Lock-in amplifiershttp://www.lockin.de/

Signals and noiseFrequency dependence of noiseLow frequency ~ 1 / fexample: temperature (0.1 Hz) , pressure (1 Hz), acoustics (10 -- 100 Hz)High frequency ~ constant = white noiseexample: shot noise, Johnson noise, spontaneous emission noiseTotal noise depends strongly on signal freqworst at DC, best in white noise regionProblem -- most signals at DC

log(Vnoise)log( f )Total noise in 10 Hz bandwidth1/f noise0White noise0.1 1 10 100 1kHzSignal at DClog(Vnoise)log( f )1/f noise0White noise0.1 1 10 100 1kHzSignal at 1 kHz10 Hz10 Hz

Lock-in amplifiersShift signal out to higher frequenciesApproach:Modulate signal, but not noise, at high freqno universal technique -- artexample: optical chopper wheel, freq modulationDetect only at modulation frequencyNoise at all other frequencies averages to zeroUse demodulator and low-pass filter

Demodulation / MixingMultiply input signal by sine waveSum and difference freq generatedCompare to signal addition -- interferenceSignal frequency close to reference freqlow freq beatDC for equal freq sine wavesDC output level depends on relative phase

Signal freq approaches ref freqBeat frequency approaches DC as signal freq approaches ref freq

1

1.05

1.1

1.15

1.2

1.25Signal freqvs ref freqReferenceMixer outputs

Phase sensitive detectionSignal freq matches reference freqReference = sin(2pft) Signal = sin(2pft + f) f is signal phase shiftProduct = cos(f) - cos(2pft) Signalphaseshift f

0 0.2 p 0.4 p

0.6 p 0.8 p pReference waveProduct waveforms-- signal times referenceDC part

Low pass filterRemoves noiseExample -- modulate above 1/f noisenoise slow compared to reference freqnoise converted to slowly modulated sine waveaverages out to zero over 1 cycleLow pass filter integrates out modulated noise leaves signal alone

Typical LIA low pass filtersFor weak signal buried in noiseIdeal low pass filter blocks all except signalApproximate ideal filter with cascaded low pass filters

Phase controlReference has phase controlCan vary from 0 to 360Arbitrary input signal phaseTune reference phase to give maximum DC output

Reference optionsOption 1 -- Internal referencebest performancestable reference freqOption 2 -- External referenceSystem generates referenceex: chopper wheelLock internal ref to system refuse phase locked loop (PLL)source of name lock-in amplifierReference Signal Mixer Lock-in amplifier System Reference Signal Mixer Lock-in amplifier System VCOIntegrate PLL

Analog mixerDirect multiplicationaccuratenot enough dynamic rangeweak signal buried in noiseSwitching mixerbig dynamic rangebut also demodulates harmonicsMultiplying mixerSwitching mixerHarmonic content of square wave11/31/51/71/9

Switching mixer designSample switching mixerBack-to-back FETsexample: 1 n-channel & 1 p-channelfeed signal to one FET, inverted signal to second FETApply square wave to gatesupper FET conducts on positive part of square wavelower FET conducts on negative partSwitching mixer circuitn-channel FET

Signals with harmonic contentOption 1: Use multi-switch mixerapproximate sine wavecancel out first few harmonic signalsOption 2: Filter harmonic content from signalbandpass filter at inputQ > 100Lock-in amp with input filter

Digital mixersDigitize input with DACMultiply in processorAdvantages:Accurate sine wave multiplicationNo DC drift in low pass filtersDigital signal enhancementProblems:Need 32 bit DAC for signals buried in noiseCannot digitize 32 bits at 100 kHz ratesDigital mixersGood for slow signalsHigh signal to noise or low accuracy

Lock-in amps in servosLock to resonance peakServos only lock to zeroNeed to turn peak into zeroTake derivative of lineshapemodulate x-voltageF(x)-voltage amplitude like derivativeUse lock-in amp to extract amplitude of F(x)DC part of mixer outputfilter with integrator, not low-pass

Take derivative with lock-inNo fundamental only 2 f signal

Lock-in amps for derivativeLock-in turns sine wave signal into DC voltageAt peak of resonanceno signal at modulation freqlock-in output crosses zeroDiscriminantuse to lock

xF(x)Input signalLock-inoutput (derivative) Zero crossingat resonance

Fabry-Perot servoLock to peak transmission of high Q Fabry-Perot etalonUse lock-in amp to give discriminantNo input bandpass -- or low Q < 2Bandpass rolloff usually 2-pole or greaterNo low pass filter -- replace with integratorLow pass filter removes noiseNeed noise to produce correctionDesign tipsreference freq must exceed servo bandwidth by factor of ~ 10but PZT bandwidth is servo limiteruse PZT resonance for modulation

Digital mixers in servosMay be okay for low precision, medium speed servoNot for fast servos -- ex: laser frequency stabilizationNot for high accuracy -- ex: laser gyroShould be excellent for slow servosEx: tele-medicine, temperature controllersDigital processing can compensate for system time delay

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