B6, General Physics Experiment II ω LC Fall Semester, 2020 Name: Team No. : Department: Date : Student ID: Lecturer’s Signature : Introduction Goals Understand a forced oscillation and resonance. Measure the natural angular frequency (ω LC ) of a RLC circuit. Understand phase shifts among v R (t), v L (t), and v C (t). Make Lissajous curves with v R (t), v L (t), and v C (t). Theoretical Backgrounds 1. RLC circuit Consider a circuit consisting of a resistor of resistance R, a capacitor of capacitance C , an inductor of inductance L and an AC emf E : E (t)= E m sin ω d t. (a) The capacitive reactance is X C = 1 ω d C . (b) The inductive reactance is X L = ω d L. (c) The impedance is Z = R + j (X L - X C )= |Z |e jφ , where j ≡ √ -1 and |Z | is the absolute value for the impedance |Z | = p R 2 +(X L - X C ) 2 and the phase φ is given by φ = arctan Im[Z ] Re[Z ] = arctan X L - X C R . (d) The current of the RLC circuit is i(t)= I Z sin(ω d t - φ). (e) The amplitude I Z for the current is I Z = E m p R 2 +(X L - X C ) 2 . 2. Resonance frequency (a) The amplitude I Z (ω d ) for the current of the RLC circuit is maximized when |Z | is minimum. (b) This happens when the imaginary part of Z vanishes: Im[Z ]= X L - X C =0. (c) Substituting the ω dependence into X L and X C , we find that ω = 1 √ LC . (d) The resonance frequency f is f = 1 2π √ LC . 3. Phasor 2020 KPOPE All rights reserved. Korea University Page 1 of 5
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B6, General Physics Experiment II ωLC Fall Semester, 2020
Name:
Team No. :
Department:
Date :
Student ID:
Lecturer’s Signature :
Introduction
Goals
� Understand a forced oscillation and resonance.
� Measure the natural angular frequency (ωLC) ofa RLC circuit.
� Understand phase shifts among vR(t), vL(t), andvC(t).
� Make Lissajous curves with vR(t), vL(t), andvC(t).
Theoretical Backgrounds
1. RLC circuit
Consider a circuit consisting of a resistor ofresistance R, a capacitor of capacitance C, aninductor of inductance L and an AC emf E :
E (t) = Em sinωdt.
(a) The capacitive reactance is
XC =1
ωdC.
(b) The inductive reactance is
XL = ωdL.
(c) The impedance is
Z = R+ j(XL −XC) = |Z|ejφ,
where j ≡√−1 and |Z| is the absolute value
for the impedance
|Z| =√R2 + (XL −XC)2
and the phase φ is given by
φ = arctanIm[Z]
Re[Z]= arctan
XL −XC
R.
(d) The current of the RLC circuit is
i(t) = IZ sin(ωdt− φ).
(e) The amplitude IZ for the current is
IZ =Em√
R2 + (XL −XC)2.
2. Resonance frequency
(a) The amplitude IZ(ωd) for the current of theRLC circuit is maximized when |Z| isminimum.
(b) This happens when the imaginary part of Zvanishes:
Im[Z] = XL −XC = 0.
(c) Substituting the ω dependence into XL andXC , we find that
ω =1√LC
.
(d) The resonance frequency f is
f =1
2π√LC
.
3. Phasor
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B6, General Physics Experiment II ωLC Fall Semester, 2020
(a) The potential differences between the twoterminals of elements are given by
vR(t) = Ri(t)
= RIZ sin(ωdt− φ),
vL(t) = Ldi(t)
dt= XLIZ cos(ωdt− φ)
= XLIZ sin(ωdt− φ+ π2 ),
vC(t) =q(t)
C= −XCIZ cos(ωdt− φ)
= XCIZ sin(ωdt− φ− π2 ),
E (t) = Em sinωdt.
(b) The above results can be understood to be theimaginary part of the complex numbers shownin the following figure.
Each arrow on the complex plane is called aphasor.
(c) Every phasor rotates counterclockwise.
(d) VR is delayed by φ in comparison with E .
(e) The phase shift φ can be expressed as
tanφ =VL − VCVR
=XL −XC
R.
4. Lissajous curve: Lissajous curve is the graph of
a system of parametric equations.
(a) We find that vR(t) and vL(t) are
vR(t) = RIZ sin(ωdt− φ),
vL(t) = XLIZ cos(ωdt− φ).
The parametric equation can be obtained asan ellipse:
v2RR2
+v2LX2L
= I2Z .
(b) We find that vL(t) and vC(t) are
vL(t) = XLIZ cos(ωdt− φ),
vC(t) = −XCIZ cos(ωdt− φ).
The parametric equation can be obtained as aline segment:
vL = −XL
XCvC .
Instrumentation
1. RLC circuit
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B6, General Physics Experiment II ωLC Fall Semester, 2020
2. Connection
Experimental Procedure
1. Excel Sheet 1 ωLC
� Find the resonance frequency ωLC .
� Fit the voltage input graphs to a sinefunction.
� Fill out the sheet 1 of the excel file.
2. Excel Sheets 2, 3 Phase Shift
� Plot the VL-VR and VL-VC Lissajous curves.
� Fill out the sheets 2, 3 to determine the phasedifferences among R and L and between Land C.
3. Excel Sheet 4 Plot the phasors of VAC , VR, VL,VC simultaneously.Interpret physical implication of the graphicalresults.
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B6, General Physics Experiment II ωLC Fall Semester, 2020
Name:
Team No. :
Department:
Date :
Student ID:
Lecturer’s Signature :
Discussion (7 points)
Problem 1Problem 2
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B6, General Physics Experiment II ωLC Fall Semester, 2020
Problem 3
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