Part I: Circuit Fundamentals | Chapter 3

AC Analysis & Phasors

Sinusoidal steady-state analysis using complex phasors, impedance, complex power, and resonance.

Sinusoidal Signals & Phasors

In AC steady-state analysis all voltages and currents are sinusoidal at the same angular frequency \( \omega \). The general sinusoidal voltage is:

\[ v(t) = V_m \cos(\omega t + \phi) \]

where \( V_m \) is the peak amplitude, \( \omega = 2\pi f \) is the angular frequency, and \( \phi \) is the phase. A phasor is the complex number that encodes amplitude and phase:

\[ \mathbf{V} = V_m e^{j\phi} = V_m\angle\phi, \qquad v(t) = \operatorname{Re}\!\left[\mathbf{V}\,e^{j\omega t}\right] \]

The key advantage: differentiation in the time domain becomes multiplication by \( j\omega \) in the phasor domain, converting differential equations into algebraic ones.

Complex Impedance

Impedance \( \mathbf{Z} = \mathbf{V}/\mathbf{I} \) generalises resistance to reactive elements. For the three basic elements:

Resistor

\[ Z_R = R \]

Real, frequency-independent. Voltage and current in phase.

Capacitor

\[ Z_C = \frac{1}{j\omega C} \]

Decreases with frequency. Current leads voltage by 90°.

Inductor

\[ Z_L = j\omega L \]

Increases with frequency. Voltage leads current by 90°.

Impedances combine exactly like resistances: series addition and parallel reciprocal addition apply directly using complex arithmetic.

Complex Power & Power Factor

For a load with impedance \( \mathbf{Z} = |\mathbf{Z}|e^{j\theta} \), the complex power is:

\[ S = \mathbf{V}\,\mathbf{I}^* = P + jQ = |\mathbf{S}|\angle\theta \]

Real power P

Watts (W). Energy consumed per second. \( P = |\mathbf{S}|\cos\theta \).

Reactive power Q

VAR. Oscillates between source and storage. \( Q = |\mathbf{S}|\sin\theta \).

Power factor

\( \text{PF} = \cos\theta = P/|S| \). Unity at resonance (pure resistive load).

Resonance

In a series RLC circuit, resonance occurs when the imaginary part of \( \mathbf{Z} \) vanishes — i.e., when \( \omega L = 1/(\omega C) \):

\[ \omega_0 = \frac{1}{\sqrt{LC}}, \qquad |\mathbf{Z}|_{\min} = R, \qquad \text{PF} = 1 \]

At resonance the impedance is purely resistive, power factor is unity, and the current is maximised. The voltage across the reactive elements can exceed the source voltage by a factor of \( Q \):\( V_L = V_C = Q \cdot V_s \) at \( \omega_0 \).

Python: RLC Bode Plot & Power Factor

Compute and plot the impedance magnitude and phase (Bode plot) of a series RLC circuit across six decades of frequency, show the resonance peak in the bandpass transfer function, and visualise power factor.

RLC Impedance Bode Plot, Resonance & Power Factor

Python

ac_phasors_bode.py121 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# Series RLC circuit: Z(jw) = R + j(wL - 1/wC)
# |Z| and phase vs frequency, resonance, and power factor

R = 100.0      # Ohm
L = 10e-3      # 10 mH
C = 100e-9     # 100 nF

omega_0 = 1.0 / np.sqrt(L * C)
f0 = omega_0 / (2 * np.pi)
Q = omega_0 * L / R
BW = omega_0 / Q   # bandwidth in rad/s

print('=== Series RLC AC Analysis ===')
print(f'R = {R:.1f} Ohm, L = {L*1e3:.1f} mH, C = {C*1e9:.1f} nF')
print(f'Resonant frequency f0 = {f0:.2f} Hz')
print(f'omega_0 = {omega_0:.2f} rad/s')
print(f'Quality factor Q = {Q:.2f}')
print(f'Bandwidth BW = {BW/(2*np.pi):.2f} Hz')

f = np.logspace(2, 6, 2000)
omega = 2 * np.pi * f

Z_R = R + 0j
Z_L = 1j * omega * L
Z_C = 1.0 / (1j * omega * C)
Z_total = Z_R + Z_L + Z_C

mag_dB = 20 * np.log10(np.abs(Z_total))
phase_deg = np.degrees(np.angle(Z_total))

# Transfer function: V_R / V_in = R / Z_total
H = R / Z_total
H_dB = 20 * np.log10(np.abs(H))
H_phase = np.degrees(np.angle(H))

# Power factor vs frequency
pf = np.cos(np.angle(Z_total))   # cos(theta)

# Impedance components
Z_L_mag = np.abs(Z_L)
Z_C_mag = np.abs(Z_C)

print()
print('=== Impedance at Key Frequencies ===')
for freq in [f0/10, f0/2, f0, f0*2, f0*10]:
    w = 2 * np.pi * freq
    Zval = R + 1j*(w*L - 1/(w*C))
    print(f'  f = {freq:.1f} Hz: |Z| = {abs(Zval):.2f} Ohm, phase = {np.degrees(np.angle(Zval)):.1f} deg, PF = {np.cos(np.angle(Zval)):.4f}')

fig, axes = plt.subplots(2, 2, figsize=(14, 10))
fig.patch.set_facecolor('#0a0a0a')
fig.suptitle('Series RLC AC Analysis', color='#d1d5db', fontsize=14, y=0.98)
for ax in axes.flat:
    ax.set_facecolor('#111827')
    for spine in ax.spines.values():
        spine.set_color('#374151')

# Plot 1: Impedance magnitude (Bode)
ax = axes[0, 0]
ax.semilogx(f, mag_dB, color='#34d399', lw=2.2, label='|Z(jf)| dB')
ax.semilogx(f, 20*np.log10(Z_L_mag), color='#6ee7b7', lw=1.2, ls='--', label='|Z_L|')
ax.semilogx(f, 20*np.log10(Z_C_mag), color='#a7f3d0', lw=1.2, ls=':', label='|Z_C|')
ax.axvline(f0, color='#fbbf24', ls='--', lw=1.5, label=f'f0 = {f0:.0f} Hz')
ax.set_xlabel('Frequency (Hz)', color='#9ca3af')
ax.set_ylabel('|Z| (dB)', color='#9ca3af')
ax.set_title('Impedance Magnitude', color='#d1d5db', pad=8)
ax.tick_params(colors='#9ca3af')
ax.legend(fontsize=8, facecolor='#1f2937', edgecolor='#374151', labelcolor='#d1d5db')
ax.grid(True, color='#1f2937', alpha=0.8, which='both')

# Plot 2: Impedance phase
ax = axes[0, 1]
ax.semilogx(f, phase_deg, color='#34d399', lw=2.2)
ax.axvline(f0, color='#fbbf24', ls='--', lw=1.5, label=f'f0 = {f0:.0f} Hz')
ax.axhline(0, color='#6b7280', lw=0.8, ls=':')
ax.axhline(90,  color='#374151', lw=0.8, ls=':')
ax.axhline(-90, color='#374151', lw=0.8, ls=':')
ax.set_xlabel('Frequency (Hz)', color='#9ca3af')
ax.set_ylabel('Phase (degrees)', color='#9ca3af')
ax.set_title('Impedance Phase', color='#d1d5db', pad=8)
ax.tick_params(colors='#9ca3af')
ax.legend(fontsize=8, facecolor='#1f2937', edgecolor='#374151', labelcolor='#d1d5db')
ax.grid(True, color='#1f2937', alpha=0.8, which='both')

# Plot 3: Transfer function H = V_R / V_in
ax = axes[1, 0]
ax.semilogx(f, H_dB, color='#34d399', lw=2.2)
ax.axvline(f0, color='#fbbf24', ls='--', lw=1.5, label=f'Resonance f0 = {f0:.0f} Hz')
bw_low  = f0 - BW/(2*2*np.pi)
bw_high = f0 + BW/(2*2*np.pi)
ax.axhline(-3, color='#f87171', ls=':', lw=1.2, label='-3 dB level')
ax.set_xlabel('Frequency (Hz)', color='#9ca3af')
ax.set_ylabel('|H| (dB)', color='#9ca3af')
ax.set_title('V_R/V_in Transfer Function', color='#d1d5db', pad=8)
ax.tick_params(colors='#9ca3af')
ax.legend(fontsize=8, facecolor='#1f2937', edgecolor='#374151', labelcolor='#d1d5db')
ax.grid(True, color='#1f2937', alpha=0.8, which='both')

# Plot 4: Power factor
ax = axes[1, 1]
ax.semilogx(f, pf, color='#34d399', lw=2.2)
ax.axvline(f0, color='#fbbf24', ls='--', lw=1.5, label=f'f0 (PF=1)')
ax.axhline(1.0, color='#6b7280', lw=0.8, ls=':')
ax.set_xlabel('Frequency (Hz)', color='#9ca3af')
ax.set_ylabel('Power Factor', color='#9ca3af')
ax.set_title('Power Factor cos(θ) vs Frequency', color='#d1d5db', pad=8)
ax.tick_params(colors='#9ca3af')
ax.set_ylim(-0.1, 1.1)
ax.legend(fontsize=8, facecolor='#1f2937', edgecolor='#374151', labelcolor='#d1d5db')
ax.grid(True, color='#1f2937', alpha=0.8, which='both')

plt.tight_layout(pad=2.0)
plt.savefig('output.png', dpi=140, bbox_inches='tight', facecolor='#0a0a0a')
print()
print('Plot saved to output.png')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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