3-1 Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts Chapter 3 Review of Basic Electrical and Magnetic.

Electrónica de Potência© 2008 José Bastos

Chapter 3 Basic Electrical and Magnetic Circuit Concepts

3-1

Chapter 3

Review of Basic Electrical and Magnetic Circuit Concepts

• Electric Circuits• Phasors• Power, Power Factor• Fourier Analysis• Inductors and Capacitors• Magnetic Circuits• Transformers



3-2

AVERAGE POWER AND RMS CURRENT

T

RMS

RMS

T

av

T

av

dtiT

I

IRdtiT

RP

iRv

dtivT

P

ivtp

0

2

2

0

2

0

1

1

, aresistênci uma apenas seja carga a caso

1

)(



3-3

Sinusoidal Steady State



3-4

Phasor Representation

0)(

fasorial notação

)0sin()0cos()(

lexponencia notação a utilizando

)0cos()(

0

)0(

VeVtv

tVjtVeVtv

tVtv

j

tj

IZ

V

ZeZ

eLRLjRj

RLarctgj

0

complexa carga uma de caso no

)/(22

Z

VI

Z

Z

R

jL

Z



3-5

Power, Reactive Power, Power Factor

II

0VVpI

qIj

Complex power S

jjj eIVeIeV 0*IVSReal power is

cosRe IVIVP p SReactive power is

22sinIm PSIVIVQ q S

Power Factor cos

cos

IV

IV

S

P

Ideally power factor should be 1.0



3-6

ExampleAn inductive load connected to a 120V, 60Hz ac source draws 1 kW at a power factor of 0.8. Calculate the capacitance required in parallel with the load in order to bring the power factor to 0.95.

)(

has one 0.95 offactor power aget order toIn

7501000

ispower complex the

750

1250

1000

load for the

22

CL

CL

QQjP

QjQjPS

j

QjPS

VAPSQ

VAPF

PS

WP



3-7

Example (end)

FV

QC

CVC

V

Z

VQ

VAQ

VAPP

QQ

PQQPS

C

CC

C

CL

CL

6.77602120

3.421

)/(1

is ecapacitanc ain power reactive but the

3.4217.328750

therefore

7.32895.0

)(

95.0)(

22

222

22

2

22



3-8

Non-sinusoidal waveforms in steady state



3-9

Non-sinusoidal waveforms in steady state (cont.)

-current drawn from power electronic equipment is highly distorted

-However

-In steady state waveforms repeat with period T=1/f

-f is the fundamental frequency (f1)

-The current signal has many harmonics (multiples) of the fundamental

-The harmonics can be calculated by Fourier Analysis



3-10

Fourier Analysis

-a non sinusoidal waveform f(t) repeating with angular frequency can be expressed as

)()sin()(1

)()cos()(1

)sin()cos(2

1)()(

1 1100

tdtntfb

tdtntfa

tnbtnaatfFtf

n

n

n nnn

nn



3-11

Fourier Analysis



3-12

Distortion in the Input Current

• Voltage is assumed to be sinusoidal

• Subscript “1” refers to the fundamental

• The phase 1 is between the voltage and the current fundamental



3-13

Line-Current Distortion(1)

11111

11

)sin()sin()(

)()()(

)sin()(

nnnsss

nsnss

ss

tnItIti

tititi

tVtv



3-14

Total Harmonic Distortion -THD

%100

%100100

as defined is distortion harmonic total

)()()(

zero, are rmsproduct te-cross theall of integrals thebecause

)(1

12

2

2/1

21

21

2

1

2/1

1

22/121

2

1

2/1

1

221

2/1

0

2

1

1

1

n

sn

s

ss

s

dist

nsnssdist

ssdist

nsnsRMS

T

ssRMS

sI

ITHD

I

II

I

ITHD

IIII

tititi

III

tiT

II



3-15

POWER FACTOR - PF

-starting with the definition of average power

DPFTHD

PF

DPFI

IPF

DPF

I

I

IV

IV

S

PPF

IVS

IVdttItVT

P

dttitvT

dttpT

P

s

s

s

s

ss

ss

ss

ss

T

ss

T

ss

T

2

1

1

1111

11

0

11111

0101

1

1

as expressed becan PF The

)cos(

is DPFfactor nt Displaceme The

)cos()cos(

asfactor power thedefine We

is Spower apparent the

)cos()sin()sin(1

zero are products cross allagain once

)()(1

)(1

1

11



3-16

Inductor and Capacitor Phasors (1)

2/1

11

)(

)(

)()()(

)(

)cos()(

Capacitor

2/

2/

C

eCC

j

Cjti

tvZ

etCvtCvjdt

tdvCti

eVvtVtv

jLC

jLL

L

tjLL



3-17

Inductor and Capacitor Phasors (2)

2/)(

)(

)()()(

)(

)cos()(

Inductor

2/

2/

LLeLjti

tvZ

etLitLijdt

tdiLtv

eIitIti

jLL

jL

tj



3-18

Phasor Representation



3-19

** Inductor and Capacitor Response

11

11

1

1

)()()(

Inductor

)()()(

Capacitor

ttdttvtiti

dt

diLv

ttdttitvtv

dt

dvCi

t

t

LLL

LL

t

t

CCC

CC



3-20

Response of L and C



3-21

Average vL and iC in steady state

-steady-state condition implies that voltage and current waveforms repeat with a time period T:

)()()()( tiTtiandtvTtv

zero bemust oltageinductor v average

01

inductor of casein

zero bemust current capacitor average

01

capacitor of casein

1

1

1

1

Tt

t

L

Tt

t

C

dtvT

dtiT



3-22

Inductor Voltage and Current in Steady State

• Volt-seconds over T equal zero.



3-23

Capacitor Voltage and Current in Steady State

• Amp-seconds over T equal zero.



3-24

Ampere’s Law

• Direction of magnetic field due to currents

• Ampere’s Law: Magnetic field along a path



3-25

Ampere’s Law

HB

iNlH

idlH

mmm

kkk

:Bflux magnetic ofdensity

circuits practicalmost For

:Hintensity of field magnetic



3-26

Direction of Magnetic Field



3-27

A

dAB

Flux Lines



3-28

Magnetic Flux

reluctance magnetic theis where

or

writtenbecan Equation

)(

cases practicalmost For

mmm

k kk

k

mmm

mmm

k kk

k

mmm

kkk

k k k kk

k

kk

kk

kk

kkk

k kk

kkkk

kkk

iN

A

l

iN

iNA

l

iNlH

A

l

A

l

A

lAB

A

lAHlH



3-29

Concept of Magnetic Reluctance

)/( Al

iN



3-30

Faraday’s Law

dt

dN

dt

Nde

)(

dt

d



3-31

Definition of Self-inductance L

dt

diLe

dt

Nde

iLN

writtenbecan )(

Equation



3-32

Inductance L

Al

NL

l

i

NL

2

A)/(

iN

coil toroidalafor

A l



3-33

Analysis of a Transformer (1)

dt

dNiRv

dt

dNiRv

22222

11111

Very simple case:

11

22

21

21 0

vN

Nv

RR



3-34


dt

di

Al

NN

dt

di

Al

NiRv

dt

di

Al

N

dt

di

Al

NiRv

vv

N

iNii

Al

iN

Al

iNiN

m

m

m

m

ll

l

l

)/()/(

)/()/(

writtenbecan and voltagesThe

with

)/(

)/(

core in theflux the and fluxes, leakage the, being

122

2

22

222

211

1

21

111

21

1

221

12211

21

22

11



3-35


11

22222

1

222222

11

1111

1111

21

2

2

22

21

2

1

simplified becan and voltagesThe

)/(;

)/(;

)/(

thatnoting

11

eN

N

dt

diLiR

dt

diL

N

N

dt

diLiRv

edt

diLiR

dt

diL

dt

diLiRv

vv

Al

NL

Al

NL

Al

NL

lm

ml

lm

ml

mll



3-36

Transformer Equivalent Circuit

Lm N2/N1 Lm



3-37

Ideal Transformers

1. R1=R2=0

2. Ll1=Ll2=0

3. Core permeability =

1112

11

1

222

2

1

1

2

2211

11

21

1

222

11

0)/(

iviN

Nv

N

Niv

N

N

i

i

iNiNAl

vN

Ne

N

Nev

ev

3-1 Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts Chapter 3 Review of Basic Electrical and Magnetic.

Documents

jos bastos chapter

review of basic electrical

reactive power

current fundamental

z slide

rms current slide

power electronic equipment

steady state waveforms