# Given three elements X, Y and Z to be connected across an ac source. With only X connected across the ac source, voltage and current are found to be in the same phase. With only element Y in the circuit, the voltage lags behind the current in phase by $\frac{\pi }{2}$,while with the element Z in the circuit, the voltage leads the current in phase by $\frac{\pi }{2}$,(a) Identify the elements X, Y and Z.(b) When all these elements are connected in series across the same source,(i) determine the power factor, and(ii) find out the condition when the circuit is in resonant state.

(a) These elements are resistor, Capacitor and inductor,

• When X connected across ac source, voltage and current are found to be in the same phase, which means X is a resistor.
• When Y connected across Ac source, Voltage lag behind current by $\dpi{80} \frac{\pi }{2}$, which means Y is capacitor.
• When Z connected across Ac source, Voltage leads the current by an angle $\dpi{80} \frac{\pi }{2}$, which means Z is an inductor.

So,         X= Resistor

Y= Capacitor

Z = Inductor

(b)

In series RLC circuit,

The importance of the circuit is

$Z=\sqrt{R^{2}+(X_{C}-X_{L})^{2}}$

$\cos \phi =\frac{R}{Z}$

(ii) At resonance

$X_{C}=X_{L}$

$X_{C}=\frac{1}{w_{L}},\; \; \; X_{L}=wL$

When, $w=w_{0}, X_{C}=X_{L}$

$X_{C}=X_{L}$    OR

$\frac{1}{w_{0}C}=w_{0}L$

Resonant frequency

$w_{0}=\frac{1}{\sqrt{LC}}$

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