Question

(a) For circuits used for transporting electric power, a low power factor implies large power loss in transmission. Explain.
(b) Power factor can often be improved by the use of a capacitor of appropriate capacitance in the circuit. Explain.

Answer 1

Hint: The transmitting power is proportional to the power factor. The power loss in the circuit is proportional to the square of the current in the circuit. Find out what will increase if there is a small power factor. The power factor can be improved by keeping the resistance and impedance the same.

Complete step by step answer:
(a) We have, the transmission power at a given voltage is expressed as,
\[P = IV\cos \phi \] …… (1)
Here, I is the current, V is the voltage and \[\cos \phi \] is the power factor.
If there is no power loss in the transmission, the power factor becomes 1. But, there always be the power loss in the transmission and the power loss in the transmission is given as,
\[{P_{loss}} = {I^2}R\]
To transmit the given power, if \[\cos \phi \] is very small, the current should be increased.

Therefore, if the current is increased, from the above expression, the power loss will also increase.

(b) We have the power factor is the ratio of the resistance to the impedance of the transmission line.
\[\cos \phi = \dfrac{R}{Z}\]
Here, R is the resistance and Z is the impedance.
The power factor will improve if \[\cos \phi \to 1\]. Therefore,
\[1 = \dfrac{R}{Z}\]
\[ \Rightarrow R = Z\]
We have the expression for the impedance of the transmission line,
\[Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \]
Here, \[{X_L}\] is the inductive reactance and \[{X_C}\] is the capacitive reactance.
As we have discussed earlier, the power factor will improve when the impedance equals the resistance of the transmission line. From the above expression, the impedance will be equal to resistance when \[{X_L} = {X_C}\].

Thus, the power factor can be improved by choosing the capacitance such that its capacitive reactance equals the inductive reactance.

Note:Students must understand the difference between capacitance and capacitive reactance. The capacitance is the ability of the capacitor to store the charge and the capacitive reactance is the resistance to the AC supply by the capacitor. We can never reach the power factor equal to 1 in practice. There is always a power loss in the transmission.