KAHIBARO
Table of Contents
When Resonance Happens in AC Circuits
In alternating current circuits, resonance is a special condition that occurs when the effects of inductance and capacitance balance each other. This happens in circuits that contain both an inductor and a capacitor, especially RLC circuits. At resonance, the circuit responds very strongly to a particular driving frequency.
To understand this chapter, it is enough to remember that inductors and capacitors oppose alternating current in different ways. The inductor has reactance $X_L = \omega L$, which increases with frequency. The capacitor has reactance $X_C = \frac{1}{\omega C}$, which decreases with frequency. Resonance occurs when these two are equal.
Resonance in an AC circuit occurs when
$$
X_L = X_C
$$
which means
$$
\omega L = \frac{1}{\omega C}
$$
Therefore the resonant angular frequency is
$$
\omega_0 = \frac{1}{\sqrt{LC}}
$$
and the resonant frequency is
$$
f_0 = \frac{1}{2\pi \sqrt{LC}}
$$
Physical Meaning of Resonance
At resonance, the inductor and capacitor exchange energy back and forth. The inductor stores energy in its magnetic field, and the capacitor stores energy in its electric field. When the driving source has exactly the right frequency, this energy exchange matches the source perfectly.
In a series RLC circuit, the inductive and capacitive reactances cancel each other. That means the circuit behaves as if only the resistance remains. As a result, the impedance becomes minimum and the current becomes maximum.
In a parallel RLC circuit, the behavior is different in detail, but the same resonant frequency idea appears. The circuit can show a strong response at one preferred frequency, often with a large impedance and a small source current.
Resonance in a Series RLC Circuit
For a series RLC circuit, the impedance is
$$
Z = \sqrt{R^2 + (X_L - X_C)^2}
$$
At resonance, since $X_L = X_C$, this becomes
$$
Z = R
$$
So the current amplitude is
$$
I = \frac{V}{R}
$$
This is the largest current the circuit can have for a given applied voltage.
The phase angle between voltage and current is also important. In general,
$$
\tan \phi = \frac{X_L - X_C}{R}
$$
At resonance,
$$
\phi = 0
$$
So the current and voltage are in phase.
For a series RLC circuit at resonance,
$$
Z = R
$$
$$
I_{\max} = \frac{V}{R}
$$
$$
\phi = 0
$$
The circuit behaves like a purely resistive circuit.
Voltage Across the Inductor and Capacitor
Even though the total voltage across the series circuit may be moderate, the voltages across the inductor and capacitor individually can become very large at resonance, especially if the resistance is small. This happens because the current is large, and each reactive element has voltage given by
$$
V_L = I X_L, \qquad V_C = I X_C
$$
At resonance, $X_L = X_C$, so $V_L$ and $V_C$ have equal magnitudes but opposite phase. They cancel in the total sum, but each one separately can be much larger than the source voltage.
This is one reason resonance is useful in some devices and dangerous in others.
Frequency Response
A resonant circuit does not respond equally to all frequencies. Near the resonant frequency, the current in a series RLC circuit is large. Far from resonance, the current is smaller because the impedance increases.
This makes the circuit frequency selective. It can be used to pick out one frequency from many. Radios and tuning circuits use this idea.
The table below summarizes the behavior of a series RLC circuit at different frequency ranges.
| Frequency condition | Comparison of reactances | Circuit behavior | Current |
|---|---|---|---|
| $f < f_0$ | $X_C > X_L$ | More capacitive | Smaller than maximum |
| $f = f_0$ | $X_C = X_L$ | Purely resistive | Maximum |
| $f > f_0$ | $X_L > X_C$ | More inductive | Smaller than maximum |
Resonance Curve
If we plot current amplitude versus frequency for a series RLC circuit, we get a peak centered at the resonant frequency. A small resistance gives a sharp peak. A larger resistance gives a broader and lower peak.
This graph is called the resonance curve. It shows how strongly the circuit responds to different frequencies.
Sharpness of Resonance
Some resonant circuits are very selective, meaning they respond strongly only in a narrow range of frequencies. Others are less selective and respond over a broader range. The sharpness depends mainly on resistance.
If the resistance is small, energy is lost slowly, so resonance is sharp. If the resistance is large, energy is dissipated more quickly, so resonance is less sharp.
A common measure of sharpness is the quality factor, or $Q$. A full treatment of $Q$ belongs to a more advanced discussion, but for a series RLC circuit it is often written as
$$
Q = \frac{\omega_0 L}{R}
$$
A larger $Q$ means a sharper resonance.
Small resistance produces sharper resonance.
Large resistance produces weaker and broader resonance.
For a series RLC circuit, a common measure is
$$
Q = \frac{\omega_0 L}{R}
$$
Bandwidth
The circuit responds most strongly near resonance, but not only at exactly one frequency. The range of frequencies over which the response remains significant is called the bandwidth.
A high $Q$ circuit has a narrow bandwidth. A low $Q$ circuit has a wide bandwidth. For tuning applications, narrow bandwidth is often useful because it helps separate nearby frequencies.
For many series RLC circuits,
$$
Q = \frac{f_0}{\Delta f}
$$
where $\Delta f$ is the bandwidth between the two half power frequencies.
At the half power frequencies, the power delivered to the resistor is half its maximum value. Since power is proportional to $I^2$, the current there is
$$
I = \frac{I_{\max}}{\sqrt{2}}
$$
Energy Exchange at Resonance
Resonance can be understood as repeated energy transfer between the capacitor and inductor. The capacitor stores energy as
$$
U_C = \frac{1}{2} C V^2
$$
and the inductor stores energy as
$$
U_L = \frac{1}{2} L I^2
$$
At resonance, energy flows back and forth between these two forms while the resistor removes some energy as heat. The AC source replaces that lost energy. If the source frequency matches the natural electrical oscillation of the circuit, the exchange becomes especially effective.
Practical Importance
Resonance is very important in electrical engineering and physics. Tuning circuits in radios use resonance to select one station. Filters use resonance to allow some frequencies through and reduce others. Oscillators and signal processing circuits also depend on resonant behavior.
At the same time, resonance can create very large currents or voltages, which may damage components if the circuit is not designed properly.
Resonance is useful for frequency selection, tuning, and filtering.
It can also cause dangerously large currents or voltages in RLC circuits.
Summary Equations
The most important formulas for AC resonance are collected here.
| Quantity | Formula |
|---|---|
| Inductive reactance | $X_L = \omega L$ |
| Capacitive reactance | $X_C = \frac{1}{\omega C}$ |
| Resonance condition | $X_L = X_C$ |
| Resonant angular frequency | $\omega_0 = \frac{1}{\sqrt{LC}}$ |
| Resonant frequency | $f_0 = \frac{1}{2\pi\sqrt{LC}}$ |
| Series RLC impedance | $Z = \sqrt{R^2 + (X_L - X_C)^2}$ |
| Impedance at resonance | $Z = R$ |
| Maximum current at resonance | $I_{\max} = \frac{V}{R}$ |
| Phase angle | $\tan\phi = \frac{X_L - X_C}{R}$ |
| Phase at resonance | $\phi = 0$ |