Question

How do you find the resistance of a star and delta connection?

Answer 1

Finding the resistance of electrical networks configured in star (Y) and delta (Δ) connections involves some specific calculations. These networks are often found in three-phase systems and their resistances can be converted from one configuration to the other. Here’s a detailed breakdown of how to determine the resistance in both configurations and how to convert between them:

### 1. **Understanding Star and Delta Configurations**

- **Star (Y) Connection**: In a star connection, each component is connected to a common central point (the neutral point). In this setup, the resistances are connected as follows:
  - R1, R2, and R3 are the resistances in the star configuration.
  
- **Delta (Δ) Connection**: In a delta connection, the resistances are connected in a loop or triangle. In this setup:
  - The resistances are denoted as \( R_{AB} \), \( R_{BC} \), and \( R_{CA} \), where each resistor is between two points of the delta connection.

### 2. **Calculating Resistance in Star and Delta Configurations**

#### **Star to Delta Conversion**

If you have resistances in a star configuration and you want to find the equivalent resistances in the delta configuration, use the following formulas:

Let the star resistances be \( R_A \), \( R_B \), and \( R_C \). The resistances in the equivalent delta configuration (\( R_{AB} \), \( R_{BC} \), and \( R_{CA} \)) can be found using:

\[
R_{AB} = \frac{R_A R_B + R_B R_C + R_C R_A}{R_C}
\]

\[
R_{BC} = \frac{R_A R_B + R_B R_C + R_C R_A}{R_A}
\]

\[
R_{CA} = \frac{R_A R_B + R_B R_C + R_C R_A}{R_B}
\]

#### **Delta to Star Conversion**

Conversely, if you have resistances in a delta configuration and need to find the equivalent star resistances, use:

Let the delta resistances be \( R_{AB} \), \( R_{BC} \), and \( R_{CA} \). The resistances in the equivalent star configuration (\( R_A \), \( R_B \), and \( R_C \)) can be found using:

\[
R_A = \frac{R_{AB} R_{CA}}{R_{AB} + R_{BC} + R_{CA}}
\]

\[
R_B = \frac{R_{AB} R_{BC}}{R_{AB} + R_{BC} + R_{CA}}
\]

\[
R_C = \frac{R_{BC} R_{CA}}{R_{AB} + R_{BC} + R_{CA}}
\]

### 3. **Example Calculations**

#### **Star to Delta Example**

Suppose you have a star configuration with resistances \( R_A = 10 \ \Omega \), \( R_B = 20 \ \Omega \), and \( R_C = 30 \ \Omega \).

Calculate the equivalent delta resistances:

\[
R_{AB} = \frac{10 \times 20 + 20 \times 30 + 30 \times 10}{30} = \frac{200 + 600 + 300}{30} = \frac{1100}{30} \approx 36.67 \ \Omega
\]

\[
R_{BC} = \frac{10 \times 20 + 20 \times 30 + 30 \times 10}{10} = \frac{200 + 600 + 300}{10} = \frac{1100}{10} = 110 \ \Omega
\]

\[
R_{CA} = \frac{10 \times 20 + 20 \times 30 + 30 \times 10}{20} = \frac{200 + 600 + 300}{20} = \frac{1100}{20} = 55 \ \Omega
\]

#### **Delta to Star Example**

Suppose you have a delta configuration with resistances \( R_{AB} = 15 \ \Omega \), \( R_{BC} = 25 \ \Omega \), and \( R_{CA} = 35 \ \Omega \).

Calculate the equivalent star resistances:

\[
R_A = \frac{15 \times 35}{15 + 25 + 35} = \frac{525}{75} = 7 \ \Omega
\]

\[
R_B = \frac{15 \times 25}{15 + 25 + 35} = \frac{375}{75} = 5 \ \Omega
\]

\[
R_C = \frac{25 \times 35}{15 + 25 + 35} = \frac{875}{75} \approx 11.67 \ \Omega
\]

### Conclusion

By using these formulas, you can convert resistances between star and delta configurations to simplify circuit analysis. This conversion is particularly useful in three-phase systems and helps in designing and analyzing electrical networks.

Answer 2

### Finding the Resistance of a Star (Y) and Delta (Δ) Connection

Both star (Y) and delta (Δ) connections are common configurations in electrical systems, particularly in three-phase power systems. Calculating the equivalent resistances for these two configurations can help in the analysis of circuits.

#### 1. **Resistance in a Star (Y) Connection**

In a star connection, each resistor is connected from one phase (line) to the neutral point. The formula for resistance between any two terminals (i.e., phase-to-phase resistance) in a star connection depends on the resistors connected between each phase and the neutral point.

- Let the three resistances in a star connection be \( R_1 \), \( R_2 \), and \( R_3 \).
  
The resistance between two phases, say between phase 1 and phase 2 (i.e., \( R_{12} \)), is calculated as:
\[
R_{12} = R_1 + R_2
\]

Similarly, for other phases:
- \( R_{23} = R_2 + R_3 \)
- \( R_{31} = R_3 + R_1 \)

This is because the current passes through two resistors when you measure between any two phases.

#### 2. **Resistance in a Delta (Δ) Connection**

In a delta connection, each resistor is connected between two phases (line-to-line). If \( R_1 \), \( R_2 \), and \( R_3 \) are the resistances between the three phases, the total resistance between two phases is more complicated.

To find the equivalent resistance for a delta connection between two terminals (phases), we use the formula derived from delta-to-star transformation:

For example, the equivalent resistance \( R_{eq} \) of a delta connection (where \( R_A \), \( R_B \), and \( R_C \) are the resistances between the phases) between any two terminals (say between terminal 1 and terminal 2) is given by:
\[
R_{eq} = \frac{R_A \cdot R_B + R_B \cdot R_C + R_C \cdot R_A}{R_C}
\]

This equation is derived from the relationship between the delta and star connections.

#### 3. **Delta-Star Transformation**

To convert from delta (Δ) to star (Y) or vice versa, you use specific transformation formulas:

##### Delta (Δ) to Star (Y):
If the resistances in the delta connection are \( R_A \), \( R_B \), and \( R_C \), the equivalent star resistances \( R_1 \), \( R_2 \), and \( R_3 \) are:
\[
R_1 = \frac{R_A \cdot R_C}{R_A + R_B + R_C}
\]
\[
R_2 = \frac{R_A \cdot R_B}{R_A + R_B + R_C}
\]
\[
R_3 = \frac{R_B \cdot R_C}{R_A + R_B + R_C}
\]

##### Star (Y) to Delta (Δ):
For the reverse conversion (star to delta), where the star resistances are \( R_1 \), \( R_2 \), and \( R_3 \), the delta resistances are given by:
\[
R_A = \frac{R_1 \cdot R_2 + R_2 \cdot R_3 + R_3 \cdot R_1}{R_1}
\]
\[
R_B = \frac{R_1 \cdot R_2 + R_2 \cdot R_3 + R_3 \cdot R_1}{R_2}
\]
\[
R_C = \frac{R_1 \cdot R_2 + R_2 \cdot R_3 + R_3 \cdot R_1}{R_3}
\]

These conversions are essential when analyzing three-phase circuits because sometimes it is easier to analyze the circuit in one configuration over the other.

### Practical Applications
- **Star Connection:** Typically used in systems where a neutral wire is required, such as distribution networks. The star configuration has a neutral point where a grounded wire is connected.
- **Delta Connection:** Common in systems where there is no need for a neutral wire, such as motors. The delta connection is often used in the high-power transmission system.

#### Summary:
- For a star connection, the resistance between two phases is the sum of the two resistances connected to the neutral.
- For a delta connection, the resistance between any two terminals is a more complex calculation involving all three delta resistances.
- You can convert between delta and star using the transformation formulas to simplify circuit analysis.

Understanding these calculations helps in designing, analyzing, and troubleshooting three-phase electrical systems.