PY3107 Experimental Physics II ock-in Amplifiers MP aughan and F Peters
Related Experiments ock-in ab ignal processing and phase sensitive detection using a lock-in amplifier
The problem The signal to noise ratio (NR) is defined as NR P P where P is the power of the signal and P N the power of the noise In decibels: NR db Our problem: signal buried in noise N P P N N 10 P log 10 PN,
The problem Amplification does not help! Eg voltage amplifier At best NR out out G in G G P P In practice, amplifier adds noise N NR in NR out G G P N P P N NR in
Noise factor Noise factor F NR NR in out 1 Noise figure NF 10 NR log 10 in NRout
olution Phase sensitive detection (using a lock-in amplifier)
The lock-in amplifier We modulate the signal with a time varying wave-form t sin t,0 and multiply this by a local oscillation to give t sin t,0 t t sin t sin t, 0,0
The lock-in amplifier Using the trig identity we have sin Asin B A BcosA cos, 1 B 1 t t cos 1,0,0,0,0 cos t t
The lock-in amplifier We now put to obtain, 1 t t cos 1,0,0,0,0 cos t This gives us a time-independent term that can be maximized by adjusting the phase of the local oscillator, until
The lock-in amplifier The high frequency term may be filtered out using a low band-pass filter (to be discussed later) Optimizing the phase, we then obtain 1 t t,0,0 ince,0 is known, we may then obtain the signal strength directly
The lock-in amplifier Modulating by a square wave A square wave may be expressed as a Fourier series where t a sin t,, 0 k1 4, k k1 k, k, k 1 a k and the wave oscillates between,0 and,0 with angular frequency
The lock-in amplifier cos cos, sin sin, ',, ', ' 1 1 ',0,0 1, ', ' 1 1 ',0,0 k k k k k k k k k k k k k k t t a a t t a a t t Multiplying this by the local oscillator square wave
The lock-in amplifier Putting as before, we obtain (after filtering out the high-frequency terms and optimizing the phase) t t a 1,0,0 k1 k
The lock-in amplifier Re-inserting the values of the coefficients, we have However, t t 8 8 1, 0,0 1 1 k k k 1 k 1, o t t, 0,0
Band pass filters
Charge carrier scattering Constant field m dv dt qe mv v qe m t / t 1 e, v D qe m
Conductivity Current density j nqv D nq E m j E, nq m
Resistivity Current oltage Resistivity j E I A, 1 j E, I A 1,
Ohm s aw I A 1, Resistance R I, A, A IR
Resistive impedance I it t I e, 0 R IZ R Z R R
Capacitance + - +ve -ve E d C E d Capacitance C C Q C
Charging a capacitor
Charging a capacitor et be constant By Kirchhoff's voltage law R IR dq dt R C Q C,, Q C
Charging a capacitor Using the integrating factor method with C 0 0, C t / RC t 1e, where RC RC
Charging a capacitor Capacitor discharges with same time constant Once charged, capacitor opposes DC current Capacitor will still pass AC when f > 1/RC
Capacitive impedance C Q C, I C IZ C it t I e, 0 dq dt Q 0 it I e i I i
Capacitive impedance C Q C I ic C IZ C Z C 1 ic
Inductance A current carrying wire has a magnetic field associated with it by virtue of Ampere s aw If the wire is wound into a coil, then the magnetic field inside the coil is the resultant of each turn of the wire
Inductance A change in the current will lead to a change in the magnetic field By Faraday s aw, this induces a field acting in opposition to the field driving the current A coil (inductance) therefore opposes changes in the current An inductance then passes DC but resists AC
Inductive impedance di dt di dt I IZ it t I e, 0 i t ii0e ii
Inductive impedance di dt ii IZ Z i
Band pass filter
Transfer functions oltage divider R C H Z ' Z ' Gain G H
Transfer functions H R R Z 1 1 ' ' i RC R 1 i HC H R, RC i H H R R
Transfer functions G R 1 R 1 1 RC 1/ G 1 RC C G R G G R R,
Band pass filter gain
ow band pass filter Input ow-frequency output only