Series, Parallel, Delta & Wye Resistive Reductions

An equivalent resistor network must preserve the voltage-current relation seen at its exposed terminals. Reduction is therefore an elimination operation: internal node voltages and branch currents disappear, but no external experiment can distinguish the reduced network from the original over the modeled range.

Series and parallel tests #

Two resistors are in series only when their shared node has no third connection, forcing the same current through both:

$$R_s=R_1+R_2.$$

They are in parallel only when both ends connect to the same two nodes, forcing the same voltage across both:

$$\frac{1}{R_p}=\frac{1}{R_1}+\frac{1}{R_2},\qquad R_p=\frac{R_1R_2}{R_1+R_2}.$$

Visual alignment on a schematic proves neither condition. Node identity does.

Delta to wye #

For a delta with edge resistances $R_{ab}$, $R_{bc}$, and $R_{ca}$, let $S=R_{ab}+R_{bc}+R_{ca}$. The equivalent wye arms are

$$R_a=\frac{R_{ab}R_{ca}}{S},\quad R_b=\frac{R_{ab}R_{bc}}{S},\quad R_c=\frac{R_{bc}R_{ca}}{S}.$$

For the reverse conversion, with wye arms $R_a,R_b,R_c$ and $P=R_aR_b+R_bR_c+R_cR_a$:

$$R_{ab}=\frac{P}{R_c},\quad R_{bc}=\frac{P}{R_a},\quad R_{ca}=\frac{P}{R_b}.$$

The formulas preserve resistance between each pair of external terminals with the third terminal left open.

Reduction strategy #

1. label nodes before looking for patterns;
2. reduce unambiguous series/parallel groups;
3. use delta-wye only when it exposes a new reduction;
4. stop when reduction would hide a requested internal current or voltage;
5. use nodal or mesh analysis when algebraic reduction grows rather than shrinks.

The open-third-terminal check is necessary but not sufficient by itself; repeat it for all three terminal pairs. Equality of those three pairwise resistances establishes the three-terminal equivalence for a passive resistive network.