Question

Derive an expressic for the average power in LCR circuit connected to A.C supply. Define power factor ?​

Answer 1

⇝ Derivation :-

❒ In LCR circuit connected to AC supply ;

》 lnstantaneous voltage is given by ,

• $\rm E = E_{\circ}sin \omega t$

》 lnstantaneous current is given by ,

• $\rm I = I_{\circ}sin( \omega t - \phi)$

Therefore,

lnstantaneous Power is ;

$\rm P = EI = [E_{\circ}sin \omega t] [I_{\circ}sin( \omega t - \phi)]$

$:\longmapsto \rm P=E_{\circ}I_{\circ}sin(\omega t)sin(\omega t-\phi)$

$\rm Expanding \: Using \: \red{sin(x-y) =sinxcosy-cosxsiny}$

$:\longmapsto \rm P=E_{\circ}I_{\circ}sin(\omega t) [sin(\omega t)cos\phi-cos(\omega t)sin\phi]$

$:\longmapsto \rm P=E_{\circ}I_{\circ}sin {}^{2} (\omega t)cos \phi - E_{\circ}I_{\circ}sin (\omega t) cos(\omega t) \sin \phi$

As We Know that,

$\large \bf \red\bigstar \: \: \orange{ \underbrace{ \underline{dW=Pdt  }}}$

⏩ Integrating Both Side ;

$:\longmapsto \rm \rm W = \int Pdt$

⏩ Putting Value of P ;

$:\longmapsto \rm W = \int^{T} _{0} E_{\circ}I_{\circ}sin ^{2} (\omega t)cos \phi \\ \rm \: \: \: \: \: \: \: \: \: \: \: \: \: - \int^{T} _{0} E_{\circ}I_{\circ}sin (\omega t) cos(\omega t) \sin \phi$

$:\longmapsto \rm W=E_{\circ}I_{\circ}cos \bigg(\dfrac{\phi}{2} \bigg) T - 0$

$:\longmapsto \rm W= \dfrac{E_{\circ}I_{\circ}cos \big({\phi} \big) T}2 - 0$

$\purple{ :\longmapsto  \underline {\boxed{{\bf  W= \dfrac{E_{\circ}I_{\circ}cos \big({\phi} \big) T}2 } }}}$

As We Know that ;

$\bf \red\bigstar \: \: \orange{ \underbrace{ \underline{ P=\frac{Work}{Time} }}}$

Therefore,

$:\longmapsto \rm P_{avg}= \dfrac{E_{\circ}I_{\circ}cos \big({\phi} \big) \cancel T}{2 \cancel T}$

$\purple{ :\longmapsto  \underline {\boxed{{\bf P_{avg}= \frac{1}{2} E_{\circ}I_{\circ}cos \big({\phi} \big) }}}}$

✧ Or We can say that ;

$\purple{ :\longmapsto  \underline {\boxed{{\bf P_{avg}= E_{rms}I_{rms}cos \big({\phi} \big)}}}}$

⇝ Power Factor :-

The ratio of the actual electrical power dissipated by an AC circuit to the product of the r.m.s. values of current and voltage.

Answer 2

Answer:

$\large{\bf{\red{\mathfrak{\dag{\underline{\underline{Questions : -}}}}}}}$

1. Derive an expression for the average power in LCR circuit connected to AC supply.
2. Define power factor.

$\large{\bf{\green{\mathfrak{\dag{\underline{\underline{Answers: -}}}}}}}$

1. $= E_{v} l_{v}cos \: theta$
2. Power factor definition i written in below at last.

$\large{\bf{\green{\mathfrak{\dag{\underline{\underline{Solution :-}}}}}}}$

1》

• For the first question refer the above given attachment for better understanding.

2》

• Power factor=It is the ratio of actual electric power dissipated by an AC circuit to the product of r.m.s values of current and voltage.

$\large{\bf{\purple{\mathfrak{\dag{\underline{\underline{Which \: are \:the \:required \: answers}}}}}}}$

Hope it helps u mate.

Thank you .