmo cos ωmt

yields

ΦCCW - Φ CW = S + 2Φmosinωmτ/2 sinωmt = s + Φmsinωmt

where the amplitude Φm=2Φmo sinωmτ/2 can be maximized by selecting a PZT modulation

frequency fm=ωm /2=1/(2τ).

The photodetected signal I1

I1=I01[1+cos(ΦCCW - ΦCW)]

is thus given by (using the Bessel's functions J)



The photodetected signal contains, in addition to a DC components, all the harmonics of

the modulating signal. The amplitude of the even harmonic components depends on cosS, as in

the basic scheme, while the odd components carry the desired sinS dependence.

Closed loop schemes with analog or digital phase ramp:

In the open loop configuration, the lock-in output signal V is given by V=V0sinS and

V0 is the fringe amplitude. A first problem of this signal is the intrinsic nonlinearity and limited

dynamic range of the sinusoidal function, which may represent a restriction in some applications.

A second issue is related to the insufficient accuracy and stability of either the fringe amplitude

and the scale factor which multiplies the rotation rate. The presence of only the analog output is

also considered a third drawback of this configuration. A closed-loop scheme has been proposed

with different implementations for solving most of the above mentioned problems. The basic

idea consists in using a feedback effect which cancel the Sagnac phase shift by adding a

controlled phase delay, thus directly proportional to the rotation rate to be detected. This solution

however was not the most appropriate in terms of maintaining reciprocity. Alternatively, the

frequency variation is simulated by a phase ramp modulation, which has to be superimposed and

synchronized to the biasing phase modulation. The analog solution, based on an analog phase

ramp (also indicated as serrodyne modulation) in addition to the sinusoidal biasing modulation,

does not represent a very efficient solution. A great improvement is obtained with the all-digital

approach based on a square wave biasing modulation and on a digital phase ramp for closed-loop

processing.

The functional block diagram of this configuration is illustrated in Figure .

Figure 6:Block diagram of closed-loop FOG



Essentially, a digital feedback loop is added to the open-loop structure previously

reported in Figure. The lock-in amplifier output is sampled and quantized yielding the error

signal, which is maintained closed to zero by the digital feedback. The sampling frequency

corresponds to the inverse of the radiation transit time τ, for the required synchronisation of the

ramp and the biasing signal. Starting from the error signal, the controller drives the phase

modulator so that it generates phase steps of amplitude equal to the Sagnac phase shift and

duration τ. The digital to analog converter automatically creates the ramp reset, by means of its

overflow. The reset step corresponds to a phase variation of 2 radian, in order to get always the

correct Sagnac phase shift. In this scheme the rotation rate is directly obtained, in a digital

format, from the error signal. Another advantage of this configuration, with respect to the analog

solution, is the phase stability during the signal recovering.

Figure 7:Functional block diagram of the closed loop configuration with digital ramp .

Applications of Fiber Optic Gyroscopes:

The fiber optic gyroscope has reached a level of practical use in navigation, guidance,

control, and stabilization of aircraft, missiles, automobiles, and spacecraft as shown in Figure 7.

The FOG performance and design requirements (such as resolution, stable scale factor,

maximum rate, frequency response, size, interface electronics, environment, etc.) have been

scaled to fulfil a broad range of applications such as route surveying and mapping, well logging,



self-guided service robots and factory floor robots, autonomous guided ground and air vehicles,

tactical missiles, guided munitions, cannon-launched vehicles, smart bombs, and seeker, missile

airframe and satellite antenna stabilization. The open-loop FOG is best suited for low-cost

applications such as gyro compassing, attitude stabilization, and pitch and roll indicators, which

require low- to moderate-performance accuracy. A number of corporations have developed low-

performance FOGs for use in automotive applications.

Program for detector current:

clc

clear all

close all

l=500;

d=30e-3;omega=(-165)*(180/pi);

phim=pi/2;

lambda=1550*(10^(-9));

c=3*(10^8);

fs=21e6;

fm=200*(10^(3));

const=((2*pi*l*d)/(lambda*c));

n=0:1/fs:0.0001;

phis=(const*omega);

phit=phim*sin(2*pi*fm*(n));

Id=(1+ cos(phis+phit));

figure(1)subplot(2,1,1);

plot(n,phit);

grid on

subplot(2,1,2);

plot(n,Id);

title('Id');

xlabel('n');

ylabel('Id');

grid on

y1=fft(phit);

m1=abs(y1);z1=length(y1)

f1=((0:z1-1)*fs)/z1;

figure(2)

subplot(2,1,1);

plot(f1,m1);

title('fft(i/p)');

xlabel('frequency');

ylabel('magnitude');



%magnitude plot

y2=fft(Id);

m2=abs(y2);

z2=length(y2)

f2=(0:z2-1)*(fs)/z2;

subplot(2,1,2);

plot(f2,m2);

title('fft(Id)');

xlabel('frequency');

ylabel('magnitude');

% phase plot

p1=unwrap(angle(y1));

figure(3);

subplot(2,1,1);

plot(f1,p1*180/pi);

p2=unwrap(angle(y2));

subplot(2,1,2);

plot(f2,p2*180/pi);title('phase');

Input sine wave and output detector current:



Fourier transform of input and output(magnitude response):



Phase response :

With rotation :

clcclear all

close all

l=500;

r=40e-3;

rot=-400:400;

lambda=1550e-9;

c=3*(10^8);

a=((4*pi*l*r)/(lambda*c));

rot1=deg2rad(rot);

b=(a*rot1);

plot(rot,b);

grid on

figure(2)

plot(rot,sin(b));

grid on



Rotation vs sagnac phase shift:



Sine wave representation:

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Basics of Gyroscope