Question

Obtain an expression for average power dissipated in a series LCR circuit.

Hint:

Power is only dissipated in the resistor. The inductor and capacitor store and release energy but don't dissipate it over a full cycle. The \(\cos(\phi)\) factor, which equals \(R/Z\), essentially tells you what fraction of the total "apparent power" (\(V_{rms}I_{rms}\)) is actually doing work in the resistor.

Answer 1

In a series LCR circuit, the instantaneous voltage \(V\) and current \(I\) are given by: \[ V = V_0 \sin(\omega t) \] \[ I = I_0 \sin(\omega t - \phi) \] where \(V_0\) and \(I_0\) are the peak voltage and current, \(\omega\) is the angular frequency, and \(\phi\) is the phase difference between the voltage and current.
The instantaneous power (\(P\)) dissipated in the circuit is the product of instantaneous voltage and current: \[ P = VI = [V_0 \sin(\omega t)][I_0 \sin(\omega t - \phi)] \] \[ P = V_0 I_0 \sin(\omega t)[\sin(\omega t)\cos(\phi) - \cos(\omega t)\sin(\phi)] \] \[ P = V_0 I_0 [\sin^2(\omega t)\cos(\phi) - \sin(\omega t)\cos(\omega t)\sin(\phi)] \] To find the average power (\(P_{avg}\)) over one complete cycle (\(T\)), we need to find the average value of this expression.
We know the average values over one cycle are:
\(\langle \sin^2(\omega t) \rangle = \frac{1}{2}\)
\(\langle \sin(\omega t)\cos(\omega t) \rangle = \langle \frac{1}{2}\sin(2\omega t) \rangle = 0\)
Therefore, the average power is: \[ P_{avg} = V_0 I_0 [ \langle \sin^2(\omega t) \rangle \cos(\phi) - \langle \sin(\omega t)\cos(\omega t) \rangle \sin(\phi) ] \] \[ P_{avg} = V_0 I_0 [ \frac{1}{2} \cos(\phi) - (0) \sin(\phi) ] \] \[ P_{avg} = \frac{V_0 I_0}{2} \cos(\phi) \] We can express this in terms of the root mean square (RMS) values, where \(V_{rms} = \frac{V_0}{\sqrt{2}}\) and \(I_{rms} = \frac{I_0}{\sqrt{2}}\). \[ P_{avg} = \left(\frac{V_0}{\sqrt{2}}\right) \left(\frac{I_0}{\sqrt{2}}\right) \cos(\phi) \] \[ P_{avg} = V_{rms} I_{rms} \cos(\phi) \] This is the expression for the average power dissipated in a series LCR circuit. The term \(\cos(\phi)\) is called the power factor.