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<h1>Capacitors</h1>
<p>Capacitors store energy in an electric field between two conductive plates separated by an insulating dielectric. At first glance they look like resistors — two-terminal passive components — but their behaviour is fundamentally different. A capacitor blocks DC and passes AC, which makes them essential for filtering, coupling, decoupling, and timing circuits. After resistors, they're the most common component you'll work with.</p>

<h2>How They Work</h2>
<p>Apply voltage across a capacitor and charge builds up on the plates, creating an electric field across the dielectric. The basic formula for capacitance:</p>
<pre>
C = Q / V

C — capacitance in farads (F)
Q — charge in coulombs (C)
V — voltage in volts (V)
</pre>
<p>The voltage-current relationship in a capacitor:</p>
<pre>
I = C × dV/dt
</pre>
<p>Current flows only when voltage is <em>changing</em>. Steady DC? No current. Rapidly changing voltage? Large current. This is the key insight: capacitors are rate-sensitive, not level-sensitive. That's why they block DC (no change) and pass AC (constant change).</p>

<h2>Capacitance Units</h2>
<p>A 1 F capacitor is enormous — physically the size of a can, and rare outside of supercapacitor applications. Real circuits use much smaller values:</p>

<table>
  <thead><tr><th>Unit</th><th>Symbol</th><th>Value</th><th>Typical use</th></tr></thead>
  <tbody>
    <tr><td>Farad</td><td>F</td><td>1 F</td><td>Supercapacitors, backup power</td></tr>
    <tr><td>Millifarad</td><td>mF</td><td>0.001 F</td><td>Large electrolytics</td></tr>
    <tr><td>Microfarad</td><td>µF</td><td>0.000001 F</td><td>Power supply filtering, audio coupling</td></tr>
    <tr><td>Nanofarad</td><td>nF</td><td>10⁻⁹ F</td><td>RF filters, timing circuits</td></tr>
    <tr><td>Picofarad</td><td>pF</td><td>10⁻¹² F</td><td>RF tuning, crystal load caps, stray capacitance</td></tr>
  </tbody>
</table>

<h2>Types</h2>

<table>
  <thead><tr><th>Type</th><th>Polarized?</th><th>Value range</th><th>Key characteristic</th><th>Watch out for</th></tr></thead>
  <tbody>
    <tr><td>Ceramic (MLCC)</td><td>No</td><td>1 pF – 100 µF</td><td>Cheap, small, widely available. The default for decoupling.</td><td>Capacitance drops significantly with applied voltage (especially X5R/X7R — check the derating curve)</td></tr>
    <tr><td>Electrolytic (aluminium)</td><td>Yes</td><td>1 µF – 100,000 µF</td><td>High capacitance per dollar. Used in PSU bulk storage.</td><td>Must get polarity right. Dry out over years. Swollen tops = end of life.</td></tr>
    <tr><td>Tantalum</td><td>Yes</td><td>0.1 µF – 1,000 µF</td><td>More stable than electrolytic, smaller for same value.</td><td>Reverse voltage or overcurrent → fire. Not a joke.</td></tr>
    <tr><td>Film (polyester/PP)</td><td>No</td><td>1 nF – 100 µF</td><td>Stable over temperature, low loss (low ESR). Good for audio and precision filters.</td><td>Larger and more expensive than ceramic for same value</td></tr>
    <tr><td>Supercapacitor (EDLC)</td><td>Yes</td><td>0.1 F – 3,000 F</td><td>Stores much more energy than standard caps. Used as battery backup.</td><td>Low voltage rating (2.5–2.7 V typical); must series-connect for higher voltages</td></tr>
  </tbody>
</table>

<h2>Charging and Discharging</h2>
<p>Through a resistor, a capacitor charges and discharges exponentially. The time constant τ (tau) = RC determines the speed:</p>
<pre>
Charging:    V(t) = V₀ × (1 - e^(-t/RC))
Discharging: V(t) = V₀ × e^(-t/RC)

τ = R × C   (in seconds, when R is in ohms and C in farads)
</pre>

<table>
  <thead><tr><th>Time elapsed</th><th>Charge (% of final voltage)</th><th>Discharge (% of initial voltage)</th></tr></thead>
  <tbody>
    <tr><td>1τ</td><td>63.2%</td><td>36.8%</td></tr>
    <tr><td>2τ</td><td>86.5%</td><td>13.5%</td></tr>
    <tr><td>3τ</td><td>95.0%</td><td>5.0%</td></tr>
    <tr><td>4τ</td><td>98.2%</td><td>1.8%</td></tr>
    <tr><td>5τ</td><td>99.3%</td><td>0.7%</td></tr>
  </tbody>
</table>

<p>After 5τ the capacitor is considered fully charged or discharged for practical purposes.</p>

<p>Example:</p>
<pre>
R = 10 kΩ, C = 100 µF
τ = 10,000 × 0.0001 = 1 second
Fully charged in ~5 seconds
</pre>

<h2>Combining Capacitors</h2>
<p>The opposite rules to resistors:</p>
<pre>
Series (reduces total):
  1/C_total = 1/C1 + 1/C2 + ...
  Two equal caps in series: C_total = C/2

Parallel (adds together):
  C_total = C1 + C2 + C3 + ...
  Two equal caps in parallel: C_total = 2C
</pre>
<p>You series-connect capacitors to increase voltage rating (the voltage divides across them). You parallel-connect to increase capacitance. When series-connecting electrolytics, voltage-balancing resistors are needed across each cap to prevent one from seeing all the voltage.</p>

<h2>ESR — Equivalent Series Resistance</h2>
<p>A real capacitor isn't purely capacitive — it has a small resistance in series (ESR) from the leads and dielectric losses. ESR matters most in switching power supplies and RF circuits:</p>

<table>
  <thead><tr><th>Type</th><th>Typical ESR</th><th>Impact</th></tr></thead>
  <tbody>
    <tr><td>Ceramic (MLCC)</td><td>1–100 mΩ</td><td>Negligible for most circuits</td></tr>
    <tr><td>Aluminium electrolytic</td><td>10 mΩ – 10 Ω</td><td>Significant in switching supplies; use low-ESR types rated for ripple current</td></tr>
    <tr><td>Tantalum</td><td>100 mΩ – 1 Ω</td><td>Better than aluminium electrolytic; watch surge current</td></tr>
    <tr><td>Film</td><td>1–100 mΩ</td><td>Low, stable — good for audio and precision work</td></tr>
  </tbody>
</table>

<h2>Common Applications</h2>

<h3>Decoupling</h3>
<p>Place a 100 nF ceramic cap between VCC and GND as close to each IC's power pin as possible. When the IC switches, it draws a sudden burst of current. The decoupling cap supplies that burst locally, preventing voltage dips from propagating to other parts of the board. This is one of the most important habits in PCB layout — it's not optional.</p>
<pre>
Rule of thumb: 100nF ceramic per IC, plus one 10µF bulk electrolytic per power rail section
</pre>

<h3>AC Coupling</h3>
<p>A capacitor in series blocks DC but passes AC. Common in audio to connect stages that operate at different DC bias points:</p>
<pre>
Stage 1 output: 2.5V DC + AC signal
                    |
                   [C]  ← blocks the 2.5V DC
                    |
Stage 2 input: 0V DC + AC signal only
</pre>

<h3>Timing (RC Circuit)</h3>
<pre>
555 timer astable oscillator (approximate):
f ≈ 1.44 / ((R1 + 2×R2) × C)

For f=1kHz with C=100nF:
(R1 + 2×R2) = 1.44 / (1000 × 0.0000001) ≈ 14,400Ω
</pre>

<h2>Common Failure Modes</h2>

<table>
  <thead><tr><th>Symptom</th><th>Likely cause</th><th>Check</th></tr></thead>
  <tbody>
    <tr><td>Swollen or leaking electrolytic</td><td>End of life, overvoltage, reversed polarity</td><td>Replace — do not continue using</td></tr>
    <tr><td>Circuit works intermittently</td><td>Dried-out electrolytic (increased ESR)</td><td>Measure ESR with a capacitor ESR meter</td></tr>
    <tr><td>Oscillator runs at wrong frequency</td><td>Wrong capacitor value; capacitance derating</td><td>Measure actual capacitance; check ceramic derating vs. voltage</td></tr>
    <tr><td>Power rail voltage drops under load</td><td>Insufficient bulk capacitance or high ESR</td><td>Increase capacitance, use low-ESR type</td></tr>
  </tbody>
</table>

<h2>References</h2>
<ul>
  <li>Horowitz &amp; Hill — <em>The Art of Electronics</em>, 3rd ed. Chapters 1 &amp; 2</li>
  <li>Scherz &amp; Monk — <em>Practical Electronics for Inventors</em>, 4th ed.</li>
  <li><a href="https://www.capacitorguide.com/" target="_blank" rel="noopener">CapacitorGuide.com ↗</a></li>
  <li><a href="https://www.murata.com/en-us/products/capacitor/mlcc/faq" target="_blank" rel="noopener">Murata — MLCC FAQ (ceramic capacitor behaviour) ↗</a></li>
</ul>
