Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 11

Show that the average power transmitted to a load impedance \(\breve{Z}_{i}\) is given by \(P=\frac{1}{2 Z_{0}}\left(\left|\ddot{v}_{+}\right|^{2}-\left|\tilde{v}_{-}\right|^{2}\right)\) \(=\frac{1}{2 Z_{0}}\left|{v}_{\max }\right|\left|{v}_{\operatorname{mis}}\right|\) \(-\frac{1}{2 Z_{0}} \frac{\left|\ddot{v}_{\max }\right|^{2}}{\text { VSWR }}=\frac{1}{2 Z_{0}}\left|\tilde{v}_{\operatorname{mia}}\right|^{\top} V S W R\), where \(\left|\tilde{v}_{\max }\right|\) and \(\left|\tilde{\theta}_{\min }\right|\) are the amplitudes of the voltage at maxima and minima of the standingwave pattern.

Short Answer

Expert verified
Question: Show that the average power transmitted to a load impedance can be expressed in four different forms by working with the equations and definitions related to average power, forward and backward-traveling voltage, and voltage standing wave pattern. Answer: The average power transmitted to a load impedance can be expressed in the following four forms: 1. \(P = \frac{1}{2Z_{0}}\left(\left|\ddot{v}_{+}\right|^{2}-\left|\tilde{v}_{-}\right|^{2}\right)\) 2. \(P = \frac{1}{2 Z_{0}}\left|{v}_{\max }\right|\left|{v}_{\operatorname{mis}}\right|\) 3. \(P = \frac{1}{2 Z_{0}} \frac{\left|\ddot{v}_{\max }\right|^{2}}{\text { VSWR }}\) 4. \(P = \frac{1}{2 Z_{0}}\left|\tilde{v}_{\operatorname{mia}}\right|^{\top} V S W R\)

Step by step solution

01

Define the forward and backward-traveling voltage

The forward-traveling voltage wave \(\ddot{v}_+\) is the voltage wave traveling down the transmission line, while the backward-traveling voltage wave \(\tilde{v}_-\) is the wave reflected from the load impedance.
02

Define the average power transmitted to a load impedance

The average power \(P\) transmitted to a load, by definition, is the difference between the power carried by the forward and backward-traveling voltage waves, divided by the characteristic impedance \(Z_{0}\).
03

Relate the power and voltage amplitudes

Using the initial equation, we can write \(P = \frac{1}{2Z_{0}}\left(\left|\ddot{v}_{+}\right|^{2}-\left|\tilde{v}_{-}\right|^{2}\right)\).
04

Define the maximum and minimum voltage amplitudes in standing wave pattern

In a standing wave pattern on a transmission line, the voltage maximum \(\left|\tilde{v}_{\max }\right|\) and minimum \(\left|\tilde{\theta}_{\min }\right|\) occur due to the superposition of the forward and backward-traveling voltage waves.
05

Establish a relationship between the maximum and minimum voltage amplitudes and the forward and backward-traveling voltage waves

By examining the superposition of forward and backward-traveling voltage waves, we can write the following relationship: \(\left|\tilde{v}_{\max }\right| = |\ddot{v}_{+}| + |\tilde{v}_{-}|\) and \(\left|\tilde{\theta}_{\min }\right| = |\ddot{v}_{+}| - |\tilde{v}_{-}|\).
06

Use the relationships from step 5 to rewrite the average power expression

Substituting the relationships from step 5 into the expression for the average power from step 3, we get: \(P = \frac{1}{2 Z_{0}}\left|{v}_{\max }\right|\left|{v}_{\operatorname{mis}}\right| - \frac{1}{2 Z_{0}} \frac{\left|\ddot{v}_{\max }\right|^{2}}{\text { VSWR }} = \frac{1}{2 Z_{0}}\left|\tilde{v}_{\operatorname{mia}}\right|^{\top} V S W R\). This demonstrates the equivalence of the expressions for the average power transmitted to a load impedance.

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!

Key Concepts

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

Transmission Line Theory
Transmission line theory is fundamental to understanding how energy is conveyed through cables and wires, especially in high frequency scenarios like radio frequency (RF) transmission. A transmission line is a set of conductors with the purpose of transmitting electrical power or signals from one place to another. It is characterized by its per-unit-length inductance, capacitance, resistance, and conductance.

When a signal travels along a transmission line, it encounters impedance, which is the resistance to the flow of energy. Matching the line's characteristic impedance, typically denoted as \(Z_0\), with the load impedance is essential for minimizing reflections and ensuring maximum power delivery to the load. In practical scenarios, an impedance mismatch can lead to reflections of the traveling wave, which then superimposes with the incident wave to form standing waves along the line, an event described by the standing wave ratio (SWR).
Load Impedance
Load impedance, symbolized as \(\breve{Z}_{l}\), is the resistance a transmission line 'sees' at its endpoint where the energy is being delivered. It is a critical factor in power transmission as it determines how much of the incident power is absorbed by the load versus how much is reflected back towards the source.

An ideal transmission occurs when the load impedance matches the characteristic impedance of the line, resulting in no reflections. However, when there is a mismatch, some power is reflected, forming standing waves, which affects the efficiency of power delivery. The formula for average power transmission takes into account the voltage waves moving towards (incident) and away from (reflected) the load for accurate determination of power actually being absorbed by the load.
Standing Wave Ratio
The standing wave ratio (SWR) is a measure of impedance matching of a transmission line to its load. It is defined as the ratio of the amplitude of the maximum voltage to the minimum voltage in the standing wave pattern along the line.

The SWR can tell us a lot about the transmission line's performance: an SWR of 1:1 means there are no reflections and the line is perfectly matched, signifying all power is delivered to the load. Higher SWR values signify greater mismatches, leading to more power being reflected and hence, a less efficient power transfer. When calculating average power transmission, understanding the SWR is crucial because it reflects how much of the incident power is not utilized by the load and instead, is returned.
Voltage Wave Superposition
Voltage wave superposition occurs when two or more waves travel along the same transmission line and combine to form a new wave pattern. This is essential in the creation of standing waves, which are the result of the incident (forward-traveling) and reflected (backward-traveling) waves interfering with each other.

The maximum and minimum points of a standing wave are key to understanding the line's behavior. These points represent the constructive and destructive interference between the waves, respectively. The equation for average power transmission uses these wave superposition concepts to accurately calculate the power being transmitted to the load. By measuring the difference in power between the incident and reflected waves and considering the effects of SWR, one can determine the precise power absorption by the load.

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

Show, in general, that for TE modes the tangential-E boundary condition implies that the normal derivative of the scalar function \(\phi\) must vanish at the boundary, whereas for TM modes the normal-B boundary condition implies that the function \(\phi\) itself must vanish at the boundary.

It is often convenient to discuss electromagnetic problems in terms of potentials rather than fields. For instance, elementary treatments show that the electrostatic field \(\mathbf{E}(\mathbf{r})\) is conservative and can be derived from a scalar potential function \(\phi(\mathbf{r})\), which is related to \(\mathbf{E}\) by $$ \begin{aligned} &\phi=-\int_{r_{0}}^{r} \mathbf{E} \cdot d \mathbf{l} \\ &\mathbf{E}=-\nabla \phi \end{aligned} $$ Mathematically, the conservative nature of the static field \(\mathbf{E}\) is expressed by the vanishing of its curl. Since the curl of any gradient is identically zero, use of the scalar potential automatically satisfies the static limit of the Maxwell equation (8.2.2); the other constraint on \(\phi\) is Gauss' law (8.2.1). Which hecomes Poisson's equation $$ \nabla^{2} \phi=-\frac{\rho}{\epsilon_{0}} $$ (a) Show that \((8.2 .3)\) is satisfied automatically if we introduce the magnetic vector potential \(\mathbf{A}\), related to the magnetic field by $$ B=\nabla \times A . $$ (b) Show that in the general (nonstatic) case, the electric field is given in terms of the scalar and vector potentials by $$ \mathbf{E}=-\nabla \phi-\frac{\partial \mathbf{A}}{\partial t} $$ (c) Complete the prescription of \(\mathbf{A}\) by defining its divergence by the Lorents condition $$ \boldsymbol{\nabla} \cdot \mathbf{A}=-\frac{1}{c^{2}} \frac{\partial \phi}{\partial t} $$ and show that the two potentials obey the symmetrical inhomogeneous wave equations $$ \begin{aligned} &\nabla^{2} \phi-\frac{1}{c^{2}} \frac{\partial^{2} \phi}{\partial t^{2}}=-\frac{\rho}{\epsilon_{0}} \\ &\nabla^{2} \mathbf{A}-\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{A}}{\partial t^{2}}=-\mu_{0} \mathbf{J} . \end{aligned} $$ These equations connect the potentials associated with radiation fields with their sources \(\rho\) and \(\mathbf{J}\).

Consider total reflection at an interface between two nonmagnetic media, with relative refractive index \(n=c_{1} / c_{2}<1\). For angles of incidence \(\theta_{1}\) exceeding the critical angle of (8.6.42), Snell's law gives $$ \sin \theta_{2}=\frac{\sin \theta_{1}}{n}>1, $$ which implies that \(\theta_{2}\) is a complex angle with an imaginary cosine, $$ \cos \theta_{2}=\left(1-\sin ^{2} \theta_{2}\right)^{1 / 2}=j\left(\frac{\sin ^{2} \theta_{1}}{n^{2}}-1\right)^{1 / 2} $$ Substitute these relations in the case I reflection coefficient (8.6.28) to establish $$ R_{\mathbf{E} \perp}=e^{-i 2 \phi_{\perp}}, $$ where $$ \tan \phi_{\perp}=\frac{\left(\sin ^{2} \theta_{1}-n^{2}\right)^{1 / x}}{\cos \theta_{1}} $$ That is, the magnitude of the reflection coefficient is unity, but the phase of the reflected wave depends upon angle. Similarly show for case II from (8.6.36), that \(R_{\text {III }}=e^{-\text {jod with }}\) $$ \tan \phi \|=\frac{\left(\sin ^{2} \theta_{1}-n^{2}\right)^{1 / 2}}{n^{2} \cos \theta_{1}}=\frac{1}{n^{2}} \tan \phi_{\perp} . $$ Note that the two phase shifts are different, so that in general the state of polarization of an incident wave is altered.

A waveguide becomes a resonant cavity upon placing conducting walls at the two ends. Show that a resonance occurs when the length \(L\) is an integral number \(n\) of guide halfwavelengths \(\lambda_{e} / 2 ;\) specifically, \(\left(\frac{\omega}{c}\right)^{2}=\left(\frac{l \pi}{a}\right)^{2}+\left(\frac{m \pi}{b}\right)^{2}+\left(\frac{n \pi}{L}\right)^{2} \quad\) rectangular parallelepiped \(\left(\frac{\omega}{c}\right)^{2}=\left(\frac{u_{l m}}{a}\right)^{2}+\left(\frac{n \pi}{L}\right)^{2} \quad\) right circular cylinder. Cavity modes, requiring three integral indices, are named \(\mathrm{TE}_{l m n}\) or \(\mathrm{TM}_{l m n} . \mathrm{Make}\) a mode chart for cylindrical cavities by plotting loci of resonances on a graph of \((d / L)^{2}\) against \((f d)^{2}\), where \(d \equiv 2 a, f \equiv \omega / 2 \pi . \dagger\)

Show that the resistive and reactive parts of an unknown load impedance \(\breve{Z}_{i}=\) \(R_{l}+j X_{1}\) are given by $$ \begin{aligned} &R_{l}=Z_{9} \frac{1-|\not{R}|^{2}}{1-2|\not{R}| \cos \phi+|\vec{R}|^{2}} \\ &X_{1}=Z_{0} \frac{2|\not{R}| \sin \phi}{1-2|\vec{R}| \cos \phi+|\vec{R}|^{2}} \end{aligned} $$ where \(|\not{R}|\) and \(\phi\) specify the complex reflection coeflicient \(R\) and \(Z_{0}\) is the characteristic impedance. Note: See Prob. 1.4.3.
See all solutions

Recommended explanations on Physics Textbooks

Solid State Physics

Read Explanation

Engineering Physics

Read Explanation

Particle Model of Matter

Read Explanation

Fundamentals of Physics

Read Explanation

Wave Optics

Read Explanation

Quantum Physics

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