Power and Power Factor in an AC Circuit

Power consumed by a resistor is dissipated in heat and not
returned to the source. This is true power. True power is the rate
at which energy is used.
Current in an AC circuit rises to peak values and diminishes to
zero many times a second. The energy stored in the magnetic
field of an inductor, or plates of a capacitor, is returned to the
source when current changes direction.
Although reactive components do not consume energy, they do
increase the amount of energy that must be generated to do
the same amount of work. The rate at which this non-working
energy must be generated is called reactive power.
Power in an AC circuit is the vector sum of true power and
reactive power. This is called apparent power. True power is
equal to apparent power in a purely resistive circuit because
voltage and current are in phase. voltage and current are also in
phase in a circuit containing equal values of inductive reactance
and capacitive reactance. If voltage and current are 90 degrees
out of phase, as would be in a purely capacitive or purely
inductive circuit, the average value of true power is equal to
zero. There are high positive and negative peak values of power,
but when added together the result is zero

True Power and  Apparent Power Formulas The formula for apparent power is  P = EI

Apparent power is measured in volt-amps (VA). True power is calculated from another trigonometric function  the cosine of the phase angle (cos θ). The formula for true power is

P = EI  cos  θ

True power is measured in watts  In a purely resistive circuit, current and voltage are in phase There is a zero degree angle displacement between current and voltage. The cosine of zero is one. Multiplying a value by one does not change the value. In a purely resistive circuit the cosine of the angle is ignored. In a purely reactive circuit, either inductive or capacitive, current and voltage are 90 degrees out of phase. The cosine of 90 degrees is zero. Multiplying a value times zero results in a zero product. No power is consumed in a purely reactive circuit

Calculating Apparent Power in a simple R-L-C Circuit In the following 120 volt circuit, current is equal to 84.9 milliamps. Inductive reactance is 100 Ω and capacitive reactance is 1100 Ω. The phase angle is -45 degrees. By referring to a trigonometric table, the cosine of -45 degrees is found to be .7071.

 

 

The apparent power consumed by the circuit is:

P = EI
P = 120 x 0.0849
P = 10.2 VA
The true power consumed by the circuit is:
P = EI cos θ
P = 120 x 0.0849 x 0.7071
P = 7.2 Watts
Another formula for true power is:
P = I2R
P = 0.08492 x 1000
P = 7.2 Watts

Power Factor

Power factor is the ratio of true power to apparent power in an
AC circuit. Power factor is expressed in the following formula

PF= True Power / Apparent Power

Power factor can also be expressed using the formulas for true
power and apparent power. The value of EI cancels out because
it is the same in the numerator and denominator. Power factor
is the cosine of the angle PF = EI cos θ / EI =  cos θ

In a purely resistive circuit, where current and voltage are in
phase, there is no angle of displacement between current and
voltage. The cosine of a zero degree angle is one. The power
factor is one. This means that all energy delivered by the source
is consumed by the circuit and dissipated in the form of heat.
In a purely reactive circuit, voltage and current are 90 degrees
apart. The cosine of a 90 degree angle is zero. The power factor
is zero. This means the circuit returns all energy it receives from
the source to the source.
In a circuit where reactance and resistance are equal, voltage
and current are displaced by 45 degrees. The cosine of a 45
degree angle is .7071. The power factor is .7071. This means the
circuit uses approximately 70% of the energy supplied by the
source and returns approximately 30%