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  • Consider the LCR circuit shown in Fig 7.6. Find the net current i and the phase of i. Show that i=frac{v}{Z}. Find the impedance Z for this circuit.

Consider the LCR circuit shown in Fig 7.6. Find the net current i and the phase of i. Show that i=\frac{v}{Z}. Find the impedance Z for this circuit.

Answers (1)

Taking Resistor andAlternating EMF circuit, i=i_{1}+i_{2}

i_{2}R=V_{m}\sin \omega t\Rightarrow i_{2}=\frac{V_{m}\sin \omega t}{R}

Taking Capacitor-Inductor and Alternating EMF circuit,

V_{m}\sin \omega t=\frac{q_{1}}{C}+L\frac{d^{2}q}{dt^{2}}

Assuming q_{1}=q_{m}\; \sin (\omega t+\phi )

\frac{dq_{1}}{dt}=\omega q_{m}\cos (\omega t+\phi )

\frac{d^{2}q_{1}}{dt^{2}}=-\omega ^{2}q_{m}\sin (\omega +\phi )

V_{m}\sin \omega t=\frac{q_{m}\sin (\omega t+\phi )}{C}+L(-\omega ^{2}q_{m}\; \sin (\omega t+\phi ))

V_{m}\sin \omega t=q_{m}\left ( \frac{1}{C}-\omega ^{2}L \right )\sin (\omega t+\phi )

q_{m}=\frac{V_{m}}{\frac{1}{C}-\omega ^{2}L}

\sin \omega t=\sin (\omega t+\phi )

\phi =0

i_{1}=\frac{dq_{1}}{dt}=\omega \left ( \frac{V_{m}}{\frac{1}{C}-\omega ^{2}L} \right )\cos (\omega t)

i=i_{1}+i_{2}=\left ( \frac{V_{m}}{\frac{1}{\omega C}-\omega L} \right )\cos (\omega t)+\frac{V_{m}(\sin \omega t)}{R}

=A\; \sin (\omega t+\phi )

A=\sqrt{\left ( \left ( \frac{V_{m}}{\left ( \frac{1}{\omega C}-\omega L \right )} \right )^{2}+\left ( \frac{V_{m}}{R} \right )^{2} \right )}=V_{m}\sqrt{\left ( \left ( \frac{V_{m}}{\left ( \frac{1}{\omega C}-\omega L \right )} \right )^{2}+\left ( \frac{1}{R} \right )^{2} \right )}

\tan \phi =\frac{R}{\frac{1}{\omega C}-\omega L}

\phi =\tan ^{-1}\left ( \frac{R}{\frac{1}{\omega C}-\omega L} \right )

 

Posted by

infoexpert23

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