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Electronic Power Conversion Review of Basic Electrical and Magnetic Circuit Concepts

2 Basic Electrical and Magnetic Circuit Concepts

3. Review of Basic Electrical and Magnetic Circuit Concepts

• Notation •  Electric circuits

•  Steady state – sinusoidal, non-sinusoidal •  Power – apparent, real, reactive •  Power factor, THD, harmonics

• Magnetic circuits •  Ampere’s law, Faraday’s law •  Magnetic reluctance and magnetic circuits •  Inductor •  Transformer


Notation

• Notation: •  Lowercase – instantaneous value e.g. vo (=v0(t) ) •  Uppercase – average or rms value e.g. Vo •  Bold – phasor, vector

•  Steady state: •  State of operation where waveforms repeat with a period T that is

specific for the system.


Instantaneous, Average Power and RMS current

•  Instantaneous power

•  Average power

•  Power to resistive load

•  I – RMS (Root-mean-square) T2

0

1I = i dtT ∫

T2

av,R0

1P = R i dtT ∫

p= vi

T T

av0 0

1 1P = pdt = v i dtT T∫ ∫

2 av,RP = R Ior

1

1

2

2

3

3

4

4

5

5

6

6

D D

C C

B B

A A

Title

Number RevisionSize

B

Date: 4-11-2008 Sheet of File: Sheet1.SchDoc Drawn By:

R

i

v

+

-

R

2

av,RVP =R


•  Example

0 1 10 4−× 2 10 4−× 3 10 4−× 4 10 4−×0

10

20v t( )

i t( )

Vav

Vrms

p t( )

Pav

t

10 5 sin(2 10 )v = V V kHz tπ+ ⋅ ⋅ ⋅1010.607

1011.25

av

rms

av

V VV VRP W

==

= Ω=

0 1 10 4−× 2 10 4−× 3 10 4−× 4 10 4−×0

10

20v t( )

i t( )

Vav

Vrms

p t( )

Pav

t


Steady-State ac Sinusoidal Waveforms

1( ) cos ( )i t = 2 I t - ω φ

0 5 10 5−× 1 10 4−×3−

2−

1−

0

1

2

3

v t( )

i t( )

ip t( )

iq t( )

t

1cosv(t) = 2 V tω

= j0V eV = -jI e φI

•  Phasor representation

φ

0 5 10 5−× 1 10 4−×3−

2−

1−

0

1

2

3

v t( )

i t( )

ip t( )

iq t( )

t

= VV 0∠ °

qjI−

pI

= I -φ∠I

ω

ip (t) = 2I p cosωt

= ( 2I cosφ)cosωt

iq (t) = 2Iq sinωt

= ( 2I sinφ)sinωt

( ) ( ) ( )p qi t i t i t= +

= jR+ j L= Ze φωZComplex impedance

= ⋅V I Z

• Resistive/inductive load


Complex, Real and Reactive Power

• Complex power [VA] * = = S je φS VI

• Apparent power S V I=

• Real average power [W]

[ ] = Re =VIcosP φS• Reactive power [VAr]

[ ] = Im =VIsinQ φS

P jQ= +S

= VV 0∠ °

qjI−

pIω

= I -φ∠I

PjQ

S

• Only Ip is responsible for power transfer from source to load!


Complex, Real and Reactive Power

•  Physical meaning •  Apparent power S influences cost and size:

•  Insulation level and magnetic core size depend on V •  Conductor size depends on I

•  Real power P – useful work and losses; •  Reactive power Q – preferably zero.


Power Factor (arbeidsfactor)

•  Power factor •  How effectively is energy transferred between source and load

•  For sinusoidal systems:

PPFS

@

cos cosP V IPFS V I

φ φ⋅ ⋅= = =⋅


Power Factor – Beer Mug Analogy

• Glass (over)sized to hold beer and foam •  Power wires (over)sized for Watts and VArs

Active Power

Reactive Power

Apparent Power

Source: Wikimedia Commons CC-BY-SA


3 3 cos 3 cosph LLP VI V Iϕ ϕ= =

cosphP VI ϕ=

3LLV V=

phS VI=

3 3 3ph LLS VI V I= =

cosPF φ=

Three Phase Circuits


Steady-State Non-Sinusoidal Waveforms

•  Periodic non-sinusoidal waveform •  Represented as sum of its harmonics

0 0.05 0.1 0.15 0.210−

5−

0

5

10

i1h t( )

i3h t( )

i5h t( )

i7h t( )

t

0 0.05 0.1 0.15 0.230−

20−

10−

0

10

20

30

i t( )

t

1st 3rd

5th

7th

i t( ) 10A sin ω1 t⋅( ) 7 A⋅ sin 3ω1 t⋅ π2−⎛⎜⎝

⎞⎟⎠⋅+ 4 A⋅ sin 5ω1 t⋅( )⋅− 2 A⋅ sin 7ω1 t⋅( )+:=


Fourier Analysis of Non-Sinusoidal Waveforms

• Non-sinusoidal periodic waveform f(t)

where {0 0

1 1

1( ) ( ) cos( ) sin( )}2h h hh h

f t = F + f t a a h t b h tω ω∞ ∞

= =

= ⋅ + ⋅ + ⋅∑ ∑

2

0

1 ( ) cos( ) ( )ha f t h t d tπ

ω ωπ

= ∫2

0

1 ( )sin( ) ( )hb f t h t d tπ

ω ωπ

= ∫

1,...,h = ∞

1,...,h = ∞

2

0 00 0

1 1 1( ) ( ) ( ) ( )2 2

T

F a f t d t f t d tT

π

ωπ

= = =∫ ∫


Line Current Distortion

1( ) sins sv t = 2 V tω

1 2 1( ) ( ( ) (( )) )s shh s ds is = i i t + i t = t+ itt i

∞

=∑

1 1 12

( ) sin( ) sin( )s s sh h hh

i t = 2 I t - + 2 I t - ω φ ω φ∞

=∑

current i = fundamental harmonic is1+ distortion current idis

0 0.05 0.1 0.15 0.220−

10−

0

10

20

v t( )

i t( )

i1h t( )

idis t( )

t

0 0.05 0.1 0.15 0.220−

10−

0

10

20

v t( )

i t( )

i1h t( )

idis t( )

t


Total Harmonic Distortion (THD)

• A measure of how much a composite current deviates from an ideal sine wave

• Caused by the way that electronic loads draw current

1 2

01

T

s s1I = i (t) dt T ∫

2 21 2s s shh

I = I + I∞=∑

2 2 21 2dis s s shh

I = I I = I ∞=

− ∑

or

and:

2

2 1

sh

h s

ITHD = I

∞

=

⎛ ⎞⎜ ⎟⎝ ⎠

∑Total harmonic distortion:

1 1 12

( ) sin( ) sin( )s s sh h hh

i t = 2 I t - + 2 I t - ω φ ω φ∞

=∑


Harmonics

• Negative effects of harmonics •  Conductor overheating •  Capacitors can be affected by heat rise increases due to power loss

and reduced life time •  Distorted voltage waveform •  Increased losses (e.g. transformers overheating) •  Fuses and circuit breakers fault operation

• Harmonic standards •  International Electrotechnical Commission Standard IEC-555 •  IEEE/ANSI Standard 519


Power Factor PPFS

@1 1

0 01 1

( ) ( ) ( )T T

s s1 1P = p t dt = v t i t dtT T∫ ∫

s sS = V I

1

1 101

sin( ) sin( )T

s sh h hh

1P = 2 V t 2 I t - dtT

ω ω φ∞=

⋅∑∫

1

1 1 1 1 1 101

sin( ) sin( ) cosT

s s s s1P 2 V t 2 I t - dt V I T

ω ω φ φ= ⋅ =∫

1 11

coscoss s1

s s s

V I IPF = V I I

φ φ=

Real power (vermogen)

Apparent power (schijnbaar vermogen)

with

and

( )sv t ( )si t

DPF


Inductor and Capacitor Phasor Diagrams

•  Phasors are only applicable to sinusoidal steady state waveforms.

jωL

+ - vL

iL

VL

IL

+ - vC

-1/jωC

VC

IC ωω


Inductor and Capacitor Response

• Time domain:

11( ) ( ) ( )

t

L L Lt

1i t = i t + v t dtL ∫ 11( ) ( ) ( )

t

c c ct

1v t = v t + i t dtC ∫

( ) LLdiv t Ldt

= ⋅ ( )C

Cdvi t Cdt

= ⋅


Inductor in Steady State Volt-second Balance

11( ) ( ) ( )

t

L L Lt

1i t = i t + v t dtL ∫

1

1

( ) 0t T

Lt

v t dt+

=∫

v(t) = T)+v(t

i(t) = T)+i(t

Steady state implies:

Net change in flux is zero

, 0L av v =


Capacitor in Steady State Amp-second balance 11( ) ( ) ( )

t

c c ct

1v t = v t + i t dtC ∫

1

1

( ) 0t T

Ct

i t dt+

=∫Net change in charge is zero

, 0C avi =

Fig. 3-9 Note: error in fig 3-9 from textbook iC in the second interval should be constant, ----- because vc has a constant derivative in that interval


Basic Magnetics Theory

c

d = i ∑∫ H l—

sec

k k m mK core Mtions windings

H l N I=∑ ∑

1i g gH l H l N i+ =

Ampere’s law (4th Maxwell’s equation)

Right-hand rule:

c


H-field, B-field and Material Properties

• Relationship between B and H given by: •  material properties •  often approximated by:

• Continuity of flux

= µB H

A

dφ = ⋅∫∫B AφclosedArea = B ⋅dA

A∫∫ = 0

Flux (by def.):

Continuity of flux: Gauss’s law of magnetism (2nd Maxwell’s equation)


Magnetic Reluctance

= µB H

( ) k k km m k k k k k km k k k kk k k k k k

l l lN I H l B A A A A

φ φ φµ µ µ

= = = = = ⋅ℜ∑ ∑ ∑ ∑ ∑

Ampere’s law:

kk

k k

l = Aµ

ℜMagnetic reluctance:

Magnetic Electrical

k ℜ R

i φv Ni


Magnetic Circuit Analysis

v i R = ⋅Magnetic Electrical

kNi φ= ⋅ℜ

m m km k

N I φ= ℜ∑ ∑ m km k

v i R =∑ ∑Ohm’s law:

Kirchhoff’s voltage law

0kkφ =∑ 0k

ki =∑ Kirchhoff’s current law


Faraday’s law

d d(N ) dBe = = NA dt dt dt

φΨ =


Inductor

• Coil self-inductance

NL dii e L

d(N ) dte = dt

φ

φ

⎫⎪⎪ → =⎬⎪⎪⎭

@

2 2NL = N ANLilN i

φµ

φ

⎫⎪ → = =⎬ ℜ⎪ℜ = ⎭

Al = µ

ℜ


Transformer •  Ideal transformer

Faraday’s law

Ampere’s law

0l = Aµ

ℜ ≈

1,2i i

iN i φ

=

= ⋅ℜ∑ 1 1 2 2N i N i φ− = ⋅ℜ

1 1dv Ndtφ= ⋅ 2 2

dv Ndtφ= ⋅

1 2

1 2

v vN N

=

1 1 2 2i N i N=Core reluctance


Transformer

• Magnetising inductance

0l = Aµ

ℜ ≠

1 1 2 2N i N i φ− = ⋅ℜ

1 1dv Ndtφ= ⋅

1 21 1 2

1

( ) m mN Nd dv i i L i

dt N dt= ⋅ − = ⋅ℜ

21

mNL =ℜ2

1 21

mNi i iN

= −

Non-zero core reluctance


Transformer •  Leakage flux

111 1 1 1 1 1 1

ldd dv = R i N R i Ndt dt dt

φφ φ⎛ ⎞+ = + +⎜ ⎟⎝ ⎠

222 2 2 2 2 2 2


φφ φ⎛ ⎞− − = − − −⎜ ⎟⎝ ⎠

1 1lφ φ φ= +

2 2lφ φ φ= − +

R1,2 – ohmic resistances of the windings

φl1,2 − leakage flux


11 1 1 1 1

22 2 2 2 2

l

l

div = R i L edtdiv = R i L edt

+ +

− − +

Equivalent Transformer Circuit

222 2 2 2 2 2 2


φφ φ⎛ ⎞− − = − − −⎜ ⎟⎝ ⎠

11 1 1 1

2 22 2 2 2

1

ml m

ml m

d di iv = R i L Ldt dt

Nd di iv = R i L Ldt N dt

+ +

− − +

/L = N iφ⋅

21 2

1m

Ni = i + iN

111 1 1 1 1 1 1


φφ φ⎛ ⎞+ = + +⎜ ⎟⎝ ⎠


Image credits

• All uncredited diagrams are from the book “Power Electronics: Converters, Applications, and Design” by N. Mohan, T.M. Undeland and W.P. Robbins.

• All other uncredited images are from research done at the EWI faculty.

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