**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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