32.1
Traveling Waves: Lossless Lines
The power lines seen along streets can be modeled as a single-phase, two-wire, lossless transmission line.
Consider a line section characterized by series inductance and shunt capacitance, with directionality from the sending to the receiving end.
Using Kirchhoff's laws, the equations for voltage and current are written and divided by Delta x. As Delta x approaches zero, equations involving partial derivatives are derived since both position and time are variables.
Laplace transforms are applied assuming zero initial conditions and simplifying the derivatives to only one variable.
Differentiating these yields linear, second-order homogeneous differential equations with respective solutions. The velocity is a function of the inductance and capacitance values.
Taking inverse Laplace transforms and applying a time shift results in functions representing voltage and current waves. These expressions represent the forward and backward traveling waves.
To evaluate the constants, the solutions are substituted in the second-order equation.
The coefficients of the exponential functions on both sides are equated, yielding the forward and backward currents in terms of the forward and backward voltages, respectively, and the characteristic impedance.
The provided content explores the behavior of traveling waves on single-phase lossless transmission lines. It begins with a single-phase two-wire lossless transmission line of length Δx, characterized by a loop inductance LH/m and a line-to-line capacitance C F/m. These parameters result in a series inductance LΔx and a shunt capacitance CΔx.
The voltage v(x,t) and current i(x,t) at any position x and time t on the line are expressed using Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL).
As Δx approaches zero, partial differential equations emerge, capturing the relationships between voltage, current, and their rates of change with respect to both time and position. Laplace transforms are applied to convert these partial differential equations into ordinary differential equations, assuming zero initial conditions.
The solutions to these equations reveal that the velocity of the traveling waves depends on the inductance and capacitance per unit length of the transmission line.
These solutions describe forward and backward waves; forward waves travel in the positive x-direction, and backward waves move in the negative x-direction. The solution incorporates a time shift and results in functions that describe voltage and current as sums of forward and backward traveling waves. These waves move in opposite directions along the transmission line, influenced by the line's inductance and capacitance. These expressions lead to the line's characteristic impedance, a function of the inductance and capacitance per unit length.
This parameter is crucial for understanding how the waves propagate along the line and interact with the terminations at each end.
The power lines seen along streets can be modeled as a single-phase, two-wire, lossless transmission line.
Consider a line section characterized by series inductance and shunt capacitance, with directionality from the sending to the receiving end.
Using Kirchhoff's laws, the equations for voltage and current are written and divided by Delta x. As Delta x approaches zero, equations involving partial derivatives are derived since both position and time are variables.
Laplace transforms are applied assuming zero initial conditions and simplifying the derivatives to only one variable.
Differentiating these yields linear, second-order homogeneous differential equations with respective solutions. The velocity is a function of the inductance and capacitance values.
Taking inverse Laplace transforms and applying a time shift results in functions representing voltage and current waves. These expressions represent the forward and backward traveling waves.
To evaluate the constants, the solutions are substituted in the second-order equation.
The coefficients of the exponential functions on both sides are equated, yielding the forward and backward currents in terms of the forward and backward voltages, respectively, and the characteristic impedance.
- Transmission Lines - Paths for transmitting energy, characterized by their inductance and capacitance.
- Terminations - Ends of transmission lines, crucial for wave propagation.
- Traveling Waves - Waveforms moving along transmission lines due to inductance & capacitance.
- Lossless Transmission Line - An ideal model of a transmission line without losses.
- Shunt Capacitance - The capacitive property of a transmission line measured across its points.
- Define Transmission Lines – Understanding transmission line parameters (e.g., inductance and capacitance).
- Contrast Lossy and Lossless Lines – Grasp differences between ideal and real models.
- Explore Traveling Waves – Examination of wave phenomena on transmission lines (e.g., wave velocity).
- Explain Impact of Shunt Capacitance – Understanding capacitive impacts on waveforms.
- Apply in Terminations Context – Studying effects of line ending on wave propagation.
- How do transmission lines affect the behavior of traveling waves?
- What differentiates a lossless transmission line from a lossy one?
- What is the role of shunt capacitance in wave propagation on transmission lines?
From Chapter 32:
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