**Problem PA.3: Complex Number Operations**Given that \( Z_1 = 10 + j5 \) and \( Z_2 = 20 - j20 \), reduce the following operations to rectangular form:1. \( Z_1 + Z_2 \)2. \( Z_1 - Z_2 \)3. \( Z_1 Z_2 \)4. \( \frac{Z_1}{Z_2} \)---**Detailed Explanation:**- **Addition (\( Z_1 + Z_2 \)):** - Combine the real parts: \( 10 + 20 = 30 \) - Combine the imaginary parts: \( 5 - 20 = -15 \) - Result: \( 30 - j15 \)- **Subtraction (\( Z_1 - Z_2 \)):** - Subtract the real parts: \( 10 - 20 = -10 \) - Subtract the imaginary parts: \( 5 - (-20) = 25 \) - Result: \( -10 + j25 \)- **Multiplication (\( Z_1 Z_2 \)):** - \( (10 + j5)(20 - j20) \) - Calculate: \( 10 \times 20 + 10 \times (-j20) + j5 \times 20 + j5 \times (-j20) \) - Simplify: \( 200 - j200 + j100 + 100 \) - Result: \( 300 - j100 \)- **Division (\( \frac{Z_1}{Z_2} \)):** - Multiply numerator and denominator by the conjugate of the denominator: \( \frac{(10 + j5)(20 + j20)}{(20 - j20)(20 + j20)} \) - Calculate the denominator: \( 20^2 + 20^2 = 800 \) - Calculate the numerator: \( 200 + j200 + j100 - 100 \) - Simplified numerator: \( 100 + j300 \) - Result: \( \frac{100 + j300}{800} = 0.125 + j0.375 \)

Answered: Given that Z1 = 10 + j5 and Z = 20 –…

Engineering

Electrical Engineering

Given that Z1 = 10 + j5 and Z = 20 – j20, reduce Z1 + Z, Z¡ – Z, ZZz, and Z¡lZz to rectangular form.

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...

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Concept explainers

Z parameter

The Z-parameters that are the impedance parameters of a two-port network are defined by expressing the two-port voltages V1 and V2 in terms of two-port currents I1 and I2. The Z-parameters are calculated by open circuiting one of the two ports at a time henceforth they are called open-circuit impedance parameters.

Question

**Problem PA.3: Complex Number Operations**

Given that \( Z_1 = 10 + j5 \) and \( Z_2 = 20 - j20 \), reduce the following operations to rectangular form:

1. \( Z_1 + Z_2 \)
2. \( Z_1 - Z_2 \)
3. \( Z_1 Z_2 \)
4. \( \frac{Z_1}{Z_2} \)

---

**Detailed Explanation:**

- **Addition (\( Z_1 + Z_2 \)):**
- Combine the real parts: \( 10 + 20 = 30 \)
- Combine the imaginary parts: \( 5 - 20 = -15 \)
- Result: \( 30 - j15 \)

- **Subtraction (\( Z_1 - Z_2 \)):**
- Subtract the real parts: \( 10 - 20 = -10 \)
- Subtract the imaginary parts: \( 5 - (-20) = 25 \)
- Result: \( -10 + j25 \)

- **Multiplication (\( Z_1 Z_2 \)):**
- \( (10 + j5)(20 - j20) \)
- Calculate: \(
10 \times 20 + 10 \times (-j20) + j5 \times 20 + j5 \times (-j20)
\)
- Simplify: \(
200 - j200 + j100 + 100
\)
- Result: \( 300 - j100 \)

- **Division (\( \frac{Z_1}{Z_2} \)):**
- Multiply numerator and denominator by the conjugate of the denominator: \(
\frac{(10 + j5)(20 + j20)}{(20 - j20)(20 + j20)}
\)
- Calculate the denominator: \(
20^2 + 20^2 = 800
\)
- Calculate the numerator:
\(
200 + j200 + j100 - 100
\)
- Simplified numerator: \(
100 + j300
\)
- Result: \( \frac{100 + j300}{800} = 0.125 + j0.375 \)

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