Chaz Taylor3/16/17-3/23/17PoudelPHY2049L .021Experiment 8.1: Series LCR resonancePurposeStudy resonance in a series inductor-capacitor-resistor (LCR) circuit by examining the voltage across the resistor as a function of frequency of the applied sine waveEquipmentL-C-R board with L=3.3 mH, C=0.39 μF, R=100Ω, PASCO 850 interface, one voltage sensor (PASCO UI-5100), computerTheorySeries LCR resonance:The amplitude of the ac current (I0)in a series LCR circuit depends on the amplitude of the applied voltage (V0)and the impedence (Z) og the LCR circuit. Notice that the upper case I0,V0,VR,VL,VCrepresent the amplitudes of the current and voltages, the lower casei, v,vR,vL,vCrepresent the instantaneous values of the current and voltages. Where XL=ω L(inductive reactance), XC=1(ωC)(capacitive reactance), R(resistance), angular frequencyω=2π f(is the linear frequency) and φ is the phase angle between the current (I) and the voltage (v)of the source. Since the impedence depends onfrequency, the current and voltages vary with frequency.

The current will be maximum when the circuit is driven at its resonance frequency:ωres=2π fres=1√LC. At resonance,XL=XC,Z=Zmin=R ,I0=Imax=V0/R,φ=0the current i(t) andvoltage v(t) are in a phase, and VR=VRMAX=V0. These results are derived based on a assumption: the inductor is ideal which has only inductive reactance without resistance. In practice, any real inductor has resistance and therefore at resonance VR=VRMAX<V0, however, this fact wont affect the present experiment.Lissajous curve:If we apply the source voltage v(t)=V0sin(ωt+φ)and the voltage across the resistor vR(t)=VRsinωtto they axis and the x axis inputs respectively of an oscilloscope, the phase difference between v and VR will produce an elliptical pattern on the screen, which is a Lissajous curve. At resonance of the series LCR circuit, φ=0, Lissajous curve becomes a straight line which is themost intuitive and accurate approach for finding the resonance frequency of the series LCR circuit.Fast Fourier transform (FFT):A fast Fourier transform (FFT) is an algorithm to compute the discreteFourier transform (DFT) and its inverse. Fourier analysis converts time or space to frequency and vice versa. FFT has been described as the “most important numerical algorithms of our life time, and is