Question

In a RLC series circuit at resonance, the value of the power factor is:
A. infinity
B. zero
C. \[\dfrac{1}{{\sqrt 2 }}\]
D. 1

Answer 1

Hint: The above problem can be resolved using the RLC circuit's fundamentals, where the resistance, inductor, and capacitor are connected in series. In the RLC series circuit, the primary condition is that the inductive reactance and the capacitive reactance are the same in magnitude, directly impacting the magnitude of the circuit's impedance. This provides the magnitude of the impedance to be dependent on the resistance of the circuit alone. Furthermore, the formula for the power factor is applied, which is given as the ratio of the resistor's resistance and the impedance of the circuit.

Complete step by step answer:
We know that, at resonance for the RLC circuit, the inductive resistance for the circuit is equal to the capacitive resistance. The mathematical relation is given as,
\[{X_L} = {X_C}\]
Then, the impedance is given as,
\[Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \]
Substituting the values as,
\[\begin{array}{l}
Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \\
Z = \sqrt {{R^2} + {{\left( {{X_L} - \left( {{X_L}} \right)} \right)}^2}} \\
Z = R
\end{array}\]
Now apply the formula for the power factor as,
\[\begin{array}{l}
\cos \phi = \dfrac{R}{Z}\\
\cos \phi = \dfrac{R}{{\left( R \right)}}\\
\cos \phi = 1
\end{array}\]
Therefore, in a RLC series circuit at resonance, the value of the power factor is 1

So, the correct answer is “Option D”.

Note:
Try to understand the concept and the construction of the RLC circuit, and fundamentals related to the circuit analysis are needed to be considered. Moreover, in an RLC circuit, the arrangements are given by the resistor's series connection, an inductor, and the capacitor. Moreover, the power factor concept is also used to resolve the given problem; the power factor is nothing but an angle that determines the phase angle within the voltage and the current flowing through the circuit.