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<h1>RLC Circuits</h1>
<p>Add an inductor to the RC circuit and you get resonance — the ability to select or reject a specific frequency. RLC circuits are the basis of radio tuning, bandpass filters, notch filters, and any application needing frequency selectivity. The interplay between capacitor (storing energy in an electric field) and inductor (storing energy in a magnetic field) creates oscillatory behaviour that neither component can produce alone, with the resistor determining how much energy is lost per cycle.</p>

<h2>Resonant Frequency</h2>
<p>At the resonant frequency, inductive reactance and capacitive reactance are equal and opposite — they cancel. What's left is purely resistive:</p>
<pre>
f₀ = 1 / (2π√(LC))

At resonance: X_L = X_C
  2πf₀L = 1/(2πf₀C)
</pre>

<table>
  <thead><tr><th>L</th><th>C</th><th>f₀</th><th>Typical application</th></tr></thead>
  <tbody>
    <tr><td>100 µH</td><td>100 pF</td><td>1.59 MHz</td><td>AM broadcast band 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 µH</td><td>47 pF</td><td>23.2 MHz</td><td>CB radio, 27 MHz ISM</td></tr>
    <tr><td>1 mH</td><td>1 µF</td><td>5.03 kHz</td><td>Audio notch filter</td></tr>
    <tr><td>10 mH</td><td>10 µF</td><td>503 Hz</td><td>Power line filter</td></tr>
    <tr><td>100 mH</td><td>100 µF</td><td>50.3 Hz</td><td>Mains frequency trap</td></tr>
  </tbody>
</table>

<h2>Series RLC Circuit</h2>
<pre>
Vin ──[R]──[L]──[C]──── GND (or to different Vout tap points)
</pre>
<p>Impedance at any frequency:</p>
<pre>
Z = R + j(X_L − X_C)
Z = R + j(2πfL − 1/(2πfC))

Magnitude: |Z| = √(R² + (X_L − X_C)²)
Phase:     θ = arctan((X_L − X_C) / R)

At f < f₀:  X_C > X_L  → circuit is capacitive, current leads voltage
At f = f₀:  X_L = X_C  → Z = R (minimum), current is maximum
At f > f₀:  X_L > X_C  → circuit is inductive, current lags voltage
</pre>

<h3>Output Taken Across Different Elements</h3>

<table>
  <thead><tr><th>Output across</th><th>Filter response</th><th>Use</th></tr></thead>
  <tbody>
    <tr><td>Resistor R</td><td>Bandpass — maximum at f₀</td><td>Frequency selection, radio IF filter</td></tr>
    <tr><td>Capacitor C</td><td>Low-pass below resonance, peaks at f₀</td><td>Voltage magnification at resonance</td></tr>
    <tr><td>Inductor L</td><td>High-pass above resonance, peaks at f₀</td><td>Voltage magnification at resonance</td></tr>
    <tr><td>L + C in series</td><td>Bandpass — sharp peak at f₀</td><td>Narrow-band filter</td></tr>
  </tbody>
</table>

<h2>Parallel RLC Circuit</h2>
<pre>
     ┌──[R]──┐
     │        │
Vin ─┤──[L]──┤── Vout
     │        │
     └──[C]──┘
</pre>
<p>At resonance, the parallel LC "tank circuit" has maximum impedance. Current from the source is minimum; circulating current within the tank can be large. This is the opposite of the series case:</p>
<pre>
Series RLC at resonance: minimum impedance (current maximum)
Parallel RLC at resonance: maximum impedance (current minimum, voltage maximum)
</pre>

<h2>Quality Factor (Q)</h2>
<p>Q measures the sharpness of the resonance — how well the circuit discriminates between f₀ and nearby frequencies:</p>
<pre>
Series RLC:   Q = (1/R) × √(L/C) = X_L/R = X_C/R = f₀/BW
Parallel RLC: Q = R × √(C/L) = R/X_L = R/X_C

Bandwidth:    BW = f₀ / Q
</pre>

<table>
  <thead><tr><th>Q factor</th><th>Bandwidth at f₀=1MHz</th><th>Selectivity</th><th>Application</th></tr></thead>
  <tbody>
    <tr><td>1</td><td>1 MHz (very wide)</td><td>Poor</td><td>Broadband matching</td></tr>
    <tr><td>10</td><td>100 kHz</td><td>Moderate</td><td>Audio bandpass filters</td></tr>
    <tr><td>50</td><td>20 kHz</td><td>Good</td><td>AM radio IF filter (455 kHz IF)</td></tr>
    <tr><td>100</td><td>10 kHz</td><td>High</td><td>Shortwave receiver front end</td></tr>
    <tr><td>10,000+</td><td>&lt;100 Hz</td><td>Very high</td><td>Quartz crystal resonator</td></tr>
  </tbody>
</table>

<h2>Voltage Magnification at Resonance</h2>
<p>In a series RLC at resonance, the voltage across L or C individually can be Q times the supply voltage:</p>
<pre>
V_L = V_C = Q × V_in   (at resonance)

Example: Q=50, V_in=1V
  Voltage across capacitor: 50V
  Voltage across inductor:  50V
  (they're equal and opposite — cancel in series, but each is large alone)

This is why high-Q RF circuits need high-voltage-rated capacitors
even when the supply voltage is low.
</pre>

<h2>Damping</h2>
<p>The resistance determines the transient behaviour — what happens when the circuit is disturbed from equilibrium:</p>

<table>
  <thead><tr><th>Damping type</th><th>Condition</th><th>Transient response</th><th>Use case</th></tr></thead>
  <tbody>
    <tr><td>Underdamped</td><td>Q &gt; 0.5 (R small)</td><td>Oscillates, rings. Exponentially decaying sine wave.</td><td>High-Q filters, oscillator tanks — ringing is wanted</td></tr>
    <tr><td>Critically damped</td><td>Q = 0.5 exactly</td><td>Fastest return to steady state with no overshoot</td><td>Control systems, step response optimisation</td></tr>
    <tr><td>Overdamped</td><td>Q &lt; 0.5 (R large)</td><td>Slow exponential return, no oscillation</td><td>Damping unwanted resonances in power supplies</td></tr>
  </tbody>
</table>

<h2>Practical Circuits</h2>

<h3>AM Radio Tank Circuit</h3>
<pre>
Antenna ──[ferrite rod inductor]──┬── to detector
                                  │
                              [variable C]  ← tune to station
                                  │
                                 GND

L ≈ 200–500 µH (ferrite rod coil)
C = 10–350 pF variable (tuning capacitor)
f₀ range covers ~540 kHz – 1700 kHz (AM band)

At 1 MHz with L=250µH: C = 1/(4π²f₀²L) = 101 pF
</pre>

<h3>60Hz Notch Filter for Audio</h3>
<pre>
Series LC in shunt path (Twin-T alternative):

Signal ──┬────────────────────────── Signal out
         │
       [L]──[C]  ← series LC to GND
         │
        GND

L and C chosen so f₀ = 60 Hz:
With L=1H (large): C = 1/(4π²×60²×1) ≈ 7 µF

In practice: use active twin-T notch (op-amp) for audio — achieves higher Q
and avoids the large, expensive, lossy 1H inductor.
</pre>

<h3>Output Filter for Switching Power Supply</h3>
<pre>
Buck converter output filter:
  L and C form a 2nd-order low-pass filter
  Switching noise (100kHz+) is attenuated; DC and slow changes pass

f_corner = 1/(2π√LC) should be << switching frequency
f_corner should be >> desired output bandwidth

Typical: f_switch=200kHz, f_corner=5-20kHz
L=10µH, C=100µF: f₀=5.03kHz → attenuates 200kHz by ~32dB
</pre>

<h2>References</h2>
<ul>
  <li>Horowitz &amp; Hill — <em>The Art of Electronics</em>, 3rd ed. Chapters 1, 6</li>
  <li>Hayt, Kemmerly &amp; Durbin — <em>Engineering Circuit Analysis</em></li>
  <li><a href="https://www.allaboutcircuits.com/textbook/alternating-current/chpt-6/resonance/" target="_blank" rel="noopener">All About Circuits — Resonance ↗</a></li>
  <li><a href="https://www.electronics-tutorials.ws/accircuits/series-resonance.html" target="_blank" rel="noopener">Electronics-Tutorials — Series Resonance ↗</a></li>
</ul>
