Name: modeling voltage and current on transmission lines
Uploaded: 2024-11-21T14:54:26.000Z
Duration: 1 min 26 s
Channel: JoVE

Consider a circuit representing a line section.
The voltage and current are measured from the receiving end, with defined series impedance and shunt admittance per unit length.
Using Kirchhoff&#39;s laws and taking limits delta x approaches zero, two linear differential equations are derived. Differentiating the voltage and substituting it into the current equation yields a second-order homogeneous equation.
The solution involves two integration constants and a propagation constant. Substituting this into the voltage equation and rearranging the denominator yields the current expression with the characteristic impedance.
After evaluating the integration constants from the boundary conditions and identifying hyperbolic functions, the ABCD parameters of the distributed line are obtained. These provide the current and voltage at any point relative to the receiving-end values.
ABCD parameters are also evaluated at the sending end. The propagation constant, a complex quantity, is dimensionless when multiplied by length and aids in determining hyperbolic functions.
These parameters apply to any line length. Approximation can be used for hand calculations involving short- and medium-length lines.

transmission-line differential equations

Transmission lines are essential components of electrical power systems. They are characterized by the distributed nature of resistance (R), inductance (L), and capacitance (C) per unit length. To analyze these lines, differential equations are employed to model the variations in voltage and current along the line.
Line Section Model
A circuit representing a line section of length &#916;x helps in understanding the transmission line parameters. The voltage V(x) and current i(x) are measured from the receiving end. The line has a series impedance z=R+j&#969;L per unit length and a shunt admittance y=G+j&#969;C per unit length. Applying Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL) to this setup, derive relationships for voltage and current variations along the line.
As &#916;x approaches zero, these relationships transform into linear, first-order, homogeneous differential equations
<img alt="Differential equation, dV(x)/dx = zi(x), mathematical formula, calculus concept." src="/files/ftp_upload/16146/16146_Equation%201.svg" style="height: 30px;" data-altupdatedat="1762173717000" data-alt="Equation1" />
<img alt="Differential equation formula di(x)/dx=yV(x) used in calculus and mathematical analysis." src="/files/ftp_upload/16146/16146_Equation%202.svg" style="height: 30px;" data-altupdatedat="1762173717000" data-alt="Equation2" />
These differential equations, in turn, describe how voltage and current change over the length of the transmission line.
Solving these differential equations involves determining the propagation constant &#947;, which encapsulates the effects of both series impedance and shunt admittance. The general solutions for voltage and current along the line include two integration constants evaluated using boundary conditions at the receiving end.
The characteristic impedance Zc is a key parameter, reflecting the intrinsic impedance of the transmission line. Using the solutions of the differential equations, the voltage and current expressed along the line in terms of hyperbolic functions, cosh (cosine hyperbolic), and sinh (sine hyperbolic), mathematical functions used to model the voltage and current distribution.
<img alt="Equation of cable voltage distribution, V(x)=VR cosh(&#947;x)+Zc iR sinh(&#947;x), mathematical symbol." src="/files/ftp_upload/16146/16146_Equation%203.svg" style="height: 14px;" data-altupdatedat="1762173717000" data-alt="Equation3" />
<img alt="Current flow equation, \( i(x) = \frac{V_R}{Z_c} \sinh(\gamma x) + i_R \cosh(\gamma x) \) formula." src="/files/ftp_upload/16146/16146_Equation%204.svg" style="height: 32px;" data-altupdatedat="1762173717000" data-alt="Equation4" />
ABCD Parameters
The ABCD parameters, or transmission line constants, are derived from these hyperbolic functions. These parameters provide a matrix representation that relates the voltage and current at any point along the line to the values at the receiving end. The ABCD parameters are crucial for understanding how signals propagate through the transmission line.
<img alt="Matrix equation diagram for electrical circuit analysis using ABCD parameters." src="/files/ftp_upload/16146/16146_Equation%205.svg" style="height: 32px;" data-altupdatedat="1762173717000" data-alt="Equation5" />
Propagation Constant
The propagation constant &#947; is a complex quantity with real and imaginary parts representing attenuation and phase shift, respectively. When multiplied by the line length, this constant becomes dimensionless and is essential for evaluating the hyperbolic functions that describe voltage and current distribution.
The precise ABCD parameters are valid for any line length and offer an exact solution for transmission line behavior. For practical hand calculations involving short- and medium-length lines, approximations may be used. These parameters comprehensively analyze transmission lines under various operating conditions, ensuring efficient and reliable power transmission.

Transmission-Line Differential Equations

Consider a circuit representing a line section.
The voltage and current are measured from the receiving end, with defined series impedance and shunt ...

Education

JoVE Core

Electrical Engineering

Unit 1

Steady-State Transmission Lines and Power Flows

steady-state-transmission-lines-and-power-flows

KEY TERMS AND CONCEPTS

modeling voltage and current on transmission lines

Transmission lines carry electric power, and their resistance, inductance, and capacitance are spread out along the line. Because these properties are distributed per unit length, voltage and current change as they move from one end of the line to the other. Differential equations are used to describe that variation.

A line-section model helps show how this works. In the model, a small length, Δx, is used to represent the line. Voltage V(x) and current i(x) are measured from the receiving end. The line also includes a series impedance, z = R + jωL, and a shunt admittance, y = G + jωC, per unit length.

Kirchhoff’s Voltage Law and Kirchhoff’s Current Law are applied to the small line section. As Δx becomes very small, the results turn into linear, first-order, homogeneous differential equations. These equations describe how voltage and current change along the transmission line.

Solving the equations gives the propagation constant, γ. This complex value combines the effects of series impedance and shunt admittance. Its real part shows attenuation, and its imaginary part shows phase shift. The product of γ and the line length is dimensionless and is used in the hyperbolic functions that describe the line behavior.

The general solutions for voltage and current include two integration constants. These constants are found from boundary conditions at the receiving end. The solutions also define the characteristic impedance, Zc, which reflects the intrinsic impedance of the transmission line. Voltage and current can then be written with cosh and sinh functions.

From these hyperbolic expressions, the ABCD parameters, also called transmission line constants, are obtained. They give a matrix form that relates the voltage and current at any point on the line to the values at the receiving end. The exact ABCD parameters work for any line length, while short- and medium-length lines may use approximations for hand calculations.

Modeling Voltage and Current on Transmission Lines

modeling-voltage-and-current-on-transmission-lines

Transmission lines carry electric power, and their resistance, inductance, and capacitance are spread out along the line. Because these properties are distributed per unit length, voltage and current change as they move from one end of the line to the other. Differential equations are used to describe that variation. A line-section model helps show how this works. In the model, a small length, Δx, is used to represent the line. Voltage V(x) and current i(x) are measured from the receiving end.

lossless lines

Lossless Lines and Voltage Behavior

Lossless transmission lines in electrical engineering are described by a purely imaginary propagation constant and a resistive characteristic impedance. Their ABCD parameters show an equivalent pi circuit with an imaginary series impedance and a shunt admittance. When the product of beta and line length is less than pi, the line has inductive series impedance and capacitive shunt admittance, which keeps it lossless. Wavelength is the physical length over which voltage or current phase changes...

maximum power flow and line loadability

Transmission Line Power Limits and Loadability

Transmission line power limits and loadability depend on voltage, angle, and line impedance. For lossy transmission lines, ABCD parameters in phasor form are used to relate sending-end and receiving-end voltages and currents. This matrix relation makes it possible to find the receiving-end current and then calculate the complex power delivered to the load. From the complex power, the real and reactive power components can be determined. For a lossless line, the equations become simpler, and...

the power flow problem and solution

Power Flow Analysis in Electric Networks

Power flow analysis helps determine how real and reactive power move through an electric network. It uses the system’s single-line diagram, which shows the buses, transmission lines, transformers, and loads. The analysis focuses on balanced three-phase steady-state operation. Each bus in the system has four key variables: voltage magnitude V k, phase angle δ k, real power P k, and reactive power Q k. Two of these four variables are treated as inputs. The power flow program solves for the...

control of power flow

Voltage and Reactive Power Control

Voltage and reactive power control help keep a power system stable and efficient. In this topic, generator settings, switched equipment, and transformer taps work together to guide how power moves through the network. A generator can be modeled with a Thevenin equivalent circuit under balanced steady-state conditions. The main terms are terminal voltage Vt, excitation voltage Eg, power angle δ, and positive-sequence synchronous reactance Xg. From this model, the generator current and the...

fast decoupled and dc powerflow

Power Flow Shortcuts for Grid Contingencies

Fast decoupled power flow and DC power flow give engineers faster ways to study a power system during contingencies such as generator outages or transmission line failures. These methods help operators get quick power flow solutions, which is important for real-time system adjustments. Fast decoupled power flow simplifies the Jacobian matrix, the set of equations used in power flow calculations, by neglecting some elements. The method drops J2 and J3 and assumes voltage magnitudes stay close...

key terms and concepts

See how distributed R, L, and C lead to transmission-line differential equations, propagation constant γ, characteristic impedance Zc, and ABCD parameters.

Modeling Voltage and Current on 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.

Modeling Voltage and Current on Transmission Lines

Description

Transcript

Tags