Question

The power factor of the $LCR$ circuit at resonance is-

Answer 1

We can solve this question by using the property of impedance at resonance. We know that the power factor is a ratio of resistance to impedance. Hence we first evaluate the impedance then using this impedance and resistance we will evaluate the power factor.

Formula used: $Z = \sqrt {{R^2} + {{\left( {{X_L}} \right)}^2} - {{\left( {{X_C}} \right)}^2}}$
$\cos \phi = \dfrac{R}{Z}$

Complete step by step answer: Here given that the circuit is $LCR$ , $LCR$ is a circuit that is made up of three different electronic components which are Inductor $(L)$ , Capacitor $(C)$ , and Resistance $(R)$ that can be connected either in series or any other combinations.
We will consider a simpler case of the $LCR$ series circuit. For an $LCR$ series circuit, we know that the impedance is defined as the difference of the inductance and capacitance. Hence given as,
$\Rightarrow Z = \sqrt {{R^2} + {{\left( {{X_L}} \right)}^2} - {{\left( {{X_C}} \right)}^2}}$
Where ${X_L}$ is the inductive reactance and ${X_C}$ is the capacitive reactance and $R$ is the component of resistance.
Now the power factor for $LCR$ circuit is defined as the ratio of resistance $R$ and impedance $Z$ , hence given as
$\Rightarrow \cos \phi = \dfrac{R}{Z}$
Now if substitution the value of impedance in the above equation then results as
$\Rightarrow \cos \phi = \dfrac{R}{{\sqrt {{R^2} + {{\left( {{X_L}} \right)}^2}} - {{\left( {{X_C}} \right)}^2}}}$ ………. $\left( 1 \right)$
But we know that the capacitive reactance and inductive reactance becomes equal to each other. Hence
$\Rightarrow {X_L} = {X_C}$
Substitution of this value in equation $\left( 1 \right)$ results in
$\Rightarrow \cos \phi = \dfrac{R}{{\sqrt {{R^2} + {{\left( {{X_L}} \right)}^2}} - {{\left( {{X_L}} \right)}^2}}}$
$\Rightarrow \cos \phi = \dfrac{R}{{\sqrt {{R^2} + 0} }}$
Therefore the value of the power factor then reduces as
$\Rightarrow \cos \phi = \dfrac{R}{R}$
$\therefore \cos \phi = 1$
Hence, the power factor of the $LCR$ circuit at resonance is $\cos \phi = 1$ .

Note: In this question, we have used the term resonance, which is defined as the phenomenon in the electrical circuit, occurs when the output of the circuit is maximum at one particular frequency and that frequency is known as the resonant frequency. Hence at the resonant frequency, the capacitive reactance and inductive reactance are equal.