Chapter 3

Analysis Techniques Using Kirchhoff's Laws

Chapter 2 gave you the laws. Applied one branch at a time they run out of road fast — the book shows a circuit and calls it “hopeless” by its second page. What rescues it is a change of unknowns: stop solving for the current in every branch and solve for the node voltages, or for the mesh currents. Two methods, one idea, and a taxonomy of awkward sources you have to learn to recognise before you write anything. Ten exercises below.

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What you need before you start

KCL, one way only. Every node equation in this chapter is written the same way, and the book is emphatic that you never improvise a second convention.

Σ i_leaving = 0 A

A current leaving counts positive. A source between two nodes appears in both node equations with opposite signs — what leaves one node enters the other.

Essential node — “a node that connects more than two branches; in other words, a current will split going through that node.” Only these get an equation. Ground is chosen first, and the sane choice is the node with the most branches on it.

The deliverable is the matrix. Nodal analysis stamps a conductance matrix, mesh analysis a resistance matrix, and both are read physically: the diagonal sums what touches that node or loop, the off-diagonal is minus what is shared.

G V = I S · V = A

R I = V Ω · A = V

Inverting it is not the skill — the book hands that to “modern computer tools”, and the calculator in the corner of this page is one. Building the system correctly is the skill.

Mesh currents are bookkeeping. They are variables you invent, always drawn clockwise here, around the smallest loops. The current in a shared element is a combination of them — and that combination is a sum when both loops run through it the same way.

When a source refuses to tell you something, name it. The current through a voltage source and the voltage across a current source are both unknowable from the element itself. In both cases the move is the same: invent a dummy variable, write the equations, then either cancel it by adding two of them together or keep it and add one constraint. Neither trick gets a special name in this book — they are the same idea twice.

voltage current power

“Always (yes, with no exceptions) apply KCL in the same consistent way no matter what circuit has been given and regardless of how tempting it may be to ‘invent a new method.’”

§3.1. The chapter's discipline in one line. Most of the errors below are not arithmetic — they are a sign convention that changed halfway through a problem.

Exercises

Each one names where it comes from in the book. § is the section sign, so “§3.2” means section 3.2; “Ex. 3.2.1” is a worked example inside that section, and “Fig.” a figure.