28.2
Lossless Lines
In electrical engineering, a lossless transmission line is characterized by a purely imaginary propagation constant and a resistive characteristic imp…
Consider a lossless line with a purely imaginary propagation constant, resistive characteristic impedance, and established ABCD parameters.
Its equivalent pi circuit has imaginary series impedance and shunt admittance. When the beta and length product is less than pi, it exhibits inductive series impedance and capacitive shunt admittance, making it lossless.
The wavelength, or distance required to change the voltage or current phase by 2-pi, is calculated using the velocity of propagation.
Surge impedance loading is the power delivered to a load resistance equal to the surge impedance.
Here, voltage remains constant along the line, and constant real power flows from the sending to the receiving end, with zero reactive power flow.
Real power delivered is determined using the rated voltage and surge impedance.
In reality, power lines are not terminated by their surge impedance, causing non-flat voltage profiles that vary depending on load conditions.
At no load, voltage increases from the sending to the receiving end, while for short circuits, it decreases to zero at the receiving end. The full-load profile is higher than the short-circuit profile.
View the full transcript and gain access to JoVE Core videos
Q1: What makes a transmission line lossless?
A lossless transmission line has a purely imaginary propagation constant and resistive characteristic impedance. Its equivalent pi circuit contains imaginary series impedance and shunt admittance. When the product of the phase constant and line length is less than pi, the line exhibits inductive series impedance and capacitive shunt admittance, ensuring no power loss occurs during signal propagation.
Q2: How is wavelength calculated on a transmission line?
Wavelength is the physical distance required for voltage or current phase to change by 2π radians. It is calculated using the signal's propagation velocity on the line. This wavelength determines the electrical length of the transmission line and its phase characteristics, which are critical for understanding how signals behave across the conductor.
Q3: What is surge impedance loading and when does it occur?
Surge impedance loading (SIL) is the power delivered when a load resistance equals the line's surge impedance. Under SIL conditions, voltage remains constant along the line, and constant real power flows from sending to receiving end with zero reactive power. This ideal condition is calculated using the rated voltage and surge impedance values.
Q4: Why do voltage profiles change under different load conditions?
Real power lines are rarely terminated by their surge impedance, causing non-uniform voltage profiles. At no load, voltage increases from sending to receiving end due to the Ferranti effect. During short circuits, voltage drops to zero at the receiving end. Full-load profiles lie between these extremes, generally higher than short-circuit but lower than no-load profiles.
Q5: What role do ABCD parameters play in lossless line analysis?
ABCD parameters describe the relationship between input and output voltages and currents on a transmission line. For lossless lines, these parameters indicate an equivalent pi circuit with imaginary series impedance and shunt admittance. They are fundamental to characterizing line behavior and predicting how electrical quantities propagate through the transmission line.
Q6: How does real power flow differ from reactive power flow on a lossless line?
On a lossless line at surge impedance loading, constant real power flows from the sending end to the receiving end while reactive power flow is zero. Real power represents actual energy delivery to the load, whereas reactive power would indicate energy oscillating back and forth. This zero reactive power condition is unique to surge impedance loading and represents optimal power transfer.
Q7: How do ABCD parameters relate to transmission line differential equations?
ABCD parameters emerge from solving transmission line differential equations, which govern voltage and current behavior along the line. These parameters encapsulate the solution to those equations and provide a compact representation of line characteristics. Understanding transmission line differential equations is essential for deriving and interpreting ABCD parameters used in lossless line analysis.