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Complex Algebra and Load Power

EE341Ali KeyhaniCircuit TheoryLecture #3

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Complex Numbers

biaz

A complex number is a quantity of the form of

Where a and b are real numbers, and

1ia: real partb: imaginary part

ii

rr00aa

bb

biaz

biaz

rr00aa

bb

biaz

biaz

z

biaz is called the conjugate of z

(1)

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Complex Numbers

)sin(cos izz iezz

zz

A complex number can also be written in phasor form:

z

- Modulus (or norm)

- Argument (or phase)

where

(2)

(3)

(4)

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Complex Numbersii

zz22 baz

ab1tan

cosza sinzb

rr00aa

bb

biaz

biaz

z

Conversion between two forms:

biaz

zz biaz

(5)

(6)

(7) (8)

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Complex Numbers Operation biaz 1 dicz 2

idbcazz )()(21

ibcadbdaczz )()(21 2

122

11 ))(( zbabiabiazz

Addition / Subtraction:

Multiplication:

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

(9)

(10)

(11)

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Complex Numbers Operation biaz 1 dicz 2

idcadbc

dcbdac

dicdicdicbia

dicbia

zz

2222

2

1

))(())((

)()(

Division:

Addition and subtraction can be easily done in regular form. While multiplication and division are a little bit complicated.

(12)

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Complex Numbers Operation 1

11iezz

)(2121

21 iezzzz

)(

2

1

2

1 21 iezz

zz

222

iezz

)sin(cos 111)(

111 ninzezz nninn

Multiplication:

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

Division:

(13)

(14)

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Electric Power

RIRVIVP 22

For DC circuit:

)(*IVIVIVS

For single-phase AC circuit:

Let IV Impedance angleImpedance angleSo

jQPjIVS )sin(cos

(15)

(16)

(17)

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Electric Power

jQPIVIVS LL ** 33

For three-phase AC circuit:

ZIV

ZIIZIIVS 2**

Since

We have

*

2**

ZV

ZVVIVS (19)

(18)

(20)

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Electric PowerFor AC circuit:

jQPjSS )sin(cos

22 QPS

PQ1tan

cosSP

sinSQ

Complex power:Complex power:

VA, kVA, MVAVA, kVA, MVA

Active power:Active power:

W, kW, MWW, kW, MW

Reactive power:Reactive power:

Var, kVar, MVarVar, kVar, MVar

(21)

(22)

(23)

(24)

(25)

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Power FactorjQPjSS )sin(cos

SPfp cos..

Lagging: Lagging: Q Q > 0, > 0, > 0, inductive> 0, inductive

Leading: Leading: Q Q < 0, < 0, < 0, capacitive< 0, capacitive

Power factor is defined as:

(26)

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Electric Load Example 1Consider 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

kVAS 300

9.0cos.. fp lagging

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Electric Load Example 1According to eqn (24),

kWSP 2709.0300cos

kVarSQ 77.1304359.0300sin

kVAjS )9.0(cos30077.130270 1

According to eqn (25),

Q > 0 because p.f. is lagging.

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Electric Load Example 2Consider 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

kWP 2408.0cos.. fp lagging

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Electric Load Example 2

kVAPS 3008.0/240cos/

kVarSQ 1806.0300sin

kVAjS )8.0(cos300180270 1

According to eqn (24),

According to eqn (25),

Q > 0 because p.f. is lagging.

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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

Electric Load Example 3

kVAS 1800.0cos.. fp leading

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Electric Load Example 3

00.0180cos SP

kVarSQ 180)0.1(300sin

kVAjS o901801800

According to eqn (24),

According to eqn (25),

Q < 0 because p.f. is leading.