Impedance – Combining Resistance and Reactance in AC

In real AC circuits, resistance and reactance never appear alone.
They combine to form impedance (Z) — the total opposition to AC current flow.

note

Impedance includes:

• Resistance (R) → dissipates energy as heat
• Reactance (X) → stores and releases energy

Impedance is measured in Ohms (Ω), but it is not added like DC resistance.

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Impedance Formula

For a circuit with resistance and reactance:

$Z = \sqrt{R^2 + X^2}$

Where:

• Z = impedance (Ohms)

• R = resistance (Ohms)

• X = reactance (Ohms)

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Example Calculation

• Resistance: 100 Ω

• Inductive reactance: 80 Ω

$Z=1002+802=16400≈128 ΩZ = \sqrt{100^2 + 80^2} = \sqrt{16400} \approx 128\ \Omega$

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Why Impedance Matters

Impedance is essential for:

• Calculating AC current → $I = V/Z$

• Designing frequency-selective filters

• Transmission line and antenna matching

• Speaker–amplifier matching

• Amplifier input and output design

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Real-World Example: Audio Systems

Speakers have an impedance (typically 4–8 Ω) that changes with frequency.
If an amplifier is not designed for that load:

• Sound quality degrades

• Power delivery drops

• Amplifier or speaker may overheat

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Impedance in Common Circuits

Circuit Type	Impedance
Pure resistor	$Z = R$
Pure inductor	$Z = X_1$
Pure capacitor	$Z = X_c$
RL circuit	$Z = \sqrt{(R^2 + X_1^2)}$
RC circuit	$Z = \sqrt{(R^2 + X_c^2)}$
RLC circuit	$Z = \sqrt{(R^2 + (X_1 − X_c)^2)}$

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Resonance in RLC Circuits

When:

$X_L=X_C$

Then:

$Z=R$

This condition is called resonance.

At resonance:

• Impedance is minimum

• Current is maximum

• Circuit becomes purely resistive

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Key Takeaway

• It combines resistance and reactance

• It depends on frequency

• It explains filters, resonance, signal shaping, and power transfer

Master impedance, and AC circuits stop being mysterious — they become predictable and powerful.