Electrical Theory: The Power Factor

REVIEW

AC electrical power consists of three types of power; apparent power, reactive power and true power. Apparent power, reactive power and true power form a right triangle. See Figure One.

True Power

True power is power dissipated by resistors.

TP = V * I * cos a

where

TP = true power
V = voltage
I = current
The angle “a” is the angle between the apparent power and the true power.

Reactive Power

Reactive power is power dissipated by inductors and capacitors.

RP = V * I * sine a

where

RP = reactive power

Apparent Power

Apparent power is measured in volt-amperes.

AP = V * I

where

AP = apparent power

According to the Pythagorean theorem

AP² = RP² + TP²

THE POWER FACTOR

The power factor is the cosine of the angle “a”. Another words, the power factor is equal to the true power divided by the apparent power.

cos a = true power/apparent power

Determining power factor in simple electrical circuits.

When the reactive power is zero, the apparent power equals the true power and the power factor is the cosine of zero degrees

cos 0^o = 1

Therefore

PF = 1

where
PF = power factor

When the true power is zero, the power factor is the cosine of 90 degrees.

cos 90^o = 0

Therefore
PF = 0

The power you pay for in your home is the apparent power. This means that you are paying for the reactive power of any inductors or capacitors in appliances in your home.

POWER FACTOR CORRECTION

Power factor correction involves the elimination of reactive power via creating a condition where the inductive reactance is equal to the capacitive reactance. The formula for the total impedance is:

Z = R + j(XL – Xc)

Where
Z = impedance
R = resistance
XL = inductive reactance
Xc = capacitive reactance

For a capacitor, voltage lags current by 90 degrees

Pc/-90^o = V/-90^o * I/0^o

Where
Pc = power dissipated due to capacitance

For an inductor, voltage leads current by 90 degrees

Pi/90^o = V/90^o * I/0^o

Where
Pi = power dissipated by inductance

The total reactive power RP is equal to the algebraic sum of the reactive power due to inductance and the reactive power due to capacitance

Rp = Pi + Pc

Rp = Pi/90^o + Pc/-90^o

If the value of Pi equals the value of Pc then the total reactive power RP is 0 watts. To understand why, see figure two. The two vectors Pi and Pc are in opposite directions and therefore the total reactive power is the algebraic difference between Pi and Pc.

If the inductive reactance XL equals the capacitive reactance Xc, then the total reactance is zero ohms. Hence the reactive power is zero watts. This means that the angle a is zero degrees. The power factor is the cosine of zero degrees.

cos a = cos 0^o = 1

EXAMPLE

The electrical circuit shown in figure three has the following components:

Capacitor 1 * 10^-2 farads

Inductor 1 henry

Resistor 100 ohms

The power source is 100 volts AC

The frequency is 1000 cycles per second

What is the power factor?

First we calculate the capacitive reactance

Xc = 1/(w*c)

where w = 2 * pi * freq
pi = 3.14
freq = frequency
c = capacitance

Xc = 1 divided by (2 * 3.14 * 10^-2)

Xc = 1 divided by (6.28 * 10^-2)

Xc = 10²/6.28 = 15.92 ohms

Then we calculate the inductive reactance

XL = w*L

where
L = inductance

XL = 6.28 * 1 = 6.28 ohms

Then we calculate the total reactance

X = XL – Xc = 6.28 – 15.92 = -9.64

where
X = total reactance

The negative sign indicates that the reactive component of the total impedance is capacitive in nature.

Hence X = 9.64 ohms

Then we calculate the impedance

Z = R – jX = 100 – j9.64

According to the Pythagorean Theorem for triangles

Z² = R² + X²

Z² = 100² + 9.64²

Z² = 10000 + 92.93 = 10092.93

Z = 100.46 ohms

The tangent of the angle is equal to the reactance divided by the resistance.

arc tan 9.64/100 = .0964

arc tan .0964 = 5.5 degrees

So the total impedance is

100.46/5.5^o

Then we calculate the current

I = V/Z = 100/(100.46/5.5^o)

I = 0.995/-5.5^o

Now we can calculate the apparent power

AP = I * V

AP = 0.995/-5.5^o * 100 = 995/-5.5^o

Now we can calculate the power factor

PF = cos 5.5^o = 0.995

References:
I have a Bachelor of Science in Electrical Engineering

Introductory Circuit Analysis Third Edition
ISBN 0-675-8559-4