Question

Show that the power factor of a series RLC circuit is$\cos \left(\varphi \right)=\frac{R}{Z}$

Short answer

We have prove that,

Power factor, $\cos \left(\varphi \right)=\frac{R}{Z}$

Required proof is given below.

Step-by-step solution

Given information

We need to find out that,

Power factor is, $\cos \left(\varphi \right)=\frac{R}{Z}$

Simplify

We know that,

Power dissipated by LCR circuit is, $P=I^{2}Z\cos \left(\varphi \right)$

Here, $I$is current in LCR circuit

And $Z$is impedance

And $\cos \left(\varphi \right)$is power factor

Also, we know that power, $P=VI$

Where $V$is voltage across resistor as in series voltage remain same across resistor, inductor and capacitor

Therefore, $V=IR$

Where $R$is resistance

Therefore,

$$\begin{gathered} P=I^{2}Z\cos \left(\varphi \right)=VI \\ {I}^{2}Z\cos \left(\varphi \right)=\left(IR\right)I \\ {I}^{2}Z\cos \left(\varphi \right)=I^{2}R \\ \cos \left(\varphi \right)=\frac{R}{Z} \end{gathered}$$

Hence we get the require expression i.e.$\cos \left(\varphi \right)=\frac{R}{Z}$