Understanding Resonance in AC Circuits — When Reactances Cancel Out

In AC circuits, frequency plays a crucial role in deciding how current flows. At one special frequency, the effects of inductors and capacitors perfectly balance each other, leading to a powerful phenomenon called resonance.

In this blog, we’ll understand what resonance is, why it happens, and how it can be visualized using a practical RLC circuit simulation.

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⚡ Resonance

Resonance in an AC circuit is the condition at which the inductive reactance (Xₗ) becomes equal to the capacitive reactance (X꜀), causing them to cancel each other out. At resonance, the circuit behaves purely resistive, and the current becomes maximum.

To understand resonance, you must understand impedance and reactance.

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🔌 What Is Impedance?

In AC circuits, current and voltage change direction continuously.

So, opposition to current is not just resistance (R). There are two more components:

• Inductive reactance (Xₗ) — comes from inductors

• Capacitive reactance (X꜀) — comes from capacitors

The total opposition in an AC circuit is called Impedance (Z):

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

Where:

• $X_L = 2\pi f L$

• $X_C = \frac{1}{2\pi f C}$

Notice something important:

• Xₗ increases with frequency

• X꜀ decreases with frequency

But there is one special frequency where their effects cancel each other.

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🎯 Resonance Condition

Resonance occurs where capacitive reactance equals inductive reactance:

$$X_L=X_{\mathrm{C<span\; style="font-family:\; georgia;"></span>}}$$

Substitute the formulas:

$$2\pi f_{0}L=\frac{1}{2\pi f_0\mathrm{C<span\; style="font-family:\; georgia;"></span>}}$$

Solving for frequency:

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

This frequency is called the resonant frequency.

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🔍 What Happens at Resonance?

At the resonant frequency $f_0$:

✔ Inductive reactance equals capacitive reactance

$$X_L-X_C=0$$

✔ Impedance becomes minimum

$$Z=R$$

The circuit behaves as if it has only resistance.

✔ Current becomes maximum

$$I=\frac{V}{\mathrm{R<span\; style="font-family:\; georgia;"></span>}}$$

✔ Voltage magnification occurs in L and C

Even though the net reactive effect is zero, energy continuously transfers between the inductor and capacitor:

• Inductor stores magnetic energy

• Capacitor stores electric energy

They keep exchanging energy, creating large voltages inside the circuit.

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🧪 LTspice Simulation of Resonance

To visualize this phenomenon, a series RLC circuit was simulated in LTspice using the following values:

• R = 50 Ω

• L = 45 mH = 0.045 H

• C = 1.36 μF = 1.36 × 10⁻⁶ F

• AC source = 230 V

• AC sweep = 100 Hz to 2000 Hz

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🔧 Circuit Explanation

The simulated circuit is a series RLC network.

All three components (R, L, and C) are connected one after another, and an AC source is applied across the combination.

The current through the resistor I(R1) is observed, which represents the current flowing through the entire series loop.

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🧮 Resonant Frequency Calculation

Step 1: Multiply L and C

$$LC = 0.045 \times 1.36 \times 10^{-6}$$
$$LC = 6.12 \times 10^{-8}$$

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Step 2: Square root of LC

$$\sqrt{LC} = \sqrt{6.12 \times 10^{-8}}$$ $$\sqrt{LC} = 2.472 \times 10^{-4}$$

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Step 3: Multiply by $2\pi$

$$2\pi \times 2.472 \times 10^{-4}$$
$$2\pi \approx 6.283$$
$$2\pi \sqrt{LC}=0.001554$$

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Step 4: Take reciprocal

$$f_0 = \frac{1}{0.001554}$$ $$f_0 \approx 643.7 \text{ Hz}$$
$$\boxed{f_0 \approx 644 \text{ Hz}}$$

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📈 Verification from Simulation

At resonance:

• $X_L = X_C$

• Net reactance $X = 0$
  

• Impedance $Z = R = 50 \Omega$
  

Current becomes:

$$I = \frac{V}{R} = \frac{230}{50} = 4.6 \text{ A}$$

The simulation peak current is very close to 4.5–4.7 A, confirming perfect resonance behavior.

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🔚 Conclusion

Resonance is a powerful condition in AC circuits where inductive and capacitive reactances cancel each other, leaving only resistance to oppose current flow. At this point, impedance is minimum and current reaches its maximum value.

Although the net reactance becomes zero, energy continuously oscillates between the magnetic field of the inductor and the electric field of the capacitor, leading to voltage magnification within the circuit. This phenomenon is the foundation of tuned circuits, filters, oscillators, and communication systems.

Understanding resonance completes the journey from resistance → reactance → impedance and opens the door to advanced AC circuit analysis.