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Chapter 3 (Mohan/UD/Robbins)
Review of Basic Electrical and
Magnetic Circuit Concepts
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Definition of basic electrical quantities
Current = Rate of change of charge
q= charge
i = dq / dt
Voltage = Amount of energy given to a unit charge v = dw / dt W= energy Power
=
p =dw / dt = (dw / dq).(dq / dt) = v.i
T
Energy =vidt
0
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3-2
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Average Power
Instantaneous power flow: p(t) = vi
= 1 Tvi dt
Average power flow: Pav
T 0
RMS Value of current
T
I = 1 i2 dt
T 0
3-3
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Sinusoidal Steady State
Vector diagram representation 3-4
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3-5
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3-6
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3-7
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3-8
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Active and Reactive Power 3-9
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3-10
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3-11
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RMS Values of typical waveforms 3-12
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Three Phase Systems
If the currents are balanced and sinusoidal, then the current through the neutral is zero.
3-13
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Three-Phase Circuits
Vector diagram representation 3-14
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Steady State in Power Electronics
Voltage wave from a motor drive inverter
Line Current Waveform
3-15
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The Basics of Fourier Analysis
Any physically realizable periodic function, f(t) = f(t+T), can be written as a
sum of sinusoids.
f(t) = a0 + Σah cos(hω)t + bh sin(hω)t
where the sum is taken over h=1 to infinity, ω= 2πT/, and the ah and bh coefficients are given by explicit integral equations,
3-16
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Fourier Coefficients
1 τ+T
a0 =
T ∫τf(tdt)
2 τ+T
ah =T ∫τ[f(t cos) hω(t dt])
2 τ+T
bh =T ∫τ[f(t sin) hω(t dt]) 3-17
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Another way of expressing a function
We can also write
f(t) = Σch cos (hωt + θn) with the sum from 0 to
infinity. This form is common in electrical engineering.
chis the component amplitude
θ is the component phase
h
1
bh
2 2 θh =tan
ch =ah +bh
ah
3-18
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Some Terminology
Each cosine term, ch cos (hωt + θh), is called a
Fourier component or a harmonic of the function f(t). We call each the nth harmonic.
The value cn is the component amplitude; θh is the component phase.
c0= ao is the dc component, equal to the average
value of f(t), c0 = <f(t)>.
The term c1 cos(ωt+ θ1) is the fundamental of f(t), while 1/T is the fundamental frequency.
3-19
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Fourier Analysis 3-20
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What About Power in non-sinusoidal circuits?
A voltage
v(t) = Σcn cos (nωt + θn),
and a current
i(t) = Σd cos (mωt + φ), m m
with the same base frequency ω.
We are interested in conversion: The energy flow over time.
This is determined by the average power flow <p(t)>.
P=((1/T) [Σcn cos(nωt + θn)][ Σdm cos(mωt + fm)] dt 3-21
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Line Current Distortion
i (t) = 2I sin ωt −φ)+ 2I sin ωt−φ) vs = 2 Vs sin ω1t s s1 1 1 sh hh
h1
is (t) =is1 (t) +ish (t)
h1 • Voltage is assumed to be sinusoidal • Subscript “1” refers to the fundamental • The angle is between the voltage and the current fundamental
3-23
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Total Harmonic Distortion
T 1
1 2 1 / 2
RMS Value of line current I s = is ( t ) dt )
T
1 0
1 / 2
I s =I s21 +I sh
2 )
h ≠1
Distortion current component i (t) i=(t) −i(t) = i (t)
dis s s1 sh
h≠1 3-24
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Total Harmonic Distortion (THD)
1/ 2
1/ 2 I [=I2
−I2]= I 2
dis s s1 sh
h1
I dis
% THD = 100 x
I s 1
I s2 − I s2 1
= 100 x
I
s 1
2
I sh
= 100 x
h 1 I s 1
I
s , peak
Crest Factor =
I s
3-25
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Power Factor – Non-sinusoidal current
1 T1 1 T1
P = p(t) dt = v (t)i (t)dt
T 0 ∫ T ∫0 s s
1 1
1 T1
P = 2V sin ωt.
2I sin(ωt −φdt)=V I cosφ
T 0 s 1 s1 11 s s1 1
1
P Vs I s1 cos φ1
I s1
=
PF =
PF =
=
cos φ S Vs Is 1
S Vs I s I s
I s 1
DPF =cosφ1
PF =
DPF
I s
1
PF = 1 +THDi2
DPF
3-26
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Inductor and Capacitor Currents Vector diagram representation
V V jπ/ 2 IL = L = L ejπ/ 2 Ic =jωCVc =ωCVc )e
j ωL ωL 3-27
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Response of L and C
dvc (t )
diL (t )
v (t ) =L
i (t ) =C
L c
dt dt
vL
t 1 i L ( t ) =i L ( t 1 ) + ∫t
v L dt t >t1
L 1
1 v ( t ) =v ( t ) +
t i dt t >t
c c 1 c 1
∫t
C 1
3-28
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Inductor Volt-second balance
The net change in inductor current or inductor voltage over one switching period is equal to 0.
v(t+T)=v(t) and i(t+T)=i(t)
vL (t ) =L di
L (t
)
dt
3-29
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Inductor Volt-second balance
vL (t ) =L di
L (t
)
dt
Integration over one complete switching period from 0 to Ts results:
1 T
iL(Ts) −iL(0)=L0svL(t)dt
The net change in inductor current over one switching period is proportional to the integral of the inductor voltage over this interval. In steady state, the initial and final values inductor current are equal
3-30
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Inductor Volt-second balance
T
s
0 =0vL (t)dt
Volt-seconds or flux-linkages
Dividing by Ts,
1 Ts
0 =
Ts 0v
L (t)dt
=v
L
The principle of inductor volt-second balance; the net volt-seconds applied to an inductor (ie. the total area) must be zero
3-31
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Capacitor Charge Balance
Amp-Second Balance
ic (t) =C dv
dtc(t)
Integrating over one switching period
1 Ts
vc (ts ) −vc(0) =C 0ic (t)dt The net change of the capacitor voltage in one switching period must be zero
1 Ts 0
=
Ts 0i
c (t)dt
=i
c
This is called the Principle of Capacitor Amp-Second Balance or Capacitor Charge Balance
3-32
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Time constant in R-C Circuit = RC =
3-33
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3-34
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Continued
1 2 Joules
Energy stored in a Capacitor = 2 CV
3-36
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3-37
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1 2 Joules
Energy stored in an Inductor = 2 LI
3-38
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Duality
These fundamental concepts with the passive elements like L and C would eventually lead to the ‘Principles of Duality’….
3-39
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Basic Magnetics
• Direction of magnetic field due to currents
Ampere’s Law
Hdl= i ∑Hkl
k =∑Nmi
m
km 3-40
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Direction of Magnetic Field 3-41
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Flemings RH and LH Rules
Right Hand Rule …..Generator principle Fore finger ---- Field direction
Middle finger ---- EMF
Thumb ---- Motion of the conductor Left Hand Rule ….. Motor Principle
Fore finger ---- Lines of flux direction Middle finger ---- Current direction Thumb ---- Motion of the conductor
3-42
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B-H Relationship; Saturation
• Definition of permeability 3-43
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Magnetic Field Units
wb Flux density = Flux per unit area = Φ/A m2
Wb / sq.m
1 gauss =104 tesla
wb 4
1 m2 = 10 gauss
Flux= mmf / reluctance
Equivalent to Ohm’s law in electrical analogy
E = -N dΦ/dt ----- Law of Electromagnetic induction L = NΦ/I Henries
3-44
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Continuity of Flux Lines
φ1+φ2 + φ3 =0 3-45
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Concept of Magnetic
Reluctance
• Flux is related to ampere-turns by reluctance 3-46
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Analogy between Electrical and Magnetic Variables
3-47
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Analogy between Equations in Electrical and Magnetic Circuits
3-48
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Magnetic Circuit and its Electrical Analog
3-49
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Faraday’s Law and Lenz’s Law 3-50
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Inductance L
• Inductance relates flux-linkage to current 3-51
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Analysis of a Transformer 3-52
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Transformer Equivalent Circuit 3-53
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Including the Core Losses 3-54
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Transformer Core
Characteristic 3-55
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Summary
Review of RMS values, average power, reactive power and power factor
Fourier Analysis-Total Harmonic Distortion
Displacement Power Factor for no-sinusoidal currents
Inductor volt-sec balance and capacitor amp-sec balance
Basics of Magnetics 3-56