To calculate the impedance between terminals $P$ and $Q$ using the star-delta transformation, we follow these steps:

Step 1: Identify the Star Network

The resistances $Z_2, Z_3, Z_5$ form a star network, and we can transform it into a delta network to simplify the calculation.

Step 2: Star-Delta Transformation

The equivalent delta impedances $Z_{23}, Z_{35}, Z_{52}$ are calculated as follows:

$Z_{23} = \frac{Z_2 Z_3 + Z_3 Z_5 + Z_5 Z_2}{Z_5}$ $Z_{35} = \frac{Z_2 Z_3 + Z_3 Z_5 + Z_5 Z_2}{Z_2}$ $Z_{52} = \frac{Z_2 Z_3 + Z_3 Z_5 + Z_5 Z_2}{Z_3}$

Substitute the values:

• $Z_2 = 14 + j8$
• $Z_3 = 4 + j3$
• $Z_5 = 11 - j10$

Calculate $Z_{23}, Z_{35}, Z_{52}$.

Step 3: Simplify the Circuit

After transforming the star to a delta network, the circuit will now have $Z_1, Z_{23}, Z_4, Z_{35}, Z_{52}$. Combine the impedances in series and parallel as follows:

1. Combine $Z_1$ with $Z_{23}$ in series.
2. Combine $Z_4$ with $Z_{35}$ in series.
3. Calculate the parallel combination of the results above.
4. Add $Z_{52}$ to the result.

Step 4: Final Impedance Between $P$ and $Q$

Combine all the equivalent impedances to find the total impedance between $P$ and $Q$.

Let me calculate the star-delta transformation and the final impedance. Importing necessary libraries for complex arithmetic import sympy as sp

Given impedances

Z2 = 14 + 8j # Impedance Z2 Z3 = 4 + 3j # Impedance Z3 Z5 = 11 - 10j # Impedance Z5

Star to Delta transformation formulas

Z23 = (Z2 * Z3 + Z3 * Z5 + Z5 * Z2) / Z5 Z35 = (Z2 * Z3 + Z3 * Z5 + Z5 * Z2) / Z2 Z52 = (Z2 * Z3 + Z3 * Z5 + Z5 * Z2) / Z3

Display the results

Z23, Z35, Z52After performing the star-delta transformation, the equivalent delta impedances are:

1. $Z_{23} = 16.24 + j16.13 \, \Omega$
2. $Z_{35} = 18.77 - j9.65 \, \Omega$
3. $Z_{52} = 56.20 - j38.40 \, \Omega$

Now, let’s simplify the circuit further by combining the impedances and calculating the total impedance between terminals $P$ and $Q$. Given other impedances Z1 = 12 - 11j # Impedance Z1 Z4 = 5 + 12j # Impedance Z4

Combine Z1 and Z23 in series

Z1_Z23_series = Z1 + Z23

Combine Z4 and Z35 in series

Z4_Z35_series = Z4 + Z35

Parallel combination of the above series results

Z_parallel = (Z1_Z23_series * Z4_Z35_series) / (Z1_Z23_series + Z4_Z35_series)

Add Z52 to the parallel combination

Z_total = Z_parallel + Z52

Display the total impedance

Z_totalThe total impedance between terminals $P$ and $Q$ is:

$Z_{PQ} = 69.13 - j36.64 \, \Omega$

This represents the equivalent impedance after applying the star-delta transformation and simplifying the circuit.

Do you need further details on any step or additional calculations?

Related Questions:

1. How is the star-delta transformation formula derived?
2. Can you explain how to combine impedances in series and parallel?
3. What are practical applications of impedance calculation in circuits?
4. How does this method compare to other techniques for solving circuits?
5. What is the physical meaning of the real and imaginary components of impedance?

Tip:

Always double-check unit consistency and keep track of complex numbers' real and imaginary parts when performing circuit transformations.