PHYSICS COURSE NAME
LAB 05
LCR CIRCUITS (RESONANCE)
Lab format: This lab is performed via an internet connection with the Remote Web-based Science
Laboratory (RWSL).
Relationship to theory: This lab corresponds to the study of LCR circuits and resonance in an electrical
circuit.
OBJECTIVES
Part I – to measure the impedance and phase angle of a LCR circuit
Part II – to investigate resonance in a LCR circuit.
EQUIPMENT LIST
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LCR circuit for RWSL
Oscilloscope - ELVIS II
Function generator – ELVIS II
Digital Multimeter – ELVIS II
INTRODUCTION
Figure 1: LCR series circuit with an AC source
The impedance of the LCR circuit shown in Figure 1 can be worked out by examining the complex or
phasor combination of voltage amplitudes across each circuit component.
𝑉𝑚𝑎𝑥 2 = (𝑉𝐿𝑚𝑎𝑥 − 𝑉𝐶𝑚𝑎𝑥 )2 + (𝑉𝑅𝑚𝑎𝑥 )2
or
𝑉𝑚𝑎𝑥 = √(𝑉𝐿𝑚𝑎𝑥 − 𝑉𝐶𝑚𝑎𝑥 )2 + (𝑉𝑅𝑚𝑎𝑥 )2
Equation 1
Equation 1can be re-written in terms of resistance and reactances as
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LAB 05
𝑉𝑚𝑎𝑥 = √(𝐼𝑚𝑎𝑥 𝑋𝐿 − 𝐼𝑚𝑎𝑥 𝑋𝐶 )2 + (𝐼𝑚𝑎𝑥 𝑅)2
Equation 2
By grouping the terms, we can solve for the impedance Z of the LCR circuit as
𝑉𝑚𝑎𝑥 = 𝐼𝑚𝑎𝑥 √(𝑋𝐿 − 𝑋𝐶 )2 + 𝑅 2
𝑍=
𝑉𝑚𝑎𝑥
= √(𝑋𝐿 − 𝑋𝐶 )2 + 𝑅2
𝐼𝑚𝑎𝑥
Equation 3
Using this expression (Equation 3) for impedance, we can get at the resonance as follows. Solving for
Imax
𝐼𝑚𝑎𝑥 =
𝑉𝑚𝑎𝑥
𝑍
=
𝑉𝑚𝑎𝑥
√(𝑋𝐿 −𝑋𝐶 )2 +𝑅2
Equation 4
Equation 4 shows that the current takes on its maximum value (called Imax o) when (𝑋𝐿 − 𝑋𝐶 ) →
0. This occurs at a specific frequency 𝜔𝑜 = 2𝜋𝑓𝑜 given by the following.
𝑋𝐿 = 𝑋𝐶
𝜔𝑜 𝐿 = 1⁄𝜔 𝐶
𝑜
𝜔𝑜 = 1⁄
√𝐿𝐶
𝑓𝑜 = 1⁄
2𝜋√𝐿𝐶
Equation 5
Here, fo is called the natural frequency of the circuit. Resonance occurs when the frequency f of the
voltage source is matched to the natural frequency of the circuit. The frequency response of an LCR
series circuit near resonance is shown in Figure 2.
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LAB 05
Figure 2: Frequency response of a LCR series circuit near resonance
The relative width of the resonance peak ∆𝑓⁄𝑓𝑜 is called the “bandwidth.” A smaller bandwidth
means a sharper peak. If f is defined as the full width of the peak when the current amplitude is at
1⁄√2 of its peak value (this corresponds to 1/2 of the peak power), then it can be shown (by an exercise
for the student) that the bandwidth of the LCR circuit is given by
∆𝑓
𝑓𝑜
1
=𝑄
Equation 6
where Q is known as the quality factor and is given by
1
𝐿
𝑄 = 𝑅 √𝐶
Equation 7
The quality factor is an important parameter of a resonance circuit. Many additional properties of the
circuit can also be related simply to Q. In light of Equation 6, a circuit exhibiting a narrow bandwidth is
also referred to as a “high Q” circuit. High Q circuits are used, for example, in radio receivers to provide
tuning.
WARNINGS

There are no special safety warnings in this lab.

If connecting your own instruments rather than RWSL, keep in mind that many instruments are
connected to each other through a common ground. Unintentionally creating a ground loop is
a common rookie mistake.
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LAB 05
PROCEDURE
Part I: Series Resonance
 Go to the RWSL-prepared circuit shown in Figure 3.
Figure 3: setup of LCR series circuit.
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We will use component values L = 25 mH and C = 0.1 F.
There is no resistor in this circuit but a resistance R is shown in the schematic diagram because
the value of R you will work with needs to represent the total resistance of the circuit from all
sources. Estimate a value for R using instrument specifications.
Set the sine wave voltage to 1.0 Volt, rms.
Calculate the theoretical natural frequency fo and the theoretical Q value of the circuit using
Equation 5 and Equation 7.
Measure Irms using the ammeter as you vary the frequency (on the ELVIS II function generator –
see Figure 4: Function generator controls and oscilloscope display) over a range of values
near resonance. You will need to make a reasonable estimate of what this range should be.
Plot a graph of Irms vs f, as in Figure 2, making sure to obtain enough data points to plot a
smooth curve, sufficient to perform the next step.
From the graph, determine values for fo and f, then calculate Q. Compare with theoretical
values obtained earlier.
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Figure 4: Function generator controls and oscilloscope display.
ANALYSIS AND/OR QUESTIONS

What factors are important in estimating R?

Describe the relationship between predicted and measured values for fo and Q.

Equation 4 predicts infinite amplitude at resonance. Discuss.
Original lab manual by John Wonghen and David Everrit.
Adapted for remote delivery by T. Sato under the Remote Science Labs for Second Year
Physics Project (2012 – 2013) funded by BCcampus.
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