Situations often arise in circuit analysis when the resistors are neither in parallel nor in series. For example, consider the bridge circuit in Fig. 2.46. How do we combine resistors R1 through R6 when the resistors are neither in series nor in parallel? Man y circuits of the type shown in Fig. 2.46 can be simplified by using three-terminal equivalent networks. These are the wye (Y) or tee (T) network shown in Fig. 2.47 and the delta (Δ) or pi (π) network shown in Fig. 2.48. These networks occur by themselves or as part of a larger network. They are used in three-phase networks, electrical filters, and matching networks.

Figure 2.46 The bridge network.


Figure 2.47 Two forms of the same network: (a) Y, (b) T.
Figure 2.48 Two forms of the same network: (a) Δ, (b) Π.



Our main interest here is in how to identify them when they occur as part of a network and how to apply wye-delta transformation in the analysis of that network

Delta to Wye Conversion


Suppose it is more convenient to work with a wye network in a place where the circuit contains a delta configuration. We superimpose a wye network on the existing delta network and find the equivalent resistances in the wye network. To obtain the equivalent resistances in the wye network, we compare the two networks and make sure that the resistance between each pair of nodes in the Δ (or Π) network is the same as the resistance between the same pair of nodes in the Y (or T) network. For terminals 1 and 2 in Figs. 2.47 and 2.48, for example,



Subtracting Eq. (2.49) from Eq. (2.47a), we obtain :


We do not need to memorize Eqs. (2.49) to (2.51). To transform a ∆ network to Y, we create an extra node n as shown in Fig. 2.49 and follow this conversion rule: 

Each resistor in the Y network is the product of the resistors in the two adjacent Δ branches, divided by the sum of the three Δ resistors.

Figure 2.49 Superposition of Y and Δ networks as an aid in transforming one to the other.

Wye to Delta Conversion

To obtain the conversion formulas for transforming a wye network to an equivalent delta network, we note from Eqs. (2.49) to (2.51) that


Dividing Eq. (2.52) by each of Eqs. (2.49) to (2.51) leads to the follo wing equations:
From Eqs. (2.53) to (2.55) and Fig. 2.49, the con version rule for Y to Δ is as follows:

Each resistor in the Δ network is the sum of all possible products of Y resistors taken two at a time, divided by the opposite Y resistor.

The Y and Δ networks are said to be balanced when 

Under these conditions, conversion formulas become

Example

Convert the Δ network in Figure to an equivalent Y network. 

Solution: