1.12 : Equivalent Resistance

In circuit analysis, situations often arise where resistors are neither in series nor parallel configurations. To tackle such scenarios, three-terminal equivalent networks like the wye (Y) (Figure 1 (a)) or tee (T) and delta (Δ) (Figure 1 (b)) or pi (π) networks come into play. These networks offer versatile solutions and are frequently encountered in various applications, including three-phase electrical systems, electrical filters, and matching networks.

The essence of these networks lies in their adaptability. They can be employed individually or integrated into more complex circuits. For instance, when dealing with a delta network but finding it more convenient to work with a wye network, a wye network can be superimposed onto the existing delta configuration. To determine the equivalent resistances within the wye network, ensuring that the resistance between each pair of nodes in the delta network equals the resistance between the same pair of nodes in the Y (or T) network is crucial.

The conversion process between these networks involves mathematical relationships that relate to the resistances. The transformation is facilitated by introducing an extra node 'n' and adhering to the conversion rule: In the Y network, each resistor is the product of the resistors in the two adjacent Δ (delta) branches, divided by the sum of the three Δ resistors. Conversely, to convert a wye network into an equivalent delta network, one can use the following conversion rule: Each resistor in the network is the sum of all possible products of Y resistors taken two at a time, divided by the opposite Y resistor.

These conversion formulas simplify for balanced networks where the resistances in both Y and Δ configurations are equal. This transformation doesn't involve adding or removing components from the circuit but substitutes mathematically equivalent three-terminal network patterns. It effectively transforms a circuit, allowing resistors to be analyzed as if they were either in series or parallel, facilitating the calculation of equivalent resistance if necessary.