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<h1>DC Circuit Analysis</h1>
<p>DC circuit analysis is about finding voltages and currents throughout a circuit powered by a constant (direct current) voltage source. The complete toolkit is small: Ohm's Law, Kirchhoff's two laws, and the series/parallel combination rules. These four tools, used methodically, can solve any purely resistive DC circuit. The harder question is which method to reach for first — this covers all of them.</p>

<h2>The Analysis Toolkit</h2>

<table>
  <thead><tr><th>Tool</th><th>What it says</th><th>Best used when</th></tr></thead>
  <tbody>
    <tr><td>Ohm's Law (V=IR)</td><td>Voltage, current, and resistance at any single component</td><td>Single-element calculations, once I or V is known</td></tr>
    <tr><td>KVL</td><td>Voltages around any closed loop sum to zero</td><td>Mesh analysis — multiple loops with unknown currents</td></tr>
    <tr><td>KCL</td><td>Currents into any node sum to zero</td><td>Nodal analysis — multiple nodes with unknown voltages</td></tr>
    <tr><td>Series/parallel reduction</td><td>Simplify groups of resistors to one equivalent</td><td>Circuits that can be collapsed step-by-step</td></tr>
    <tr><td>Thevenin's theorem</td><td>Any two-terminal circuit = V_th + R_th in series</td><td>Analysing the effect of a changing load</td></tr>
    <tr><td>Norton's theorem</td><td>Any two-terminal circuit = I_N in parallel with R_N</td><td>Parallel current source representation</td></tr>
    <tr><td>Superposition</td><td>Linear circuits with multiple sources: solve each independently, then add</td><td>Multi-source circuits, checking individual contributions</td></tr>
  </tbody>
</table>

<h2>Simple Series Circuit</h2>
<pre>
    +──[R1=1kΩ]──[R2=2kΩ]──+
    |                       |
   [5V]                     |
    |                       |
    +───────────────────────+

Total R = 1k + 2k = 3kΩ
I = 5V / 3000Ω = 1.667 mA (same through both)

V_R1 = 1.667 mA × 1kΩ = 1.667V
V_R2 = 1.667 mA × 2kΩ = 3.333V
KVL check: 1.667 + 3.333 = 5V ✓
</pre>

<h2>Simple Parallel Circuit</h2>
<pre>
    +────[R1=6Ω]────+
    |               |
   [12V]   [R2=12Ω] |
    |               |
    +───────────────+

R_parallel = (6 × 12) / (6 + 12) = 4Ω
I_total = 12V / 4Ω = 3A

I_R1 = 12V / 6Ω = 2A
I_R2 = 12V / 12Ω = 1A
KCL check: 2 + 1 = 3A ✓
</pre>

<h2>Thevenin's Theorem</h2>
<p>Any linear two-terminal circuit can be replaced by a single voltage source (V_th) in series with a single resistor (R_th). This simplification is enormously useful when you want to analyse the effect of connecting different loads to a circuit — instead of redoing the full analysis for every load, you find the Thevenin equivalent once, then V and I for any load is trivial.</p>

<pre>
Procedure:
  1. Remove the load from the two terminals
  2. V_th = open-circuit voltage at those terminals
  3. Zero all independent sources (V sources → short, I sources → open)
  4. R_th = resistance seen looking into the terminals

Then with any load R_L:
  I_L = V_th / (R_th + R_L)
  V_L = I_L × R_L = V_th × R_L / (R_th + R_L)
</pre>

<h3>Thevenin Example</h3>
<pre>
        R1=4Ω       Terminals: A, B
12V ────[  ]────A
         |
        [R2=8Ω]
         |
        ─B──────────── (GND)

Step 1 — V_th (open circuit, no load between A and B):
  Voltage divider: V_th = 12V × 8/(4+8) = 8V

Step 2 — R_th (zero the 12V source = short circuit):
  R_th = R1 ∥ R2 = (4 × 8)/(4 + 8) = 2.67Ω

Thevenin equivalent: 8V source in series with 2.67Ω

With a 10Ω load:
  I_L = 8V / (2.67 + 10) = 0.631A
  V_L = 0.631 × 10 = 6.31V

With a 1Ω load:
  I_L = 8V / (2.67 + 1) = 2.18A
  V_L = 2.18V
</pre>

<h2>Norton's Theorem</h2>
<pre>
Norton equivalent = current source I_N in parallel with R_N

I_N = V_th / R_th   (or short-circuit current at the terminals)
R_N = R_th          (same resistance)

From the example above: I_N = 8V / 2.67Ω = 3A, R_N = 2.67Ω

Thevenin ↔ Norton conversions:
  V_th = I_N × R_N
  I_N = V_th / R_th
  R_N = R_th (identical)
</pre>

<h2>Superposition</h2>
<p>For circuits with multiple independent sources: analyse the contribution of each source separately, zero the others, then sum the results. Works for voltage and current but <em>not</em> directly for power (power is non-linear).</p>

<pre>
Circuit: two voltage sources V1 and V2 with resistors R1, R2, R3

Step 1: Zero V2 (replace with short wire). Find I' due to V1 alone.
Step 2: Zero V1 (replace with short wire). Find I'' due to V2 alone.
Step 3: I_total = I' + I''  (algebraic sum — watch directions)
</pre>

<h3>Superposition Example</h3>
<pre>
      R1=4Ω        R2=6Ω
V1=12V──[  ]──●──[  ]──V2=6V
              |
             [R3=12Ω]
              |
             GND

Contribution of V1 alone (V2 shorted):
  R2 ∥ R3 = (6×12)/(6+12) = 4Ω
  R_total = R1 + 4 = 8Ω
  I1' = 12/8 = 1.5A through R1
  V_node' = 1.5 × 4 = 6V
  I_R3' = 6/12 = 0.5A (downward)

Contribution of V2 alone (V1 shorted):
  R1 ∥ R3 = (4×12)/(4+12) = 3Ω
  R_total = R2 + 3 = 9Ω
  I2'' = 6/9 = 0.667A through R2
  V_node'' = 0.667 × 3 = 2V
  I_R3'' = 2/12 = 0.167A (downward)

Total current through R3: I_R3 = 0.5 + 0.167 = 0.667A
Total node voltage: 6 + 2 = 8V
</pre>

<h2>Source Transformations</h2>
<p>A convenient shortcut that converts between Thevenin (voltage source + series resistor) and Norton (current source + parallel resistor) forms:</p>

<table>
  <thead><tr><th>From</th><th>To</th><th>Transformation</th></tr></thead>
  <tbody>
    <tr><td>Thevenin (V_th, R_th)</td><td>Norton</td><td>I_N = V_th / R_th, same R in parallel</td></tr>
    <tr><td>Norton (I_N, R_N)</td><td>Thevenin</td><td>V_th = I_N × R_N, same R in series</td></tr>
  </tbody>
</table>

<h2>Real Sources</h2>
<p>Ideal voltage sources don't exist — batteries and power supplies have internal resistance. A real battery is its open-circuit voltage in series with an internal resistance r:</p>
<pre>
V_terminal = V_open_circuit − I × r

Battery: V_oc = 12V, r = 0.5Ω
Load = 10Ω:
  I = 12 / (0.5 + 10) = 1.143A
  V_terminal = 12 − 1.143 × 0.5 = 11.43V

Under heavy load (1Ω):
  I = 12 / (0.5 + 1) = 8A
  V_terminal = 12 − 8 × 0.5 = 8V  ← significant sag
</pre>

<h2>References</h2>
<ul>
  <li>Hayt, Kemmerly &amp; Durbin — <em>Engineering Circuit Analysis</em>, 8th ed.</li>
  <li>Nilsson &amp; Riedel — <em>Electric Circuits</em>, 10th ed.</li>
  <li><a href="https://www.allaboutcircuits.com/textbook/direct-current/" target="_blank" rel="noopener">All About Circuits — DC textbook ↗</a></li>
  <li><a href="https://www.electronics-tutorials.ws/dccircuits/dcp_1.html" target="_blank" rel="noopener">Electronics-Tutorials — DC circuits ↗</a></li>
</ul>
