Linearity is what makes every theorem work

Kirchhoff alone will always solve a circuit, but on a large network the algebra becomes punishing. The theorems that follow are shortcuts, and every one of them rests on linearity.

• Homogeneity (scaling): multiply the input by k and the output is multiplied by the same k.
• Additivity: the response to a sum of inputs equals the sum of the individual responses.
• A circuit is linear only if both hold. A resistor qualifies, since v = iR is linear in both directions.
• Power does not. p = i²R is quadratic, so no theorem in this chapter may be applied to power directly.

Using linearity directly

Scaling is a working technique, not just a property.

• Assume a convenient output, usually io = 1 A, and work backwards to the source it would require.
• Compare that with the real source value: the ratio is the scale factor k.
• Multiply your assumed answer by k. This turns a ladder network into simple arithmetic.

Source transformation

Two circuits are equivalent when their terminal v-i behaviour is identical, and that is all equivalence means here.

• A voltage source vs in series with R behaves exactly like a current source is = vs/R in parallel with the same R.
• The current source arrow points toward the positive terminal of the voltage source it replaces.
• Transform repeatedly, combining series and parallel elements between transforms, until the circuit collapses to a single loop.
• The transformation is undefined when R is zero or infinite: an ideal voltage source has no series resistance to move.