Impedance

Part of Electrical Theory

The complete opposition to AC current flow — combining resistance, capacitive reactance, and inductive reactance into a single practical concept.

Why This Matters

Ohm’s law (V = IR) works perfectly for DC circuits and for AC circuits containing only resistors. But the moment you add a coil (inductor) or a capacitor to an AC circuit, Ohm’s law needs an upgrade. That upgrade is impedance.

Impedance explains why a transformer draws almost no current unloaded but huge current when loaded. It explains why certain frequencies pass through a filter while others are blocked. It explains why motors have different behaviors at startup versus running speed. In practical terms, impedance is what you’re dealing with every time you work with AC power and reactive components.

For rebuilding work, this matters most when: winding coils for generators and motors, building transformers, designing simple radio circuits, and troubleshooting AC equipment that behaves unexpectedly.

Resistance vs. Reactance vs. Impedance

Resistance (R): Opposition to current that dissipates energy as heat. Present in both AC and DC. Doesn’t depend on frequency. Unit: ohms (Ω).

Reactance (X): Opposition to current that stores and returns energy (no heat). Only present in AC circuits. Depends on frequency. Has two types:

• Inductive reactance (X_L): from coils and motors
• Capacitive reactance (X_C): from capacitors

Impedance (Z): The combined total opposition to AC current. Includes both resistance and reactance. Unit: ohms (Ω), but it’s a complex quantity — has both magnitude and phase angle.

The relationship: Z = √(R² + X²) for a simple series circuit with one reactive component.

Inductive Reactance

An inductor (coil of wire) opposes changes in current. The faster current tries to change (higher frequency), the more the inductor opposes it. This is inductive reactance:

X_L = 2π × f × L

Where:

• X_L = inductive reactance in ohms
• f = frequency in hertz
• L = inductance in henries

Key behavior: Inductive reactance increases linearly with frequency.

• At DC (0 Hz): X_L = 0 — coil is just resistance of the wire
• At 50 Hz with L = 1H: X_L = 314 ohms
• At 100 Hz with L = 1H: X_L = 628 ohms (doubled)
• At 1000 Hz with L = 1H: X_L = 6,280 ohms

Practical example: A motor coil with 1H inductance and 10Ω wire resistance, running at 50 Hz:

• X_L = 314 Ω
• Z = √(10² + 314²) ≈ 314 Ω (resistance is negligible)
• The motor’s opposition to current is almost entirely from inductance, not wire resistance

This is why motors can have thin wire yet carry substantial current while running — the inductance limits current. At startup (near-zero frequency, motor not spinning), inductance doesn’t help and motors draw 5–10× running current.

Capacitive Reactance

A capacitor opposes changes in voltage. The faster voltage tries to change (higher frequency), the less the capacitor opposes it. This is capacitive reactance:

X_C = 1 / (2π × f × C)

Where:

• X_C = capacitive reactance in ohms
• f = frequency in hertz
• C = capacitance in farads

Key behavior: Capacitive reactance decreases with increasing frequency — opposite to inductors.

• At DC (0 Hz): X_C = infinite — no current flows through a capacitor
• At 50 Hz with C = 100μF: X_C = 32 ohms
• At 100 Hz with C = 100μF: X_C = 16 ohms (halved)
• At 1000 Hz with C = 100μF: X_C = 1.6 ohms

Practical example: A capacitor-start motor uses a capacitor to shift phase on one winding, creating starting torque. The capacitor’s reactance at 50 Hz is calculated to produce the required current and phase shift.

Series Impedance Calculation

For a series RLC circuit (resistor, inductor, capacitor in series):

Z = √(R² + (X_L - X_C)²)

Note: X_L and X_C partially cancel each other. If X_L = X_C, they cancel completely — this is resonance.

Example: Series circuit at 50 Hz with R = 10Ω, L = 100mH, C = 100μF

1. X_L = 2π × 50 × 0.1 = 31.4 Ω
2. X_C = 1/(2π × 50 × 0.0001) = 31.8 Ω
3. X_L - X_C = 31.4 - 31.8 = -0.4 Ω (nearly cancelled!)
4. Z = √(10² + 0.4²) ≈ 10 Ω
5. At this frequency, the circuit behaves almost like a pure resistor

This is resonance — used in radio receivers to select specific frequencies.

Parallel Impedance

Parallel reactive circuits are more complex. The total impedance of parallel components follows the same general pattern as resistors, but with complex numbers due to phase.

For practical work, the key result is: at resonance in a parallel LC circuit, impedance is very HIGH (opposite of series resonance where impedance is very low).

Configuration	At Resonance
Series RLC	Minimum impedance — passes that frequency
Parallel LC (tank circuit)	Maximum impedance — blocks that frequency

Radio tuning uses parallel resonant circuits to select one frequency (high impedance at tuned frequency rejects detuned signals).

Phase Angle and Power Factor

When impedance has a reactive component, voltage and current are out of phase. The phase angle (φ) tells you by how much:

tan(φ) = X / R

Where X is the net reactance (X_L - X_C).

The power factor is: PF = cos(φ) = R / Z

Phase angle	Power factor	Meaning
0°	1.0	Pure resistance — all power is real
45°	0.707	Equal resistance and reactance
90°	0.0	Pure reactance — no real power

Why power factor matters: Your generator produces apparent power (V × I). Only the real power component (V × I × cos φ) does useful work. With PF = 0.7, you need 43% more current (and thicker wires, larger generator) to deliver the same useful power compared to PF = 1.0.

Inductive loads (motors, transformers) have lagging power factor. Capacitors can correct this — adding capacitance to a predominantly inductive system brings the combined power factor closer to 1.0.

Impedance Matching

For maximum power transfer between a source and a load, their impedances should match. This is critical in audio amplifiers and radio transmitters but matters less in power distribution.

Transformer as impedance transformer: A transformer with turns ratio N transforms impedance by N². A transformer with 10:1 turns ratio transforms impedance by 100:1. This allows a high-impedance microphone to couple efficiently into a low-impedance amplifier input.

Practical impedance matching:

• Loudspeaker to amplifier: typically 4–16 Ω speaker, amplifier output designed to match
• Antenna to transmitter: antenna impedance 50–75 Ω, transmitter output to match
• Microphone to preamplifier: often transformer-coupled for impedance matching

Measuring Impedance in the Field

Without an LCR meter, impedance can be estimated:

Current method:

1. Apply known AC voltage V to unknown impedance
2. Measure current I
3. Z = V / I (in ohms)

Bridge method (Wheatstone bridge variant):

1. Build a bridge with two known resistors and the unknown impedance
2. Balance the bridge by adjusting a variable resistor
3. At balance, the unknown equals the ratio of the known components
4. More accurate but requires a null detector (galvanometer)

Resonance method:

1. Connect unknown inductance or capacitance with a known complementary component
2. Sweep frequency (vary generator speed or use variable LC)
3. Find resonant frequency where response peaks (series) or dips (parallel)
4. Calculate: at resonance, X_L = X_C, so L = 1/(4π²f²C) or C = 1/(4π²f²L)

Understanding impedance bridges the gap between simple DC circuit theory and the more complex world of AC machinery, filters, and communication systems — all essential in a more advanced rebuilding phase.