Resonance

Resonance - PHY 132 LAB: Resonance in LRC circuit...

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PHY132 Labs ( © P. Bennett, JCHS) -1- 08/11/04 PHY 132 LAB: Resonance in LRC circuit 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 sinewave 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 maximum 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. Resonant circuits are used in radio and television transmitters. If the frequency is above about 0.5 megaHz, the circuit will transmit electromagnetic waves into space travelling at the speed of light. Cell phones operate at about 900 megaHz, television is at about 100 MHz, and AM radio at about 1 MHz. Frequencies below about 40 MHz are reflected by the ionosphere, and bounce around the earth, so these "long-wave" bands are best for world-wide communication. Higher frequencies travel along straight line-of-sight paths into outer space, so they are used to communicate with satellites. At low frequencies (below about 12 kHz) the alternating voltages can be heard by the human ear if connected to a speaker. Resonant circuits can thus be used to generate tones for a synthesiser (electronic keyboard). 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 Kirchoff’s law (sum of instantaneous voltages around a closed loop is zero): ν i i = 0 eq. 1
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PHY132 Labs ( © P. Bennett, JCHS) -2- 08/11/04 This takes the form L di ( t ) dt + 1 C i ( t ) dt + i ( t ) R = v in ( t ) eq. 2 The terms correspond to voltage on: the coil (Faraday’s Law V = -Ldi/dt), capacitor (V c =Q/C), and resistor (V R =iR). This equation has a relatively simple solution provided v in (t) is a sinewave. Thus we assume v in (t) = V 0 sin( ω t) eq. 3 where V 0 is a constant amplitude fixed by the power supply. Current flows with the same frequency (of course!), but maybe different phase, and is given by
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Resonance - PHY 132 LAB: Resonance in LRC circuit...

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