Worked examples: splitting a source current

Two problems — a two-branch split and a three-branch bank. The two-branch case uses the closed-form formula; the three-branch case shows why the voltage-then-Ohm shortcut is usually easier than the resistance-form divider.

Example 1 — two-branch split

A 6 A source current enters a fan-out node that splits between $R_1 = 4\,\Omega$ and $R_2 = 12\,\Omega$. Find $I_1$ and $I_2$.

1. Apply the two-branch formula. Remember the flip — R₁'s share has R₂ on top.
   $$I_1 = 6 \cdot \frac{12}{4 + 12} = 6 \cdot \frac{12}{16} = 4.5\,\text{A}$$
   $$I_2 = 6 \cdot \frac{4}{4 + 12} = 6 \cdot \frac{4}{16} = 1.5\,\text{A}$$
2. KCL check. $I_1 + I_2 = 4.5 + 1.5 = 6\,\text{A} = I_{src}$. ✓
3. Sanity check — which branch should win? R₁ is smaller, so R₁ should carry more current. 4.5 A vs. 1.5 A — yes, three times more through the smaller resistor, which matches the 3:1 conductance ratio (R₂/R₁).

Example 2 — three-branch bank, voltage shortcut

A source forces 12 A into a three-branch parallel bank: $R_1 = 2\,\Omega$, $R_2 = 4\,\Omega$, $R_3 = 6\,\Omega$. Find each branch current.

1. Parallel equivalent.
   $$1/R_{total} = 1/2 + 1/4 + 1/6 = 6/12 + 3/12 + 2/12 = 11/12$$
   $R_{total} = 12/11 \approx 1.091\,\Omega$.
2. Voltage across the bank. $V = I_{src} \cdot R_{total} = 12 \cdot (12/11) = 144/11 \approx 13.09\,\text{V}$. Every branch sees this same voltage — same reason KCL says every branch shares the fan-out node.
3. Branch currents via Ohm's Law.
   $$I_1 = V/R_1 = 13.09/2 \approx 6.54\,\text{A}$$
   $$I_2 = V/R_2 = 13.09/4 \approx 3.27\,\text{A}$$
   $$I_3 = V/R_3 = 13.09/6 \approx 2.18\,\text{A}$$
4. KCL check. $I_1 + I_2 + I_3 \approx 6.54 + 3.27 + 2.18 \approx 12\,\text{A}$. The small rounding in the second decimal washes out; they sum back to the source.

Notice again: the 2 Ω branch hogs the current (6.54 A, about 55 %), and the 6 Ω branch takes the smallest slice. That's the conductance distribution — $G_k / G_{total}$ — showing up as the expected fractions (6/11, 3/11, 2/11).