Name: wave speed on lossless transmission lines
Uploaded: 2024-11-21T14:54:41.000Z
Duration: 1 min 27 s
Channel: JoVE

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&#39;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.

traveling waves: lossless lines

Traveling Waves: Lossless Lines

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 &#916;x, characterized by a loop inductance LH/m and a line-to-line capacitance C F/m. These parameters result in a series inductance L&#916;x and a shunt capacitance C&#916;x.
<img alt="Electrical circuit diagram, showing inductance L&#916;x and capacitance C&#916;x for wave propagation analysis." src="/files/ftp_upload/16175/16175_figure_001.svg" style="height: 125px;" data-altupdatedat="1762173717000" data-alt="Figure1" />
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).
<img alt="Circuit theory equations, describing voltage and current changes, mathematical formula, educational." src="/files/ftp_upload/16175/16175_PT_Equation_1v2.svg" style="height: 70px;" data-altupdatedat="1762173717000" data-alt="Equation1" />
As &#916;x 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.
<img alt="Differential equations in circuit modeling, diagram showing voltage and current derivatives." src="/files/ftp_upload/16175/16175_PT_Equation_2v2.svg" style="height: 70px;" data-altupdatedat="1762173717000" data-alt="Equation2" />
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.
<img alt="Resonant frequency formula \(v = 1/\sqrt{LC}\), used in LC circuit analysis." src="/files/ftp_upload/16175/16175_PT_Equation_3v4.svg" style="height: 30px;" data-altupdatedat="1762173717000" data-alt="Equation3" />
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.
<img alt="Capacitance formula: Zc=&#8730;(L/C), electrical impedance equation." src="/files/ftp_upload/16175/16175_PT_Equation_4v4.svg" style="height: 30px;" data-altupdatedat="1762173717000" data-alt="Equation4" />
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 ...

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JoVE Core

Electrical Engineering

Unit 1

Transmission Lines: Transient Operation

KEY TERMS AND CONCEPTS

wave speed on lossless transmission lines

Traveling waves on a lossless transmission line are shaped by the line’s inductance and capacitance. The model starts with a single-phase, two-wire line of length Δx. It has loop inductance L H/m and line-to-line capacitance C F/m. A short section of the line can then be treated as a series inductance LΔx and a shunt capacitance CΔx.

Voltage v(x,t) and current i(x,t) are written for any position x and time t along the line. Kirchhoff’s Voltage Law, or KVL, is used for the voltage relation. Kirchhoff’s Current Law, or KCL, is used for the current relation. As Δx becomes very small, these relations turn into partial differential equations that link voltage and current to their changes in time and position.

Laplace transforms are then applied to these equations. This step assumes zero initial conditions and converts the partial differential equations into ordinary differential equations. The solutions show that wave speed depends on the inductance and capacitance per unit length of the transmission line. The voltage and current can be written as sums of forward and backward traveling waves.

The forward wave moves in the positive x-direction. The backward wave moves in the negative x-direction. A time shift appears in the solution, which helps describe how the waves travel along the line. These expressions also lead to the characteristic impedance, which depends on the line inductance and capacitance per unit length and helps describe how the waves interact with terminations at each end.

Wave Speed on Lossless Transmission Lines

wave-speed-on-lossless-transmission-lines

Traveling waves on a lossless transmission line are shaped by the line’s inductance and capacitance. The model starts with a single-phase, two-wire line of length Δx. It has loop inductance L H/m and line-to-line capacitance C F/m. A short section of the line can then be treated as a series inductance LΔx and a shunt capacitance CΔx. Voltage v(x,t) and current i(x,t) are written for any position x and time t along the line. Kirchhoff’s Voltage Law, or KVL, is used for the voltage relation.

boundary conditions: lossless lines

Transmission Line Reflections at Both Ends

Single-phase, two-wire, lossless transmission lines show how voltage and current waves reflect at both ends of a circuit. In this model, the line has a length, a surge impedance, and a wave velocity. These are set by the line’s inductance and capacitance per unit length. A home electrical wiring system can be treated this way in a simple circuit model. At the receiving end, the boundary condition says that the voltage equals the receiving-end impedance times the current. This relationship is...

bewley lattice diagram

Tracing Transient Waves with a Bewley Diagram

The Bewley lattice diagram organizes voltage-wave reflections in transmission-line transients. It helps students follow how a signal moves along a line and bounces back from each end. The diagram was developed by L. V. Bewley. The vertical axis shows transient time, and the horizontal axis shows the line position x. Diagonal lines show traveling waves. These lines make the wave motion easier to read because each segment marks a wave moving through the line. Each reflection is found by...

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Signal Attenuation and Power Losses

Transmission lines with series resistance and shunt conductance can weaken signals, change wave shape, and waste power. These effects are called attenuation, distortion, and power losses. They matter in long power lines and other practical transmission systems. When constant resistance and conductance are present, the voltage and current equations change. The propagation constant shows that the wave includes both forward-traveling and backward-traveling parts. As the wave moves along the line,...

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key terms and concepts

Learn how inductance and capacitance set wave speed on a lossless transmission line, and how forward and backward waves lead to characteristic impedance.

Wave Speed on Lossless Transmission Lines | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

Wave Speed on Lossless Transmission Lines

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