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<h1>AC Circuit Analysis</h1>
<p>AC circuits are more complex than DC because the voltage and current are constantly changing — and capacitors and inductors respond to that change in ways resistors don't. The key insight that makes AC analysis tractable is <em>impedance</em>: a generalisation of resistance that works for all three passive components, using complex numbers to track both magnitude and phase. Once you have impedance, Ohm's Law and Kirchhoff's laws apply in exactly the same form.</p>

<h2>The AC Waveform</h2>
<pre>
V(t) = V_peak × sin(ωt + φ)

V_peak — peak voltage (amplitude)
ω = 2πf — angular frequency in radians/second
f — frequency in Hz
φ — phase angle (radians)
T = 1/f — period
</pre>

<table>
  <thead><tr><th>Quantity</th><th>Symbol</th><th>Formula</th><th>Example (UK mains)</th></tr></thead>
  <tbody>
    <tr><td>Peak voltage</td><td>V_peak</td><td>V_RMS × √2</td><td>325 V</td></tr>
    <tr><td>RMS voltage</td><td>V_RMS</td><td>V_peak / √2 ≈ 0.707 × V_peak</td><td>230 V</td></tr>
    <tr><td>Peak-to-peak</td><td>V_pp</td><td>2 × V_peak</td><td>650 V</td></tr>
    <tr><td>Frequency</td><td>f</td><td>1/T</td><td>50 Hz</td></tr>
    <tr><td>Period</td><td>T</td><td>1/f</td><td>20 ms</td></tr>
    <tr><td>Angular frequency</td><td>ω</td><td>2πf</td><td>314 rad/s</td></tr>
  </tbody>
</table>

<p>RMS (Root Mean Square) is the effective value — the DC equivalent that delivers the same power. For power calculations, always use RMS values.</p>

<h2>Reactance</h2>
<p>Reactance is the frequency-dependent opposition to current flow in capacitors and inductors:</p>
<pre>
Capacitive reactance:  X_C = 1 / (2πfC)   (ohms)
Inductive reactance:   X_L = 2πfL          (ohms)
</pre>

<table>
  <thead><tr><th>Component</th><th>At low frequency</th><th>At high frequency</th><th>Intuition</th></tr></thead>
  <tbody>
    <tr><td>Capacitor</td><td>High X_C (blocks)</td><td>Low X_C (passes)</td><td>Blocks DC, passes AC — more so at higher frequencies</td></tr>
    <tr><td>Inductor</td><td>Low X_L (passes)</td><td>High X_L (blocks)</td><td>Passes DC, blocks AC — more so at higher frequencies</td></tr>
    <tr><td>Resistor</td><td>Constant</td><td>Constant</td><td>Frequency-independent</td></tr>
  </tbody>
</table>

<h3>Reactance Examples</h3>

<table>
  <thead><tr><th>Component</th><th>Value</th><th>@ 100 Hz</th><th>@ 1 kHz</th><th>@ 10 kHz</th></tr></thead>
  <tbody>
    <tr><td>Capacitor</td><td>100 nF</td><td>15,915 Ω</td><td>1,592 Ω</td><td>159 Ω</td></tr>
    <tr><td>Capacitor</td><td>10 µF</td><td>159 Ω</td><td>15.9 Ω</td><td>1.59 Ω</td></tr>
    <tr><td>Inductor</td><td>10 mH</td><td>6.28 Ω</td><td>62.8 Ω</td><td>628 Ω</td></tr>
    <tr><td>Inductor</td><td>100 µH</td><td>0.063 Ω</td><td>0.628 Ω</td><td>6.28 Ω</td></tr>
  </tbody>
</table>

<h2>Impedance</h2>
<p>Impedance (Z) combines resistance and reactance into one complex quantity. Ohm's Law applies: V = I × Z</p>
<pre>
Z = R + jX          (j = imaginary unit = √−1)

Series RC:  Z = R − jX_C         (capacitor lags)
Series RL:  Z = R + jX_L         (inductor leads)
Series RLC: Z = R + j(X_L − X_C)

Magnitude: |Z| = √(R² + X²)
Phase angle: θ = arctan(X / R)
</pre>

<h3>Phase Relationships</h3>

<table>
  <thead><tr><th>Component</th><th>Current vs. Voltage phase</th><th>Memory aid</th></tr></thead>
  <tbody>
    <tr><td>Resistor</td><td>In phase (θ = 0°)</td><td>—</td></tr>
    <tr><td>Capacitor</td><td>Current leads voltage by 90°</td><td>ICE — I before E in C</td></tr>
    <tr><td>Inductor</td><td>Current lags voltage by 90°</td><td>ELI — E before I in L</td></tr>
  </tbody>
</table>

<h2>Resonance</h2>
<p>In a series RLC circuit, X_L and X_C cancel when they are equal. The frequency where this happens is the resonant frequency — at which impedance is purely resistive (minimum Z) and current is at maximum:</p>
<pre>
At resonance: X_L = X_C
  2πf₀L = 1/(2πf₀C)

Resonant frequency: f₀ = 1 / (2π√LC)

Q factor (quality): Q = (1/R) × √(L/C)    (series RLC)
  High Q → sharp resonance, narrow bandwidth
  Low Q → broad, gentle resonance
</pre>

<table>
  <thead><tr><th>L</th><th>C</th><th>Resonant frequency</th><th>Application</th></tr></thead>
  <tbody>
    <tr><td>100 µH</td><td>100 pF</td><td>1.59 MHz</td><td>AM radio tuning</td></tr>
    <tr><td>10 µH</td><td>100 pF</td><td>5.03 MHz</td><td>Shortwave receiver</td></tr>
    <tr><td>1 mH</td><td>1 µF</td><td>5,032 Hz</td><td>Audio crossover</td></tr>
    <tr><td>10 mH</td><td>10 µF</td><td>503 Hz</td><td>Notch filter</td></tr>
  </tbody>
</table>

<h2>Power in AC Circuits</h2>
<pre>
Real power (P):       P = V_RMS × I_RMS × cos(θ)   [watts — does real work]
Reactive power (Q):   Q = V_RMS × I_RMS × sin(θ)   [VAR — stored/returned, no net work]
Apparent power (S):   S = V_RMS × I_RMS             [VA — total current drawn]

Power factor: PF = cos(θ) = P / S    (1.0 is perfect, 0 is purely reactive)
</pre>

<table>
  <thead><tr><th>Load type</th><th>Power factor</th><th>θ</th><th>Real power consumed</th></tr></thead>
  <tbody>
    <tr><td>Pure resistor (heater, bulb)</td><td>1.0</td><td>0°</td><td>100% of apparent</td></tr>
    <tr><td>Induction motor (typical)</td><td>0.7–0.9</td><td>25–45°</td><td>70–90% of apparent</td></tr>
    <tr><td>Pure inductor (ideal)</td><td>0.0</td><td>90°</td><td>0% — no real power consumed</td></tr>
    <tr><td>Switching supply with PFC</td><td>0.95–0.99</td><td>~10°</td><td>Near-ideal efficiency</td></tr>
  </tbody>
</table>

<p>Poor power factor means drawing more current for the same real work — causes larger wire losses and transformer heating. Industrial sites pay penalties for low power factor and install capacitor banks to correct it.</p>

<h2>Basic Filters</h2>

<table>
  <thead><tr><th>Filter type</th><th>Circuit</th><th>Cutoff frequency</th><th>Passes</th><th>Blocks</th></tr></thead>
  <tbody>
    <tr><td>RC low-pass</td><td>R series, C to GND</td><td>f_c = 1/(2πRC)</td><td>Low frequencies</td><td>High frequencies</td></tr>
    <tr><td>RC high-pass</td><td>C series, R to GND</td><td>f_c = 1/(2πRC)</td><td>High frequencies</td><td>Low frequencies</td></tr>
    <tr><td>RL low-pass</td><td>L series, R to GND</td><td>f_c = R/(2πL)</td><td>Low frequencies</td><td>High frequencies</td></tr>
    <tr><td>RL high-pass</td><td>R series, L to GND</td><td>f_c = R/(2πL)</td><td>High frequencies</td><td>Low frequencies</td></tr>
    <tr><td>LC band-pass</td><td>Series L, parallel C</td><td>f₀ = 1/(2π√LC)</td><td>Band around f₀</td><td>Above and below</td></tr>
  </tbody>
</table>

<pre>
RC low-pass example: R=10kΩ, C=100nF
f_c = 1 / (2π × 10,000 × 0.0000001) = 159 Hz

Frequencies well below 159Hz pass with negligible attenuation
Frequencies well above 159Hz are attenuated at -20dB/decade (-6dB/octave)
At exactly 159Hz: -3dB (voltage out = 0.707 × voltage in)
</pre>

<h2>References</h2>
<ul>
  <li>Horowitz &amp; Hill — <em>The Art of Electronics</em>, 3rd ed. Chapters 1–2</li>
  <li>Hayt, Kemmerly &amp; Durbin — <em>Engineering Circuit Analysis</em></li>
  <li><a href="https://www.electronics-tutorials.ws/accircuits/ac-waveform.html" target="_blank" rel="noopener">Electronics-Tutorials — AC Waveforms ↗</a></li>
  <li><a href="https://www.allaboutcircuits.com/textbook/alternating-current/" target="_blank" rel="noopener">All About Circuits — AC textbook ↗</a></li>
</ul>
