Page 207 - Instrumentation Reference Book 3E

Doppler anemometry 191

second arm and be reflected back along its path By setting A($,) = B(q5,) = R, by adjusting the
by a silvered reflecting end coating deposited on voltage on the pzt modulator to give 4, = 2.82
the fiber end face. This back-reflected light will radians, the output of the instrument is then given
also be directed to the photodetector where it will by the reduced form of:
interfere coherently with the light that is Doppler
shifted by the vibrating surface. By applying a S(t) = R COS (2w,t + 6) (12.41)
high frequency sinusoidal voltage to the piezo-
electric modulator of lOOkHz the phase q5(t) of which represents a carrier of frequency 2wm that is
the light will be modulated due to the modulated phase modulated by the vibrating surfkce
interferometer arm imbalance as: 8 = 47rL,(t)/X,. An electronic phase tracker is
4m used to follow the phase movement induced by
4(t) = -Lo sin (wmt) (12.35) the vibrating surface. To achieve this objective,
A0 the phase modulated carrier (PMC) square wave
where Lo is the peak fiber length extension due to output is monitored against the reference wave
the pzt element. Similarly, light reflected back derived from the pzt master oscillator. The phase
from the vibrating target will also be phase modu- of the PMC relative to the reference is measured
lated, but at a somewhat lower frequency due to by use of the master clock running at 25MHz
the displacement effect of the vibrating surface with a 1/126 measurement increment using the
modulating the arm imbalance L, of the inter- rising edges of the two waveforms as trigger
ferometer, giving rise to a second time-dependent points. By storing two successive phase measure-
phase component 8(t) of the form: ments in buffer memories, the direction of the
surface displacement can be determined by com-
4?iL,(t)
@(f) = - (12.36) parison of their two most significant bits. The
A0 resolution of the system corresponds to about 3"
If the two arms of the interferometer are equal, in phase measurement or 2.47nm (Le. A0/126).
then Ls(r) will represent the time-dependent dis- The noise floor of the processing system was
placement of the vibrating surface. The output determined to be equivalent to approximately
photodiode current Ipd will then have the form, 1 nm in surface displacement. Since the gated out-
from the transfer function of the Michelson inter- put of the detector produces two such PMC sig-
ferometer, of: nals moving in opposite directions, it is possible
to process both in order to halve the resolution of
zpd = A { 1 + K cos [e@) + do sin wmt]} the system. The performance of the system is
(12.37) limited by the maximum slew rate that can be
satisfactorily processed due to the bandwidth limi-
Using standard trigonometric identities this can tation of the band pass fdter of about 30 kHz that
be expressed as:
must accept only the second harmonic 2w,
zpd =A{1 +K[cos€Jcos(q5sinwmt) (about 200 kHz) of the pzt modulation frequency,
- sinOsin(4, sinw,t)]j (12.38) and reject the adjacent fundamental wm and the
third harmonic 3wm. Using a bandwidth of
and can be expanded in terms of a Bessel function 30 IrHz, this corresponded to a displacement
series: amplitude of 1 pm at 5 kHz and down to 0.5 mm
displacement amplitude at 10 kHz vibration fre-
Ipd(b) OC
quency. Results indicate that an error in surface
M amplitude of less than 1 percent was present over
COS&- a 2 kHz bandwidth for a scaled 1 g surface accel-
eration; this rises to <3 percent over a 5kHz
bandwidth when measured against a reference
grade piezoelectric accelerometer under standard
conditions.
(12.39)
.
where Jo(d0), J1(q5~), . . , Jzn(q5,j are Bessel fun0
tions of the first order of argument 4,. 12.4.4.3 Frequency modulated laser diode-
The output signal IM is first gated with a 5050 vibration monitorimg
mark space ratio square wave and then band-pass In this method (Laming et ai. 1985) a frequency
filtered at a frequency of twice the modulation modulated single-mode laser diode is used as the
frequency, that is at 2wm, to give: source element. No separate frequency shifter
S(t) = [A(&,) cos e cos (24dmf) element is required and the set-up is suitable as
- B(4,) sin 8 COS (hmt)] (12.40) a hand-held vibration monitor that is insensitive
to external vibrations. It is claimed to be robust,