The complex power of a circuit in sinusoidal steady state is defined as
\(
(1) \ \overline{S} \, = \, P \, + \, jQ
\)
where P is the real power and Q is the reactive power given by
\(
(2) \ P \, = \, V_{eff} \cdot I_{eff} \, \cos(\Theta_{vi})
\
(3) \ Q \, = \, V_{eff} \cdot I_{eff} \, \sin(\Theta_{vi})
\)
where Veff and Ieff are the effective voltage and effective current -- also known as RMS voltage and current -- respectively. The phase angle, θvi, is the phase angle difference between the voltage and the current (i.e., phase angle of the voltage relative to the current)
\(
(4) \ \Theta_{vi} \, = \, \left( \Theta_v \, - \, \Theta_i \right)
\)
Like any complex quantity, S can be written in polar form:
\(
(5) \ \overline{S} \, = \, \left| \overline{S} \right| \angle{\Theta} \, = \, S \angle{\Theta}
\)
In terms of the real and imaginary components, this is
\(
(6) \ \left| \overline{S} \right| \, = \, S \, = \, \sqrt{ P^2 \, + \, Q^2 }
\
(7) \ \angle{\Theta} \, = \, \tan^{-1} \left( \frac{Q}{P} \right)
\)
The complex (i.e., phasor) representation for the current and the voltage are
\(
(8) \ \overline{V}_{eff} \, = \, V_{eff }\, \angle{\Theta_v}
(9) \ \overline{I}_{eff} \, = \, I_{eff} \, \angle{\Theta_i}
\)
If we multiply these two quantities together, the magnitude would be the product of Veff and Ieff, which we want, but we would obtain the sum of the phase angles, whereas we need the difference. This can be easily remedied by noting that the conjugate of a complex quantity has the same magnitude but the additive inverse (i.e., negative) of the phase. Thus
\(
(10) \ \overline{I^*}_{eff} \, = \, I_{eff} \, \angle{-\Theta_i}
\)
Therefore
\(
(11) \ \overline{S} \, = \, \overline{V}_{eff}\, \overline{I^*}_{eff}
\
(12) \ \left| \overline{S} \right| \, = \, S \, = \, V_{eff} \cdot I_{eff}
\
(13) \ \arg \left( \overline{S} \right) \, = \, \Theta_{v} \,- \, \Theta_{i} \, = \, \Theta_{vi}
\)
You can substitute (2) and (3) into (6) to obtain (12) and you can substitute (2) and (3) into (7) and, by way of (4), obtain (13).
The magnitude of the complex power, S, given in (12) is known as the apparent power. It is simply the product of the effective voltage and the effective current without regard to the impact of the phase angle between them, and thus says nothing about how power is split between real power and reactive power.
At this point we can write the real and reactive power in terms of the complex power in exponential (or polar) form
\(
(14) \ P \, = \, Re \left{ \overline{S} \right} \, = \, \left| \overline{S} \right| \, \cos(\Theta_{vi}) \, = \, S \cdot \cos(\Theta_{vi})
\
(15) \ Q \, = \, Im \left{ \overline{S} \right} \, = \, \left| \overline{S} \right| \, \sin(\Theta_{vi}) \, = \, S \cdot \sin(\Theta_{vi})
\)
We now define the power factor, pf, to be
\(
(16) \ pf \, = \, \cos(\Theta_{vi})
\)
Using the trig identity
\(
(17) \sin^2(\alpha) \, + \, \cos^2(\alpha) \, = \, 1
\)
We can express the reactive factor, rf, as
\(
(18) \ rf \, = \, \sin(\Theta_{vi}) \, = \, \sqrt{1 \, - \, {pf}^2}
\)
We can then use the power factor (and generally do not use the reactive factor) to express (14) and (15) as follows
\(
(19) \ P \, = \, \left| \overline{S} \right| \, pf \, = \, S \cdot pf
\
(20) \ Q \, = \, \left| \overline{S} \right| \, rf \, = \, \left| S \right| \, \sqrt{1 \, - \, {pf}^2} \, = \, S \cdot \sqrt{1 \, - \, {pf}^2}
\)
A word of caution is in order about what complex power IS and what it is NOT.
Complex power IS a mathematical construct -- a gimmick, if you will -- that lets us package the concepts of real, reactive, and apparent power as well as phase angles and power factor, into one simple, easy to use quantity. That's quite an accomplishment, when you think about it; but at the end of the day it is merely a mathematical gimmick and has no intrinsic physical meaning, not even to the same degree that a voltage or current phasor do.
Which brings us to what complex power is NOT -- namely, it is NOT a phasor.
When we write voltage and current phasors, such as given by (8) and (9), it is effectively a shorthand notation (another mathematical gimmick, in some respects) for a sinusoidally varying function of time, namely
\(
(21) \ \overline{A} \, = \, \left| A \right| \angle{\Theta} \ \Leftrightarrow \ a(t) \, = \, A \cdot \cos(\omega t + \Theta )
\)
We cannot use this same notion to use the complex power to write down the instantaneous power in a circuit as a function of time, We can't do it. It's not a phasor, it is a completely different gimmick that just happens to also use complex quantities to express something.
Finally, recall that for a resistor the relationship between power and the restance can be expressed a number of ways, such as
\(
(22) \ P \, = \, V \cdot I \, = \, \frac{V^2}{R} \, = \, I^2 R
\)
Now recall how convenient it was that DC concepts for finding the voltages and currents in resistive circuits carried over into the analysis of linear circuits in steady state sinusoidal operation by just replacing resistances with complex impedances and voltage/current signals with phasors. It would be nice if this same thing worked for finding power in such circuits, but it doesn't. Fundamentally, the reason is because power is not a linear function. But the corresponding relationships are still very easy to arrive at in light of (11) and the fact that the relationships between voltage, current, and impedace ARE linear.
\(
(11) \ \overline{S} \, = \, \overline{V}_{eff}\, \overline{I^*}_{eff}
\
(23) \ \overline{V} = \overline{I} \overline{Z}
\
(24) \ \overline{I} = \frac{\overline{V}}{ \overline{Z}}
\
(25) \ \overline{I*} = \frac{\overline{V*}}{ \overline{Z*}}
\)
Noting that, for any complex quantity, we have
\(
(26) \ \overline{A}\overline{A^*} \, = \, { \left| \overline{A} \right| }^2 \, = \, A^2
\)
Thus, we can substitute (23) into (11) and get
\(
(27) \ \overline{S} \, = \, I^2_{eff}\, \overline{Z}
\)
While this does conform to our intuitive instinct that the DC power relations should carry across, when we subtitute (24) into (11) we get
\(
(28) \ \overline{S} \, = \, \frac{V^2_{eff}}{\overline{Z^*}}
\)
\(
(1) \ \overline{S} \, = \, P \, + \, jQ
\)
where P is the real power and Q is the reactive power given by
\(
(2) \ P \, = \, V_{eff} \cdot I_{eff} \, \cos(\Theta_{vi})
\
(3) \ Q \, = \, V_{eff} \cdot I_{eff} \, \sin(\Theta_{vi})
\)
where Veff and Ieff are the effective voltage and effective current -- also known as RMS voltage and current -- respectively. The phase angle, θvi, is the phase angle difference between the voltage and the current (i.e., phase angle of the voltage relative to the current)
\(
(4) \ \Theta_{vi} \, = \, \left( \Theta_v \, - \, \Theta_i \right)
\)
Like any complex quantity, S can be written in polar form:
\(
(5) \ \overline{S} \, = \, \left| \overline{S} \right| \angle{\Theta} \, = \, S \angle{\Theta}
\)
In terms of the real and imaginary components, this is
\(
(6) \ \left| \overline{S} \right| \, = \, S \, = \, \sqrt{ P^2 \, + \, Q^2 }
\
(7) \ \angle{\Theta} \, = \, \tan^{-1} \left( \frac{Q}{P} \right)
\)
The complex (i.e., phasor) representation for the current and the voltage are
\(
(8) \ \overline{V}_{eff} \, = \, V_{eff }\, \angle{\Theta_v}
(9) \ \overline{I}_{eff} \, = \, I_{eff} \, \angle{\Theta_i}
\)
If we multiply these two quantities together, the magnitude would be the product of Veff and Ieff, which we want, but we would obtain the sum of the phase angles, whereas we need the difference. This can be easily remedied by noting that the conjugate of a complex quantity has the same magnitude but the additive inverse (i.e., negative) of the phase. Thus
\(
(10) \ \overline{I^*}_{eff} \, = \, I_{eff} \, \angle{-\Theta_i}
\)
Therefore
\(
(11) \ \overline{S} \, = \, \overline{V}_{eff}\, \overline{I^*}_{eff}
\
(12) \ \left| \overline{S} \right| \, = \, S \, = \, V_{eff} \cdot I_{eff}
\
(13) \ \arg \left( \overline{S} \right) \, = \, \Theta_{v} \,- \, \Theta_{i} \, = \, \Theta_{vi}
\)
You can substitute (2) and (3) into (6) to obtain (12) and you can substitute (2) and (3) into (7) and, by way of (4), obtain (13).
The magnitude of the complex power, S, given in (12) is known as the apparent power. It is simply the product of the effective voltage and the effective current without regard to the impact of the phase angle between them, and thus says nothing about how power is split between real power and reactive power.
At this point we can write the real and reactive power in terms of the complex power in exponential (or polar) form
\(
(14) \ P \, = \, Re \left{ \overline{S} \right} \, = \, \left| \overline{S} \right| \, \cos(\Theta_{vi}) \, = \, S \cdot \cos(\Theta_{vi})
\
(15) \ Q \, = \, Im \left{ \overline{S} \right} \, = \, \left| \overline{S} \right| \, \sin(\Theta_{vi}) \, = \, S \cdot \sin(\Theta_{vi})
\)
We now define the power factor, pf, to be
\(
(16) \ pf \, = \, \cos(\Theta_{vi})
\)
Using the trig identity
\(
(17) \sin^2(\alpha) \, + \, \cos^2(\alpha) \, = \, 1
\)
We can express the reactive factor, rf, as
\(
(18) \ rf \, = \, \sin(\Theta_{vi}) \, = \, \sqrt{1 \, - \, {pf}^2}
\)
We can then use the power factor (and generally do not use the reactive factor) to express (14) and (15) as follows
\(
(19) \ P \, = \, \left| \overline{S} \right| \, pf \, = \, S \cdot pf
\
(20) \ Q \, = \, \left| \overline{S} \right| \, rf \, = \, \left| S \right| \, \sqrt{1 \, - \, {pf}^2} \, = \, S \cdot \sqrt{1 \, - \, {pf}^2}
\)
A word of caution is in order about what complex power IS and what it is NOT.
Complex power IS a mathematical construct -- a gimmick, if you will -- that lets us package the concepts of real, reactive, and apparent power as well as phase angles and power factor, into one simple, easy to use quantity. That's quite an accomplishment, when you think about it; but at the end of the day it is merely a mathematical gimmick and has no intrinsic physical meaning, not even to the same degree that a voltage or current phasor do.
Which brings us to what complex power is NOT -- namely, it is NOT a phasor.
When we write voltage and current phasors, such as given by (8) and (9), it is effectively a shorthand notation (another mathematical gimmick, in some respects) for a sinusoidally varying function of time, namely
\(
(21) \ \overline{A} \, = \, \left| A \right| \angle{\Theta} \ \Leftrightarrow \ a(t) \, = \, A \cdot \cos(\omega t + \Theta )
\)
We cannot use this same notion to use the complex power to write down the instantaneous power in a circuit as a function of time, We can't do it. It's not a phasor, it is a completely different gimmick that just happens to also use complex quantities to express something.
Finally, recall that for a resistor the relationship between power and the restance can be expressed a number of ways, such as
\(
(22) \ P \, = \, V \cdot I \, = \, \frac{V^2}{R} \, = \, I^2 R
\)
Now recall how convenient it was that DC concepts for finding the voltages and currents in resistive circuits carried over into the analysis of linear circuits in steady state sinusoidal operation by just replacing resistances with complex impedances and voltage/current signals with phasors. It would be nice if this same thing worked for finding power in such circuits, but it doesn't. Fundamentally, the reason is because power is not a linear function. But the corresponding relationships are still very easy to arrive at in light of (11) and the fact that the relationships between voltage, current, and impedace ARE linear.
\(
(11) \ \overline{S} \, = \, \overline{V}_{eff}\, \overline{I^*}_{eff}
\
(23) \ \overline{V} = \overline{I} \overline{Z}
\
(24) \ \overline{I} = \frac{\overline{V}}{ \overline{Z}}
\
(25) \ \overline{I*} = \frac{\overline{V*}}{ \overline{Z*}}
\)
Noting that, for any complex quantity, we have
\(
(26) \ \overline{A}\overline{A^*} \, = \, { \left| \overline{A} \right| }^2 \, = \, A^2
\)
Thus, we can substitute (23) into (11) and get
\(
(27) \ \overline{S} \, = \, I^2_{eff}\, \overline{Z}
\)
While this does conform to our intuitive instinct that the DC power relations should carry across, when we subtitute (24) into (11) we get
\(
(28) \ \overline{S} \, = \, \frac{V^2_{eff}}{\overline{Z^*}}
\)
