A short transmission line, having its line-impedance angle as Θ, is delivering a given power at the receiving-

Question

A short transmission line, having its line-impedance angle as Θ, is delivering a given power at the receiving-end at a lagging power-factor angle of Φ. Which one of the following is a set of conditions for which this line will have maximum and zero regulation ?

Examoogle · Accepted Answer

Option C: $\Phi = \Theta$ and $\Phi + \Theta = \frac{\pi}{2}$

Explanation & Concepts:
Voltage regulation of a short transmission line is defined as the change in receiving-end voltage from no-load to full-load. Maximum regulation occurs when the load power factor angle is equal to the line impedance angle ($\Phi = \Theta$), and zero regulation occurs when the load power factor angle makes the voltage drop exactly compensate for the phase shift ($\Phi + \Theta = \frac{\pi}{2}$ for leading power factor).

Working Principle: The voltage drop across a short transmission line is approximated by $V_S - V_R \approx I(R \cos \Phi + X \sin \Phi)$. For maximum regulation, we differentiate this expression with respect to $\Phi$ and set it to zero, resulting in $	an \Phi = X/R = 	an \Theta$, hence $\Phi = \Theta$. For zero regulation, the component of the voltage drop along the vector $V_R$ must be zero, leading to the condition $\Phi + \Theta = \pi/2$.

Key Formulas: $V_{reg} = \frac{V_{NL} - V_{FL}}{V_{FL}} 	imes 100$; $V_S \approx V_R + I(R \cos \Phi + X \sin \Phi)$

Key Points: Voltage regulation indicates the sensitivity of the output voltage to changes in load.; Maximum voltage regulation occurs under lagging power factor conditions where $\Phi = \Theta$.; Zero regulation typically occurs with a leading power factor load.; For a short line, $R$ and $X$ are the primary parameters influencing the regulation.

Advantages: Simple mathematical model for short lines.; Predicts voltage stability under varying load conditions.

Disadvantages: Does not account for shunt capacitance effects.; Accuracy decreases significantly for long transmission lines.

Applications: Distribution network planning.; Short-distance industrial power feeders.

Additional Info: The impedance angle $\Theta = 	an^{-1}(\frac{X}{R})$.. Option A and B represent incorrect pairings or invalid trigonometric identities.

Answer: C is correct — Maximum regulation occurs at $\Phi = \Theta$ and zero regulation occurs at $\Phi + \Theta = \frac{\pi}{2}$.

Exam Tip: Remember: For short lines, regulation is positive for lagging loads, zero for specific leading loads, and negative (voltage rise) for very high leading power factor loads.

Examoogle · Answer

Option A: $\Phi = \Theta$ and $\Phi - \Theta = \frac{\pi}{2}$

Examoogle · Answer

Option B: $\Phi - \Theta = \frac{\pi}{2}$ and $\Phi = \Theta$

Examoogle · Answer

Option D: $\Phi + \Theta = \frac{\pi}{2}$ and $\Phi = \Theta$

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