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State University of New York at Stony Brook Department of Electrical and Computer Engineering ESE 211 Electronics Laboratory A 2011 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ Lab 5: Frequency response of RC and LR circuits 1. Objectives 1. Measure the magnitude and phase frequency responses of RC and LR circuits. 2. Determine the time constant of the 1st order filter circuit from its magnitude response. 3. Become familiar with the procedure and the automated program for frequency response measurements. 2. Introduction The Laplace transform (1) simplifies the circuit analysis by replacing time-dependent integral-differential equations with algebraic equations. By Laplace transform the temporal signal waveform of interest v(t) is replaced with complex amplitude V(s) that can be found after solving these algebraic equations. Then the inverse Laplace transform is applied to V(s) and signal waveform of interest can be found. In other word the circuit analysis in time-domain is replaced with circuit analysis in s-domain, where s is complex variable. V t vt exp s t dt (1) 0 For linear circuits (circuits which parameters does not depend on magnitudes of the signals) the output signal complex amplitude V0(s) can be represented by product of the input signal complex altitude VIN(s) and the circuit transfer function T(s). VO s T s VIN s (2) The circuit transfer function T(s) is completely determined by circuit parameters, i.e. values of R, L and C. Experimentally, the magnitude and phase angle of the transfer function are obtained by measuring the frequency responses of the circuit. In this case Laplace transforms formalism is reduced to less general Fourier transform formalism with j*ω in place of s in (1) and (2). The angular frequency ω is equal to 2πf, where f is frequency in Hz. In this frequency domain analysis the circuit response to harmonic excitation of variable frequency is measured or calculated. In other words, the voltage (or current) phasors are used, i.e. VO(ω), VIN(ω) and T(ω) are dealt with. Parameters for linear circuits such a time constant, etc. (or small-signal parameters for non-linear circuits) are easier to obtain in frequency-domain rather than in time-domain, because the system response due to nonideal properties of a signal generator, a scope or a DMM can be measured separately and subtracted from the overall output response. Alternatively, determination of the circuit time constant in time–domain would require 1 State University of New York at Stony Brook Department of Electrical and Computer Engineering ESE 211 Electronics Laboratory A 2011 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ either a rectangular input signal with small rise time or a complicated deconvolution analysis for the output signal. The transfer functions for the 1st order integrating circuit (low-pass filter, LP) shown in Figure 1a or differentiating circuit (high–pass filter, HP) shown in Figure 1b can be presented in the following forms TLP ( ) 1 1 j and THP ( ) j 1 j (3) , where the time constants are = RC and = L/R for RC and RL circuits, correspondingly. In this lab the time constants of both types of circuits will be determined from the array of measured data by fitting to the model dependences (3). Presence of only one fitting parameter, i.e. just τ, will ensure high confidence in the obtained results. 1k V1 1k V1 V V 100n 10mH 0 0 (a) (b) Figure 1 3. Preliminary lab Calculate the time constant for 1st order circuits with R = 1 k, C = 0.1 F and L =10 mH. Sketch the schematics for four possible 1st order circuits containing either R and C or R and L elements. For each of them draw asymptotic magnitude and phase responses (use logarithmic scale for frequency). The magnitude response should be presented in dB scale, indicate the characteristic frequencies and -3 dB points. If you are in doubts, simulate the responses with PSPICE. 4. Experiment The experiments will be performed with a 1 kΩ resistor, a 0.1 µF capacitor and a 10 mH inductor. These are nominal values. Measure the real values of each of them and calculate the expected time constants and corresponding cut-off frequencies for the two circuits in Figure 1. Low-pass RC filter 1. Assemble the circuit in Figure 1 (a). First measure manually the amplitude response using the signal generator and the DMM in AC mode. The measurement should be performed in a frequency range listed in 2 State University of New York at Stony Brook Department of Electrical and Computer Engineering ESE 211 Electronics Laboratory A 2011 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ Table 1. At each frequency, measure the input amplitude Vin and the output amplitude Vout. The input is the output of the signal generator and the output is shown on the figure. Calculate the ratio of output and input amplitudes in dB scale and determine the – 3 dB cut-off frequency. Compare with the calculated value. f (Hz) 10 100 1k 10 k 100 k 1M 10 M Vin (V) Table 1 Vout (V) Vout / Vin (dB) Δφ (º) 2. Perform the phase response measurements manually using an oscilloscope. Attach the two probes to the input and output; measure the phase shift Δφ between them and present in Table 1. 3. To perform automated measurements, run the program FREQUENCYresponse.vxe and PHASEresponse.vxe. The program will take a moment to load excel. The program will prompt the user for parameters before starting the frequency sweep. Choose a reasonable frequency range and step (it takes about 1 second to scan 1 point) based on the manual measurements. Once the sweep is complete the program will prompt the user to save the excel data. Note: Closing the phase response window will close the excel window; save your data when prompted to do so! In Excel, calculate the amplitude response in dB scale and plot against frequency. Use logarithmic scale for frequency. Plot the phase response in another figure. Determine the cut-off frequency of this low-pass filter and compare with the one calculated before. High-pass LR filter 4. Assemble the circuit in Figure 1 (b). Perform the automated measurements as in part 3. Choose a frequency range and step properly to be able to resolve the cut-off frequency (use your calculations as a first approximation). Plot the amplitude and phase response similar like you did for RC circuit in part 3 above. Determine the cut-off frequency and compare with the calculated value. Note: A nominal value of the inductance in the kit is 10 mH. It is wound in three sections to reduce the parasitic capacitance due to electric field interaction between the adjacent turns. The inductor also has an equivalent DC resistance which you can measure with DMM. Thus, performance of a real inductor will be affected by R and C so that the actual impedance deviates from the purely inductive. At low frequency, the equivalent impedance approaches the DC resistance value. At moderate frequencies the inductor would 3 State University of New York at Stony Brook Department of Electrical and Computer Engineering ESE 211 Electronics Laboratory A 2011 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ behave like series connection of this resistance and actually inductance. At high frequency, the parasitic capacitance will create a parallel (LR)-C resonant circuit. Well above that frequency, the inductor will behave like a capacitor. Report Perform PSPICE simulation of frequency responses for both circuits in Figure 1. Compare it with the measured data and explain the observed difference. Improve the model for LR circuit incorporating a resistance in series with L and a capacitance in parallel to RL. The resistance value should be taken equal to the measured with DMM DC resistance of the inductor. Try small capacitance values in the range of 20 - 200 pF to obtain the best fit to the measured phase response for the circuit. The report should include the lab goals, short description of the work, the experimental and simulated data presented in plots, the data analysis and comparison followed by conclusions. Please follow the steps in the experimental part and clearly present all the results of measurements. 4