Experiment 7 – October 9, 2012
RLC Resonance

Purpose
Study resonance in a series resistor-inductor-capacitor (RLC) circuit by examining the current through the circuit as a function of the frequency of the applied voltage.

Theory
The amplitude of the AC current (I0) in series RLC circuit depends on the amplitude of the applied voltage (V0) and the impedance (Z).

Where XL =ωL (inductive reactance), XC = 1/ωC (capacitave reactance), R (resistance), angular frequency ω = 2πf (f is linear frequency) and is the phase angle between the current (i) and voltage (v) pf the source. Impedance depends on frequency so the current varies with frequency. The current will be maximum when the circuit is driven at its resonance frequency, ωres = 2πfres = 1 / √LC. At resonance, XL = XC, Zmin = R, I0 = Imax = V0/ R, = 0. The current (i) and voltage (v) are in phase and Vr = V0. Three characteristics of resonance include: Vr reaches maximum value VRmax (at ω ≠ ωres VR < VRmax), the voltage v and voltage vR are in phase in which their peaks (V0 and Vr) are lined up, and the trace of vr and v in xy-plot is a straight line.

Data

Table 1 | R (Ω) | C (µF) | x2(s) | x3(s) | T (period)| x3 – x2| | fres (Hz) | fave.res (Hz) | Lave(mh) | 1 | 100 | 107.0 | .02859 | .04668 | .01809 | 55.2 | 55.2 | .00207 | | | | .02354 | .04162 | .01808 | 55.3 | | | | | | .02088 | .03896 | .01808 | 55.3 | | | 2 | 100 | 10.40 | .00332 | .00851 | .00519 | 192.6 | 192.3 | .00355 | | | | .00255 | .00779 | .00524 | 190.8 | | | | | | .0040 | .00556 | .00516 | 193.7 | | | 3 | 100 | 1.02 | .00734 | .01059 | .00325 | 307.6 | 307.9 | .00898 | | | | .00713 | .01037 | .00324 | 308.6 | | | | | | .00707 | .01032 | .00325 | 307.6 | | |

Table 2 x2(s) | .0023 | .0021 | .0052 | .0047 | .0045 | .0104 | .0104 | .0090 | .0042 | .0167 | .0090 | .0417 | 0 | x3(s) | .0033 | .0034 | .0068 | .0067 | .0069 | .0136 | .0152 | .0142 | .0110 | .0262 | .0414 | .0920 | 0 | T(s) | .001 | .0013 | .0016 | .002 | .0024 | .0032 | .0048 | .0052 | .0068 | .0095 | .0324 | .0503 | 0 | f(Hz) | 1000 | 769.2 | 625 | 500 | 416.6 | 312.5 | 208.3 | 192.3 | 147.1 | 105.2 | 30.86 | 19.88 | 0 | y2(V) | -0.66 | -0.83 | -1.06 | -1.23 | -1.39 | -1.60 | -1.77 | -1.73 | -1.70 | -1.51 | -1.11 | -0.40 | 0 | y3(V) | 0.66 | 0.86 | 1.05 | 1.22 | 1.39 | 1.59 | 1.74 | 1.74 | 1.72 | 1.55 | 1.10 | 0.43 | 0 | V(V) | 1.32 | 1.69 | 2.11 | 2.45 | 2.78 | 3.19 | 3.51 | 3.47 | 3.42 | 3.06 | 2.21 | 0.83 | 0 |

Voltage vs Frequency

Questions 1. 1. Vr reaches maximum value VRmax (at ω ≠ ωres VR < VRmax).
2. The voltage v and voltage vR are in phase in which their peaks (V0 and Vr) are lined up.
3. The trace of vr and v in xy-plot is a straight line. 2.