Correct Answer

Concept & PrincipleElectricalBasic Electrical

Option A

$Z^2 = R^2 + X_L^2$

Quick Summary: In an RL series circuit, the total impedance $Z$ is the vector sum of resistance $R$ and inductive reactance $X_L$. Because the voltage drop across the resistor and inductor are in quadrature ($90^\circ$ phase difference), the impedance follows the Pythagorean theorem: $Z = \sqrt{R^2 + X_L^2}$, which implies $Z^2 = R^2 + X_L^2$.

💡 Explanation

In an RL series circuit, the total impedance $Z$ is the vector sum of resistance $R$ and inductive reactance $X_L$ . Because the voltage drop across the resistor and inductor are in quadrature ( $90°$ phase difference), the impedance follows the Pythagorean theorem: $Z = \sqrt{R^2 + X_L^2}$ , which implies $Z^2 = R^2 + X_L^2$ .

🔢 Key Formulas

$Z = \sqrt{R^2 + X_L^2}$ — Definition of impedance magnitude

$X_L = 2\pi f L$ — Inductive reactance formula

⚙️ Working Principle

The total impedance represents the opposition offered by an AC circuit to current flow. In a series RL circuit, the resistor $R$ offers opposition along the real axis, while the inductor $X_L$ offers opposition along the imaginary (positive j) axis. The magnitude of the resultant phasor is found using the vector addition rule, yielding the impedance triangle where $Z$ is the hypotenuse.

📌 Key Points

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Resistance $R$ is in-phase with current.
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Inductive reactance $X_L$ leads current by $90°$ .
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Impedance is a complex quantity represented as $\bar{Z} = R + jX_L$ .
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The power factor angle $\phi$ is given by $\tan^{-1}(\frac{X_L}{R})$ .

✅ Advantages

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Simple linear relationship for series components.
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Allows phase angle calculation between voltage and current.

❌ Disadvantages / Limitations

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Inductive reactance is frequency-dependent, making impedance variable with supply frequency.
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Causes a lagging power factor in the circuit.

🛠️ Applications / Uses

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Filter circuits (Low-pass).
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Chokes for fluorescent lamps.
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Induction motor windings modeling.

📄 Additional Information

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Option B is incorrect because it includes capacitive reactance ( $X_C$ ) instead of inductive reactance.
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Option C is only true for purely resistive circuits ( $X_L=0$ or $f=0$ ).
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The impedance $Z$ is measured in Ohms ( $\Omega$ ).

📊 Diagram / Illustration

✅

A is correct — The impedance of an RL series circuit is the vector magnitude of resistance and inductive reactance, satisfying the relation $Z^2 = R^2 + X_L^2$ .

Core Concepts Used

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Phasor Algebra Impedance Triangle Inductive Reactance

💡 EXAM TIP

Always verify if the circuit is series or parallel. In series circuits, we add impedances; in parallel, we add admittances ( $Y = \sqrt{G^2 + B^2}$ ).

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