PY3107 Experimental Physics II

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Rosalyn Bennett
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1 PY3107 Experimental Physics II ock-in Amplifiers MP aughan and F Peters

2 Related Experiments ock-in ab ignal processing and phase sensitive detection using a lock-in amplifier

3 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,

4 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

5 Noise factor Noise factor F NR NR in out 1 Noise figure NF 10 NR log 10 in NRout

6 olution Phase sensitive detection (using a lock-in amplifier)

7 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

8 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

9 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

10 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

11 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

12 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

13 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

14 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

15 Band pass filters

16 Charge carrier scattering Constant field m dv dt qe mv v qe m t / t 1 e, v D qe m

17 Conductivity Current density j nqv D nq E m j E, nq m

18 Resistivity Current oltage Resistivity j E I A, 1 j E, I A 1,

19 Ohm s aw I A 1, Resistance R I, A, A IR

20 Resistive impedance I it t I e, 0 R IZ R Z R R

21 Capacitance + - +ve -ve E d C E d Capacitance C C Q C

22 Charging a capacitor

23 Charging a capacitor et be constant By Kirchhoff's voltage law R IR dq dt R C Q C,, Q C

24 Charging a capacitor Using the integrating factor method with C 0 0, C t / RC t 1e, where RC RC

25 Charging a capacitor Capacitor discharges with same time constant Once charged, capacitor opposes DC current Capacitor will still pass AC when f > 1/RC

26 Capacitive impedance C Q C, I C IZ C it t I e, 0 dq dt Q 0 it I e i I i

27 Capacitive impedance C Q C I ic C IZ C Z C 1 ic

28 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

29 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

30 Inductive impedance di dt di dt I IZ it t I e, 0 i t ii0e ii

31 Inductive impedance di dt ii IZ Z i

32 Band pass filter

33 Transfer functions oltage divider R C H Z ' Z ' Gain G H

34 Transfer functions H R R Z 1 1 ' ' i RC R 1 i HC H R, RC i H H R R

35 Transfer functions G R 1 R 1 1 RC 1/ G 1 RC C G R G G R R,

36 Band pass filter gain

37 ow band pass filter Input ow-frequency output only

Solutions to these tests are available online in some places (but not all explanations are good)...

Solutions to these tests are available online in some places (but not all explanations are good)... The Physics GRE Sample test put out by ETS https://www.ets.org/s/gre/pdf/practice_book_physics.pdf OSU physics website has lots of tips, and 4 additional tests http://www.physics.ohiostate.edu/undergrad/ugs_gre.php