Chapter 14 – Basic Elements and Phasors

Lecture 17

by Moeen Ghiyas

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Chapter 14 – Basic Elements and Phasors

Average Power & Power Factor

Complex Numbers

Math Operations with Complex Numbers

We know for any load

v = Vm sin(ωt + θv)

i = Im sin(ωt + θi)

Then the power is defined by

Using the trigonometric identity

Thus, sine function becomes

Putting above values in

We have

The average value of 2nd term is zero over one cycle, producing

no net transfer of energy in any one direction.

The first term is constant (not time dependent) is referred to as

the average power or power delivered or dissipated by the load.

Since cos(–α) = cos α,

the magnitude of average power delivered is independent of whether v

leads i or i leads v.

Ths, defining θ as equal to | θv – θi |, where | | indicates that only the

magnitude is important and the sign is immaterial, we have average

power or power delivered or dissipated as

The above eq for average power can also be written as

But we know Vrms and Irms values as

Thus average power in terms of vrms and irms becomes,

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For resistive load,

We know v and i are in phase, then |θv - θi| = θ = 0°,

And cos 0° = 1, so that

becomes

or

For inductive load ( or network),

We know v leads i, then |θv - θi| = θ = 90°,

And cos 90° = 0, so that

Becomes

Thus, the average power or power dissipated by the ideal

inductor (no associated resistance) is zero watts.

For capacitive load ( or network),

We know v lags i, then |θv - θi| = |–θ| = 90°,

And cos 90° = 0, so that

Becomes

Thus, the average power or power dissipated by the ideal

capacitor is also zero watts.

Power Factor

In the equation,

the factor that has significant control over the delivered power

level is cos θ.

No matter how large the voltage or current, if cos θ = 0, the

power is zero; if cos θ = 1, the power delivered is a maximum.

Since it has such control, the expression was given the name

power factor and is defined by

For situations where the load is a combination of resistive and

reactive elements, the power factor will vary between 0 and 1

In terms of the average power, we know power factor is

The terms leading and lagging are often written in conjunction

with power factor and defined by the current through load.

If the current leads voltage across a load, the load has a leading

power factor. If the current lags voltage across the load, the load

has a lagging power factor.

In other words, capacitive networks have leading power factors,

and inductive networks have lagging power factors.

EXAMPLE - Determine the average power delivered to network

having the following input voltage and current:

v = 150 sin(ωt – 70°) and i = 3 sin(ωt – 50°)

Solution

EXAMPLE - Determine the power factors of the following loads,

and indicate whether they are leading or lagging:

Solution:

EXAMPLE - Determine the power factors of the following loads,

and indicate whether they are leading or lagging:

Solution:

Application of complex numbers result in a technique for finding

the algebraic sum of sinusoidal waveforms

A complex number represents a point in a two-dimensional

plane located with reference to two distinct axes.

The horizontal axis is called the real axis, while the vertical axis

is called the imaginary axis.

The symbol j (or sometimes i) is

used to denote the imaginary

component.

Two forms are used to represent a

complex number:

rectangular and polar.

Rectangular Form

Polar Form

Polar Form

θ is always measured counter-clockwise

(CCW) from the positive real axis.

Angles measured in the clockwise direction

from the positive real axis must have a

negative sign

EXAMPLE - Sketch the following complex numbers in the

complex plane:

a. C = 3 + j 4 b. C = 0 - j 6

EXAMPLE - Sketch the following complex numbers in the

complex plane:

c. C = -10 - j20

EXAMPLE - Sketch the following complex numbers in the

complex plane:

EXAMPLE - Sketch the following complex numbers in the

complex plane:

Rectangular to Polar

Polar to Rectangular

Angle determined to be

associated carefully with the

magnitude of the vector as per

the quadrant in which complex

number lies

EXAMPLE - Convert the following from polar to rectangular form:

Solution:

EXAMPLE - Convert the following from rectangular to polar form:

C = - 6 + j 3

Solution:

EXAMPLE - Convert the following from polar to rectangular form:

Solution

Let us first examine the symbol j associated with imaginary

numbers. By definition,

The conjugate or complex conjugate is found

by changing sign of imaginary part in rectangular form

or by using the negative of the angle of the polar form.

Rectangular form,

Polar form,

(conjugate)

(conjugate)

Addition Example (Rectangular)

Add C1 = 3 + j 6 and C2 = -6 + j 3.

Solution

Subtraction Example (Rectangular)

Solution

Imp Note

Addition or subtraction cannot be performed in polar

form unless the complex numbers have the same

angle θ or unless they differ only by multiples of 180°.

Addition Example (Polar)

Subtraction Example (Polar)

Average Power & Power Factor

Complex Numbers

Math Operations with Complex Numbers

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