28.1
Transmission-Line Differential Equations
Transmission lines are essential components of electrical power systems. They are characterized by the distributed nature of resistance (R), inductanc…
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'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.
View the full transcript and gain access to JoVE Core videos
Q1: How are transmission line differential equations derived from Kirchhoff's laws?
Applying Kirchhoff's Voltage Law and Kirchhoff's Current Law to a line section of length Δx yields relationships for voltage and current variations. As Δx approaches zero, these relationships transform into linear, first-order homogeneous differential equations that describe how voltage and current change along the transmission line's length.
Q2: What is the propagation constant and why is it important in transmission line analysis?
The propagation constant γ is a complex quantity with real and imaginary parts representing attenuation and phase shift, respectively. When multiplied by line length, it becomes dimensionless and is essential for evaluating hyperbolic functions that describe voltage and current distribution throughout the transmission line.
Q3: What role does characteristic impedance play in transmission line equations?
Characteristic impedance Zc reflects the intrinsic impedance of the transmission line and emerges from solving the differential equations. It appears in the current expression derived from the voltage equation and is fundamental to understanding how the line responds to applied signals and boundary conditions.
Q4: How do ABCD parameters relate voltage and current along a transmission line?
ABCD parameters, derived from hyperbolic functions cosh and sinh, provide a matrix representation relating voltage and current at any point along the line to receiving-end values. These transmission line constants are valid for any line length and offer exact solutions for transmission line behavior under various operating conditions.
Q5: What are the key distributed parameters that characterize a transmission line?
Transmission lines are characterized by distributed resistance (R), inductance (L), capacitance (C), and conductance (G) per unit length. These combine into series impedance z=R+jωL and shunt admittance y=G+jωC, which determine how voltage and current propagate through the line and affect maximum power flow and line loadability.
Q6: Why are approximations used for short- and medium-length transmission lines?
Precise ABCD parameters derived from hyperbolic functions are valid for any line length but involve complex calculations. For practical hand calculations on short- and medium-length lines, approximations simplify analysis while maintaining reasonable accuracy for design and operation of power systems.
Q7: How do integration constants and boundary conditions determine transmission line solutions?
Solving the second-order homogeneous differential equation yields two integration constants. These constants are evaluated using boundary conditions at the receiving end, where voltage and current are measured. This process establishes the complete voltage and current expressions along the entire transmission line.