Frequency domain analysis of linear circuits using synchronous detection

Transcription

1 Frequency doma analysis of lear circuits usg synchronous detection Introduction In this experiment, we will study the behavior of simple electronic circuits whose response varies as a function of frequency. Our ma objective this lab is to troduce you to synchronous detection a technique for characterizg systems that exhibit complex response. By complex response, we mean that the response of a system to a mono-frequency drive contas two components: a component that is -phase with the drive (-phase component) and a component that is 9 out-of-phase with the drive (quadrature component.) One way to obta the frequency response is to drive the system at a sgle frequency and measure the response of the system at this frequency. The frequency can then be swept to obta the full frequency response of the system. Lock- detectors measure the component of the signal that has a defite phase relationship to a reference signal ( most cases, the reference would be the signal used to drive the system.) Lock- amplifiers have two outputs an -phase or X-output, that measures the put signal that is -phase with the reference, and a quadrature or Y-output, that measures the signal that is 9 out of phase with the reference. In this lab, we will use the Stanford esearch Systems S83 lock- amplifier. A great troduction to lock- detection is given the S83 manual, which is cluded at the end of this manual. As an example of a system that exhibits a frequency dependent response, consider a damped driven harmonic oscillator described by the followg differential equation d x dx γ ω x Ft () dt + + (1.1) dt j t where, Ft () ae ω is the drive at frequency ω. The steady-state solution to the above equation is given by For a real, we have ( ω ω ) + jωγ xt () Ft () (. (1.) ω ω ) + ( γω ) 1

2 a( ω ω ) aγω e[ x( t)] cos t+ sωt ( ω ω ) ( γω) ( ω ω ) ( γω) ω + + (1.3) jωt ecall, e cosωt jsωt and j 1. We see that that the response of the system to a pure cose drive ( e[ Ft ( )] acosωt) contas both a cose term (-phase with drive) and a se term (9 out-of-phase.) In general, we can express the system response to a mono-frequency drive at frequency ω terms of a complex response function H( ω) H ( ω) + jh ( ω). I x() t H( ω) F() t (1.4) For the case of the damped driven harmonic oscillator, xt () ( ω ω ) + jωγ H ( ω) Ft () ( ω ω ) + ( γω) ( ω ω ) H ( ω) ( ω ω ) + ( γω) γω H I ( ω) ( ω ω ) + ( γω). (1.5) In practice, H ( ω) is measured by drivg the system at a sgle frequency and comparg the phase of the response relative to the drive. The coefficients of the se and cose terms (1.3) can be obtaed by performg the followg calculation. πω H ( ω) e[ x( t)] cos t dt a ω πω HI ( ω) e[ x( t)]sωt dt a (1.6) As we shall see, a lock- amplifier performs the operations specified by (1.6) to obta the coefficients of the -phase and quadrature response as a function of frequency. In so dog, it allows us to fd H ( ω ) and H ( ) I ω. A functional representation of a lock- detector is shown Fig. 1. All lock- amplifiers are made up of three basic blocks: (1) a phase-locked-loop (PLL) circuit, () mixers that multiply the put signal and the phase and quadrature clock signals, and (3) low-pass filters at the output of the mixers. The PLL circuit produces a susoidal clock output that has the same frequency and phase as the reference signal. The reference signal is sent to phase-shifter that produc-

3 Figure 1: Schematic of lock- detection. es a clock signal that is 9 out of phase with respect to the reference. The two mixers multiply each clock signal with the put signal. The output of the mixers conta a constant term and a term at frequency ω. H ω ωt+ H ω ωt ωt 1 1 ( 1+ cos ωt) H( ω) + s ωthi( ω) In-phase mixer output: ( )cos I( )cos s Quadrature mixer output: H( ω)cosωts ωt+ HI( ω)s ωt 1 1 s ωth( ω) + ( 1 cos ωt) HI( ω) A low-pass filter is used to remove the high frequency signal at ω and only pass the constant term. In this way, the filters perform the same function as tegratg over one period. (1.7) 3

4 Frequency Doma Measurement of Complex Impedance (A) C (B) (C) limit V V out C V V V out L V out C Figure 1: Different LC circuits for frequency analysis: (A) high-pass filter; (B) low-pass filter; (C) series LC resonant circuit. In your analysis, it will be helpful to recall the followg expressions for the complex impedance for a capacitor and ductor: ZL jω L. Zc j ω C and In this lab, you will analyze the frequency response of the circuits shown Fig. 1 usg lock- detection. Let s beg by derivg the response function for each circuit. Vout H ( ω ) (1.8) V As an example, we ll outle the derivation for the high-pass filter shown Fig. 1(A). In order to fd the voltage across, we first apply Kirchoff s laws to fd the current passg through. V it () (1.9) Z Here, Z is the impedance of the series combation of Zc j ω C and. j Z (1.1) ω C and Vout i() t V (1.11) j 4

5 thus, Vout H( ω ) ( ω C+ j) V ( ω C) + 1 H ( ) ( ω ) ω C + ( ) 1 H I ( ω ) ω C + ( ) 1 (1.1) Or, polar form H ( ω ) ( ω C) H I ( ω) 1 1 Θω ( ) tan tan H( ω) (1.13) We can see from (1.13) why this circuit is called a high-pass filter. If ω is smaller than a cutoff frequency given by ω 1 C, then V V < 1. Far above this frequency, V out c V 1. Thus, the circuit filters out low frequencies and allows high frequencies to pass. In this exercise, we will use a lock- amplifier to measure both the real and imagary parts of H ( ω ). To see how this is accomplished, it is convenient to represent the put voltage and output voltages as the real part of complex quantities. out Workg through some algebra, we fd jωt V () t e[ V e ] jωt V () t e[ V e H( ω)] out V () t V cosωt Vout ( t) V [( )cos ω t+ s ωt] ( ω C) + 1 (1.14) (1.15) While the put signal is only proportional to cos ω t, the output contas both cos ω t and s ω t terms. The lock- amplifier has two puts: 1) a reference signal V () t that has the same frequency and phase of the drive signal V () t and ) the signal to be analyzed Vout () t. The -phase (or X-output) of the lock- amplifier will thus be proportional to the MS-amplitude of the cos ω t part and the quadrature (or Y output) will be proportional to the MS-amplitude of the s ωt part. ref 5

6 V V X Y V V ( ) ω C + ( ) 1 ω C + ( ) 1 (1.16) The factor of 1 is there because the output of the lock- amplifier gives root-meansquare (MS) not peak amplitudes. H ( ω) can be found directly from (1.16). H ( ω ) ( VX + jvy) ( ω C+ j) (1.17) V ( ω C) + 1 The response function for parts (B) and (C) are found usg the procedure outled above. (A) High-pass filter H( ω ) ( ω C+ j) ( ω C) + 1 (B) Low-pass filter H ( ω) 1 j ω + ( C) 1 (C) LC resonant circuit 1 + jωl H ( ω) + limit 1 ω LC j Table 1: esponse functions for the circuits Fig. 1. Measurement Procedure The S83 comes with a se-wave function generator output which you will use to drive the circuit. Note that the S83 function generator output is V-MS, so you do not need to clude the factor of is (1.16) or (1.17). When the lock- reference is set to Internal, the reference is locked to the se wave output of the lock-. V out should be connected to the A-put of the S83. The se wave generator of the S83 has 5 W output impedance, thus the magnitude of the impedance of your circuit should be greater than 5 W over the frequency range of the measurement order to ensure that the voltage across the circuit is close to the output voltage of the function generator. Part I: For each circuit parts (A) and (B), pick an C combation that gives a cutoff frequency f 1πC between 1-5 khz. Keep the value of between.5-1 kω to en- c sure the circuit impedance is much greater than the output impedance of the S83 function generator. Place the circuit components on the breadboard, then connect the 6

7 V to the se wave output of the S83 and V out to the A-put. For part (C), use a limitg resistor limit.1mω, L -5 mh (air filled ductor), C 1. μf, and 5 Ω. Drive the circuit with 1. V-rms from the function generator. The sensitivity of the lock- should be set to 1 V/V and the time constant to 3 ms - 1 db. These values are meant to be guideles. You are encouraged to experiment with the lock- settgs 1. Quadrature channel: Y-output X (V-rms).1.1 1E-3 In-phase channel: X-output (A) Equation y Vrms/(1+(*pi*x*t)^) Y (V-rms) (B) Equation y Vrms**pi*x*t/(1+(*pi*x*t)^) Adj. -Square E-4 Adj. -Square Value Standard Error Vx Vrms E-4 Vx t E E Value Standard Error Vy Vrms E-4 Vy t 5.945E E-8 1E Frequency (Hz) Frequency (Hz) 1 Magnitude: Sqrt(V y +V x ) Phase angle: 18*tan -1 (V y /V x )/π (Volts).1.1 1E-3 (C) Equation y Vrms/sqrt(1+(*pi*x*t)^) Adj. -Square Value Standard Error Vr Vrms E-4 Vr t 5.914E E Frequency (Hz) Phase angle (deg) (D) Equation y Vrms*18*atan(-*pi*x*t)/pi Adj. -Square Value Standard Error Vtheta Vrms Vtheta t 5.485E E Frequency (Hz) Figure : Lock- measurements taken for a low-pass filter with 995 Ω and C 5.37 nf. (A) X-output proportional to H ( ω ), (B) Y-output proportional to H I ( ω ), (C) and (D) are derived from the measured data. fact, the ma pot of this exercise is to familiarize you to lock- detection. The sensitivity sets the ga of the amplifier after the low-pass filters. The time constant sets the cutoff frequency of the low-pass filters. You will notice that the larger the time constant, the more stable the readgs appear. However, larger time constants also mean that the output responds more slowly to any changes. When sweepg frequency, it is important 7

8 to have the time constant of the lock- to be faster than the sweep rate so that the output can track the changg response. Set the two outputs to X and Y. Both the X and Y output values can be directly read off from the front panel display. You will measure the real and imagary parts of the response function by manually sweepg the frequency and recordg the X and Y outputs from the lock- amplifier. Sample data taken for the low-pass filter (Fig. 1(B)) is shown Fig.. Input the data to Orig and make log-log plot for each channel. Use the nonlear fit option to fit the data to the expected form. Calculate the magnitude and phase of the response function from your data and fit these also usg the expected form. All fit functions can be obtaed directly from Table 1. Measure the, C and L usg the impedance analyzer and compare to the fitted data. Part II: The lock- amplifier can also be used to measure the response at a harmonic of the drive frequency. This feature is particularly useful when characterizg non-lear systems which create harmonic distortion. As an troduction to this 1..5 feature of the lock-, you will use. the lock- amplifier to analyze the harmonic content of a square wave. A periodic waveform can be Figure 3: Square waveform expressed as a sum of ses and coses whose frequencies are multiples of the fundamental frequency. The frequency content of the square waveform shown Fig. 3 can be analyzed by expandg it terms of a Fourier series. a πnt πnt Ft () + ancos + bns n 1 T n 1 T Here, T is the period of the waveform, and the Fourier coefficients are given by (1.18) 8

9 T π nt an F()cos t dt T T T π nt bn F()s t dt T T a T T F() t dt (1.19) 1 kω V out Use the Wavetek function generator to synthesize the square waveform with a 5% duty cycle. Set the function to bipolar square wave with a peak amplitude of.1 V and a frequency of 1 Hz. Connect the function output across a 1 Figure 4: Setup for square waveform measurement. kω resistor and connect output voltage to the A-put of the S83 lock- amplifier. For this measurement, the reference signal must be supplied by the Wavetek. Connect the Snyc output of the Wavetek to the reference put of the lock- and set the reference source to external. Select the harmonic number by pressg the Harm button. You can select harmonics up to a maximum frequency of 1 khz. Measure the X and Y output for harmonics up to N. Calculate the Fourier coefficients for the square wave analytically usg Eq. (1.19) and compare to your measurement. 9

10 eport 1. From your data part I, calculate the real and imagary parts as well as the magnitude and phase of the response function for the three circuits. Make a log-log plot for each component and fit it usg the derived functions given Table 1.. Compare the parameters obtaed from your fits to the expected value. eport all errors the fitted values and compare this error to your uncertaty the measured values of, L and C. 3. In part II, plot of amplitudes obtaed from the lock- amplifier vs. 1/n (n is the harmonic number) for a square waveform for the first harmonics. Calculate the Fourier coefficients for the square waveform and compare to your data. 1