Question

In a series $LR$ circuit, power of $400W$ is dissipated from a source of $250V,50Hz$. The power factor of the circuit is $0.8$. In order to bring the power factor to unity, a capacitor of value $C$ is added in series to the $L$ and $R$. Taking the value of $C$ as $\left(\frac{n}{3\mathrm{\pi }}\right)\mu F$, then the value of $n$ is __________.

Answer 1

Step 1: Given data

Power, $P=400W$

Source voltage,$V_{rms}=250V$

Frequency, $f=50Hz$

Power factor,$\cos \varphi =0.8$

Step 2: To find

We have to find the value of $n$ in the equation $C=\left(\frac{n}{3\mathrm{\pi }}\right)\mu F$

Step 3: Finding Impedance, $Z$

We know, power is given by,

$$\begin{gathered} P=V_{rms}{I}_{rms}\cos \varphi \\ \Rightarrow P=V_{rms}\frac{V_{rms}}{Z}\cos \varphi \hspace{0.17em}(\because I_{rms}=\frac{V_{rms}}{Z}) \\ \Rightarrow Z=\frac{{V_{rms}}^2}{P}\cos \varphi \end{gathered}$$

Where $I_{rms}$ is rms current and $Z$ is the impedance of the circuit.

Substituting the values,

$$\begin{gathered} Z=\frac{{250}^2}{400}\times 0.8 \\ \Rightarrow Z=125\Omega \end{gathered}$$

Step 4: Finding resistance$\left(R\right)$, inductive reactance$\left(X_L\right)$ and capacitive reactance$\left(X_C\right)$

Also,

$$\begin{gathered} \cos \varphi =\frac{R}{Z} \\ \Rightarrow R=Z\cos \varphi \\ \Rightarrow R=125\times 0.8 \\ \Rightarrow R=100\Omega \end{gathered}$$

Impedance,

$$\begin{gathered} Z=\sqrt{R^2+X_L^2} \\ \Rightarrow X_L=\sqrt{Z^2-R^2} \\ \Rightarrow X_L=\sqrt{{125}^2-{100}^2} \\ \Rightarrow X_L=\sqrt{{125}^2-{100}^2} \\ \Rightarrow X_L=75\Omega \end{gathered}$$

Since the power factor is unity, $X_L=X_C=75\Omega$

Step 5: Finding capacitance $C$ and hence $n$

$$\begin{gathered} X_C=75\Omega \\ \Rightarrow \frac{1}{C\omega }=75 \\ \Rightarrow \frac{1}{2\mathrm{\pi f}C}=75\left(\because \omega =2\mathrm{\pi f}\right) \\ \Rightarrow \frac{1}{2\pi \times 50\times C}=75\left(\because f=50Hz\right) \\ \Rightarrow C=\frac{1}{2\mathrm{\pi }\times 50\times 75} \\ \Rightarrow C=\frac{400}{3\mathrm{\pi }}\mu F \end{gathered}$$

Since, $C=\frac{400}{3\mathrm{\pi }}\mu F$, we get $n=400$

The value of $n$ is $400$