Special cases for power in an AC circuit

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    Special cases for power in an AC circuit


1. Pure resistive circuit

Here the voltage and current are in same phase, I. E., phie = 0 and cos(phie) = 1

Pav = Erms × Irms × 1 = Erms × Irms = Erms ×                                                                         Erms/R

2. Pure inductive circuitcircuit

Here the voltage leads the current in phase by π/2, I. E., phie = π/2

Pav = Erms × Irms cos(π/2) = 0

Thus the average power consumed in an inductive circuit over a complete cycle is zero. 

3. Pure capacitive circuit

Here the voltage lags behind the current in phase by π/2, I. E., phie = -π/2

Pav = Erms × Irms cos(-π/2) = 0

Thus the average power consumed in a capacitive circuit over a complete cycle is also zero. 

4. Series LCR circuit 

For a series LCR - circuit, 
Pav = Erms × Irms × cos(phie), where    phie = 1/tan((xL -xC)/R) 

So, phie may have a non - zero value for series LR-, RC- and LCR- circuits. Some power is consumed in such circuits, but only in the resistor R. 

5. Power dissipated at resonance in LCR-circuit


At resonance, xL - xC, and phie = 0. So cos(phie) = 1, and pav = Erms Irms. That is, maximum power is dissipated in the circuit (through R) at resonance.


                         Power factor

The power factor is defined as the ratio of true power to the apparent power of an AC circuit. It is equal to the cosine of the phase angle between current and boltage in the AC circuit. It is given by

      Cos(phie) = true power/ apparent power = pav/Vrms Irms

For a series LCR circuit, power factor is

Cos(phie) = R/Z = R/[R×R + (wL - 1/wC) (wL - wC)] where, denominator bracket has power of 1/2.

(b) for a purely inductive or capacitive circuit,  phie = 90 

Thus the power factor assumes the minimum value for a purely inductive or capacitive circuit. 
For a purely resistive circuit, phie = 0 degree       power factor = cos 0 = 1

Thus the power factor assumes the maximum value for a purely resistive circuit. 

             

                         Wattless current


The current in an AC circuit is said to be wattless if the average power consumed in the circuit is zero. It is the component Irms sin(phie) of the alternating current. In an inductive or capacitive AC circuit, the phase difference between voltage and current, phie = +-π/2 power factor, cos(phie) = 0 and so the current is wattless. 

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