Q.In a series LCR circuit, the peak current $I_{max}$ is plotted against the driving angular frequency $\omega$. The plot is a resonance (bell-shaped) curve: the vertical axis $I_{max}$ is marked in amperes at $0.5$ and $1.0$, and the horizontal axis $\omega$ is marked in rad/s at $0.5,\ 1.0,\ 1.5$ and $2.0$. Starting near zero, the current rises to a peak of about $1.0$ A close to $\omega_0\approx 1.2$ rad/s and then falls back towards zero. Find the bandwidth of this resonance curve and indicate the half-power points on it.

Resonance in AC Circuits: Why the Key Formulas Hold

Resonance in an AC circuit occurs when the inductive reactance ($X_L$) and capacitive reactance ($X_C$) exactly cancel each other out. Let's build the understanding step-by-step.

1. The Core Condition for Resonance

Consider a series RLC circuit (resistor $R$, inductor $L$, capacitor $C$) driven by an AC voltage source $V = V_0 \sin(\omega t)$.

The total impedance $Z$ of the series combination is:

where:

• $X_L = \omega L$ (inductive reactance)
• $X_C = \frac{1}{\omega C}$ (capacitive reactance)
• $j = \sqrt{-1}$

Why resonance happens:

The circuit "wants" to let maximum current flow. The opposition to current comes from both resistance and reactance. But reactance can be negative (capacitive) or positive (inductive). When they are equal in magnitude but opposite in sign, they cancel:

This is the fundamental condition — not a formula to memorize, but a logical consequence of impedance minimization.

2. Deriving the Resonant Frequency

From $X_L = X_C$:

Multiply both sides by $\omega$:

Thus:

Since $\omega = 2\pi f$, the resonant frequency in hertz is:

Why this makes sense:

• A larger $L$ or $C$ means the circuit takes longer to "oscillate" — lower frequency.
• A smaller $L$ or $C$ means faster oscillations — higher frequency.
• The product $LC$ controls the natural time scale of the circuit.

3. What Happens at Resonance — Key Consequences

(a) Impedance is Minimum (Purely Resistive)

At resonance, $X_L - X_C = 0$, so:

Why: The reactive parts cancel, leaving only the resistance. The circuit behaves like a pure resistor.

(b) Current is Maximum

From Ohm's law for AC:

At resonance, $Z = R$ (minimum possible), so current is maximum:

Why: The opposition to current is smallest when reactance cancels.

(c) Voltage Across L and C Can Be Very Large

The voltage across the inductor:

The voltage across the capacitor:

Since $X_L = X_C$ at resonance, $V_L = V_C$ in magnitude, but they are 180° out of phase — they cancel each other in the loop.

Why this is important:

If $R$ is small, $V_L$ and $V_C$ can be many times larger than the source voltage $V$. This is called voltage magnification — a key concept for tuned circuits and filters.

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