18 lock in amplifiers
TRANSCRIPT
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Signals and noiseFrequency dependence of noise• Low frequency ~ 1 / f
– example: temperature (0.1 Hz) , pressure (1 Hz), acoustics (10 -- 100 Hz)
• High frequency ~ constant = white noise– example: shot noise, Johnson noise, spontaneous emission
noise• Total noise depends strongly on signal freq
– worst at DC, best in white noise region• Problem -- most signals at DC
log(Vnoise)
log(f )
Noise amplitude
1/f noise
0
White noise
0.1 1 10 100 1kHz
log(
Vno
ise)
log(f )
Total noise in 10 Hz bandwidth
1/f noise
0
White noise
0.1 1 10 100 1kHz
Signal at DC
log(
Vno
ise)
log(f )
1/f noise
0
White noise
0.1 1 10 100 1kHz
Signal at 1 kHz
10 Hz
10 Hz
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Lock-in amplifiers• Shift signal out to higher frequencies• Approach:• Modulate signal, but not noise, at high freq
– no universal technique -- art– example: optical chopper wheel, freq modulation
• Detect only at modulation frequency– Noise at all other frequencies averages to zero– Use demodulator and low-pass filter
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Demodulation / Mixing• Multiply input signal by sine wave• Sum and difference freq generated• Compare to signal addition -- interference• Signal frequency close to reference freq
– low freq beat– DC for equal freq sine waves– DC output level depends on relative phase
Two sine waves
Product
Sum
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Signal freq approaches ref freq• Beat frequency approaches DC as signal freq approaches ref freq
1
1.05
1.1
1.15
1.2
1.25
Signal freqvs ref freq
Reference
Mix
er o
utpu
ts
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Phase sensitive detection• Signal freq matches reference freq• Reference = sin(2ft) • Signal = sin(2ft + )
– is signal phase shift• Product = cos() - cos(2ft)
Signalphaseshift
0 0.2 0.4
0.6 0.8
Reference wave
Prod
uct w
avef
orm
s--
sign
al ti
mes
refe
renc
eDC part
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Low pass filterRemoves noise• Example -- modulate above 1/f noise
– noise slow compared to reference freq– noise converted to slowly modulated sine wave– averages out to zero over 1 cycle
• Low pass filter integrates out modulated noise – leaves signal alone
Reference
Input Output Mixer Low pass
filter Buffer
Lock-in amplifier Demodulated signal
After mixer
Vol
tage
time
After mixer & low pass
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Typical LIA low pass filters• For weak signal buried in noise• Ideal low pass filter blocks all except signal• Approximate ideal filter with cascaded low pass filters
18 db/oct
12 db/oct
6 db/oct
Ideal
loggain
frequency
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Phase control• Reference has phase control• Can vary from 0 to 360°• Arbitrary input signal phase• Tune reference phase to give maximum DC output
Reference
Phaseshift
Input Output Mixer
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Reference options• Option 1 -- Internal reference
– best performance– stable reference freq
• Option 2 -- External reference• System generates reference
– ex: chopper wheel• Lock internal ref to system ref
– use phase locked loop (PLL)– source of name “lock-in amplifier”
Reference
Signal Mixer
Lock-in amplifier System
Reference
Signal Mixer
Lock-in amplifier System
VCO
Integrate
PLL
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Analog mixer• Direct multiplication
– accurate– not enough dynamic range– weak signal buried in noise
• Switching mixer– big dynamic range– but also demodulates harmonics
Multiplying mixer
Switching mixer
Harmonic content of square wave
1
1/31/5 1/7 1/9
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Switching mixer design• Sample switching mixer• Back-to-back FETs
– example: 1 n-channel & 1 p-channel– feed signal to one FET, inverted signal to second FET
• Apply square wave to gates– upper FET conducts on positive part of square wave– lower FET conducts on negative part
Switching mixer circuit
pn
Signal voltage
source draingate
bias
n-channel FET
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Signals with harmonic content• Option 1: Use multi-switch mixer
– approximate sine wave– cancel out first few harmonic signals
• Option 2: Filter harmonic content from signal– bandpass filter at input– Q > 100
Lock-in amp with input filter
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Digital mixers
• Digitize input with DAC• Multiply in processor• Advantages:
– Accurate sine wave multiplication– No DC drift in low pass filters– Digital signal enhancement
• Problems:– Need 32 bit DAC for signals buried in noise– Cannot digitize 32 bits at 100 kHz rates
• Should be excellent for slow servos– Ex: tele-medicine, temperature controllers– Digital processing can compensate for certain system time delays ?
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Lock-in amps in servos• Lock to resonance peak
– Servos only lock to zero– Need to turn peak into zero
• Take derivative of lineshape– modulate x-voltage– F(x)-voltage amplitude like derivative
• Use lock-in amp to extract amplitude of F(x)– “DC” part of mixer output– filter with integrator, not low-pass
x
F(x)
Take derivative with lock-in
No fundamental• only 2 f signal
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Lock-in amps for derivative• Lock-in turns sine wave signal into DC voltage• At peak of resonance
– no signal at modulation freq– lock-in output crosses zero
• Discriminant– use to lock
x
F(x)
Input signal
Lock-inoutput
(derivative)
Zero crossingat resonance
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Effect of modulation on lineshape• Start with resonance lineshape• Intensity vs PZT voltage: I = I0 exp( -V2)
• Modulate voltage: V= V0 sin (2 f t)
• Modified lineshape
• Analog to numerical derivatives• Derivative is: I’ = I(V+ V) - I(V) / V
– Set V = 1• Modulation replaces V= V0 sin (2 f t)• Derivative is sine wave part
– Assumes is V0 small
V
I
t
t
V
I
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Modulation amplitude
0.05 linewidth
0.1
0.2
0.5 linewidth
1
2
Effect of modulation amplitude• For large modulation amps
– Distortion and broadening• Modulation like a noise source
– Always use minimum necessary
Expanded scan
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Modulation amplitude
0.1 linewidth
0.2
0.5 linewidth
1
2
Mixer outputs• Maximum mixer output
– modulation ~ 1 linewidth– saturates and broadens
Mixer out0.1 linewidth
0.2
0.5
1
2
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Fabry-Perot servo• Lock to peak transmission of high Q Fabry-Perot etalon• Use lock-in amp to give discriminant
– No input bandpass -- or low Q < 2• Bandpass rolloff usually 2-pole or greater
– No low pass filter -- replace with integrator• Low pass filter removes noise• Need noise to produce correction
• Design tips– reference freq must exceed servo bandwidth by factor of ~ 10– but PZT bandwidth is servo limiter– use PZT resonance for modulation
Acoustic noise
Laser
Fabry-Perot
PD LIA
Sum& HV
reference