# Delta-star and Star-delta Transformations

### Introduction

By using Kirchhoff’s laws, mesh-current analysis, nodal analysis or the super- position theorem, currents and voltages in many network can be determined as shown in chapters 70 to 72. Thevenin’s and Norton’s theorems, introduced in chapter 73, provide an alternative method of solving networks and often with considerably reduced numerical calculations. Also, these latter theorems are especially useful when only the current in a particular branch of a complicated network is required. Delta-star and star-delta transformations may be applied in certain types of circuit to simplify them before application of circuit theorems.

#### Delta and Star Connections

The network shown in Figure 74.1(a) consisting of three impedances ZA, ZB and ZC is said to be p-connected. This network can be redrawn as shown in Figure 74.1(b), where the arrangement is referred to as delta-connected or mesh-connected.

The network shown in Figure 74.2(a), consisting of three impedances, Z1, Z2 and Z3, is said to be T-connected. This network can be redrawn as shown in Figure 74.2(b), where the arrangement is referred to as star-connected.

Delta-star Transformation

It is possible to replace the delta connection shown in Figure 74.3(a) by an equivalent star connection as shown in Figure 74.3(b) such that the impedance

measured between any pair of terminals (1– 2, 2– 3 or 3– 1) is the same in star as in delta. The equivalent star network will consume the same power and operate at the same power factor as the original delta network. A delta-star transformation may alternatively be termed ‘n to T transformation’.

The star section shown in Figure 74.3(b) is equivalent to the delta section shown in Figure 74.3(a) when

In another example, the equivalent circuit impedance across terminals AB in the network of Figure 74.5 is determined as follows:

The network of Figure 74.5 is redrawn, as in Figure 74.6, showing more clearly the part of the network 1, 2, 3 forming a delta connection. This may he transformed into a star connection as shown in Figure 74.7

Star-delta Transformation

It is possible to replace the star section shown in Figure 74.10(a) by an equivalent delta section as shown in Figure 74.10(b). Such a transformation is also known as ‘T to n transformation’.

The delta section shown in Figure 74.10(b) is equivalent to the star section shown in Figure 74.10(a) when

In another example, the delta-connected equivalent network for the star- connected impedances shown in Figure 74.12 is determined as follows: Figure 74.13(a) shows the network of Figure 74.12 redrawn and Figure 74.13(b) shows the equivalent delta connection containing impedances ZA, ZB