Training Electrical Engineering AC Circuits — Phasors & Impedance
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AC Circuits — Phasors & Impedance

24 min Electrical Engineering

AC Circuits — Phasors & Impedance

Alternating-current analysis uses phasors — complex numbers that encode amplitude and phase — enabling the use of algebraic methods instead of differential equations.

Impedance

$$Z_R = R, \qquad Z_L = j\omega L, \qquad Z_C = \frac{1}{j\omega C}$$

where $j = \sqrt{-1}$, $\omega = 2\pi f$ is the angular frequency.

Generalized Ohm's Law

$$\mathbf{V} = \mathbf{I} \cdot Z$$

All DC analysis techniques (KVL, KCL, voltage divider, etc.) work with complex impedances.

RMS Values

$$V_{\text{rms}} = \frac{V_{\text{peak}}}{\sqrt{2}}, \qquad P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} \cos\phi$$

where $\phi$ is the phase angle between voltage and current.

Example 1

An RL series circuit has $R = 30\,\Omega$ and $L = 0.1$ H at $f = 60$ Hz. Find the impedance magnitude and phase.

$\omega = 2\pi(60) = 377$ rad/s

$Z_L = j(377)(0.1) = j37.7\,\Omega$

$$Z = 30 + j37.7 \implies |Z| = \sqrt{30^2 + 37.7^2} = 48.2\,\Omega$$

$$\phi = \arctan(37.7/30) = 51.5°$$

Example 2

An RC series circuit: $R = 100\,\Omega$, $C = 10\,\mu$F, $f = 1{,}000$ Hz, $V_s = 10$ V rms. Find the current.

$Z_C = 1/(j \cdot 2\pi \cdot 1000 \cdot 10^{-5}) = -j15.92\,\Omega$

$|Z| = \sqrt{100^2 + 15.92^2} = 101.3\,\Omega$

$$I = 10/101.3 = 98.7 \text{ mA}$$

Example 3

An RLC series circuit: $R = 50\,\Omega$, $L = 0.2$ H, $C = 50\,\mu$F. Find the resonant frequency.

At resonance, $Z_L = -Z_C$:

$$f_0 = \frac{1}{2\pi\sqrt{LC}} = \frac{1}{2\pi\sqrt{0.2 \times 50 \times 10^{-6}}} = 50.3 \text{ Hz}$$

At resonance, $Z = R = 50\,\Omega$ (purely resistive; current is maximum).

Practice Problems

1. An inductor $L = 0.05$ H at $f = 50$ Hz. Find $Z_L$.
2. A capacitor $C = 100\,\mu$F at $f = 60$ Hz. Find $|Z_C|$.
3. RL series: $R = 20\,\Omega$, $L = 0.1$ H, $f = 50$ Hz. Find $|Z|$ and phase.
4. An AC source delivers 120 V rms to a load with $Z = 60\angle 30°$ Ω. Find the current magnitude and power factor.
5. Find the resonant frequency for $L = 0.5$ H, $C = 20\,\mu$F.
6. At resonance, what is the impedance of a series RLC circuit if $R = 25\,\Omega$?
7. $V_{\text{peak}} = 170$ V. Find $V_{\text{rms}}$.
8. A parallel RL circuit: $R = 100\,\Omega \parallel Z_L = j50\,\Omega$. Find the equivalent impedance.
9. Power factor $= \cos\phi$. If $\phi = 45°$, $V = 240$ V, $I = 5$ A, find average power.
10. Design a series RLC band-pass filter centered at 1 kHz with $L = 10$ mH. Find $C$.
11. Quality factor $Q = \omega_0 L/R$. For $L = 0.1$ H, $R = 10\,\Omega$, $f_0 = 100$ Hz, find $Q$.
12. The bandwidth of a series RLC circuit is $BW = R/L$. For $R = 20\,\Omega$, $L = 0.2$ H, find BW in Hz.
Show Answer Key

1. $Z_L = j2\pi(50)(0.05) = j15.71\,\Omega$

2. $|Z_C| = 1/(2\pi \cdot 60 \cdot 10^{-4}) = 26.5\,\Omega$

3. $Z_L = j31.42$; $|Z| = \sqrt{400 + 987.6} = 37.3\,\Omega$; $\phi = 57.5°$

4. $I = 120/60 = 2$ A; PF $= \cos 30° = 0.866$

5. $f_0 = 1/(2\pi\sqrt{0.5 \times 2 \times 10^{-5}}) = 50.3$ Hz

6. $Z = R = 25\,\Omega$

7. $V_{\text{rms}} = 170/\sqrt{2} = 120.2$ V

8. $Z_{\text{eq}} = (100 \cdot j50)/(100 + j50) = (j5000)/(100+j50)$. Multiply by conjugate: $|Z| = 44.7\,\Omega$ at $63.4°$

9. $P = 240 \times 5 \times \cos 45° = 848.5$ W

10. $C = 1/(4\pi^2 f_0^2 L) = 1/(4\pi^2 \times 10^6 \times 0.01) = 2.53\,\mu$F

11. $Q = 2\pi(100)(0.1)/10 = 6.28$

12. $BW = 20/0.2 = 100$ rad/s $= 15.9$ Hz

DC Circuit Analysis — Kirchhoff's Laws Transfer Functions, Poles & Zeros