Circuit Theory/Source Injection

Source injection is a trick used when parallel/series combinations don't exist in a circuit.

Source injection helps find the overall impedance of a circuit from Vs/Is. Vs and Is are found in the phasor/complex frequency domain using node or mesh analysis.

Suppose have a complicated circuit, want to break it up into smaller sections, find a clump of passive devices in the middle of it, want to find their equivalent, but can not combine in series or parallel.

This type of problem has to be done with source injection (or Y Δ Transform .. but that is later).

One is really not physically injecting anything into the circuit, one is just pretending so that node analysis can be used to find the overall real (in phasor/complex frequency domain) impedance.

For the circuit above, the node analysis shows that the overall current is:
 * $$I_s = 0.2542735043$$

This means that:
 * $$R_{eq} = \frac{V_s}{I_s} = \frac{1}{0.2542735043} = 3.9328 ohms$$