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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: \(I_s(z,t) = -\frac{2V_0}{Z_0}\sin(\omega t)\sin(\beta z)\)

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Key Concepts

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

Sinusoidal Voltage Wave
A sinusoidal voltage wave is an electrical wave that oscillates in a sine pattern. Its formula is defined by its amplitude, frequency, and phase. In mathematical terms, it can be expressed as \( V(z, t) = V_0\cos(\omega t - \beta z) \), where \( V_0 \) is the amplitude, \( \omega \) is the angular frequency, and \( \beta \) is the phase constant. This wave moves forward along the transmission line.

Important characteristics of a sinusoidal voltage wave include:
  • Amplitude \( V_0 \): Maximum peak value of the wave.
  • Frequency \( \omega \): How often the wave oscillates per unit time.
  • Phase constant \( \beta \): Determines the frequency and direction of the wave along the line.
Understanding how these elements interact is fundamental to analyzing signals on transmission lines, especially in applications involving alternating current (AC) systems.
Reflection in Transmission Line
Reflection in a transmission line occurs when a traveling wave hits a boundary and bounces back along the medium. In the context of the exercise, when the sinusoidal voltage wave reaches the open load end, it reflects with zero phase shift. The reflected wave travels in the opposite direction from the incident wave.

When it reflects, the wave equation changes direction but retains the same form, described by \( V'(z, t) = V_0\cos(\omega t + \beta z) \). This negated \( \beta \) factor indicates the reversal of direction.
  • Reflection occurs due to impedance mismatches or open/short circuits.
  • The reflected wave can interfere with incoming waves, creating a standing wave pattern.
Reflection is crucial because it impacts how power is distributed in the transmission line and creates points of constructive and destructive interference.
Characteristic Impedance
Characteristic impedance, denoted as \( Z_0 \), is a property of a transmission line that defines the relationship between voltage and current at any point along the line. It plays a critical role in determining how waves reflect and transmit across boundaries.

A line's characteristic impedance can be defined as the ratio between the amplitudes of voltage and current travelling waves, expressed as \( I(z,t) = \frac{V(z,t)}{Z_0} \). The expression helps derive current wave patterns from known voltage wave patterns.
  • Characteristic impedance is uniform along the line if the line is lossless.
  • Misalignment in \( Z_0 \) can cause reflections, influencing standing wave patterns.
  • Proper matching of \( Z_0 \) ensures efficient power transmission without reflections.
Characteristic Impedance is a fundamental concept in understanding energy transfer in transmission systems.
Trigonometric Identities in Waves
Trigonometric identities are mathematical tools that simplify the analysis of wave patterns. In this context, they help us express the standing wave pattern in a more straightforward form.

The step-by-step solution used the identity \( \cos(A) - \cos(B) = -2\sin\left(\frac{A + B}{2}\right)\sin\left(\frac{A - B}{2}\right) \), which transformed the individual cosine terms into a product of sine terms.
  • Simplification is achieved through identities, allowing easier computation and understanding.
  • Key identities assist in deducing phase differences and amplitude variations in interference patterns.
This aids in predicting the behavior of waves in the transmission line, especially when designing systems for minimized wave reflections and maximizing signal integrity.

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Most popular questions from this chapter

An absolute measure of power is the \(\mathrm{dBm}\) scale, in which power is specified in decibels relative to one milliwatt. Specifically, \(P(\mathrm{dBm})=10 \log _{10}[P(\mathrm{~mW}) / 1 \mathrm{~mW}]\). Suppose that a receiver is rated as having a sensitivity of \(-20 \mathrm{dBm}\), indicating the mimimum power that it must receive in order to adequately interpret the transmitted electronic data. Suppose this receiver is at the load end of a \(50-\Omega\) transmission line having \(100-\mathrm{m}\) length and loss rating of \(0.09 \mathrm{~dB} / \mathrm{m}\). The receiver impedance is \(75 \Omega\), and so is not matched to the line. What is the minimum required input power to the line in \((a) \mathrm{dBm},(b) \mathrm{mW} ?\)

A transmission line having primary constants \(L, C, R\), and \(G\) has length \(\ell\) and is terminated by a load having complex impedance \(R_{L}+j X_{L}\). At the input end of the line, a dc voltage source, \(V_{0}\), is connected. Assuming all parameters are known at zero frequency, find the steady-state power dissipated by the load if \((a) R=G=0 ;(b) R \neq 0, G=0 ;(c) R=0\), \(G \neq 0 ;(d) R \neq 0, G \neq 0 .\)

A \(100-\Omega\) lossless transmission line is connected to a second line of \(40-\Omega\) impedance, whose length is \(\lambda / 4\). The other end of the short line is terminated by a \(25-\Omega\) resistor. A sinusoidal wave (of frequency \(f\) ) having \(50 \mathrm{~W}\) average power is incident from the \(100-\Omega\) line. \((a)\) Evaluate the input impedance to the quarter-wave line. (b) Determine the steady-state power that is dissipated by the resistor. \((c)\) Now suppose that the operating frequency is lowered to one-half its original value. Determine the new input impedance, \(Z_{i n}^{\prime}\), for this case. \((d)\) For the new frequency, calculate the power in watts that returns to the input end of the line after reflection.

A sinusoidal wave on a transmission line is specified by voltage and current in phasor form: $$V_{s}(z)=V_{0} e^{\alpha z} e^{j \beta z} \quad \text { and } \quad I_{s}(z)=I_{0} e^{\alpha z} e^{j \beta z} e^{j \phi}$$ where \(V_{0}\) and \(I_{0}\) are both real. (a) In which direction does this wave propagate and why? \((b)\) It is found that \(\alpha=0, Z_{0}=50 \Omega\), and the wave velocity is \(v_{p}=2.5 \times 10^{8} \mathrm{~m} / \mathrm{s}\), with \(\omega=10^{8} \mathrm{~s}^{-1}\). Evaluate \(R, G, L, C, \lambda\) and \(\phi\).

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.
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