AC Power analysis

Apparent Power and Power Factor

Apparent power is a measure of alternating current (AC) power that is computed by multiplying the root-mean-square (rms) current by the root-mean-square voltage. 

it is the magnitude of a complex power S

The apparent power is measured by VA (volt ampere) it is also the product of the rms value.

in previous formula:

                                                      P = 1/2 Vm Im cos(θv - θi)

and

P = Vrms Irms cos(θv - θi) = S cos(θv - θi)

S = Vrms Irms

Power factor is define as the ratio of the real power flowing to the load, to the apparent power in the circuit.

                                                          pf = P/S = cos(θv - θi)

Example:

Obtain the power factor and the apparent power of a load whose impedance is Z=60+j40Ω  when the applied voltage is v(t) = 150 cos(377t + 10) V.

I = V/Z  = (150 ∠ 10)/(60 + j40 Ω) = 2.08 ∠-23.69 A

S = Vrms Irms = (120/√2)(2.08/√2) = 156 VA

pf = cos(θv - θi) = cos(10 - (-23.69)) = 0.832



Complex Power

complex power defined as:

                                                                  S = P + jQ

where:

P = the real part and measured in watt (W)

Q = is the reactive power it is measured by volt-ampere reactive (VAR)

S = is the complex power measured by volt-ampere (VA)

Complex power can be calculated by multiplying the Vrms of a circuit by the complex conjugate of the total circuit current.

                                                                     S = VI*

example:

Z = 25 + j15Ω  = 29.15∠ 30.69

I = Vrms/Z = 10/29.15∠ 30.69 = 342.99∠ -30.964 mA = 

I* = 342.99∠30.964 mA

Therefore:

S = VI*

S = (10∠0)(342.99∠30.964 mA)

S = 2.94 + j1.76 VA = 3.49∠ 30.964 VA

This is the power triangle the expression for the complex power.