Question

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 \((a) R=G=0 ;(b) R \neq 0, G=0 ;(c) R=0\), \(G \neq 0 ;(d) R \neq 0, G \neq 0 .\)

Short answer

In summary, the steady-state power dissipated by the load in each scenario is given by: (a) Scenario 1 (R = G = 0): \(P_{dissipated} = (|I|)^{2} * R_{L}\), where \(I = \frac{V_{0}}{R_{L} + jX_{L}}\) (b) Scenario 2 (R ≠ 0, G = 0): \(P_{dissipated} = (|I|)^{2} * R_{L}\), where \(I = \frac{V_{0}}{R_{L} + jX_{L} + j \omega L - \frac{j}{\omega C}}\) (c) Scenario 3 (R = 0, G ≠ 0): \(P_{dissipated} = (|I|)^{2} * R_{L}\), where \(I = \frac{V_{0}}{R_{L} + jX_{L} + \frac{j}{\omega C} + j \omega L \frac{1}{1 + j \omega C R_{L}}}\) (d) Scenario 4 (R ≠ 0, G ≠ 0): \(P_{dissipated} = (|I|)^{2} * R_{L}\), where \(I = \frac{V_{0}}{R_{L} + jX_{L} + \frac{j}{\omega C} + j \omega L + \frac{1}{\omega C} \frac{1}{1 + j \omega (C R_{L} + G R_{L}^{2})}}\) The power dissipation in each scenario depends on the specific values of R, G, L, C, 𝑅𝐿, 𝑋𝐿, and 𝑉₀.

Step-by-step solution

Determine the input impedance

To find the input impedance, apply the formula: \(Z_{in} = R_{L} + jX_{L}\) We will calculate \(Z_{in}\) for each scenario separately.

Calculate the current in each scenario

The current in each scenario can be calculated by using Ohm's law: \(I = \frac{V_{0}}{Z_{in}}\) We will also calculate the current in each scenario separately.

Calculate the steady-state power dissipation

The steady-state power dissipation for each scenario can be calculated using the following formula: \(P_{dissipated} = (|I|)^{2} * R_{L}\) Now we need to find the power dissipation for each scenario separately.

(a) Scenario 1: R = G = 0

In this case, the input impedance is: \(Z_{in} = R_{L} + jX_{L}\) The current for this scenario is: \(I = \frac{V_{0}}{R_{L} + jX_{L}}\) And steady-state power dissipated is: \(P_{dissipated} = (|I|)^{2} * R_{L}\)

(b) Scenario 2: R ≠ 0, G = 0

In this case, the input impedance is: \(Z_{in} = R_{L} + jX_{L} + j \omega L - \frac{j}{\omega C}\) The current for this scenario is: \(I = \frac{V_{0}}{R_{L} + jX_{L} + j \omega L - \frac{j}{\omega C}}\) And steady-state power dissipated is: \(P_{dissipated} = (|I|)^{2} * R_{L}\)

(c) Scenario 3: R = 0, G ≠ 0

In this case, the input impedance is: \(Z_{in} = R_{L} + jX_{L} + \frac{j}{\omega C} + j \omega L \frac{1}{1 + j \omega C R_{L}}\) The current for this scenario is: \(I = \frac{V_{0}}{R_{L} + jX_{L} + \frac{j}{\omega C} + j \omega L \frac{1}{1 + j \omega C R_{L}}}\) And steady-state power dissipated is: \(P_{dissipated} = (|I|)^{2} * R_{L}\)

(d) Scenario 4: R ≠ 0, G ≠ 0

In this case, the input impedance is: \(Z_{in} = R_{L} + jX_{L} + \frac{j}{\omega C} + j \omega L + \frac{1}{\omega C} \frac{1}{1 + j \omega (C R_{L} + G R_{L}^{2})}\) The current for this scenario is: \(I = \frac{V_{0}}{R_{L} + jX_{L} + \frac{j}{\omega C} + j \omega L + \frac{1}{\omega C} \frac{1}{1 + j \omega (C R_{L} + G R_{L}^{2})}}\) And steady-state power dissipated is: \(P_{dissipated} = (|I|)^{2} * R_{L}\)

Key concepts

Transmission Line Theory

Understanding transmission line theory is essential for anyone looking to grasp the principles of electrical engineering. A transmission line is a conductor designed to carry electrical signals over a long distance with minimal losses. This concept is not just about the wires themselves but also encompasses the electrical characteristics and behaviors that arise due to the line's physical properties and geometry.

One way to visualize transmission lines is by imagining them as pipes carrying water, where the water flow represents the electrical current and the pressure correlates to the voltage. In this analogy, any impedance would be like restrictions or changes in the pipe's diameter that affect the flow. In electrical terms, these impedances can come from the natural properties of the materials the line is made of (resistance) or the line's design and surrounding environment (capacitance and inductance).

To analyze the performance of a transmission line, engineers use primary constants such as inductance (L), capacitance (C), resistance (R), and conductance (G), which collectively determine how a signal is transmitted along the line. By studying these properties and how they interact with each other, engineers can optimize transmission lines for a variety of electrical systems.

Complex Impedance

Complex impedance is a fundamental concept in the realm of alternating current (AC) circuits, where resistance alone is not enough to describe the opposition to current flow. Complex impedance takes into account both resistance (R) and reactance (X), combining them into a single quantity that has both magnitude and phase.

The complex impedance, commonly denoted as (Z), is represented as a sum of its real part (resistance) and imaginary part (reactance):
\[\begin{equation}Z = R + jXg\end{equation}\]
Here, the 'j' signifies the imaginary unit, and it is used instead of the traditional 'i' to prevent confusion with electrical current denoted as (I). Reactance itself is further divided into two types: inductive reactance (XL), which is positive and arises from inductors, and capacitive reactance (XC), which is negative and comes from capacitors. The complex impedance is crucial in calculating how a circuit will respond to a given AC signal, influencing current flow and voltage drop across various components.

Ohm's Law

Ohm's law is a foundational principle in the study of electricity, outlining the direct proportionality between the voltage across a conductor and the current flowing through it, provided the temperature remains constant. The formula is elegantly simple:
\[\begin{equation}V = IRg\end{equation}\]
Here, (V) represents the voltage, (I) is the current, and (R) is the resistance. What's powerful about Ohm's law is that it allows for the prediction of one quantity if the other two are known. This becomes particularly useful when analyzing electric circuits, where understanding the relationship between voltage, current, and impedance is essential for designing and troubleshooting.

In scenarios involving AC currents and complex impedances, Ohm's law is extended from resistance to impedance, and the formula adapts to:
\[\begin{equation}V = IZg\end{equation}\]
Even with this extension, the core idea remains that current passing through a circuit element is directly related to voltage and inversely related to impedance.

Primary Constants of Transmission Lines

The primary constants of a transmission line describe its characteristics that directly affect signal transmission quality. These constants include resistance (R), inductance (L), capacitance (C), and conductance (G):

• Resistance (R) is the opposition to current flow due to the conductor material and cross-sectional area.
• Inductance (L) entails the magnetic field created around the conductor as current flows through it, which opposes changes in current.
• Capacitance (C) is the ability of the line to store charge when a voltage exists between two conductors.
• Conductance (G) measures the ease with which an electric current can pass through a segment of the insulator between conductors.

These constants not only help in predicting the performance of the line over different frequencies but also in understanding how power is distributed and dissipated within the system. In the exercise provided, we consider these constants to find the steady-state power dissipated by the load under various conditions. Each primary constant plays a vital role in determining both the input impedance seen by the source and how much power the load ultimately consumes.