An Introduction to Resonance in the RLC Circuits - Experiment 10 | PHY 132 (LAB), Lab Reports of Physics

Download An Introduction to Resonance in the RLC Circuits - Experiment 10 | PHY 132 (LAB) and more Lab Reports Physics in PDF only on Docsity! -1- PHY 132 – Summer 2000 LAB 10: Resonance in LRC Circuit1 Introduction In this lab we will measure the steady-state behavior of a resonant system. Specifically we will look at the forced response of a series LRC circuit to a sine wave input. This builds on the previous lab where we looked at the phase behavior of each component in the same circuit. There also is a close link with mechanical resonant behavior, which many of you studied in PHY122, in context of “damped oscillations”. We will see that the power transfer to the load resistor is maximal at resonant frequency. We will also look at the transient behavior, and analyze the digitized waveform to estimate the time constant τ and the resonance quality factor Q. Theory: R C V_in L Fig. 1 Generic series LRC circuit. Consider the series LRC circuit as shown in Fig. 1. We start with Kirchhoff’s law (sum of instantaneous voltages around a closed loop is zero): ∑ = j jv 0 Eq. 1 This takes the form )()()( 1)( tvRtidtti Cdt tdi L in=++ ∫ Eq. 2 1 Adapted by R. J. Jacob from P. Bennett, PHY-132 Lab Manual (ASU) -2- The terms correspond to voltage on: the coil (Faraday’s Law V = -Ldi/dt), capacitor (Vc=Q/C), and resistor (VR=iR). This equation has a relatively simple solution provided vin(t) is a sine wave. Thus we assume vin(t) = V0sin(ω t) Eq. 3 where V0 is a constant amplitude fixed by the power supply. Current flows with the same frequency (of course!) and is given by i(t) = I(w)* sin(ω t - Φ) Eq. 4 Notice that the amplitude depends on frequency, and is given by Ohm’s law as I(w) = V0/XLRC Eq. 5 where the reactance of the series circuit XLRC is given by (XLRC) 2 = R2 + (XL-XC) 2 Eq. 6 Putting in for XL and XC we then have 2/1 2 2 0 1 )(               −+ = C LR V I ω ω ω Eq. 7 It is useful to look now at the time-averaged power consumed in the circuit. This is given by <P(w)> = <i2(t)R> = ½I0 2R or               −+ = 2 2 2 0 1 2/ )( C LR RV P ω ω ω Eq. 8 This function is sharply peaked at the resonant frequency ω 0 defined by XL=XC or ω 0L = 1/(ω 0C) or ω 0 2LC=1 Eq. 9 -5- Fig. 2 Circuit used for measuring LRC behavior. The circuit we will use is shown in Fig. 2. Note that the coil is non-superconducting and has an unavoidable non-zero internal resistance “r”. This circuit element is only experimentally accessible as the combination (L+r). The internal “r” contributes to the total circuit resistance R in the equations above. Thus we have R = Rload +r Eq. 19 Scope ch #2 senses the input voltage, while scope ch #1 senses the series current, given by voltage across the series resistor Rload. This is also measured by the DVM (rms values, of course) set on the AC voltage scale (NOT current scale!). You can calculate the current from Ohm’s law. Procedure 1. Before connecting the circuit, measure the coil internal resistance “r” using the DVM. 2. Connect the circuit of Fig. 2 using Rload = 50 ohms, C = 0.1 µfd and L = 85mH. Use the “Low ohms” output of the signal generator. Get both waves showing on the scope simultaneously. We will use the DVM for most of the data, but its useful to “see” the waveforms on the scope, nonetheless. Input a sine wave about 1 Vrms at 2.0 kHz. (Temporarily move the DVM over to the input). R_load C L r Scope ch#1Scope ch#2 Sig. Gen. Ground DVM -6- 3. Move the DVM back to the resistor. Find and record the resonant frequency f0 where the current through Rload is maximum. Take 3 independent readings to get ±. 4. Measure the current vs. frequency I(w) as you tune through resonance. Take about 15 points total, going on both sides of resonance, far enough away so I0 drops to ~ 20% of its value at resonance. 5. (transient) Change the input to a square wave about 30 Hz. The output wave (scope ch#1) should show a damped sine wave starting with each up/down of the square- wave. Open the setup file “phy132/lab12” and capture the waveform. You need not export this data file, we will do all analysis by eyeball of the v(t) plot. Analysis 1. Compare your resonant frequency f0 (procedure part 3) with theory. Component values are accurate to R(1%), C(10%) and L(5%). 2. Working in GA, calculate and plot P(ω ) derived from current measurements. Mark the FWHM points on the P(ω ) curve, and determine Q from this. 3. Fit P(ω ) to determine Q, and w0 and R. You should put in Vrms as a known constant. See notes in “fits.pdf”. 4. Do eyeball fits for Q, and w’ based on print-out of the transient waveform v(t). 5. Compare your values (and errors) of Q and ω 0 from the various methods.