UNIVERSITY OF MASSACHUSETTS DARTMOUTH DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING ECE 202 CIRCUIT THEORY 2 RLC SERIES-RESONANT CIRCUIT INTRODUCTION A series-resonant RLC circuit is shown in Figure 1. L 100m H C Rinductor 100 Ohm 0.022uF Vsource Rexternal 330 Ohm 10 V 1kHz 0Deg Figure 1. A series resonant RLC circuit. The circuit “resonates” at a single frequency determined by the expression f 0 1 2 1 Hz LC 2 0 At the resonant frequency, the reactance of the capacitor and the reactance of the inductor are equal and opposite, and, in effect, cancel each other out. This can be shown by looking at the total series impedance of the circuit, Ztotal series at ω = ω0 Ztotal series = Rexternal + Rinductor + jω0L + 1/jω0C Ztotal series = Rexternal + Rinductor + jω0L – j/ω0C since (1/j = -j) If ω0L = 1/ω0C Ztotal series = Rexternal + Rinductor = R This leaves the total “equivalent” impedance of the series circuit to be purely resistive! 1 PRELIMINARY DESIGN Submit a copy before you come into the lab! 1. Each group member will design and simulate a series RLC circuit that resonates at a frequency equal to f N 2500Hz 0 where N is the last three non-zero digits of the group member’s UMD ID Number. The formula for the resonant frequency is f 0 1 2 1 Hz LC 2 0 Since there are 2 unknowns, L and C, you must choose 1 of them. For this case, each student will use a 100 mH inductor. The “quality factor”, Q, for the circuit is given by the expression Q 0 B L CR 2 where B = the bandwidth of the circuit in radians/sec. For this circuit, we will design for a Q = 5. One last note. The inductor has a “built-in” resistance, RL. When you get the inductor, measure the DC resistance with your digital multimeter. Subtract this measured value of resistance from the series resistance necessary to obtain Q = 5. For both the capacitor and resistor, use the standard component value that is closest to your calculated value. 2. Once you have designed your circuit, construct it in Multisim and run a simulation using the Bode Plotter. Measure the resonant frequency and compare the measured value with the desired design value. Measure both the lower and upper -3dB frequencies and calculate the bandwidth B. 3. Calculate the value of Q and compare with the design value. 4. Make any modifications to the component values that you feel might be necessary and simulate again until you are satisfied that your results are reasonable. Your preliminary design and simulation results will be your ticket into the lab! 2 LABORATORY PROCEDURE / RESULTS 1. Experimentally determine the resonant frequency for your circuit by measuring the voltage across the resistor and adjusting the frequency of the function generator until that voltage is at its maximum value. 2. Set up a table in your lab notebook to record the values of the amplitude and phase shift of the voltage across the resistor with respect to the input voltage at frequencies of f=0.1f0 1 decade below f0 f=0.5 f0 1 octave below f0 f=f0 at resonance f=2f0 1 octave above f0 f=10f0 1 decade above f0 Be sure to keep the input voltage constant at 1 volt peak! 3. If time permits, shut down your circuit and “exchange” the positions of the resistor and the capacitor. Apply a 1 volt peak input voltage at the resonant frequency and measure the voltage (amplitude and phase) across the capacitor. Compare this voltage with the input voltage and comment on what you observed. 4. Each group member will submit Bode Plots of both the Gain and Phase of the output voltage as a function of frequency. Compare with the Bode Plots from Multisim. Comment on how well the experimentally obtained results compare with the theoretical results from Multisim. 3