Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 10: Problem 4

A sinusoidal voltage wave of amplitude \(V_{0}\), frequency \(\omega\), and phase constant \(\beta\) propagates in the forward \(z\) direction toward the open load end in a lossless transmission line of characteristic impedance \(Z_{0}\). At the end, the wave totally reflects with zero phase shift, and the reflected wave now interferes with the incident wave to yield a standing wave pattern over the line length (as per Example 10.1). Determine the standing wave pattern for the current in the line. Express the result in real instantaneous form and simplify.

Short Answer

Expert verified
Question: Determine the standing wave pattern for the current in a lossless transmission line with an open load end, given a sinusoidal voltage wave of amplitude \(V_0\), frequency \(\omega\), and phase constant \(\beta\). Express the result in real instantaneous form. Answer: The standing wave pattern for the current in the line is given by the expression: \(I_s(z,t) = -\frac{2V_0}{Z_0}\sin(\omega t)\sin(\beta z)\).

Step by step solution

01

Write down the given sinusoidal voltage wave expression

We are given a sinusoidal voltage wave of amplitude \(V_0\), frequency \(\omega\), and phase constant \(\beta\). The expression for the voltage wave is: \(V(z, t) = V_0\cos(\omega t - \beta z)\)
02

Calculate the reflected voltage wave expression

Since the wave totally reflects with zero phase shift, the expression for the reflected voltage wave will be the same as the incident wave but with the opposite direction of propagation: \(V'(z, t) = V_0\cos(\omega t + \beta z)\)
03

Use the characteristic impedance to find the current expressions for incident and reflected waves

We know that the current in a transmission line is given by: \(I(z,t) = \frac{V(z,t)}{Z_0}\) So, the incident current wave is: \(I_i(z,t) = \frac{V(z,t)}{Z_0} = \frac{V_0}{Z_0}\cos(\omega t - \beta z)\) And the reflected current wave is: \(I_r(z,t) = -\frac{V'(z,t)}{Z_0} = -\frac{V_0}{Z_0}\cos(\omega t + \beta z)\) (The reflected current expression has a minus sign because the current direction is opposite to that of the incident current.)
04

Determine the standing wave pattern for the current

To find the standing wave pattern for the current, we can add the incident and reflected current expressions: \(I_s(z,t) = I_i(z,t) + I_r(z,t) = \frac{V_0}{Z_0}\cos(\omega t - \beta z) - \frac{V_0}{Z_0}\cos(\omega t + \beta z)\)
05

Simplify the expression and express it in real instantaneous form

We use the trigonometric identity: \(\cos(A) - \cos(B) = -2\sin(\frac{A + B}{2})\sin(\frac{A - B}{2})\) So, we get: \(I_s(z,t) = -\frac{2V_0}{Z_0}\sin\left(\frac{(\omega t - \beta z) + (\omega t + \beta z)}{2}\right)\sin\left(\frac{(\omega t - \beta z) - (\omega t + \beta z)}{2}\right)\) Simplify the expression: \(I_s(z,t) = -\frac{2V_0}{Z_0}\sin(\omega t)\sin(-\beta z)\) Finally, the standing wave pattern for the current in the line is: \(I_s(z,t) = -\frac{2V_0}{Z_0}\sin(\omega t)\sin(\beta z)\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A lossless transmission line is \(50 \mathrm{~cm}\) in length and operates at a frequency of \(100 \mathrm{MHz}\). The line parameters are \(L=0.2 \mu \mathrm{H} / \mathrm{m}\) and \(C=80 \mathrm{pF} / \mathrm{m}\). The line is terminated in a short circuit at \(z=0\), and there is a load \(Z_{L}=50+j 20 \Omega\) across the line at location \(z=-20 \mathrm{~cm} .\) What average power is delivered to \(Z_{L}\) if the input voltage is \(100 \angle 0^{\circ} \mathrm{V} ?\)

The characteristic impedance of a certain lossless transmission line is \(72 \Omega\). If \(L=0.5 \mu \mathrm{H} / \mathrm{m}\), find \((a) C ;(b) v_{p} ;(c) \beta\) if \(f=80 \mathrm{MHz} .(d)\) The line is terminated with a load of \(60 \Omega\). Find \(\Gamma\) and \(s\).

The parameters of a certain transmission line operating at \(\omega=6 \times 10^{8} \mathrm{rad} / \mathrm{s}\) are \(L=0.35 \mu \mathrm{H} / \mathrm{m}, C=40 \mathrm{pF} / \mathrm{m}, G=75 \mu \mathrm{S} / \mathrm{m}\), and \(R=17 \Omega / \mathrm{m}\). Find \(\gamma, \alpha, \beta, \lambda\), and \(Z_{0}\)

In the transmission line of Figure \(10.20, R_{g}=Z_{0}=50 \Omega\), and \(R_{L}=25 \Omega\). Determine and plot the voltage at the load resistor and the current in the battery as functions of time by constructing appropriate voltage and current reflection diagrams.

Two lossless transmission lines having different characteristic impedances are to be joined end to end. The impedances are \(Z_{01}=100 \Omega\) and \(Z_{03}=25 \Omega\). The operating frequency is \(1 \mathrm{GHz}\). \((a)\) Find the required characteristic impedance, \(Z_{02}\), of a quarter-wave section to be inserted between the two, which will impedance-match the joint, thus allowing total power transmission through the three lines. \((b)\) The capacitance per unit length of the intermediate line is found to be \(100 \mathrm{pF} / \mathrm{m}\). Find the shortest length in meters of this line that is needed to satisfy the impedance-matching condition. ( \(c\) ) With the three-segment setup as found in parts \((a)\) and \((b)\), the frequency is now doubled to \(2 \mathrm{GHz}\). Find the input impedance at the line-1-to-line- 2 junction, seen by waves incident from line \(1 .(d)\) Under the conditions of part \((c)\), and with power incident from line 1 , evaluate the standing wave ratio that will be measured in line 1 , and the fraction of the incident power from line 1 that is reflected and propagates back to the line 1 input.
See all solutions

Recommended explanations on Physics Textbooks

Geometrical and Physical Optics

Read Explanation

Torque and Rotational Motion

Read Explanation

Radiation

Read Explanation

Classical Mechanics

Read Explanation

Astrophysics

Read Explanation

Conservation of Energy and Momentum

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free