Memorial University of Newfoundland
Department of Physics and Physical Oceanography
Physics 2055 Laboratory
AC Resonance
The total impedance of a circuit with L, C and R in series is given by
Z=
q
R2 + (XL − XC )2
(1)
or, in complex notation, Z = R + jωL − 1/ jωC. The corresponding phase angle φ between the
current and voltage is given by
XL − XC
(2)
tan φ =
R
so that when XL > (or <) XC , the current lags behind (or leads) the applied voltage. When
XL = XC , the phase angle between current and voltage will be zero; the frequency at which this
occurs is called the resonant frequency. The resonant frequency is, therefore, given by
1
ω◦ = √
LC
or
f◦ =
1
√
2π LC
(3)
Imax
Imax
2
∆ω
ωo
ω
Figure 1: Resonance Curve
At this point the net reactance of the circuit is zero and the impedance of the circuit is just
√
|Z| = R. The current in the circuit is a maximum, falling to 1/ 2 of the maximum value at the half
power points.
The shape of the curve is usually described by a dimensionless parameter known as the Quality
1
Factor, given by
Q=
ω◦
ω◦ L
=
∆ω
R
(4)
Procedure
1. Construct the series circuit as shown with R ≈ 200 Ω, C ≈ 22 nF and L ≈ 85 mH. Remember
that the inductor has an associated resistance which must be measured and included in the
total resistance of the circuit.
VX
VL
r
VC
L
C
R
VR
Figure 2: Series Resonant Circuit
2. Measure accurately the values of R, C and L which you use, and calculate the resonant
frequency f◦ and the quality factor Q of the circuit, so that if VR is the voltage measured
across the load resistor, the gain is given by
R
Gain = p
(R + r)2 + (XL − XC )2
(5)
3. Set the signal generator to output a ∼ 500 Hz sine wave with a p-p amplitude of approximately 5 V. Be careful to keep this input signal constant throughout the experiment. Use the
oscilloscope to measure the voltage across R as the frequency is increased past resonance.
4. Plot gain vs. frequency and explain the shape of the graph. Use your graph to estimate f◦
and Q. How well do these values agree with your predictions from Equations (3), (4) and
2
(5)?
5. Increase the load resistance to 1 kΩ and repeat the measurements of gain and frequency. Plot
this curve on the same axes as the previous plot and account for any difference between the
two resonance curves.
6. The voltage across the LC combination is expected to be zero at resonance. Measure it
using a DMM, and also measure the voltage across the inductor and the capacitor separately.
Comment briefly on what you find.
Phase in an LCR circuit
For the same range of frequencies studied, use the oscilloscope to measure the phase difference
between the input and output signals. (Do this for one value of load resistance only.) Plot your
experimental data and the curve predicted from Eq (2), and explain what happens to the phase
• at low frequencies
• at high frequencies
• at resonance.
3