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Complex Algebra and Load Power

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Complex Algebra and Load Power

EE341

Ali Keyhani

Circuit Theory

Lecture #3

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

A complex number can also be written in phasor form:

(2)

(3)

(4)

where

- Modulus (or norm)

- Argument (or phase)

b

a

0

r

Conversion between two forms:

i

(5)

(6)

(8)

(7)

Addition / Subtraction:

(9)

Multiplication:

(10)

(11)

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

Division:

(12)

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

division are a little bit complicated.

Multiplication:

(13)

Division:

(14)

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

For DC circuit:

(15)

For single-phase AC circuit:

(16)

Let

Impedance angle

So

(17)

Since

We have

(18)

(19)

For three-phase AC circuit:

(20)

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 is defined as:

(26)

Lagging: Q > 0, > 0, inductive

Leading: Q < 0, < 0, capacitive

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

According to eqn (24),

According to eqn (25),

Q > 0 because p.f. is lagging.

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

According to eqn (24),

According to eqn (25),

Q > 0 because p.f. is lagging.

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

According to eqn (24),

According to eqn (25),

Q < 0 because p.f. is leading.