Chapter 3 (Mohan/UD/Robbins)

Review of Basic Electrical and

Magnetic Circuit Concepts

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

3-2

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

Sinusoidal Steady State

Vector diagram representation 3-4

3-5

3-6

3-7

3-8

Active and Reactive Power 3-9

3-10

3-11

RMS Values of typical waveforms 3-12

Three Phase Systems

If the currents are balanced and sinusoidal, then the current through the neutral is zero.

3-13

Three-Phase Circuits

Vector diagram representation 3-14

Steady State in Power Electronics

Voltage wave from a motor drive inverter

Line Current Waveform

3-15

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

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

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

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

Fourier Analysis 3-20

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

3-22

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

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

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

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

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

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

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

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

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

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

Time constant in R-C Circuit = RC =

3-33

3-34

3-35

Continued

1 2 Joules

Energy stored in a Capacitor = 2 CV

3-36

3-37

1 2 Joules

Energy stored in an Inductor = 2 LI

3-38

Duality

These fundamental concepts with the passive elements like L and C would eventually lead to the ‘Principles of Duality’….

3-39

Basic Magnetics

• Direction of magnetic field due to currents

Ampere’s Law

Hdl= i ∑Hkl

k =∑Nmi

m

km 3-40

Direction of Magnetic Field 3-41

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

B-H Relationship; Saturation

• Definition of permeability 3-43

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

Continuity of Flux Lines

φ1+φ2 + φ3 =0 3-45

Concept of Magnetic

Reluctance

• Flux is related to ampere-turns by reluctance 3-46

Analogy between Electrical and Magnetic Variables

3-47

Analogy between Equations in Electrical and Magnetic Circuits

3-48

Magnetic Circuit and its Electrical Analog

3-49

Faraday’s Law and Lenz’s Law 3-50

Inductance L

• Inductance relates flux-linkage to current 3-51

Analysis of a Transformer 3-52

Transformer Equivalent Circuit 3-53

Including the Core Losses 3-54

Transformer Core

Characteristic 3-55

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