Direct Branch-Current Analysis with KCL & KVL

Branch-current analysis is the literal formulation of a circuit: one unknown current per branch, conservation equations from topology, and voltage-current equations from the elements. It makes every assumption visible and works for networks that are awkward to reduce, at the cost of solving more simultaneous equations.

Equation count #

For a connected network with $b$ branches and $n$ nodes, choose:

• $n-1$ independent KCL equations;
• $b-n+1$ independent KVL equations.

The total is $b$, exactly the number of branch currents. Express each resistor voltage as $R_k I_k$ and each source voltage with the sign implied by the loop traversal.

Procedure #

1. assign arbitrary current arrows $I_1,\ldots,I_b$;
2. mark voltage polarity consistently, normally using the passive sign convention for loads;
3. write KCL at all but one node;
4. choose independent loops, preferably from a spanning tree plus one link at a time;
5. substitute element laws into KVL;
6. solve the linear system and retain the signs of the answers;
7. verify unused node/loop equations and the power balance.

In matrix form the DC resistor problem becomes

$$MI=s,$$

where rows of $M$ contain incidence coefficients from KCL and signed resistances from KVL. A negative current simply reverses the assigned arrow.

Why the method is still useful #

Nodal analysis minimizes voltage unknowns; mesh analysis minimizes loop-current unknowns in planar circuits. Direct branch analysis is preferable when the required outputs are branch currents, the topology is small, or an audit must show exactly which KCL and KVL equations define the result.

Large residuals in unused equations usually expose a sign or transcription error earlier than inspecting the numerical solution does.

Direct analysis keeps the negative current rather than redrawing the arrow. Substituting the solved currents into an independent loop equation and a signed power balance turns the example into a complete audit rather than a single linear solve.