Consider a single-phase, two-wire, lossless transmission line terminated by an impedance at the receiving end and a source with Thevenin voltage and impedance at the sending end. The line, with length, has a surge impedance and wave velocity determined by the line's inductance and capacitance.
At the receiving end, the boundary condition states that the voltage equals the product of the receiving-end impedance and current. This relationship is expressed as a function of the incident and reflected voltage waves in the Laplace domain. The receiving-end voltage reflection coefficient characterizes how the wave reflects at the receiving end.
At the transmitting end, the boundary condition involves the difference between the source voltage and the product of the sending-end impedance and the current. This boundary condition leads to an equation for the incident voltage wave in terms of the sending-end reflection coefficient. This coefficient indicates the proportion of the wave reflected towards the source due to impedance mismatch.
The complete solutions for voltage and current along the transmission line are derived using the established boundary conditions and the reflection coefficients. These solutions incorporate both the incident and reflected waves, demonstrating how they combine to form the overall voltage and current at any point on the line. These solutions account for the effects of reflections at both the sending and receiving ends.
The line's characteristic impedance is derived from its inductance and capacitance per unit length, and these parameters also determine the wave velocity. By integrating these boundary conditions and parameters, a comprehensive understanding of the behavior of traveling waves on single-phase lossless transmission lines is obtained.
Home electrical wiring systems can be conceptually represented as a single-phase, two-wire, lossless transmission line.
These begin at the transmitting end, where power enters from the main grid with a specific voltage and impedance. At the receiving end, appliances and devices, each with a unique impedance, utilize this power as loads.
Boundary conditions are set to dictate how the voltage and current behave at both ends of the wiring system.
The receiving-end voltage equals impedance times its current. Solving this equation introduces the receiving-end voltage reflection coefficient, where transit time is the time a wave takes to travel the length of the wire.
Utilizing the established voltage and current relationships, modified expressions are obtained.
The transmitting end's boundary condition is set by the gap between the source voltage and the impedance-current product.
This relationship yields an expression in terms of the forward-traveling voltage wave.
Upon solving this expression, a coefficient reflecting the voltage at the sending end is established.
With the integration of this into the voltage and current relationships, a comprehensive solution is derived.
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