lecture 12 (ac power) - overhead

University of British Columbia Elec Machines & Electronics

EECE 365 – Winter 2012 Lecture 12: AC Power Systems

Nathan Ozog © 2013 of 12

AC Power Systems

Power Grid




Basic Components

Generators

In BC, Mainly Hydro is most common elsewhere

Transformers Step-up

Step-down Transmission

Lines Cables

Transport power over long distances Switches

Circuit Breakers – To clear faults Disconnects – To isolate equipment for safety

Customers Distribution (Loads)

Industrial Commercial

Residential

Generated energy must equal consumed energy

Transmission & Distribution of Levels Transmission ~ 138 to 765 kV

Sub-transmission ~ 34 to 138 kV Distribution ~ 4 to 34 kV

Service / Customer level ~ 240, 208, 120 V




BC Power Grid Long and radial

Long high voltage transmission lines required




BC Power Grid

Long and radial Long high voltage transmission lines required

5L11

5L12

5L13

5L29

5L31

5L51

5L52

5L71

5L72

5L75

5L77

5L1

5L2

2L112

2L277 / 71L

Alberta Electric

System Operator

(AESO)

5L91

2L113

2L293




WECC We are part of the Larger “Western Electricity Coordinating Council”

Jurisdiction




AC Phasors Consider an inductive circuit

Using Kirchoff’s Voltage law:

Steady State Solution

( )

( )+=

+=

tEte

tIti

m

m

ω

ω

cos)(

cos)(

( )tVtv m ωcos)( =

dt

diLri

rieritv

+=

+=+=)(




AC Phasors

Consider a capacitive circuit

Using Kirchoff’s Voltage law:

In both cases only magnitude and phase are required

stays constant

Steady State Solution

( )+= tIti m ωcos)(

( )+= tVtv mcc ωcos)( ,

( )tVtv m ωcos)( =

cvritv +=)(

dt

dvC

ri c== ( )tv

rvrdt

dvC c

c 11=+⇒




AC Phasors Review of the complex plane

Note All vectors rotate at the same speed ω !

Only the amplitudes and their phase differences are important

Phasor: A complex number that represents the amplitude and phase of a

sinusoid

Time Domain

( )θω ±tC cos

Phasor Representation

( ) ( )tVjtVtv mm ωω sincos)( ⋅+=

( ) ( )φωφω −⋅+−= tIjtIti mm sincos)(

)()(

)(

φω

ω

−=

=

tjm

tjm

eIti

eVtv

Euler’s Identity ( ) ( )tjte tj ωωω sincos +=

φφω −∠≡= −m

tjm IeIti )()(

0)( ∠≡= mtj

m VeVtv ω

( )θω ±tC sin




AC Phasors

Can be described in:

Polar: φ∠mC

Rectangular: jBA+

Conversion:

Rectangular � polar

=mC =φ

Polar � rectangular

=

=

B

A

Addition of Phasors Use rectangular notation

[ ] [ ]=+++=+ 2211 jBAjBAjBA tt

So 21

21

BBB

AAA

t

t

+=

+=

Multiplication of Phasors

[ ] [ ]=∠⋅∠=∠ 2211 φφφ CCC tt

Division of Phasors

∠=∠∠

=∠22

11

φφ

φC

CC tt




Complex Impedances

C

L

Linear Passive Elements

RRZ =

Complex Impedance

( ) oo 9090 ∠=∠== LXLLjZ ωω

oo 901

9011

−∠

=−∠

==

cXCCjZ

ωω

zmim

vm ZI

V

I

VZ θ

θθ

∠==∠∠

==




2

2

0

2

0

2

0

2

0

1

)(11

)(11

)(1

)(1

rms

V

T

T

T

T

ave

vr

dttvTr

dttvTr

dtr

tv

T

dttPT

P

rms

=

=

⋅=

=

=

∫

∫

∫

∫

4434421

2

2

0

2

0

2

0

2

0

)(1

)(1

)(1

)(1

rms

i

T

T

T

T

ave

ri

dttiT

r

dttiT

r

rdttiT

dttPT

P

rms

=

=

⋅=

=

=

∫

∫

∫

∫

4434421

Root Mean Square (RMS) Quantities

Value equivalent to the DC voltage (current) that when applied to a resistor dissipates the same average amount of power as the given AC

Consider AC Power across a resistor r

For different types of waveforms

AC Quantities usually described in RMS

( )∫=T

rms dttvT

V0

2)(

1 ( )∫=T

rms dttiT

I0

2)(

1

( )vrms tVtv ϕω +⋅= cos2)(

( )irms tIti ϕω +⋅= cos2)(

vrmsVV ϕ∠=~

vrmsII ϕ∠=~




( )

( )

( ) ( ) ( )

=

+==

+⋅=

⋅=

ϕωω

ϕω

ω

ttVIvitp

tIti

tVtv

rms

rms

coscos2

cos2)(

cos2)(

AC Power Equations

If we consider a partially resistive and partially reactive load:

Average (real) power over one cycle

Apparent power

Reactive power

Power Factor (pf)

Reactive Power is the power that is “ ” to the source

Power Factor the ratio of power to power High power factor is preferable

( )S

P

VI

P=== ϕcosPF

( )ϕsinVIQ =

22 QPVIS +== [ ] [ ] [ ]MVAkVAVA ,,

[ ] [ ] [ ]MVARkVARVAR ,,

( )ϕcosVIP = [ ] [ ] [ ]MWkWW ,,

lecture 12 (ac power) - overhead

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