Animated Solution for Physics - Electromagnetic Induction: In an AC-circuit, an inductor, a capacitor and a resistor are connected in series with $X_L=R=X_C$. Impedance of this circuit is

When Reactances Cancel Out

Imagine you are observing a tug-of-war where both teams are pulling with exactly the same force. What happens to the rope? It doesn't move horizontally; the opposing forces perfectly cancel each other out. A beautifully similar phenomenon occurs in Alternating Current (AC) circuits, and this problem is a perfect illustration of it.

We are given a series LCR circuit—a circuit containing an inductor ($L$), a capacitor ($C$), and a resistor ($R$). The problem presents a very special condition: the inductive reactance ($X_L$), the capacitive reactance ($X_C$), and the resistance ($R$) are all equal in magnitude. Mathematically, this is written as $X_L=X_C=R$.

To find the total opposition to the flow of alternating current, which we call impedance ($Z$), we cannot simply add the resistance and reactances algebraically. Because voltage and current are out of phase in inductors and capacitors, we must add them like vectors (phasors).

The general formula for the impedance of a series LCR circuit is derived from the Pythagorean theorem applied to the phasor diagram:

Here, $R$ represents the real, energy-dissipating resistance, while $(X_L-X_C)$ represents the net reactive, energy-storing opposition. Notice the minus sign! This is because the voltage across an inductor leads the current by $90^{\circ }$, while the voltage across a capacitor lags by $90^{\circ }$. They are exactly $180^{\circ }$ out of phase, meaning they directly oppose each other.

Now, let's substitute our special given condition into the master equation. We know that $X_L=R$ and $X_C=R$.

Look at the term inside the parentheses. $R-R$ is exactly zero! The inductive 'pull' and the capacitive 'pull' are perfectly matched, resulting in a net reactance of zero.

Resonance

The math tells us that the total impedance of the circuit is simply equal to its resistance, $R$. But what does this mean physically?

This specific state, where $X_L=X_C$, is known as electrical resonance. In this state, the inductor and capacitor are exchanging energy back and forth perfectly. The AC power source doesn't have to supply any net energy to the reactive components; it only 'sees' the resistor.

Because the impedance $Z$ is at its absolute minimum value (since the $(X_L-X_C{)}^2$ term can never be negative, its minimum is zero), the current flowing through the circuit will be at its maximum. Furthermore, because the circuit behaves as if it is purely resistive, the total voltage and the current are perfectly in phase with each other.