Question

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

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

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: \(V(z, t) = V_0\cos(\omega t - \beta z)\)

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: \(V'(z, t) = V_0\cos(\omega t + \beta z)\)

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.)

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: \(I_s(z,t) = I_i(z,t) + I_r(z,t) = \frac{V_0}{Z_0}\cos(\omega t - \beta z) - \frac{V_0}{Z_0}\cos(\omega t + \beta z)\)

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

Key concepts

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.