## What is star delta conversion?

Find the input resistance between points A and B of the following resistive network:

It will be a challenging task to solve for R_{2}, R_{4}, and R_{6} so that we can apply series or parallel formulas to get the resultant resistance. In order to solve networks using Kirchhoff’s laws, mesh current analysis, or nodal voltage analysis, we may face great difficulty because of many simultaneous equations.

Our priority should be to simplify the network as much as possible before solving it. This is where Star to Delta or Delta to Star conversion comes into the picture. A standard 3-phase network can take two forms- a star network, which is represented by Y or Wye, and a delta network, which is represented by Δ.

In electronics, star-delta circuits are referred to as T-π circuits.

## Delta star transformation

The above circuit is a delta configuration. To convert the delta circuit into an equivalent star network, use these formulas. To visualize while calculating the values of star-connected resistances, use this figure.

The values for R_{1}, R_{2}, and R_{3,} derived from these transformation formulas are,

${R}_{1}=\frac{{R}_{a}{R}_{b}}{{R}_{a}+{R}_{b}+{R}_{c}}$ (1)

${R}_{2}=\frac{{R}_{b}{R}_{c}}{{R}_{a}+{R}_{b}+{R}_{c}}$ (2)

${R}_{3}=\frac{{R}_{c}{R}_{a}}{{R}_{a}+{R}_{b}+{R}_{c}}$ (3)

By observing the equtiona (1), (2), and (3) of star conversion, we can conclude that the equivalent star resistance arm between two delta sides is determined by the two-product of parallel resistors divided by the sum of all resistors. In electrical networks, we can represent the delta network in π shape as the diagram below:

## Star delta transformation

The above configuration is a star configuration. To convert a star circuit into an equivalent delta network, it is important to visualize the resultant resistors as the following figure.

The values for R_{a}, R_{b}, and R_{c} derived from these transformation equations:

${R}_{a}={R}_{1}+{R}_{3}+\frac{{R}_{1}{R}_{3}}{{R}_{2}}$ (4)

${R}_{b}={R}_{1}+{R}_{2}+\frac{{R}_{1}{R}_{2}}{{R}_{3}}$ (5)

${R}_{c}={R}_{2}+{R}_{3}+\frac{{R}_{2}{R}_{3}}{{R}_{1}}$ (6)

By observing the above equations of delta conversion, we can see that the equivalent delta resistance between any two-star terminals is given by the sum of both the star resistances plus the product of both these resistances divided by the third-star arm resistance.

In the star conversion, the denominator contained the sum of all the delta resistors, while in delta conversion, we do not see a sum of resistances in the denominator.

In electrical networks, we can represent the delta network in a T shape as the diagram below:

## Example

Now that we know how to simplify the star or delta connected network, we can work on the first problem in figure 1(a). The R_{2}, R_{4}, and R_{6 }resistors are in the delta-connected network. If we convert that to a star network, then the resultant network will look like this.

By using equations (1), (2), and (3), one can find the values of equivalent star network resistor values.

${R}_{a}=\frac{8\times 4}{18}=16/9\Omega $

${R}_{b}=\frac{6\times 4}{18}=12/9\Omega $

${R}_{c}=\frac{8\times 6}{18}=24/9\Omega $

Here, the resultant circuit is visible as, (R_{b} series R_{3}) || (R_{c} series R_{6})

After solving,

$\left(\frac{12}{9}+8\right)\parallel \left(\frac{24}{9}+4\right)=\left(\frac{84}{9}\right)\parallel \left(\frac{60}{9}\right)=\frac{35}{9}$

Now the resultant circuit can be seen as this,

The resultant circuit has 3 points. Therefore, the equivalent resistance value across terminals A and B is

$\left(\frac{16}{9}\right)+\left(\frac{35}{9}\right)=\frac{87}{9}\Omega $

In the example, we did a Y-Δ transformation or a delta transformation to simplify the analysis of an electrical network. Similarly, we can do delta star transformation to simplify circuits. Once we get the star equivalent circuit, we can solve that with other series or parallel circuits.

The star-delta (Y-Δ) and relevant combinations are required when working with transformer windings.

The above image shows an example of the Δ-Y connection, which means Δ in primary winding and Y in a secondary winding of the transformer. Transformers have primary and secondary three-phase windings for stepping up or stepping down the voltage.

The primary and secondary winding of a transformer can be set up in 4 different ways.

- Y-Y
- Δ-Y
- Δ-Δ
- Y-Δ

For transmission purposes, secondary winding should be in a delta connection; while for distribution purposes, secondary winding should be in star connection. We use the star connection for distribution because we get a neutral terminal at the center of the star connection.

## Impedance

While solving circuits, you may encounter the word impedance. What is it?

In DC circuits, inductors act as closed-circuit while capacitors act as open circuits. Hence, these circuits can only be explained using resistance. In AC circuits, the combined resistive effect of resistors, capacitors, and inductors makes impedance. Although both resistance and impedance are denoted by R and Z, respectively, the unit we use for both impedance and resistance is Ω.

## Formulas

- If two resistors R
_{1}and R_{2 }are in series then equivalent resistance (R_{eq}) is given by ${R}_{eq}={R}_{1}+{R}_{2}$ - If two resistors R
_{1}and R_{2 }are in parallel then equivalent resistance (R_{eq}) is given by ${R}_{eq}=\frac{{R}_{1}{R}_{2}}{{R}_{1}+{R}_{2}}$

## Context and Applications

This topic is included in the curriculum of an undergraduate degree that includes the study of basic electrical and electronics such as electrical engineering, electronics and computer engineering, electronics and communication engineering, and so on.

- Bachelors in Technology (Electronics)
- Bachelors in Technology (Electronics and Communication)
- Masters in Technology (Electronics)
- Masters in Technology (Electronics and Communications)

## Practice Problems

- A bridge network ABCD has arms AB, BC, CD, and DA of resistances 1Ω, 1Ω, 2Ω, and 1Ω, respectively. If the detector AC has a resistance of 1Ω determined by star/delta transformation then the network resistance is viewed from the battery terminals of?

- 1.86 Ω
- 1.53 Ω
- 2.09 Ω
- 1.18 Ω

Answer: Option d

Explanation: Using equations 1, 2, and 3, if one transforms DAC, which is a delta configuration to star configuration, they will get one resistor in series with one parallel circuit. Solving these, one will get 1.18 Ω resistance across the battery terminals.

2. What is the unit of inductance?

- Farad
- Candela
- Henry
- Ohm

Answer: Option c

Explanation: Henry is a unit of inductance.

- Farad is a unit of Capacitance.
- Candela is a unit of luminous intensity.
- Ohm is a unit of resistance.

3. Which of the following is used to represent star-delta in electronics?

- Y-Δ
- *-¥
- T-π
- Þ-▲

Answer: Option c

Explanation: T-π is used to represent star-delta in electronics.

4. If two resistors R_{1} and R_{2 }are in parallel then what will be the equivalent resistance value?

- (R
_{1}+R_{2})/(R_{1}R_{2}) - (R
_{1}R_{2})/(R_{1}+R_{2}) - (R
_{1}R_{2})/(R_{1}) - (R
_{1})/(R_{2})

Answer: Option b

Explanation: The reciprocal of the equivalent value of two parallel resistors is the sum of the reciprocal of both the resistors. Consequently, after solving, one gets (R_{1}R_{2})/(R_{1}+R_{2}).

5. If the apparent power of a generator is 100kVA and the power factor is 60%. Find the actual power.

- 60kVA
- 600kW
- 600W
- 6kW

Answer: Option a

Explanation: Actual power = Actual power x power factor = 100kVA x 0.6 = 60kVA

## Common Mistakes

- The formulas for a star to delta and vice versa may be confusing at the start. Consequently, it is recommended for the students to practice with arbitrary, but using the simple values in the initial stage. Practice can help to overcome this confusion. Also, visualizing the transformations provides clarity about the denominator resistor values.

- Impedance can be confused with reactance. Impedance is the resistive effort of the circuit, while reactance is the combined effort of inductors and capacitors on the circuit output.

## Related Concepts

- Kirchhoff’s laws
- Thevenin theorem
- Norton’s theorem
- Superposition theorem

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