Apparent Power and Power Factor

Today we went over all kinds of definitions of power regrading AC circuits. Unlike DC circuit, AC circuit has different values of voltage and current at different moments. Using the original definition for DC circuits on AC circuits only shows the instantaneous power. It does not tell us enough information about the circuit. Then we introduced the idea of effective voltage and current, which is equal to Vrms and Irms respectively.

We define apparent power as the product of Vrms magnitude and Irms magnitude. Since the voltage might be leading or lagging the current, there might be a phase difference between them. We represent the phase difference by using the power factor.

Since voltage and current have a real part and a imaginary part. We define complex power as the combination of both. The apparent power can be considered as the magnitude of the complex power, and the power factor is the cosine value of the angle. In rectangular format, the real term is called the real power, the imaginary term is called the reactive power.

Then we did a lab to test the new definitions of power. Here is a schematic of the circuit.

We applied different resistance values to the load resistor and measured them in the oscilloscope.

10Ω

47Ω

100Ω

Then we compared the experimental values with theoretical values

10Ω

47Ω

100Ω

Summary: We concluded a list of useful formulas regarding power in AC circuits.

RMS Value of a Sinusoid
Vrms =Vp /√2 
Irms =Ip /√2

Impedance
Z = R + j X = |Z|∠θ
|Z| = (R2 + X2)1/2
θ = tan−1(X/R)
Note: θ is the angle of the load impedance (We have suppressed the subscript z.)

Ohm’s Law in Frequency Domain V = IZ
Vp = Ip |Z|
Vrms = V / √ 2
Irms =I/√2
Vrms = Irms |Z|
θv −θi =θ
θ > 0 when X > 0 (Inductive impedance) θ < 0 when X < 0 (Capacitive impedance)

Average Power (W)
P = Vrms Irms cosθ = Irms2 R = (Vrms2 cosθ)/|Z|

Power Factor
pf=cosθ=R/(R2 +X2)1/2, 1≥pf≥0.
If θ > 0 (inductive impedance), θi < θv, pf lagging 
If θ < 0 (capacitive impedance), θi > θv, pf leading 
If θ = 0 for purely resistive load and the pf is unity

Reactive Power (VAR) Q = Vrms Irms sinθ = Irms2 X
Apparent Power (VA) S=Vrms Irms =Irms2 |Z|=Vrms2 /|Z|
Complex Power (VA)
S=Vrms (Irms)*=Vrms Irms ∠θ=P+jQ=Irms2 Z=(Vrms)2/Z*