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Average Power In LCR Circuit | Average Power In LCR Circuit Derivation Class 12

Average Power In LCR Circuit :- Suppose the following alternating voltage is applied to a series LCR circuit :-

$\displaystyle E={{E}_{0}}sin\text{ }\omega t$ …..(1)

If the alternating current lags behind the alternating voltage by a phase angle ϕ, then

$\displaystyle I={{I}_{0}}sin\left( \omega t-\phi \right)$ …..(2)

Instantaneous power in the circuit,

$\displaystyle P=E\times I=\left[ {{E}_{0}}sin\omega t \right]\times \left[ {{I}_{0}}sin\left( \omega t-\phi \right) \right]$

$\displaystyle \Rightarrow P={{E}_{0}}{{I}_{0}}sin\omega t\times \left[ sin\omega tcos\phi -cos\omega tsin\phi \right]$

$\displaystyle \Rightarrow P={{E}_{0}}{{I}_{0}}si{{n}^{2}}\omega tcos\phi -{{E}_{0}}{{I}_{0}}sin\omega tcos\omega tsin\phi$

$\displaystyle \Rightarrow P={{E}_{0}}{{I}_{0}}si{{n}^{2}}\omega tcos\phi -\frac{{{E}_{0}}{{I}_{0}}}{2}\left( 2sin\omega tcos\omega t \right)sin\phi$

$\displaystyle \Rightarrow P={{E}_{0}}{{I}_{0}}si{{n}^{2}}\omega tcos\phi -\frac{{{E}_{0}}{{I}_{0}}}{2}sin2\omega tsin\phi$ …..(3)

The average power in an LCR circuit is defined as the mean value of the instantaneous power (given by equation 3) taken over one complete cycle.

Hence, the average power (P_avg.),

$\displaystyle {{P}_{avg.}}=\left\langle {{E}_{0}}{{I}_{0}}si{{n}^{2}}\omega tcos\phi -\frac{{{E}_{0}}{{I}_{0}}}{2}sin2\omega tsin\phi \right\rangle$

$\displaystyle \Rightarrow {{P}_{avg.}}=\left\langle {{E}_{0}}{{I}_{0}}si{{n}^{2}}\omega tcos\phi \right\rangle -\left\langle \frac{{{E}_{0}}{{I}_{0}}}{2}sin2\omega tsin\phi \right\rangle$

$\displaystyle \Rightarrow {{P}_{avg.}}=\left( {{E}_{0}}{{I}_{0}}cos\phi \right)\times \left\langle si{{n}^{2}}\omega t \right\rangle -\left[ \frac{{{E}_{0}}{{I}_{0}}}{2}sin\phi \right]\times \left\langle sin2\omega t \right\rangle$

$\displaystyle \Rightarrow {{P}_{avg.}}=\left( {{E}_{0}}{{I}_{0}}cos\phi \right)\times \left( \frac{1}{2} \right)-\left[ \frac{{{E}_{0}}{{I}_{0}}}{2}sin\phi \right]\times \left( 0 \right)$

$\displaystyle \Rightarrow {{P}_{avg.}}=\frac{{{E}_{0}}{{I}_{0}}}{2}cos\phi =\frac{{{E}_{0}}}{\sqrt{2}}\frac{{{I}_{0}}}{\sqrt{2}}cos\phi$

$\displaystyle \therefore {{P}_{avg.}}={{E}_{rms}}{{I}_{rms}}cos\phi ={{E}_{rms}}\left( \frac{{{E}_{rms}}}{Z} \right)cos\phi$ …..(4)

Equation (4) expresses the average power (P_avg.) in a series LCR circuit, which can also be written as follows :-

Average Power (P_avg.) = Virtual emf (RMS value) × virtual current × cos ϕ

Thus, over a complete cycle, the average power in LCR circuit is equal to the product of the RMS voltage, RMS current, and the cosine of the phase angle ϕ between the voltage and the current.

Equation (4) is valid for all alternating circuits, for example :-

1. 1. In a series RL circuit, $\displaystyle Z=\sqrt{{{R}^{2}}+X_{L}^{2}}$ and $\displaystyle cos\phi =\frac{R}{Z}$
  2. In a series RC circuit, $\displaystyle Z=\sqrt{{{R}^{2}}+X_{C}^{2}}$ and $\displaystyle cos\phi =\frac{R}{Z}$
  3. In a series LC circuit, $\displaystyle Z={{X}_{L}}-{{X}_{C}}$ and $\displaystyle \phi ={{90}^{\circ }}$
  4. In a series RLC circuit, $\displaystyle Z=\sqrt{{{R}^{2}}+{{\left( {{X}_{L}}-{{X}_{C}} \right)}^{2}}}$ and $\displaystyle cos\phi =\frac{R}{Z}$