 # Introduction to AC Circuits

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## 3.5 Star-Delta and Delta-Star Conversion in Three-Phase AC Circuits

In this book, the three-phase ac systems are considered as a balanced circuit, made up of a balanced three-phase source, a balanced line, and a balanced three-phase load. Therefore, a balanced system can be studied using only one-third of the system, which can be analyzed on a line to neutral basis.

The star-delta (Y-Δ) or delta-star (Δ-Y) conversion (Fig. 3-15) is required in three-phase ac systems to simplify the circuits and ease their analysis. If a three-phase supply or a three-phase load is connected in delta, it can be transformed into an equivalent star-connected supply or load. After the analysis, the results are converted back into their original delta equivalent. Figure 3-15. Impedance circuits that are equivalent in relationship to terminals a, b, and c: (a) star-connected and T-connected impedances, and (b) delta-connected and π-connected impedances.

The complex delta-star or star-delta conversion formulas are given next. These are based on the electric circuits shown in Fig. 3-15.

Equation 3.35 Equation 3.36 Equation 3.37 Equation 3.38 Equation 3.39 Equation 3.40 where Z is the complex impedance, Z = R ± jX.

Since the load is balanced, the impedance per phase of the star-connected load will be one-third of the impedance per phase of the delta-connected load. Hence the equivalent impedances can be given by

Equation 3.41 One of the common uses of these transformations is in power system transmission line modeling and in three-phase transformer analysis. Circuit analysis involving three-phase transformers under balanced conditions can be performed on a per-phase basis. When Δ-Y or Y-Δ connections are present, the parameters refer to the Y side. In Δ-Δ connections, the Δ-connected impedances are converted to equivalent Y-connected impedances.

#### 3.5.1 Virtual Instrument Panel

The objective of the following VI is to study these transformation concepts and provide an easy calculation tool using the complex impedances. The front panel of Star Delta Transformations.vi is given in Fig. 3-16 and is capable of transforming balanced or unbalanced three-phase impedance loads. Figure 3-16. Front panel and brief user guide of Star Delta Transformations.vi.

#### 3.5.2 Self-Study Questions

Open and run the custom-written VI named Star Delta Transformations.vi in the Chapter 3 folder, and investigate the following questions.

 1: Set all impedances equal and perform Δ-Y and Y-Δ transformations, then repeat the transformations for unequal impedances, and verify the results analytically. 2: The circuit shown in Fig. 3-17 is called an unbalanced Wheatstone Bridge. Find the equivalent resistance between terminals A and D, which then can be used to calculate the source current for a given supply voltage. Figure 3-17. Sample circuit for question 2. A2: Answer: 20.94 Ω Hint: Use Δ-Y transformation to simplify the circuit.