Circuit Theory/Complex Power

Complex Power

Just like the other values of voltage, current, and resistance, power also has a complex phasor quantity that we are going to become familiar with. Complex power is denoted with a $\mathbb{S}$ symbol. It is calculated as such:

[Complex Power]

$\mathbb{S} = \frac{\mathbb{V}\mathbb{I}^*}{2}$

Where the quantity $\mathbb{I}^*$ denotes the complex conjugate of the phasor current. To get the complex conjugate of $\mathbb{I}$, we have two formulas:

Given: $\mathbb{I} = A + jB = M \angle \phi$

• $\mathbb{I}^* = A - jB$ (rectangular)
• $\mathbb{I}^* = M \angle -\phi$ (polar)

There is more information on complex conjugation of phasors in the Appendix.

Apparent Power

If we take the magnitude of our Complex power variable, we get the following:

[Apparent Power]

$|\mathbb{S}| = \frac{|\mathbb{V}\mathbb{I}^*|}{2}$

Where $|\mathbb{S}|$ is called the apparent power. It is this quantity that we can measure.

Average and Reactive Power

Let us break up our voltage and current phasors for a moment:

$\mathbb{V} = M_v\angle\phi_v\quad$, and $\quad\mathbb{I} = M_i\angle\phi_i$

if we plug those two values into our equation for complex power, above, we get the following:

$\mathbb{S} = \frac{(M_v\angle\phi_v)(M_i\angle - \phi_i)}{2} = \frac{M_vM_i}{2}\angle(\phi_v - \phi_i)$

We can then convert this quantity into rectangular form where:

$\mathbb{S} = P + jQ$

[Average Power]

$P = \frac{M_vM_i}{2}\cos(\phi_v - \phi_i)$

[Reactive Power]

$Q = \frac{M_vM_i}{2}\sin(\phi_v - \phi_i)$

We call P the Average Power and Q the Reactive Power. We will discuss these quantities later.

Units

Unlike impedance, power quantities do not all share the same units. We list the units for each type of power, below:

Time-Domain Power 

Watts (W)

Average Power

Watts (W)

Complex Power 

Volt-Amps (VA)

Reactive Power

Volt-Amps Reactive (VAR)

Technically, all these units are equatable, but they are named differently to distinguish their use in different contexts.

Power and Impedance

Complex power can be expressed in terms of impedance and complex current, using the following formula:

$\mathbb{S} = \left(\frac{I_m^2}{2}\right)\operatorname{Re}(\mathbb{Z}) + j\left(\frac{I_m^2}{2}\right)\operatorname{Im}(\mathbb{Z})$

If the element in question is a resistor, the reactive power delivered will be 0. Likewise, if the element is a capacitor or an inductor, the average power delivered will be zero. If the impedance is complex, then the delivered power will be complex.

Conservation of Power

Power in a circuit is conserved. Therefore, the following equation holds true:

[Conservation of Power]

$\sum_{circuit} \frac{\mathbb{V}\mathbb{I}^*}{2} = 0$

Remember that sources supply power, and that impedance elements (resistors, capacitors and inductors) absorb power.

Power Factor

The relationship between the average power, and the apparent power is called the power factor. Power factor is given the variable p_f, and is calculated as such:

[Power Factor]

p_f = cos(ϕ_v − ϕ_i)

There is also a quantity called the power-factor angle, which is equal to the differences in phase angle between the current and the voltage:

p_fangle = ϕ_v − ϕ_i

Since the cosine is an even function, the following values are equal:

cos(ϕ_v − ϕ_i) = cos(ϕ_i − ϕ_v)

This means that to be able to accurately calculate the phase angles of the current and the voltage from the power factor, we need an additional specifier of either leading or lagging.

Lagging 

The phase angle of the voltage is greater than the phase angle for the current.

ϕ_V > ϕ_I

Leading 

The phase angle of the current is greater than the phase angle for the voltage.

ϕ_V < ϕ_I

a visual depiction of the relationship between P, Q, and the angle φ

Maximum Transfer Theorem

Similarly to DC power, AC power has its own maximum power transfer theorem that can be expressed in terms of phasors.

Given a Thevenin equivalent source with an impedance $\mathbb{Z}_{Thevenin}$, maximum power transfer is attained when the load impedance is:

$\mathbb{Z}_{thevenin} = \mathbb{Z}_{load}^*$

In plain English, the source impedance must be the complex conjugate of the load impedance to attain maximum power transfer.