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<h1>Series and Parallel Circuits</h1>
<p>Series and parallel are the two fundamental ways to connect components. They behave differently in almost every way — resistance, current, voltage, and failure modes all work differently in each topology. Real circuits combine both; knowing how to recognise and simplify each combination is the core skill of circuit analysis.</p>

<h2>Side-by-Side Comparison</h2>

<table>
  <thead><tr><th>Property</th><th>Series</th><th>Parallel</th></tr></thead>
  <tbody>
    <tr><td>Current</td><td>Same through every component</td><td>Divides — each branch carries its own current</td></tr>
    <tr><td>Voltage</td><td>Divides — each component drops a portion</td><td>Same across every component</td></tr>
    <tr><td>Total resistance (R)</td><td>R = R1 + R2 + … (always increases)</td><td>1/R = 1/R1 + 1/R2 + … (always decreases)</td></tr>
    <tr><td>Total capacitance (C)</td><td>1/C = 1/C1 + 1/C2 + … (decreases)</td><td>C = C1 + C2 + … (increases)</td></tr>
    <tr><td>Total inductance (L)</td><td>L = L1 + L2 + … (increases)</td><td>1/L = 1/L1 + 1/L2 + … (decreases)</td></tr>
    <tr><td>Component failure (open)</td><td>Breaks entire circuit — all components go off</td><td>Only that branch fails — others keep working</td></tr>
    <tr><td>Component failure (short)</td><td>Remaining components share increased voltage</td><td>Collapses the shared rail — blows supply fuse</td></tr>
    <tr><td>Common example</td><td>Old Christmas lights (one fails, all go out)</td><td>Household wiring (one appliance off, others unaffected)</td></tr>
  </tbody>
</table>

<h2>Series Circuits</h2>
<p>Components connected end-to-end so current has only one path through the circuit:</p>
<pre>
    +──[R1]──[R2]──[R3]──+
    |                    |
   [V]                   |
    |                    |
    +────────────────────+
</pre>

<p><strong>Total resistance:</strong></p>
<pre>
R_total = R1 + R2 + R3

Example: R1=2Ω, R2=3Ω, R3=5Ω → R_total = 10Ω
</pre>

<p><strong>Current:</strong> Same everywhere in the loop:</p>
<pre>
I = V / R_total = 10V / 10Ω = 1A  (flows through R1, R2, and R3)
</pre>

<p><strong>Voltage drops:</strong> Each resistor drops voltage proportional to its resistance (voltage divider):</p>
<pre>
V_R1 = I × R1 = 1A × 2Ω = 2V
V_R2 = I × R2 = 1A × 3Ω = 3V
V_R3 = I × R3 = 1A × 5Ω = 5V

KVL check: 2 + 3 + 5 = 10V ✓ (equals supply voltage)
</pre>

<h3>Voltage Divider Formula</h3>
<pre>
Two resistors R1 (top) and R2 (bottom), supply V_in:

V_out = V_in × R2 / (R1 + R2)

Useful for: scaling voltages, setting bias points, generating reference voltages
Limitation: only accurate when load impedance >> R2
</pre>

<h2>Parallel Circuits</h2>
<p>Components connected across the same two nodes — current has multiple simultaneous paths:</p>
<pre>
    +────+────+────+
    |    |    |    |
   [V]  [R1] [R2] [R3]
    |    |    |    |
    +────+────+────+
</pre>

<p><strong>Voltage:</strong> Same across all branches (both nodes shared).</p>

<p><strong>Total resistance:</strong></p>
<pre>
1/R_total = 1/R1 + 1/R2 + 1/R3
(result is always less than the smallest individual resistor)

Two resistors shortcut: R_total = (R1 × R2) / (R1 + R2)
</pre>

<p><strong>Current per branch:</strong></p>
<pre>
Example: V=12V, R1=6Ω, R2=4Ω, R3=12Ω

I_R1 = 12/6  = 2A
I_R2 = 12/4  = 3A
I_R3 = 12/12 = 1A

I_total = 2 + 3 + 1 = 6A

Verify: R_total = 1/(1/6 + 1/4 + 1/12) = 1/(2/12 + 3/12 + 1/12) = 12/6 = 2Ω
I = 12/2 = 6A ✓
</pre>

<h3>Current Divider Formula</h3>
<pre>
Two parallel resistors, total current I_total:

I_R1 = I_total × R2 / (R1 + R2)
I_R2 = I_total × R1 / (R1 + R2)

(note: each branch gets the OTHER resistor in the numerator — because higher
resistance means less current in that branch)
</pre>

<h2>Mixed (Series-Parallel) Circuits</h2>
<p>Real circuits combine both. The technique is to simplify from the inside out:</p>
<pre>
+──[R1=10Ω]──+──────────────+
             |              |
            [R2=20Ω]      [R3=30Ω]
             |              |
+────────────+──────────────+
Supply: 24V

Step 1: Find R2 ∥ R3:
  R_parallel = (20 × 30) / (20 + 30) = 600/50 = 12Ω

Step 2: R1 in series with R_parallel:
  R_total = 10 + 12 = 22Ω

Step 3: Total current from supply:
  I_total = 24V / 22Ω = 1.09A

Step 4: Voltage across parallel section:
  V_parallel = I_total × 12Ω = 13.1V

Step 5: Currents in each parallel branch:
  I_R2 = 13.1V / 20Ω = 0.655A
  I_R3 = 13.1V / 30Ω = 0.436A
  Check: 0.655 + 0.436 ≈ 1.09A ✓
</pre>

<h2>Practical Examples</h2>

<table>
  <thead><tr><th>Application</th><th>Connection</th><th>Why</th></tr></thead>
  <tbody>
    <tr><td>LED strings (common in older lights)</td><td>Series</td><td>One driver powers many LEDs — but one failure kills the string</td></tr>
    <tr><td>LED strips (modern)</td><td>Parallel sections</td><td>Failure of one LED doesn't kill the rest</td></tr>
    <tr><td>Battery banks (increase voltage)</td><td>Series</td><td>Voltages add: 3 × 3.7V = 11.1V (Li-ion pack)</td></tr>
    <tr><td>Battery banks (increase capacity)</td><td>Parallel</td><td>Current capacity adds, voltage stays same</td></tr>
    <tr><td>House wiring</td><td>Parallel</td><td>Each outlet/light gets full mains voltage independently</td></tr>
    <tr><td>Speaker crossover network</td><td>Mixed</td><td>High-pass (series cap) to tweeter, low-pass (series inductor) to woofer, both in parallel across amp output</td></tr>
  </tbody>
</table>

<h2>References</h2>
<ul>
  <li>Horowitz &amp; Hill — <em>The Art of Electronics</em>, 3rd ed. Chapter 1</li>
  <li><a href="https://www.allaboutcircuits.com/textbook/direct-current/chpt-5/series-parallel-combination-circuits/" target="_blank" rel="noopener">All About Circuits — Series-Parallel Combinations ↗</a></li>
  <li><a href="https://www.khanacademy.org/science/physics/circuits-topic" target="_blank" rel="noopener">Khan Academy — Circuits ↗</a></li>
</ul>
