Question

In an RLC series circuit shown in the figure, the readinf of voltmeter $V_1$ and $V_2$ are $100V$ and $120V$, respectively. The source voltage is $130V$. For this situation, mark out the correct statement(s).

• A. Voltage across resistor, inductor and capacitor are $50V$, $50\sqrt{3}V$ and $120+50\sqrt{3}V$ respectively
• B. Voltage across resistor, inductor and capacitor are $50V$, $50\sqrt{3}V$ and $120-50\sqrt{3}V$ respectively
• C. Pwer factor of the circuit is $5/13$
• D. The circuit is capacitive in nature

Answer 1

Answer: (A), (C), (D)

The circuit is capacitive in nature

Given,
$V_1=100V$, ( it reads between inductor and resistor)
$V_2=120V$, (it reads between inductor and capacitor)
Source $V=130V$
Let the potential difference across resistor, inductor and capacitor is $V_R,V_L,V_C$ respectively.
this voltage can be represented on the phasor diagram.

$\begin{array}{}{V}^2=V_R^2+(V_L-V_C{)}^2& \text{(1)}\end{array}$


Case 1: As given voltage across inductor and resistor is $100V$ as we know the phase difference between them is $\frac{\pi }{2}$. So,
$\begin{array}{}{V}_R^2+V_L^2={100}^2& \text{(2)}\end{array}$

Case 2: As given voltage across inductor and capacitor is $120V$ as we know the phase difference between them is $\$\pi$. So these can be directly subtracted,
$\begin{array}{}|V_L-V_C|=120& \text{(3)}\end{array}$
Note: this can be $V_C-V_L$ thats why we are using mode for this case we assume $V_L>V_C$

From equation 1 and 3
${130}^2=V_R^2-{120}^2$
$V_R^2=2500$
$V_R=50V$
Now, by using $V_R$ in equation 2
${50}^2+V_L^2={100}^2$
$V_L=50\sqrt{3}V$
Substitute $V_L$ in equation 3
$50\sqrt{3}-V_C=120$
$V_C=-120+50\sqrt{3}V$

So, $V_C$ is coming negative so it means our assumption is wrong so $V_C>V_L$
$V_C-V_L=120$
$V_C=(120+50\sqrt{3})V$

Now for power factor,

$\cos \varphi =\frac{R}{Z}=\frac{50}{130}=\frac{5}{13}$
But directly we also know that $Z=130V$. So without this phasor diagram also we can directly calulate power factor.

As a potential difference across the capacitor is more than the potential difference across the inductor so it's a capacitive circuit. So option D is also correct