Resonance in RLC Circuits

Resonance is the tendency of a system to oscillate with greater amplitude at some frequencies—in an RLC series circuit, it occurs at $\nu_0 = \frac{1}{\sqrt{2 \pi LC}}$.

Key Points

• Resonance condition of an RLC series circuit can be obtained by equating XL and XC, so that the two opposing phasors cancel each other.

• At resonance, the effects of the inductor and capacitor cancel, so that Z=R, and Irms is a maximum.

• Higher-resistance circuits do not resonate as strongly compared to lower-resistance circuits, nor would they be as selective in, for example, a radio receiver.

Terms

• The opposition to the change in flow of current in an alternating current circuit, due to inductance and capacitance; the imaginary part of the impedance.

• Root mean square: a statistical measure of the magnitude of a varying quantity.

• A measure of the opposition to the flow of an alternating current in a circuit; the aggregation of its resistance, inductive and capacitive reactance. Represented by the symbol Z.

Figures

1. RLC Series Circuit

An RLC series circuit with an AC voltage source. f is the frequency of the source.

2. Current vs. Frequency

A graph of current versus frequency for two RLC series circuits differing only in the amount of resistance. Both have a resonance at f0, but that for the higher resistance is lower and broader. The driving AC voltage source has a fixed amplitude V0.

Resonance is the tendency of a system to oscillate with greater amplitude at some frequencies than at others. Frequencies at which the response amplitude is a relative maximum are known as the system's resonance frequencies. To study the resonance in an RLC circuit, as illustrated in Figure 1, we can see how the circuit behaves as a function of the frequency of the driving voltage source.

Combining Ohm’s law, Irms=Vrms/Z, and the expression for impedance Z from

$Z = \sqrt{R^2 + (X_L - X_C)^2}$ gives

$I_{rms} = \frac{V_{rms}}{\sqrt{R^2 + (X_L - X_C)^2}}$,

where Irms and Vrms are rms current and voltage, respectively. The reactances vary with frequency $\nu$, with XL large at high frequencies and XC large at low frequencies given as:

$X_L = 2\pi \nu L, X_C = \frac{1} {2\pi \nu C}$).

At some intermediate frequency $\nu_0$, the reactances will be equal and cancel, giving Z=R —this is a minimum value for impedance, and a maximum value for Irms results. We can get an expression for $\nu_0$ by taking XL=XC. Substituting the definitions of XL and XC yields:

$\nu_0 = \frac{1}{2\pi \sqrt{LC}}$

$\nu_0$is the resonant frequency of an RLC series circuit. This is also the natural frequency at which the circuit would oscillate if not driven by the voltage source. At $\nu_0$, the effects of the inductor and capacitor cancel, so that Z=R, and Irms is a maximum. Resonance in AC circuits is analogous to mechanical resonance, where resonance is defined as a forced oscillation (in this case, forced by the voltage source) at the natural frequency of the system.

The receiver in a radio is an RLC circuit that oscillates best at its $\nu_0$. A variable capacitor is often used to adjust the resonance frequency to receive a desired frequency and to reject others. Figure 2 is a graph of current as a function of frequency, illustrating a resonant peak in Irms at $\nu_0 = f_0$. The two curves are for two different circuits, which differ only in the amount of resistance in them. The peak is lower and broader for the higher-resistance circuit. Thus higher-resistance circuits do not resonate as strongly, nor would they be as selective in, for example, a radio receiver.

Key Term Glossary

AC
Alternating current.
Appears in these related concepts:
amplitude
The maximum absolute value of some quantity that varies.
Appears in these related concepts:
capacitor
An electronic component capable of storing an electric charge, especially one consisting of two conductors separated by a dielectric.
Appears in these related concepts:
circuit
A pathway of electric current composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow. T
Appears in these related concepts:
current
The time rate of flow of electric charge.
Appears in these related concepts:
frequency
The quotient of the number of times n a periodic phenomenon occurs over the time t in which it occurs: f = n / t.
Appears in these related concepts:
impedance
A measure of the opposition to the flow of an alternating current in a circuit; the aggregation of its resistance, inductive and capacitive reactance. Represented by the symbol Z.
Appears in these related concepts:
inductor
A passive device that introduces inductance into an electrical circuit.
Appears in these related concepts:
Inductor
A device or circuit component that exhibits significant self-inductance; a device which stores energy in a magnetic field.
Appears in these related concepts:
Law
A concise description, usually in the form of a mathematical equation, used to describe a pattern in nature
Appears in these related concepts:
ohm
in the International System of Units, the derived unit of electrical resistance; the electrical resistance of a device across which a potential difference of one volt causes a current of one ampere; symbol: Ω
Appears in these related concepts:
oscillate
To swing back and forth, especially if with a regular rhythm.
Appears in these related concepts:
oscillation
the act of oscillating or the state of being oscillated
Appears in these related concepts:
reactance
The opposition to the change in flow of current in an alternating current circuit, due to inductance and capacitance; the imaginary part of the impedance.
relative
Expressed in relation to another item, rather than in complete form.
Appears in these related concepts:
resistance
The opposition to the passage of an electric current through that element.
Appears in these related concepts:
resonance
The increase in the amplitude of an oscillation of a system under the influence of a periodic force whose frequency is close to that of the system's natural frequency.
Appears in these related concepts:
rms
Root mean square: a statistical measure of the magnitude of a varying quantity.
Appears in these related concepts:
rms current
the root mean square of the current, Irms=I0/√2 , where I0 is the peak current, in an AC system
Appears in these related concepts:
series
A number of things that follow on one after the other or are connected one after the other.
Appears in these related concepts:
voltage
The amount of electrostatic potential between two points in space.