﻿ Electrical Power Equations- Watts, Power Factor, THD

# A PRIMER ON WORK AND AC POWER IN ELECTRICAL CIRCUIT

## DEFINITIONS AND MATH EQUATIONS FOR WATT, VA, POWER FACTOR AND THD

Power in general is defined as the rate of energy change with time. Energy is referred to as the capacity for doing physical work. Work is done when an object is moved by a force. Numerically work done by any force when an object moves from a point A to a point B is specified as linear (or path) integral of the force component in the direction of motion times distance moved:
 (1)
Note that in general is a scalar product of two vectors and , which is equal to
, where φ is an angle between these vectors.

How do we get from here the familiar expression for electric power: p=V×i ?

Let's start with electric force. If a point charge Q is placed in electric field it will experience a force, which is proportional to the amount of Q. This force per unit charge is called electric field:
 (2)
We presume here that Q has a negligible size and does not distort the field.

Combining (1) and (2) yields the expression for work in electric field to move a charge Q from a point A to a point B:
 (3)

From (3) the power required to move charge Q in electric field from a point A to a point B is:
 , (4)
where is the rate of charge flow past a given area called electric current.

The linear integral in (4) is called voltage (V) or potential difference between points A and B:
 (5)
Substituting (5) into (4) yields an expression for the instantaneous electric power:

 P=V×i, (6)
where V and i are instantaneous values of voltage and current.

Note that generally, any linear integral is a function of the path from A to B. In electrostatic fields however, the integral (5) does not depend on the path and is the function only of the electric field and coordinates of points A and B. Likewise, work in electrostatic field does not depend on the path. Particularly, the work to move a charge around any closed loop (i.e. when A=B) is zero. Fields in which work does not depend on path are called conservative (or potential) fields.

Electrostatic fields are generated by electric charges that remain in rest or are moving with a constant speed. Therefore, such fields present mainly academic interest since no electronic circuit would work without accelerating the charges. Only changing fields are usable in electrical systems. In such fields integral (5) along a closed loop is no longer zero due to the presence of variable magnetic field:
 , (7)
where Ψ- magnetic flux through the closed loop, - magnetic flux density through the loop, Ac – area of the closed loop.
As the result, the linear integral in (5) generally depends on the pass from a point A to a point B. Therefore the voltage between any two points A and B is no longer defined. Strictly speaking, in variable electric fields we may only talk about voltage between points A and B along a given path. For different paths between the same points A and B the voltage (5) may be different. If magnetic flux is the same through entire surface Ac formed by the loop, then (7) can be simplified:
 , (8)
where Ac is the area of the loop.

We see from (8) that in order to minimize the effects of changing magnetic field we need to reduce the rate of change of magnetic field dB/dt and/or the area Ac of the loop. If their product is small enough we may neglect it and consider the electric field quasi-potential with the familiar definition of voltage.

In AC circuits all quantities in (4), (5) and (6) are continuously varying and are functions of time t. The average value of the power over certain period of time T is given by
 , (9)
Average power (9) is called active or real power, or simply watts.

The AC values are often stated as root-mean-square (RMS). The RMS value of any variable X(t) is generally defined by
 (10)
The product of RMS voltage and current Vrms×Irms is called apparent power (or volt-amps).

The ratio between watts and apparent power is called power factor. It shows how well the electricity generator is utilized and for example how much real power you can get from your home's wall outlet:
 (11)
Note that any periodic non-sinusoidal current can be presented by Fourier transform:
 (12)
For a pure sinusoidal voltage V(t), substituting (12) into (9) gives active power as:
 (13)
It can be shown that for n≥2: , that is with a pure sinewave voltage source active power is supplied only by the first (fundamental) harmonic of the current:
 (14)
Similarly, it can be shown that in a general case of a non-sinusoidal voltage, net energy is transmitted to the load only by the harmonics of voltage and current that have the same frequency.

For a sinusoidal voltage we derive from (14):
 , (15)
where Vpk and I1pk- peak (maximum) values of the voltage and fundamental harmonic of the current respectively, ω - angular frequency (in radian/sec), φ - is the phase angle (in radians) between the fundamental harmonic of the current and the voltage.

It can be derived from (10) that for any pure sine wave signal X(t): .
Then the expression for active power (15) can be rewritten as:
 , (16)
where Vrms- RMS value of the voltage, I1rms – RMS value of fundamental harmonic of the current.

By comparing (11) and (16) we derive the power factor equation for a sinusoidal voltage:
 , (17)
where φ - is the phase angle between the 1st harmonic of the current and the voltage.

In practice, the phase shift between voltage and current is caused by capacitors and inductances, while higher current harmonics are caused by non-linear components like rectifiers. The ratio between apparent power associated with higher order harmonics and apparent power associated with fundamental harmonic is called Total Harmonic Distortion (THD). For sinusoidal input voltage:
 , (18)
where Inrms- RMS value of the n-th harmonic of the current.

For a periodic current from (12):
 ,
If Io=0 (which is usually the case in AC lines unless you use an input single-ended rectifier), then from (18) and (19) THD can be expressed as:
 (20)
Finally, by combining expressions (17) and (20) we can also derive the relationship between PF and THD:
 (21)
Note that in electronic circuits, both PF and THD can be improved by using an SMPS with power factor correction.

If you wonder how to place right-align equation numbers in a document, check my post on numbering equations with chapter numbers.

<------------------------------------------------------------------------------------------------------------------------------------------------------->