AC Circuits and Electrical Resonance
Department of Physics & Astronomy
Texas Christian University, Fort Worth, TX
June 29, 2016
1
Introduction
Consider an AC circuit containing a resistor, an inductor, and a capacitor connected in series, as
seen in Fig. 1 (left). Remember that the same current flows through all three elements. Since the
current is common to all elements, we will take it as a reference, and will measure voltages across
the resistor, the capacitor and the inductor with respect to the current. It is convenient to present
the results in the form of a graph in which the horizontal axis represents the current, see Fig. 1
(right). The voltage across the resistor is given by Ohm’s law
VR = IR
(1)
and is in phase with the current. Thus, VR is displayed on the x-axis. The voltage across the
inductor,
VL = ωLI
(2)
leads the current by 90◦ , and it will be presented along the positive y-axis. The voltage across the
capacitor,
I
VC =
(3)
ωC
lags the current by 90◦ , and is also presented on the vertical axis.
To obtain the resultant voltage, ε, we need to add voltages VR , VL , and VC as vectors. The
vector addition is illustrated in Fig. 1 (right). Because the vectors VR and VC or VL form a right
triangle, ε may be found from
ε2 = VR2 + (VL − VC )2 .
(4)
Substituting Eqs. (1)–(3) into Eq. (4), leads to
s
2
1
ε = I R2 + ωL −
= IZ,
ωC
(5)
p
where Z = R2 + (ωL − 1/ωC)2 , and is called the impedance of the circuit. The phase difference
between the current, I, and the line voltage, ε, is given by
tan φ =
ωL −
R
1
1
ωC
.
(6)
Figure 1: AC circuits. For an AC circuit containing a resistor, an inductor and a capacitor in
series (left), the voltages across the three components are represented in a phase diagram (right).
From Eq. (6) it is seen that for ωL = 1/ωC the phase difference, φ, is zero and the impedance,
Z, equals R. This condition
1
ωL =
(7)
ωC
is called resonance, and it can be reached by changing any of the quantities ω, C, or L. In this
experiment, to reach resonance, you will change ω, and observe what happens to the voltage across
the resistor. When Z equals R, the current is maximum and the circuit is said to be in resonance.
In this experiment you will find resonance by changing the frequency of the system. ω measured
in radians per second, and is related to the frequency, measured in cycles per second as ω = 2πf .
You can compare your measured value to one calculated using the resonance condition (ωL =
1/ωC).
2
Equipment
Function generator, LRC circuit, voltmeter, wires
3
Procedure
1. Set the function generator to produce a sinusoidal function. Set the frequency to 10 kHz.
2. Connect in series one capacitor, one inductor and one resistor on the circuit board. Connect
the circuit board to the function generator.
3. Connect the voltmeter across the resistor. Make sure that it is set to read AC current.
4. Calculate the value of frequency at which you expect resonance to occur using the equation,
ω=√
2
1
.
LC
5. Starting at a frequency below what you calculated, change the frequency in steps, larger steps
are fine initially, but once the voltage starts to increase rapidly, take smaller steps. Record
the voltage across the resistor as a function of frequency.
6. Repeat the procedure for a different combination of capacitor, inductor and resistor. Make
sure you note the values of capacitance, resistance, and inductance.
4
Report
Make sure your reports include the following:
1. Include graphs of voltage as a function of frequency. Indicate your calculated resonance
frequency on the graph.
2. Estimate the resonance frequency from your graph and calculate the percent difference between your graph value and your calculated value. Comment on why there might be discrepancies.
3