Power consumed in a Series LCR Circuit

Power consumed in a Series LCR Circuit

Let the alternating e.m.f. applied to LCR circuit be

\begin{align*} E &= E_0\sin\: \omega t \dots (i) \\ \end{align*}

If the alternating current developed lags behind the applied e.m.f. by a phase angle \begin{align*}\: \theta , \\ \text {then} \\ I &= I_0 \sin (\omega t - \theta ) \dots (ii) \\ \text {Power at instant t is given by} \\ \frac {d\omega }{dt} &= EI \\ &= E_0\sin\: \omega t \times I_0 \sin (\omega t - \theta ) \\ &= E_0I_0 \sin \omega t \:(\sin \omega t \cos \theta - \cos \omega t \sin \theta ) \\ &= E_0I_0 \sin^2\omega t \cos \theta – E_0I_0\sin \omega t \: \cos \: \omega t\sin \: \theta \\ &= E_0I_0 \sin ^2\omega t \cos \theta - \frac {E_0I_0}{2} \sin \: 2\omega t\:\sin \: \theta \\ \end{align*}

If this instantaneous power is assumed to remain constant for a small constant for a small time dt, then small amount of work done in this time is

\begin{align*} dW = \left ( E_0I_0 \sin ^2 \omega t \cos \theta - \frac {E_0I_0}{2} \sin \: 2\omega t \sin \: \theta \right )dt \\ \end{align*}

Total work done over a complete cycle is obtained by integrating above equation from 0 to T.

\begin{align*} \text {i.e.} \: W &= \int _0^T E_0I_0 \sin^2 \omega t \: \cos \theta \: dt - \int _0^T \frac {E_0I_0}{2} \sin \: 2\omega T \: dt \dots (iii) \\ \text {or,} \: W &= E_0I_0 \: \cos \:\theta \int _0^T \sin ^2 \omega t \: dt - \frac {E_0I_0}{2} \sin \: \theta \:\int _0^T \sin \: 2\: \omega t \: dt \\ \text {Now,} \\\end{align*}

\begin{align*} \int _0^T \sin^2\omega t \: dt &= \int _0^T \left (\frac {1-\cos \: 2\omega t}{2} \right )dt \\ &= \frac 12 \left [ \int _0^T dt - \int _0^T \cos \:2\omega t \: dt \right ] \\ &= \frac 12 [T -0] \\ &= \frac T2 \dots (iv) \\ \text {and} \: \int _0^T \sin \: 2\omega t \: dt &= 0 \dots (v) \\ \text {Using equation} \: (iv) \: \text {and} \: (V) \: \text {in equation} \: (iii), \\ \text {we get} \\ \end{align*}

\begin{align*} W &= E_0I_0 \: \cos \: \theta \frac T2 – 0 \\ W &= \frac {E_0I_0 \cos \: \theta}{2} \\ \therefore \end{align*} average power in the inductive circuit over a complete cycle\begin{align*} \\ P &= \frac WT \\&= \frac {E_0I_0 \cos \: \theta }{T} \frac T2 \\ &= \frac {E_0}{\sqrt 2} \frac {I_0}{\sqrt 2} \cos \: \theta \\ P &= E_vI_v \: \cos \theta \dots (iv) \\ \end{align*}

Hence, average power over a complete cycle in an inductive circuit is the product of virtual e.m.f., virtual current and cosine of the phase angle between the voltage and the current. The quantity \(\cos \theta \) is called power factor.

Here, P is called true power and E_vI_v is called apparent power or virtual power.

Special cases:

1. A.C. circuit containing resistor only:
   In such a circuit, phase angle, \(\theta = 0^o\), so true power dissipated \((P) = E_vI_v \cos \:0^o\: =E_vI_v\)
   \begin{align*}\text {or,} \: \text {True power} &= \text {apparent power} \\ \text {Power loss} &= \text {Product of virtual value of voltage and current} \\ \end{align*}
2. A.C. circuit having pure inductor or capacitor only:
   In such a circuit, the phase angle, between voltage and current is \(\frac {\pi}{2} \).

\begin{align*} \theta &= \frac {\pi }{2} \\ \therefore \text {Power dissipated, P} = E_vI_v\: \cos \left (\frac {\pi }{2} \right ) = 0 \\ \end{align*}
Thus, no power loss takes place in a circuit having pure inductor or capacitor only.

Power Factor of an A.C. Circuit

The ratio of true power and apparent power in an a.c. circuit is called power factor of the circuit.

\begin{align*}\text {i.e. Power factor,}\: \cos \theta = \frac {P}{E_vI_v} = \frac {P}{P_v} \\ \end{align*}

Power factor \(cos \: \theta \) is always positive and not more than 1.

1. For circuit having pure reistance, \(\cos \: \theta = 1\)
2. For circuit having pur inductor or pure capacitor, \(\cos \: \theta = 0\: \:\: [\because \theta = \frac {\pi }{2}]\)
3. For RC circuit, \(\cos \: \theta = \frac {R}{\sqrt {R^2} + \frac {1}{\omega ^2C^2}}\)
4. For LR circuit, \(\cos \: \theta = \frac {R}{\sqrt {R^2 + L^2\omega ^2}}\)
5. For LCR circuit, \(\cos \: \theta = \frac {R}{\sqrt {R^2 + \left ( L\omega - \frac {1}{C\omega }\right )^2}} \)

Wattless and Wattful Current

In pure inductor or an ideal capacitor, \(\theta = 90^o\), so average power consumed in a pure inductor or capacitor, \(P = E_vI_v \cos 90^o = 0^o\).

The current through pure inductor or capacitor consumes no power for its maintenance in the circuit is called idle wattles current.

At resonance, \( X_L = X_C \) and \(\theta = 0^o \).

$$\therefore \cos \: \theta = \cos \: 0^o = 1$$

In this case, maximum power is dissipated. The current through resistance R which consumes power for its maintenance in the circuit is called wattful current.

Choke Coil

In a d.c. circuit, current can be reduced by using suitable resistance. While using resistance, electrical power is lost in the form of heat across the resistance. But in a case of a.c. circuit, choke coil can be used to reduce current without losing power by it. The reason is alternating e.m.f leads the current by phase angle \(\pi/2\) and average power consumed will be

$$P = E_vL_V \: \cos \: \pi/2 = 0$$

So an inductance used to control current in an a.c. a circuit without much loss of energy is called choke coil.

Reference

Manu Kumar Khatry, Manoj Kumar Thapa, Bhesha Raj Adhikari, Arjun Kumar Gautam, Parashu Ram Poudel.Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.