Understanding AC Short Transmission Lines: Steady-State Operation and Phasor Diagram for Resistive Loads
In electrical power systems, transmission lines play a critical role in delivering power from generating stations to distribution networks. Among these, short transmission lines are those with lengths typically less than 80 km (50 miles). These lines are often modeled using lumped parameters, as their distributed capacitance and inductance effects are negligible. This article explores the steady-state operation of AC short transmission lines and provides insights into the phasor diagram for a resistive load.
Steady-State Operation of Short Transmission Lines
In steady-state operation, the voltage and current in a short transmission line are assumed to be sinusoidal and constant over time. The line is represented by its series impedance, Z=R+jX, where:
- R is the resistance of the line,
- X is the inductive reactance of the line.
For short lines, the shunt capacitance is ignored, simplifying the analysis. The sending-end voltage (Vs) and current (Is) are related to the receiving-end voltage (VR) and current (IR) by the following equations:
Vs= VR+IRZ
IS=IR
Here, VR and IR are the voltage and current at the load, respectively.
Phasor Diagram for a Resistive Load
When the load connected to the transmission line is purely resistive, the current IRIR is in phase with the receiving-end voltage VR. This simplifies the phasor diagram and provides a clear visualization of the voltage drop across the line.
- Receiving-End Voltage (VR): This is the reference phasor, drawn along the positive real axis.
- Load Current (IR): Since the load is resistive, IR is in phase with VR and is drawn along the same axis.
- Voltage Drop (IRZ): The voltage drop across the line is the product of the load current and the line impedance. It has two components:
- Resistive drop (IRR): In phase with IR.
- Inductive drop (IRX): Leads IR by 90 degrees.
- Sending-End Voltage (VS): This is the vector sum of VR and IRZ. It is slightly higher in magnitude and phase-shifted compared toVR.
The phasor diagram visually demonstrates how the sending-end voltage compensates for the resistive and inductive drops in the line to maintain the required voltage at the load.
Key Takeaways
- Short transmission lines are modeled using series impedance, ignoring shunt capacitance.
- For resistive loads, the current is in phase with the receiving-end voltage.
- The phasor diagram clearly shows the relationship between sending-end and receiving-end voltages, highlighting the impact of line impedance.
Understanding these concepts is essential for designing and analyzing power systems, ensuring efficient and reliable power delivery. For a more detailed explanation and step-by-step tutorial, watch this video on our YouTube channel or click here for more.
