Definition Power Triangle and Complex Power AC Circuits

Considerable effort has been expended over the years to express power relations as simply as possible. Power engineers have coined the term complex power, which they use to find the total effect of parallel loads.

Complex power is important in power analysis because it contains all the information pertaining to the power absorbed by a given load.

Make sure to read what is ac circuit first.

There are several types of power in ac circuit:

  1. Maximum average power transfer
  2. Voltage and current RMS
  3. Power factor and apparent power
  4. Power triangle and power complex
  5. Power ac conservation

Complex Power

Consider the ac load in Figure.(1).

Complex Power Formula
Figure 1. The voltage and current phasors associated with a load

Given the phasor form V = Vm∠θv and I = Im∠θi of voltage v(t) and current i(t), the complex powerS absorbed by the ac load is the product of the voltage and the complex conjugate of the current, or

Complex Power Formula
(1)

assuming the passive sign convention (see Figure.1). In terms of the rms values,

Complex Power Formula
(2)

where

Complex Power Formula
(3)

and

Complex Power Formula
(4)

Thus we may write Equation.(2) as

Complex Power Formula
(5)

We notice from Equation.(5) that the magnitude of the complex power is the apparent power; hence, the complex power is measured in volt-amperes (VA).

Also, we notice that the angle of the complex power is the power factor angle.

The complex power may be expressed in terms of the load impedance Z. The load impedance Z may be written as

Complex Power Formula
(6)

Thus, Vrms = ZIrms. Substituting this into Equation.(2) gives

Complex Power Formula
(7)

Since Z = R + jX, Equation.(7) becomes

Complex Power Formula
(8)

where P and Q are the real and imaginary parts of the complex power; that is,

Complex Power Formula
(9)
Complex Power Formula
(10)

P is the average or real power and it depends on the load’s resistance R.

Q depends on the load’s reactance X and is called the reactive (or quadrature) power.

Comparing Equation.(5) with (8), we notice that

Complex Power Formula
(11)

The real power P is the average power in watts delivered to a load; it is the only useful power. It is the actual power dissipated by the load.

The reactive power Q is a measure of the energy exchange between the source and the reactive part of the load.

The unit of Q is the volt-ampere reactive (VAR) to distinguish it from the real power, whose unit is the watt.

It represents a lossless interchange between the load and the source. Notice that :

  1. Q = 0 for resistive loads (unity pf)
  2. Q < 0 for capacitive loads (leading pf)
  3. Q > 0 for inductive loads (lagging pf)

Thus,

Complex power (in VA) is the product of the rms voltage phasor and the complex conjugate of the rms current phasor. As a complex quantity, its real part is real power P and its imaginary part is reactive power Q.

Introducing complex power enables us to obtain the real and reactive powers directly from voltage and current phasors.

Complex Power Formula
(12)

This shows how the complex power contains all the relevant power information in a given load.

It is a standard practice to represent S, P, and Q in the form of a triangle, known as the power triangle, shown in Figure.(2a).

This is similar to the impedance triangle showing the relationship between Z, R, and X, illustrated in Figure.(2b).

Complex Power Formula
Figure 2. (a) Power triangle, (b) impedance triangle

The power triangle has four items – the apparent/complex power, real power, reactive power, and the power factor angle. Given two of these items, the other two can easily be obtained from the triangle.

As shown in Figure.(3), when S lies in the first quadrant, we have an inductive load and a lagging pf.

When S lies in the fourth quadrant, we have a capacitive load and a leading pf.

It is also possible for the complex power to lie in the second or third quadrant.

This requires that the load impedance have a negative resistance, which is possible with active circuits.

Complex Power Formula
Figure 3. Power triangle

Complex Power Examples

For better understanding let us review examples below :
1. The voltage across a load is v(t) = 60 cos(ωt – 10o) V and the current through the element in the direction of the voltage drop is i(t) =  1.5 cos(ωt + 50o) A.

Find : (a) the complex and apparent powers, (b) the real and reactive powers, and (c) the power factor and the load impedance.

Solution :
(a) For the rms values of the voltage and current, we write

Complex Power Formula

The complex power is

Complex Power Formula

The apparent power is

Complex Power Formula

(b) We can express the complex power in rectangular form as

Complex Power Formula

Since S = P + jQ, the real power is

Complex Power Formula

while the reactive power is

Complex Power Formula

(c) The power factor is

Complex Power Formula

It is leading, because the reactive power is negative. The load impedance is

Complex Power Formula

which is a capacitive impedance.

2. A load Z draws 12 kVA at a power factor of 0.856 lagging from a 120-V rms sinusoidal source.

Calculate:

(a) the average and reactive powers delivered to the load,

(b) the peak current, and

(c) the load impedance.

Solution:
(a) Given that, pf = cos θ = 0.856, we obtain the power angle as θ = cos-1 0.856 = 31.13o. If the apparent power is S = 12,000 VA, then the average or real power is

Complex Power Formula

while the reactive power is

Complex Power Formula

(b) Since the pf is lagging, the complex power is

Complex Power Formula

From SVrmsI*rms, we obtain

Complex Power Formula

Thus Irms = 100∠31.13o and the peak current is

Complex Power Formula

(c) The load impedance

Complex Power Formula

which is an inductive impedance.