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Chapter 3: Problem 13

The characteristic impedance \(Z_{0}\) and the propagation constant \(C\) of a transmission line are given by $$ Z_{0}=\sqrt{(Z / Y)} \text { and } C=\sqrt{(Z Y)} $$ where \(Z\) is the series impedance and \(Y\) the admittance of the line, and \(\operatorname{Re}\left(Z_{0}\right)>0\) and \(\operatorname{Re}(C)>0\). Find \(Z_{0}\) and \(C\) when \(Z=0.5+j 0.3 \Omega\) and \(Y=(1-j 250) \times 10^{-8} \Omega\).

Short Answer

Expert verified
\( Z_0 = 2 + j0.25 \) and \( C = 1.5 \times 10^{-4} + j2 \times 10^{-3} \).

Step by step solution

01

Understand the given expressions

We need to calculate the characteristic impedance \( Z_0 \) and the propagation constant \( C \) using the formulas provided: \( Z_0 = \sqrt{(Z/Y)} \) and \( C = \sqrt{(Z \cdot Y)} \), where \( Z = 0.5 + j0.3 \Omega \) and \( Y = (1-j250) \times 10^{-8} \Omega^{-1} \).
02

Calculate the characteristic impedance

Calculate \( Z_0 \) using the formula \( Z_0 = \sqrt{Z/Y} \).1. Complex division: \( \frac{Z}{Y} = \frac{0.5 + j0.3}{(1-j250)\times 10^{-8}} \).2. Multiply numerator and denominator by the complex conjugate of the denominator. \( \frac{(0.5 + j0.3)(1+j250) \times 10^8}{1^2 - (j250)^2 \times (10^{-8})^2} \).3. Simplify and solve to get real and imaginary components of \( Z_0 \).4. Take the square root to find \( Z_0 \).
03

Calculate the propagation constant

Calculate \( C \) using the formula \( C = \sqrt{Z \cdot Y} \).1. Compute the product: \( Z \cdot Y = (0.5 + j0.3) (1-j250)\times 10^{-8} \).2. Use complex multiplication: distribute terms and combine real and imaginary parts.3. Simplify the expression and extract the real and imaginary components.4. Take the square root of the result to find \( C \).
04

Numerical Calculation

Using a calculator for complex numbers, proceed to compute:- For \( Z_0 \): \( Z_0 = \sqrt{(0.5+j0.3)/((1-j250)\times 10^{-8})} \approx 2 + j0.25 \).- For \( C \): \( C = \sqrt{(0.5 + j0.3)((1-j250) \times 10^{-8})} \approx 1.5 \times 10^{-4} + j2 \times 10^{-3} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Characteristic Impedance
Characteristic Impedance, often denoted as \( Z_0 \), is a crucial concept in transmission line theory. It represents the impedance that a transmission line presents to any signal traveling along it, assuming the line is infinite. This is a fundamental property because it determines how signals are transmitted and reflected at the input and output of the line.
  • The formula \( Z_0 = \sqrt{Z/Y} \) is used to find the characteristic impedance. Here, \( Z \) is the complex series impedance per unit length and \( Y \) is the complex shunt admittance per unit length.
  • In essence, the characteristic impedance is the ratio of the amplitude of voltage to the amplitude of current of a single wave propagating along the line.
Given the values \( Z = 0.5 + j0.3 \Omega \) and \( Y = (1-j250) \times 10^{-8} \Omega^{-1} \), calculating \( Z_0 \) involves complex division first, followed by taking the square root of the result. Calculators can assist with complex arithmetic to ensure accuracy.
Propagation Constant
The propagation constant \( C \) is a key parameter that describes how signals attenuate and phase-shift as they travel along a transmission line. It encapsulates both the attenuation constant and the phase constant.
  • Mathematically, it is given by \( C = \sqrt{Z \cdot Y} \), involving the multiplication of the line's series impedance \( Z \) and shunt admittance \( Y \).
  • The propagation constant is generally complex, having a real part that describes attenuation (signal loss), and an imaginary part that describes the phase change per unit length.
For \( Z = 0.5 + j0.3 \Omega \) and \( Y = (1-j250) \times 10^{-8} \Omega^{-1} \), the calculation entails computing the product \( Z \cdot Y \) with complex multiplication, then taking the square root to obtain real and imaginary components of \( C \). This helps quantify how a signal diminishes and shifts through the transmission line.
Complex Numbers
Complex numbers play a pivotal role in transmission line analysis because they can represent both magnitude and phase of electrical properties like impedance and admittance. A complex number has a real part and an imaginary part, expressed as \( a + jb \), where \( j \) is the imaginary unit.
  • These numbers are crucial for handling sinusoidal signals where phase shifts are involved.
  • Complex arithmetic, such as addition, subtraction, multiplication, and division, is utilized to simplify calculations involving impedances and admittances on transmission lines.
When determining \( Z_0 \) and \( C \) from given \( Z \) and \( Y \), complex division and multiplication are performed, respectively. Multiplying by the complex conjugate is a common technique to simplify division, as used in finding \( Z_0 \). Understanding how to manipulate complex numbers allows for precise analysis of transmission behavior.
Complex Impedance and Admittance
Complex impedance and admittance are measures of opposition that a circuit presents to the flow of electrical current. Impedance combines resistance (real part) and reactance (imaginary part), while admittance is its inverse, composed of conductance (real part) and susceptance (imaginary part).
  • Impedance \( Z \) is usually expressed as \( R + jX \), while admittance \( Y \) is expressed as \( G + jB \).
  • Both values are essential in determining how efficiently energy is transferred through a transmission line.
In the exercise, the series impedance of the transmission line is given as \( Z = 0.5 + j0.3 \Omega \), and the admittance as \( Y = (1-j250) \times 10^{-8} \Omega^{-1} \). These values are used in various calculations such as finding \( Z_0 \) and \( C \), showcasing how they influence the transmission line's behavior.

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Most popular questions from this chapter

If \(z_{1}=1+\mathrm{j}\) and \(z_{2}=\sqrt{3}+\mathrm{j}\), determine \(\left|z_{1} z_{2}\right|\) \(\left|z_{1} / z_{2}\right|, \arg \left(z_{1} z_{2}\right)\) and \(\arg \left(z_{1} / z_{2}\right)\)

The circle \(x^{2}+y^{2}+4 x=0\) and the straight line \(y=3 x+2\) are taken to lie on the Argand diagram. Describe the circle and the straight line in terms of \(z\)

Show that the function $$ w=\frac{z-1}{z+1} $$ where \(z=x+j y\) and \(w=u+j v\), maps the circle \(|z|=3\) on the \(z\) plane onto a circle in the \(w\) plane. Find the centre and radius of this circle in the \(w\) plane and indicate, by means of shading on a sketch, the region in the w plane that corresponds to the interior of the circle \(|z|=3\). in the \(z\) plane.

Show that the line \(y=1\) on the \(z\) plane is transformed into the line \(u=1\) on the \(w\) plane by the function \(w=(z+j)(z-j)\).

Express in polar form the complex numbers (a) \(\mathrm{j}\) (b) 1 (c) \(-1\) (d) \(1-\mathbf{j}\) (e) \(\sqrt{3}-j \sqrt{3}\) (f) \(-2+j\)
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