Linear Circuit Experiment (MAE171a) Prof: Raymond de Callafon

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1 Linear Circuit Experiment (MAE171a) Prof: Raymond de Callafon TA: Younghee Han tel. (858) / , class information and lab handouts will be available on MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 1

2 Main Objectives of Laboratory Experiment: modeling, building and debugging of op-amp based linear circuits for standard signal conditioning Ingredients: modeling of standard op-amp circuits signal conditioning with application to audio (condensor microphone as input, speaker as output) implementation & verification of op-amp circuits sensitivity and error analysis Background Theory: Operational Amplifiers (op-amps) Linear circuit theory (resistor, capacitors) Ordinary Differential Equations (dynamic analysis) Amplification, differential & summing amplifier and filtering MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 2

3 Outline of this lecture Linear circuits & purpose of lab experiment Background theory - op-amp - linear amplification - single power source - differential amplifier - summing circuit - filtering Laboratory work - week 1: microphone and amplification - week 2: mixing via difference and adding - week 3: filtering and power boost Summary MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 3

4 Linear Circuits & Signal Conditioning Signal conditioning crucial for proper signal processing. Applications may include: Analog to Digital Conversion - Resolution determined by number of bits of AD converter - Amplify signal to maximum range for full resolution Noise reduction - Amplify signal to allow processing - Filter signal to reduce undesired aspects Feedback control - Feedback uses reference r(t) and measurement y(t) - Compute difference e(t) =r(t) y(t) - Amplify, Integrate and or Differentiate e(t) (PID control) Signal generation - Create sinewave of proper frequency as carrier - Create blockwave of proper frequency for counter - etc. etc. MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 4

5 Purpose of Lab Experiment In this laboratory experiment we focus on a (relatively simple) signal conditioning algorithms: amplification, adding/difference and basic (at most 2nd order) filtering. Objective: to model, build and debug op-amp based linear circuits that allow signal conditioning algorithms. We apply this to an audio application, where the signal of a condenser microphone needs to amplified, mixed and filtered. Challenge: single source power supply of 5 Volt. Avoid clipping/distortion of amplified, mixed and filtered signal. Aim of the experiment: insight in op-amp based linear circuits build and debug (frustrating) compare theory (ideal op-amp) with practice (build and test) verify circuit behavior (simulation/pspice) MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 5

6 Background Theory - op-amp op-amp = operational amplifier more precise definition: DC-coupled high-gain electronic voltage amplifier with differential inputs and a single output. DC-coupled: constant (direct current) voltage at inputs results in a constant voltage at output differential inputs: two inputs V and V + and the difference V δ = V + V is only relevant high-gain: V out = G(V + V )whereg>>1. MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 6

7 Background Theory - op-amp Ideal op-amp (equivalent circuit right): input impedance: R in = i in =0 output impedance: R out =0 gain: V δ =(V + V ), V out = GV δ, G = rail-to-rail: V S V out V S+ Ideal op-amp (block diagram below) V + ideal op-amp + V δ V out G V MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 7

8 Background Theory - op-amp Infinite input impedance (R in = ) useful to minimize load on sensor/input. Zero output impedance (R out =0)useful to minimize load dependency and obtain maximum output power. Rail-to-rail operation to maximize range of output V out between negative source supply V S and positive source supply V S+. But why (always) infinite gain G? Obviously: V out = V S+ if V + >V 0 if V + = V V S if V + <V not very useful with any (small) noise on V + or V. MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 8

9 Background Theory - op-amp Usefulness of op-amp with high gain G only by feedback! Consider open-loop behavior: V out = GV δ, where V δ = V + V and create a feedback of V out by choosing to make V = KV out V δ = V + KV out Then V out = GV δ = GV + GKV out allowing us to write V out = G 1+GK V + MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 9

10 Background Theory - op-amp So, with the feedback V = KV out we obtain V out = ideal op-amp G 1+GK V + V + + V δ V out G V K In case G we see: V out = 1 K V + Don t care what gain G is, as long as it is LARGE Make sure K is well-defined and accurate If 0 <K<1thenV + is nicely amplified to V out by 1/K MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 10

11 Background Theory - op-amp Amplification 1/K by feedback K of ideal high gain op-amp: V + ideal op-amp + V δ V out G V K Series of and R 2 leads to voltage divisor on V given by: V = + R 2 V out = KV out, 0 <K 1 and with ideal high gain op-amp we get lim V out = G lim G G 1+GK V + = 1 K V + = + R 2 V + = ( 1+ R 2 MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 11 ) V +

12 Background Theory - non-inverting amplifier (voltage follower) Our first application circuitry: V out = ( 1+ R 2 ) V in So-called voltage follower in case = (not present) and R 2 =0 where V out = V in but improved output impedance! Quick (alternative) analysis based on V + = V and i + = i =0: Since i =0andseries, R 2 we have V = V out + R 2 Hence V in = V + = + R 2 V out V out = + R 2 V in = ( 1+ R 2 ) V in MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 12

13 Background Theory - inverting amplifier Similar circuit but now negative sign: V out = R 2 V in Quick (alternative) analysis based on V + = V and i + = i =0: With V = V + =0andi = 0, Kirchhoff s Current Law indicates V in + V out =0 R 2 Hence V out R 2 = V in V out = R 2 V in MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 13

14 Background Theory - effect or rail (source) voltages V out = ( 1+ R ) 2 V in V out = R 2 V in Formulae are for ideal op-amp with boundaries imposed by negative source supply V S and positive source supply V S+ V S V out V S+ (rail-to-rail op-amp) Single voltage power supply with V S+ = V cc and V S = 0 (ground): Limits use of inverting amplifier (V out < 0 not possible) Limits use of large gain R 2 / (V out >V cc not possible) Design challenge: 0 <V out <V cc to avoid clipping of V out. MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 14

15 Background Theory - effect or rail (source) voltages V out = ( 1+ R ) 2 V in V out = R 2 V in Single voltage power supply with V S+ = V cc and V S = 0 (ground) complicates amplification of V in (t) =a sin(2πft) as a <V in (t) <a(both positive and negative w.r.t. ground). Example: audio application (as in our experiment). To ensure 0 <V out <V cc provide offset compensation V in (t) =a sin(2πft)+a to ensure V in (t) > 0 and use non-inverting amplifier. MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 15

16 Background Theory - differential amplifier Instead of amplifying one signal, amplify the difference: V out = + R 2 R 3 + R 4 R4 V 2 R 2 V 1 Difference or differential amplifier is found by inverting amplifier and adding signal to V + via series connection of R 3 and R 4. Analysis: With i + =0theseriesorR 3 and R 4 leads to V + = R 4 R 3 +R 4 V 2 With V = V + and Kirchhoff s Current Law we have Hence or V out = R 2 V 1 R 4 R 3 +R 4 V 2 + V out R 4 R 3 +R 4 V 2 R 2 =0 R 4 R 3 + R 4 V 2 + R 4 R 3 + R 4 V 2 R 2 V 1 V out = + R 2 R 3 + R 4 R4 V 2 R 2 V 1 MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 16

17 Background Theory - differential amplifier Choice = R 3 and R 2 = R 4 reduces to V out = + R 2 R 3 + R 4 R4 V 2 R 2 V 1 V out = R 2 (V 2 V 1 ) create a difference/differential amplifier. Further choice of = R 3, R 2 = R 4 and R 2 = yields V out = V 2 V 1 and computes the difference between input voltages V 1 and V 2. NOTE: V 2 >V 1 for a single voltage power supply with V S+ = V cc and V S = 0 (ground) to avoid clipping of V out against ground. MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 17

18 Background Theory - more advanced differential amplifiers Difference amplifier does not have high input impedance (loading of sensors). Better design with voltage followers: With we have R 2 = R 4 R 3 V out = R 2 (V 2 V 1 ) R 5 is used to adjust offset (balance) MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 18

19 Background Theory - more advanced differential amplifiers Even better differential amplifier that has a variable gain is a so-called instrumentation amplifier: Setting all resistors R i = R, i =1, 2,...5 except R var,makes ( V out = 1+ 2R ) (V 2 V 1 ) R var High input impedance and variable gain via an (external) resistor R var makes this ideal for the amplification of (non-grounded) instrumentation signals. Instrumentation amplifiers are made & sold as a single chip. MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 19

20 Background Theory - inverting summing amplifier Inverting amplifier can also be extended to add signals: V out = R 4 ( V1 + V 2 R 2 + V 3 R 3 Analysis follows from Kirchhoff s Current Law for the input of the op-amp: With V = V + we have V =0 With i =0 wehave Hence ) V out R 4 + V 1 + V 2 R 2 + V 3 R 3 =0 V out = R 4 creating a weighted sum of signals. ( V1 + V 2 + V ) 3 R 2 R 3 MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 20

21 Background Theory - inverting summing amplifier The choice = R 2 = R 3 reduces to V out = R 4 ( V1 + V 2 R 2 + V 3 R 3 V out = R 4 (V 1 + V 2 + V 3 ) simply amplifying the sum of the signals. Oftentimes extra resistor R 5 is added: 1 = R 5 R 2 R 3 R 4 to account for possible small (bias) input currents i 0, i + 0. This ensures V out remains sum, without bias/offset. ) MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 21

22 Background Theory - inverting summing amplifier Inverting summing amplifier: V out = R 4 ( V1 + V 2 R 2 + V 3 R 3 and for = R 2 = R 3 : V out = R 4 (V 1 + V 2 + V 3 ) has a limitation for single source voltage supplies: ) Single voltage power supply with V S+ = V cc and V S =0: Limits use of inverting summer (V out < 0 not possible) Limits use of large gain R 4 / (V out >V cc not possible) clipping of V out will occur if sum of input signals is positive. MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 22

23 Background Theory - non-inverting summing amplifier Based on a non-inverting amplifier signals can also be summed: V out = Analysis: ( 1+ R 4 R 3 ) R1 R 2 + R 2 ( V1 + V 2 R 2 due to i =0we havev = R 3 R 3 +R 4 V out Due to V + V and i + = 0 with Kirchhoff s Current Law: ) Hence and V 1 R 3 R 3 +R 4 V out V 1 + V 2 R 2 = V out = ( ( 1 1+ R 4 R 3 + V R 2 ) ) R1 R 2 + R 2 R 3 R 3 +R 4 V out R 2 =0 R 3 R 3 + R 4 V out ( V1 + V ) 2 R 2 MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 23

24 Background Theory - non-inverting summing amplifier The choice = R 2 reduces V out = ( 1+ R 4 R 3 ) R1 R 2 + R 2 ( V1 + V 2 R 2 to ( V out = 1+ R ) 4 V1 + V 2 R 3 2 indicating amplifciation of the sum V 1 and V 2 if R 4 R 3. Further choice of also = R 2 = R 3 = R 4 leads to V out = V 1 + V 2 indicating a simple summation of V 1 and V 2. Unlike inverting summing amplifier, no extra resistor can be added to compensate for bias input current. Not desirable: source impedance part of gain calculation... ) MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 24

25 Background Theory - filtering So far, all circuits were build using op-amps and resistors. When building filters, mostly capacitors are used as negative, positive or grounding elements. Interesting phenomena: resistor value of capacitor depends on frequency of signal. Analysis for capacitor: capacitance C is ratio between charge Q and applied voltage V : C = Q V Since charge Q(t) at any time is found by flow of electrons: we have Q(t) = V (t) = Q(t) C t τ=0 i(τ)dτ = 1 C t τ=0 i(τ)dτ MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 25

26 Background Theory - filtering Application of Laplace transform to yields V (t) = Q(t) C = 1 C t V (s) = 1 Cs i(s) τ=0 i(τ)dτ Hence we can define the impedance/resistence of a capacitor as R(s) = V (s) i(s) = 1 Cs With Fourier analysis we use s = jω and we find the frequency dependent equivalent resistor value of a capacitor : R(jω)= 1 jcω This value will allow analysis of op-amp circuits based on resistors (as we have done so far) MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 26

27 Background Theory - filtering Consider simple 1st oder RC-filter with voltage follower op-amp. Due to i + =0wehave V + (jω)= 1 C 1 jω + C 1 V in (jω) 1 jω With V out = V = V + we have V out (jω)= 1 C 1 jω +1 V in(jω) This is a 1st order low-pass filter with a cut-off frequency ω c = 1 C 1 rad/s or f c = 1 2π C 1 Hz MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 27

28 Background Theory - filtering Consider circuit of non-inverting amplifier where is now series of and C 1. Equivalent series resistance is given by + 1 jc 1 ω Application of gain formula for non-inverting amplifier yields: V out (jω)= We can directly see: 1+ R jc 1 ω V in (jω) For low frequencies ω 0 we obtain a Voltage follower with V out = V in For high frequencies ω we obtain our usual non-inverting amplifier V out (jω)= ( 1+ R 2 ) Vin (jω) MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 28

29 Background Theory - filtering Transition between low and high frequency can be studies better by writing V out (s) =G(s)V in (s) whereg(s) isatransfer function. This allows us to write as V out (s) = making ( V out (s) = 1+ R 2C 1 s C 1 s R C 1 s ) V in (s) V in (s) = ( + R 2 )C 1 s +1 V in (s) C 1 s +1 G(s) = ( + R 2 )C 1 s +1 C 1 s +1 MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 29

30 Background Theory - filtering The transfer function G(s) = ( + R 2 )C 1 s +1 C 1 s +1 has the following properties: single pole at p 1 = 1 C and found by solving R 1 1 C 1 s+1 = 0. single zero at z 1 = 1 and found by solving (R ( +R 2 )C R 2 )C 1 s +1=0. DC-gain of 1 and found by substitution s = 0 in G(s). Related to the final value theorem for a step input signal v in (t): lim t V out(t) = lim s V out (s) = lim s G(s) 1 s 0 s 0 s = lim G(s) s 0 where 1 s is the Laplace transform of the step input v in(t). and found by com- High frequency gain of +R 2 R =1+ R 2 1 puting s. MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 30

31 Background Theory - filtering The transfer function G(s) = ( + R 2 )C 1 s +1 C 1 s +1 is a first order system where zero z 1 = 1 <p ( +R 2 )C 1 1 = 1 This indicates G(s) isaleadfilter. C 1. Easy to study in Matlab: >> R2=100e3;R1=10e3;C1=10e-6; >> G=tf([(R1+R2)*C1 1],[R1*C1 1]); >> bode(g) Magnitude (db) Bode Diagram of FIlter 600 Phase (deg) Frequency (rad/sec) MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 31

32 Background Theory - filtering Amplification lead filter circuit with R V out (jω)= jc 1 V in (jω) 1 ω will be used to strongly amplify a small high frequent signal but maintain (follow) the DC-offset. From the previous analysis we see: Gain at DC (ω =0) issimply1. Gain at higher frequencies approaches 1 + R 2 MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 32

33 Background Theory - filtering Another fine filter: 2nd order low pass Butterworth filter Pass-band frequency of 1kHz 2nd order 1kHz Butterworth filter is a standard 2nd order system where V out (s) =G(s)V in (s) ω 2 n G(s) = s 2 +2βω n 2+ωn 2 with ω n =2π 1000, β = 1/ MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 33

34 Background Theory - filtering where V out (s) =G(s)V in (s) ω 2 n G(s) = s 2 +2βω n 2+ωn 2 with ω n =2π 1000, β = 1/ means: 0 Bode Diagram Butterworth Filter well damped filter -40dB/dec above 1kHz Magnitude (db) >> [num,den]=butter(2,2*pi*1000, s ); >> G=tf(num,den); >> bode(g) Phase (deg) Frequency (Hz) MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 34

35 Laboratory Work - week 1 Audio application: generate input signal via MIKE-74 electret microphone build a DC-bias circuit for the microphone to measure (sound) pressure variations Measure the DC-bias (offset) voltages Display and analyse the time plots generated by the microphone MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 35

36 Laboratory Work - week 1 Build non-inverting amplifier using quad LM324 op-amp Measure bias voltages Experiments: determine gain for different resistor values and avoid clipping on output signal V out. Measure the frequency response of your amplifier. Connect amplifier to microphone Provide off-set of microphone signal to avoid clipping Create lead filter for DC-gain of 1, and high frequency gain of 100. MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 36

37 Laboratory Work - week 2 Build none inverting summing amplifier Experiments: verification of operation (adding of signals) Bias voltage adjustment Experimental verification of bias effects. MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 37

38 Laboratory Work - week 2 Build differential/difference amplifier Experiments: verification of operation (difference of signals) Bias voltage adjustment Gain adjustments and experimental verification Combine microphone & amplifier circuit from week 1 with difference amplifier to allow mixing of signals. Measure bias voltages Demonstrate mixing of sine wave signal and microphone signal without distortions MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 38

39 Laboratory Work - week 3 completed amplified/mixed micro- Starting point of week 3: phone signal. V out will now be filtered and (optional) power boosted for speaker output. MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 39

40 Laboratory Work - week 3 Create active low pass filter with cutt-off frequency of 1kHz. Demonstrate filter by measuring amplitude of output signal for sine wave excitation of different frequencies Connect filter to circuit of week 2 (microphone, amplifier and differential amplifier for mixing) Demonstrate filter by measuring microphone and filtered microphone signal Optional: add power boost to circuitry and connect speaker. MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 40

41 Summary (relatively simple) signal conditioning algorithms: amplification, adding/difference and basic (most 2nd order) filtering Challenge: single source power supply of 5 Volt. Avoid clipping/distortion of amplified, mixed and filtered signal. insight in op-amp based linear circuits by building and debuging compare theory (ideal op-amp) with practice (build and test) experimentally verify gain of circuitry for error/statistical analysis: measure gain for different resistor (of the same value) audio application on a single voltage power supply shows amplification of small signals and careful filter design to maintain DC (off-set) GOOD LUCK MAE171a Linear Circuit Experiment, Winter 2009 R.A. de Callafon Slide 41

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