Question

A current source sends a current \[I-{{i}_{0}}\cos (\omega t)\] when connected across an unknown load, it gives a voltage output of\[v={{v}_{0}}\sin [\omega t+\dfrac{\pi }{4}]\] across that load. then the voltage across the current source may be brought in phase with the current through it by
(A) Connecting an inductor in series with the load
(B) connecting a capacitor in series with the load
(C) connecting an inductor in parallel with the load
(D) connecting a capacitor in parallel with the load

Answer 1

Hint: In this problem, we are given sinusoidal values of voltage and alternating current. It is clear from the given equation that they are out of phase with each other. We know in an LCR circuit current leads the voltage in the capacitor and it lags behind the voltage in the inductor.

Complete step by step answer: Here, the given equations of voltage and current are \[I-{{i}_{0}}\cos (\omega t)\]and \[v={{v}_{0}}\sin [\omega t+\dfrac{\pi }{4}]\]
On comparing them we find that the current leads the voltage by a phase of \[\dfrac{\pi }{4}\]. We know in case of capacitor current leads the voltage. Thus, the unknown load is a capacitor.
Now we want to bring both the current and the voltage in phase and in order to do that we need to connect some component in which current lags behind the voltage. We know that in case the inductor current lags behind the voltage by a phase of \[\dfrac{\pi }{4}\]. Thus, an inductor needs to be inserted in series in the given circuit. Hence, the correct option is (A)
Note: we can also approach this problem as;
\[v={{v}_{0}}\sin [\omega t+\dfrac{\pi }{4}]\]
\[\begin{align}
  & v={{v}_{0}}\sin [\omega t+\dfrac{\pi }{4}+\dfrac{\pi }{4}-\dfrac{\pi }{4}] \\
 & v={{v}_{0}}\sin [\omega t+\dfrac{\pi }{2}-\dfrac{\pi }{4}] \\
 & v={{v}_{0}}\cos [\omega t-\dfrac{\pi }{4}] \\
\end{align}\]
The current equation can be written as
\[\begin{align}
  & I-{{i}_{0}}\cos (\omega t) \\
 & I+{{i}_{0}}\cos (\omega t+\pi ) \\
\end{align}\]
Now on seeing both voltage and current equation it is clear that ac current leads the ac voltage across the load. Hence, an unknown load is a capacitor and to bring both in phase an inductor needs to be attached in series in the given circuit.