Kirchhoff's Rules for Multi-Loop Circuits
The junction rule is charge conservation and the loop rule is energy conservation, and together they give enough equations to solve any network. Sign conventions are the only real difficulty.
What a learner can do afterwards
- Assigns currents and signs consistently, then writes independent junction and loop equations
- Solves a two-loop circuit with two sources for every branch current
- Interprets a negative current as a direction guessed the wrong way round
1 · Read
The junction rule is charge conservation: currents entering a junction equal currents leaving it, so the sum in equals the sum out. The loop rule is energy conservation: the voltage changes around any closed loop sum to zero. Crossing a battery from minus to plus counts positive, and crossing a resistor along the current drops the potential by I times R.
Take a single 12.00 V loop with 1.00, 2.00 and 3.00 ohm resistors. The loop rule gives 12 minus I times 1 minus I times 2 minus I times 3 equals zero. That solves to I equals 12 divided by 6, which is 2.00 A through every part.
For a two-loop circuit with two sources, first draw an arrow for each branch current, such as I1, I2 and I3. One junction gives I1 equals I2 plus I3, and the second junction only repeats it, so use one junction equation. Then write one loop equation per loop and solve the three equations together. A negative answer is not a failure: it means the true direction runs opposite your arrow.
Fix a travel direction before writing each loop equation. A resistor crossed with the current contributes minus I R, and crossed against the current contributes plus I R. Multi-loop circuits cannot be reduced to simple series and parallel groups, so write the full set of equations.
One junction equation plus one loop equation per loop gives every branch current, and a minus sign only flips the arrow.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.