Physics · Ch 7 — Alternating Current

Resonance

7.6.2

Resonance

What is Resonance?

Resonance is a phenomenon that occurs in any system that has a natural frequency — the frequency at which it prefers to oscillate. When such a system is driven by an external source at a frequency close to its natural frequency, the amplitude of oscillation becomes very large.

Example: A child on a swing. The swing has a natural frequency. If the child pulls the rope at the same frequency as the swing's natural frequency, the swing goes higher and higher.

In a series RLC circuit, the current amplitude depends on the driving frequency $\omega$ . The circuit has a natural frequency at which the inductive and capacitive effects cancel each other out. This is the resonant frequency.

The Condition for Resonance

In a series RLC circuit, the impedance $Z$ is given by:

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

where:

$X_L = \omega L$ is the inductive reactance.
$X_C = \frac{1}{\omega C}$ is the capacitive reactance.

The current amplitude $i_m$ is:

i_m = \frac{v_m}{Z} = \frac{v_m}{\sqrt{R^2 + (X_L - X_C)^2}}

At a particular frequency $\omega_0$ , the inductive and capacitive reactances become equal:

X_L = X_C

This is the condition for resonance. When this happens:

The impedance is minimum: $Z = R$ .
The current amplitude is maximum: $i_m = \frac{v_m}{R}$ .

The Resonant Frequency

Setting $X_L = X_C$ gives:

\omega_0 L = \frac{1}{\omega_0 C}

Solving for $\omega_0$ :

\omega_0 = \frac{1}{\sqrt{LC}}

This is the resonant angular frequency. The corresponding resonant frequency $f_0$ (in Hz) is:

f_0 = \frac{\omega_0}{2\pi} = \frac{1}{2\pi\sqrt{LC}}

Key Observations from the Textbook

Figure 7.14 shows how $i_m$ $i_{m}$ varies with $\omega$ $ω$ for a circuit with $L = 1.00 \text{ mH}$ $L = 1.00 mH$ , $C = 1.00 \text{ nF}$ $C = 1.00 nF$ , and $v_m = 100 \text{ V}$ $v_{m} = 100 V$ .
- For $R = 100\ \Omega$ , the peak current is $i_m = 100/100 = 1.0 \text{ A}$ .
- For $R = 200\ \Omega$ , the peak current is $i_m = 100/200 = 0.5 \text{ A}$ .
- The resonant frequency $\omega_0$ is $1.00 \times 10^6 \text{ rad/s}$ .
The current amplitude at resonance is inversely proportional to $R$ . A smaller $R$ gives a sharper, higher peak.

Why Resonance is Important: Tuning

Resonant circuits are used in tuning (e.g., in radios and TVs).

The antenna picks up signals from many stations at different frequencies.
The tuning circuit (a series RLC circuit) is driven by these signals.
By varying the capacitance $C$ (or inductance $L$ ), we change the resonant frequency $\omega_0$ of the circuit.
When $\omega_0$ matches the frequency of a desired station, the current amplitude for that signal becomes maximum. …

Figure 7.14Variation of i_m with ω for two cases: (i) R = 100 Ω, (ii) R = 200 Ω, L = 1.00 mH.

Fig. 7.14 — Variation of i_m with ω for two cases: (i) R = 100 Ω, (ii) R = 200 Ω, L = 1.00 mH.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

The figure is a resonance curve for a series RLC circuit. It plots the current amplitude $i_m$ (in amperes) on the vertical axis against the angular frequency $\omega$ (in Mrad/s) on the horizontal axis.

Horizontal axis ( $\omega$ ): Marked from 0.5 to 2.0 Mrad/s. The resonant frequency $\omega_0 = 1.0$ Mrad/s is clearly indicated.
Vertical axis ( $i_m$ ): Ranges from 0.0 to 1.0 A.
Two curves:
- Curve (i) ( $R = 100\ \Omega$ ): A tall, narrow peak reaching $i_m \approx 1.0$ A at $\omega_0$ .
- Curve (ii) ( $R = 200\ \Omega$ ): A shorter, broader peak reaching $i_m \approx 0.5$ A at the same $\omega_0$ .
Both curves fall symmetrically to near zero away from $\omega_0$ , showing that current is maximum only at resonance.

Physical idea: The circuit exhibits resonance when the inductive reactance $X_L$ equals the capacitive reactance $X_C$ . At this frequency, the impedance $Z$ is minimum ( $Z = R$ ), so the current amplitude is maximum. A smaller $R$ gives a sharper, higher peak; a larger $R$ gives a broader, lower peak.

Key formula: The current amplitude is

i_m = \frac{v_m}{Z} = \frac{v_m}{\sqrt{R^2 + (X_L - X_C)^2}}

where $X_L = \omega L$ , $X_C = 1/(\omega C)$ , and $v_m$ is the source voltage amplitude. At resonance,

$\omega_0 = \frac{1}{\sqrt{LC}} \quad \text{and} \quad i_m = \frac{v_m}{R}.$ …

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