3. Basic Electronical and Magnetic Circuit Concepts
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Transcript of 3. Basic Electronical and Magnetic Circuit Concepts
1 Challenge the future 1 Challenge the future
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
3 Basic Electrical and Magnetic Circuit Concepts
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.
4 Basic Electrical and Magnetic Circuit Concepts
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
5 Basic Electrical and Magnetic Circuit Concepts
• 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
6 Basic Electrical and Magnetic Circuit Concepts
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
7 Basic Electrical and Magnetic Circuit Concepts
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!
8 Basic Electrical and Magnetic Circuit Concepts
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.
9 Basic Electrical and Magnetic Circuit Concepts
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⋅ ⋅= = =⋅
10 Basic Electrical and Magnetic Circuit Concepts
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
11 Basic Electrical and Magnetic Circuit Concepts
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
12 Basic Electrical and Magnetic Circuit Concepts
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⋅( )+:=
13 Basic Electrical and Magnetic Circuit Concepts
Fourier Analysis of Non-Sinusoidal Waveforms
• Non-sinusoidal periodic waveform f(t)
where {0 0
1 1
1( ) ( ) cos( ) sin( )}2h h h
h hf 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
= = =∫ ∫
14 Basic Electrical and Magnetic Circuit Concepts
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
15 Basic Electrical and Magnetic Circuit Concepts
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 2
1 2s s shhI = I + I∞
=∑2 2 2
1 2dis s s shhI = 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∞
=∑
16 Basic Electrical and Magnetic Circuit Concepts
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
17 Basic Electrical and Magnetic Circuit Concepts
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
18 Basic Electrical and Magnetic Circuit Concepts
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
19 Basic Electrical and Magnetic Circuit Concepts
Inductor and Capacitor Response
• Time domain:
11( ) ( ) ( )
t
L L Lt
1i t = i t + v t dtL ∫ 1
1( ) ( ) ( )t
c c ct
1v t = v t + i t dtC ∫
( ) LL
div t Ldt
= ⋅ ( ) CC
dvi t Cdt
= ⋅
20 Basic Electrical and Magnetic Circuit Concepts
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 =
21 Basic Electrical and Magnetic Circuit Concepts
Capacitor in Steady State Amp-second balance 1
1( ) ( ) ( )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
22 Basic Electrical and Magnetic Circuit Concepts
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
23 Basic Electrical and Magnetic Circuit Concepts
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 A
φclosedArea = B ⋅dAA∫∫ = 0
Flux (by def.):
Continuity of flux: Gauss’s law of magnetism (2nd Maxwell’s equation)
24 Basic Electrical and Magnetic Circuit Concepts
Magnetic Reluctance
= µB H
( ) k k km m k k k k k k
m 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
25 Basic Electrical and Magnetic Circuit Concepts
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
26 Basic Electrical and Magnetic Circuit Concepts
Faraday’s law
d d(N ) dBe = = NA dt dt dt
φΨ =
27 Basic Electrical and Magnetic Circuit Concepts
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
28 Basic Electrical and Magnetic Circuit Concepts
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
29 Basic Electrical and Magnetic Circuit Concepts
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
30 Basic Electrical and Magnetic Circuit Concepts
Transformer • Leakage flux
111 1 1 1 1 1 1
ldd dv = R i N R i N
dt dt dtφφ φ⎛ ⎞+ = + +⎜ ⎟⎝ ⎠
222 2 2 2 2 2 2
ldd dv = R i N R i Ndt dt dt
φφ φ⎛ ⎞− − = − − −⎜ ⎟⎝ ⎠
1 1lφ φ φ= +
2 2lφ φ φ= − +
R1,2 – ohmic resistances of the windings
φl1,2 − leakage flux
31 Basic Electrical and Magnetic Circuit Concepts
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
ldd dv = R i N R i Ndt dt dt
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
ldd dv = R i N R i N
dt dt dtφφ φ⎛ ⎞+ = + +⎜ ⎟⎝ ⎠
32 Basic Electrical and Magnetic Circuit Concepts
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.