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Lock-in amplifiers

VIEWS: 16 PAGES: 20

									Lock-in amplifiers

 http://www.lockin.de/
                                 Signals and noise
   Frequency dependence of noise                                    Total noise in 10 Hz bandwidth
   • Low frequency ~ 1 / f
         – example: temperature (0.1 Hz) , pressure                            Signal at DC
           (1 Hz), acoustics (10 -- 100 Hz)
                                                                              1/f noise
   • High frequency ~ constant = white noise




                                                      log(Vnoise)
         – example: shot noise, Johnson noise,
           spontaneous emission noise                                 10 Hz                   White noise
   • Total noise depends strongly on signal freq
         – worst at DC, best in white noise region
                                                       0
   • Problem -- most signals at DC                                                                    log( f )
                                                               0.1 1          10 100 1kHz
                     Noise amplitude
                                                                               Signal at 1 kHz
                     1/f noise
                                                                              1/f noise


                                                      log(Vnoise)
log(Vnoise)


                                   White noise                                                White noise

                                                                                 10 Hz
                                                       0
        0
                                                               0.1 1          10 100 1kHz             log( f )
            0.1 1   10 100 1kHz          log( f )
                  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
                 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

Product



Two sine waves




    Sum
                   Signal freq approaches ref freq
                • Beat frequency approaches DC as signal freq approaches ref freq

                                                             Reference
                                                                     Signal freq
                                                                     vs ref freq
                                                                     1

                                                                     1.05
Mixer outputs




                                                                     1.1

                                                                     1.15

                                                                     1.2

                                                                     1.25
          Phase sensitive detection
• Signal freq matches reference freq
• Reference = sin(2pft)
• Signal = sin(2pft + f)
    – f is signal phase shift
• Product = cos(f) - cos(2pft)

      DC part
                                              Signal
                                              phase
          Reference wave                      shift f

                                               0
                  -- signal times reference




                                              0.2 p
                  Product waveforms




                                              0.4 p

                                              0.6 p
                                              0.8 p
                                               p
                           Low pass filter
    Removes 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
                                                                        Demodulated signal
                   Lock-in amplifier
                                                                          After mixer
              Mixer                         Low pass
Input                                        filter  Output
                                Buffer

                                                                        After mixer & low pass




                                                              Voltage
                      Reference                                          time
        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

                               Ideal




                                       6 db/oct




                                 12 db/oct



                log
                gain            18 db/oct
                         frequency
                         Phase control
•   Reference has phase control
•   Can vary from 0 to 360°
•   Arbitrary input signal phase
•   Tune reference phase to give maximum DC output

                            Mixer
               Input                     Output




             Reference
                             Phase
                             shift f
                        Reference options
                                           System                  Lock-in amplifier
• Option 1 -- Internal reference                                Mixer
    – best performance                                 Signal
    – stable reference freq
• Option 2 -- External reference
• System generates reference                                            Reference
    – ex: chopper wheel
• Lock internal ref to system ref
    – use phase locked loop (PLL)           System                 Lock-in amplifier
    – source of name “lock-in amplifier”                        Mixer
                                                       Signal


                                           Reference                VCO        PLL



                                                                         Integrate
                             Analog mixer
• Direct multiplication                 Multiplying mixer
    – accurate
    – not enough dynamic range
    – weak signal buried in noise
• Switching mixer
    – big dynamic range
    – but also demodulates harmonics      Switching mixer




   Harmonic content of square wave

            1


                1/3
                      1/5   1/7   1/9
             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

                                                              n-channel FET
                                                                   bias
                                                         source                drain
                                                                        gate
                                                                    n
                                                                    p
                                                                  Signal
                                                                  voltage
     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
                        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 ?
             Lock-in amps in servos
•   Lock to resonance peak                         Take derivative with lock-in
     – Servos only lock to zero
     – Need to turn peak into zero
•   Take derivative of lineshape
     – modulate x-voltage
     – F(x)-voltage amplitude like derivative
                                                                         No fundamental
•   Use lock-in amp to extract amplitude of F(x)
                                                                         • only 2 f signal
     – “DC” part of mixer output
     – filter with integrator, not low-pass




                     F(x)



                                                           x
    Lock-in amps for derivative
• Lock-in turns sine wave signal into DC voltage
• At peak of resonance
    – no signal at modulation freq                              Input signal
    – lock-in output crosses zero
• Discriminant
    – use to lock

                                     F(x)


                                                            x
                                                                    Lock-in
                                                                    output
                                                                  (derivative)


                                            Zero crossing
                                            at resonance
     Effect of modulation on lineshape
                                               I
• Start with resonance lineshape
• Intensity vs PZT voltage: I = I0 exp( -V2)



                                                                   V
• Modulate voltage: V= V0 sin (2 p f t)
                                                       V

• Modified lineshape
                                                           t
• Analog to numerical derivatives
• Derivative is: I’ = I(V+ DV) - I(V) / DV
    – Set DV = 1
• Modulation replaces DV= V0 sin (2 p f t)         I
• Derivative is sine wave part
    – Assumes is V0 small

                                                               t
          Effect of modulation amplitude
•   For large modulation amps
     – Distortion and broadening                      Modulation amplitude
•   Modulation like a noise source
     – Always use minimum necessary                   0.05 linewidth


                                                      0.1


                                                      0.2


                                                                        0.5 linewidth


                                                                             1


                                                                             2


                                      Expanded scan
                           Mixer outputs
                                                 Modulation amplitude
• Maximum mixer output
   – modulation ~ 1 linewidth                    0.1 linewidth
   – saturates and broadens
                                                 0.2


                                                           0.5 linewidth


                                                                 1

                                                                 2


                                           Mixer out
                                           0.1 linewidth

                                           0.2


                                                 0.5


                                                  1

                                                  2
                      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

                            Fabry-Perot

          Laser                                  PD        LIA

                    Acoustic noise                            reference

                                               Sum
                                               & HV

								
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