Theory — Kirchhoff’s Rules
Kirchhoff’s rules are conservation laws for DC circuits:
charge conservation at junctions (KCL) and energy conservation around loops (KVL).
They are the standard way to solve multi-branch circuits by converting the circuit into a linear system.
1) Junction rule (KCL)
At any node (junction), the algebraic sum of currents is zero:
\( \sum I = 0 \).
A common sign convention is “currents entering the node are positive, leaving are negative” (or vice versa—just be consistent).
2) Loop rule (KVL)
Around any closed loop, the algebraic sum of potential changes is zero:
\( \sum \Delta V = 0 \).
For resistors, the drop is typically written as
\( \Delta V = -IR \)
when traversing in the direction of current.
For an ideal source, traversing from \(-\) to \(+\) gives
\( +V \).
Single-loop series example
One battery \(V\) and series resistors \(R_1, R_2, \dots\) give:
\( V - I(R_1+R_2+\cdots)=0 \),
so
\( I = \dfrac{V}{R_{eq}} \)
with
\( R_{eq}=R_1+R_2+\cdots \).
Two-loop mesh (shared resistor)
For two meshes with a shared resistor \(R_S\), define mesh currents \(I_1\) and \(I_2\).
The shared branch current is \(I_S = I_1 - I_2\) (sign depends on your direction choice).
The loop equations become a 2×2 linear system:
If a solved current is negative, it simply flows opposite to the assumed arrow direction. The physics is still consistent.
University extension
With capacitors/inductors you use impedance and complex algebra (AC steady-state), or differential equations (transients).
The “matrix setup → solve system” pattern remains the same.
