Complex algebra and load power
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Complex Algebra and Load Power. EE341 Ali Keyhani Circuit Theory Lecture #3. Complex Numbers. i. b. b. . . a. a. 0. 0. r. r. A complex number is a quantity of the form of. (1). Where a and b are real numbers, and. a : real part b : imaginary part.

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Complex Algebra and Load Power

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Complex algebra and load power

Complex Algebra and Load Power

EE341

Ali Keyhani

Circuit Theory

Lecture #3


Complex numbers

Complex Numbers

i

b

b

a

a

0

0

r

r

A complex number is a quantity of the form of

(1)

Where a and b are real numbers, and

a: real part

b: imaginary part

is called the conjugate of z


Complex numbers1

Complex Numbers

A complex number can also be written in phasor form:

(2)

(3)

(4)

where

- Modulus (or norm)

- Argument (or phase)


Complex numbers2

Complex Numbers

b

a

0

r

Conversion between two forms:

i

(5)

(6)

(8)

(7)


Complex numbers operation

Complex Numbers Operation

Addition / Subtraction:

(9)

Multiplication:

(10)

(11)

A complex number times its conjugate  the square of its modulus.


Complex numbers operation1

Complex Numbers Operation

Division:

(12)

Addition and subtraction can be easily done in regular form. While multiplication and

division are a little bit complicated.


Complex numbers operation2

Complex Numbers Operation

Multiplication:

(13)

Division:

(14)

Multiplication and division are much easier to be done in phasor form.


Electric power

Electric Power

For DC circuit:

(15)

For single-phase AC circuit:

(16)

Let

Impedance angle

So

(17)


Electric power1

Electric Power

Since

We have

(18)

(19)

For three-phase AC circuit:

(20)


Electric power2

Electric Power

For AC circuit:

(21)

Complex power:

VA, kVA, MVA

Active power:

W, kW, MW

Reactive power:

Var, kVar, MVar

(22)

(24)

(23)

(25)


Power factor

Power Factor

Power factor is defined as:

(26)

Lagging: Q > 0,  > 0, inductive

Leading: Q < 0,  < 0, capacitive


Electric load example 1

Electric Load Example 1

Consider a three-phase 480V, 300kVA load with p.f. = 0.9 lagging, what is the active, reactive, and complex power of the load?

Solution:

Known:

To compute: P, Q, from S

lagging


Electric load example 11

Electric Load Example 1

According to eqn (24),

According to eqn (25),

Q > 0 because p.f. is lagging.


Electric load example 2

Electric Load Example 2

Consider a three-phase 480V,240kW load with pf = 0.8 lagging, what is the active, reactive, and complex power of the load?

Solution:

Known:

To be compute:Q, S from P

lagging


Electric load example 21

Electric Load Example 2

According to eqn (24),

According to eqn (25),

Q > 0 because p.f. is lagging.


Electric load example 3

Electric Load Example 3

Consider a three-phase 480V,180kVA load with pf = 0.0 leading, what is the active, reactive, and complex power of the load?

Solution:

Known:

Compute be:P, Q, from S

leading


Electric load example 31

Electric Load Example 3

According to eqn (24),

According to eqn (25),

Q < 0 because p.f. is leading.


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