28.1
Ecuaciones diferenciales de líneas de transmisión
Las líneas de transmisión son componentes esenciales de los sistemas de energía eléctrica. Se caracterizan por la naturaleza distribuida de la resiste…
Considere un circuito que representa una sección de línea.
El voltaje y la corriente se miden desde el extremo receptor, con impedancia en serie definida y admitancia de derivación por unidad de longitud.
Usando las leyes de Kirchhoff y tomando límites delta x tiende a cero, se derivan dos ecuaciones diferenciales lineales. Diferenciando el voltaje y sustituyéndolo en la ecuación actual se obtiene una ecuación homogénea de segundo orden.
La solución implica dos constantes de integración y una constante de propagación. Sustituyendo esto en la ecuación de voltaje y reorganizando el denominador, se obtiene la expresión de corriente con la impedancia característica.
Después de evaluar las constantes de integración de las condiciones de contorno e identificar las funciones hiperbólicas, se obtienen los parámetros ABCD de la línea distribuida. Estos proporcionan la corriente y el voltaje en cualquier punto en relación con los valores del extremo receptor.
Los parámetros ABCD también se evalúan en el extremo de envío. La constante de propagación, una cantidad compleja, es adimensional cuando se multiplica por la longitud y ayuda a determinar las funciones hiperbólicas.
Estos parámetros se aplican a cualquier longitud de línea. La aproximación se puede utilizar para cálculos manuales que involucran líneas cortas y medianas.
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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.