AC Circuit Parameters: Impedance

AC Circuit Parameters: Impedance

1. What Impedance Is

Impedance ($Z$, ohm $Ω$) = complex opposition to AC current, generalizing resistance $R$ to include reactance $X$ from $L$ and $C$:

• Resistance:

(real, no phase)

• Inductive reactance:

,

• Capacitive reactance:

, magnitude

Magnitude

, phase

($θ>0$ inductive lagging, $θ<0$ capacitive leading).

Ohm's law for AC phasors:

with $Z$ complex.

2. Why Complex

$j$ captures 90° phase shift: inductor voltage leads current 90°, capacitor lags 90°. $R$ has 0°.

3. Series Combinations

• $R$+$L$ series:

,

,

lagging.

• $R$+$C$ series:

,

leading.

• $R$+$L$+$C$ series:

, resonance when

⇒

(pure resistive, $θ=0$, minimum $|Z|$).

4. Worked Examples

(a) Series RL: R=3Ω, X_L=4Ω ⇒ Z=3+j4=5∠53.1° Ω, I= V/Z ⇒ I lags V 53.1° ✓

(b) Series RC: R=4Ω, X_C=3Ω ⇒ Z=4 -j3=5∠-36.9° ⇒ I leads V 36.9° ✓

(c) Series RLC R=10Ω, L=0.1H, C=100µF at f=50Hz: X_L=31.4Ω, X_C=31.8Ω ⇒ Z≈10 -j0.4≈10∠-2.3° near resonance (almost resistive) ✓

(d) Need resonance f0=1/(2π√(LC)) ⇒ where X_L=X_C ⇒ Z=R minimal, current maximal.

5. Exam Points

• Define $Z=R+jX$, $X_L=ωL$, $X_C=1/ωC$ — 5-7 marks, very frequent (almost certain).
• Show $Z$ magnitude/phase and $V=I·Z$ — 5 marks, frequent.
• Compute $Z$ for RL/RC/RLC series — 7 marks, very frequent.
• Explain resonance $X_L=X_C$ ⇒ $Z=R$ — 3-5 marks, frequent.