# Power in Series LCR Circuits

## Power in Series LCR Circuits

Alternating current plays a central role in the system for distributing, converting and using electrical energy, so it is important to look at power relationships in ac circuits. The power in an electric current is the rate at which electrical energy is consumed in the circuit. Let an emf E = E₀ sinωt be applied to a series LCR circuit.

The current in the circuit is I = I₀ sin (ωt + φ)

Where,

φ = Phase difference between current and voltage.

The instantaneous power is given by: P = E x I = E₀I₀ sinωt sin (ωt + φ)

If the average power consumed in the cycle from time t₁ to t₂ is Pavg, then:

$${{P}_{avg}}({{t}_{2}}-{{t}_{1}})=\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{Pdt}$$,

Average power over one cycle, i.e. t₁ = 0 to t₂ = T be given by:

$${{P}_{avg}}T=\int\limits_{0}^{T}{{{E}_{0}}{{I}_{0}}\sin \omega t\,\sin (\omega t+\phi )dt}$$,

$$\Rightarrow\,\,{{P}{avg}}T={{E}{0}}{{I}{0}}\int\limits{0}^{T}{\left( {{\sin}^{2}}\omega t\,\cos \phi +\frac{\sin 2\omega t}{2}\sin \phi\right)}dt$$,

$$\Rightarrow \,\,{{P}_{avg}}T=\frac{{{E}_{0}}{{I}_{0}}}{2}\cos \phi T$$ $$\left[ \because \,\,\,\,\int\limits_{0}^{T}{{{\sin }^{2}}\omega t\,dt=\frac{T}{2}\,\,\,\,\,\And \,\,\,\,\,\int\limits_{0}^{T}{\sin 2\omega t\,dt=0}} \right]$$,

Pavg = EVIV cos φ = Papp cos φ

Where,

Papp = Apparent power or virtual power = EVIV.

cos φ = Power factor = Pavg/ Papp.

Also,

Pavg = EVIV cos φ = (IVZ) IV R/Z = IV²R

For pure resistive circuit: φ = 0 and Z = R.

Pavg = EVIV = (IVZ) IV = IV²Z = IV²R.

For pure inductive circuit: φ = – π/2

Pavg = 0

Therefore, the average power over a complete cycle of ac through an ideal inductor is zero. Actually, whatever energy is needed in building up current in inductance is returned back during the decay of current.

For pure capacitive circuit: φ = π/2.

Pavg = 0.

In this case too, the average power is zero. Actually, whatever energy is needed in building up the voltage across capacitor is returned to the source during discharging of capacitor. So average power consumed in pure inductive or pure capacitive circuit is zero. So, there is no loss of energy in the inductor or capacitor. But in a resistance, loss of energy occurs. It can’t store the energy like inductor or capacitor.

The current through pure L or C, which dissipates no power is called idle current or wattles current. L and C are most suitable for controlling the current in ac circuits.