1/450

10:10~12:40 80% 20%.

Fundamentals of Electrical Drives, Andre Veltman, Duco W.J. Pulle, R.W. de Doncker,Springer, 2007.

Advanced Electrical Drives - Analysis, Modeling, Control, Rik De Doncker, Duco W.J. Pulle, and Andre Veltman, Springer Science+Business Media B.V. 2011.

()

2/450

Fitzgerald & Kingsley's Electric Machinery, Stephen Umans, McGraw-Hill Education, 7th Ed., Jan. 28, 2013.

Fundamentals of Electrical Drives,Andre Veltman, Duco W.J. Pulle, R.W. de Doncker, Springer, 2007.

Advanced Electrical Drives - Analysis, Modeling, Control,Rik De Doncker, Duco W. J. Pulle, and Andre Veltman, Springer Science+Business Media B.V. 2011.

Fundamentals of Electrical Drives, G. K. Dubey, Alpha Science International, Ltd, March 30th 2001.

3/450

Fundamentals of Electrical DrivesAndre Veltman, Duco W.J. Pulle, R.W. de Doncker, Springer, 2007.

Chapter 1 IntroductionChapter 2 Simple Electro-Magnetic Circuits Chapter 3 Transformers Chapter 4 Three-Phase Circuits Chapter 5 Concepts of Real and Reactive Power Chapter 6 Space Vector Based Transformer Models Chapter 7 Introduction to Electrical Machines Chapter 8 Voltage Source Connected Synchronous MachinesChapter 9 Voltage Source Connected Asynchronous Machines Chapter 10 Direct Current Machines Chapter 11 Analysis of a Simple Drive SystemAppendix A: Concept of Sinusoidal Distributed Windings Appendix B: Generic Module Library

4/450

Advanced Electrical Drives - Analysis, Modeling, Control,Rik De Doncker, Duco W. J. Pulle, and Andre Veltman, Springer Science+Business Media B.V. 2011.

Chapter 1 Modern Electrical Drives: An OverviewChapter 2 Modulation Techniques for Power Electronic ConvertersChapter 3 Current Control of Generalized LoadChapter 4 Drive PrinciplesChapter 5 Modeling and Control of DC MachinesChapter 6 Synchronous Machine Modeling ConceptsChapter 7 Control of Synchronous Machine DrivesChapter 8 Induction Machine Modeling Concepts Chapter 9 Control of Induction Machine DrivesChapter 10 Switched Reluctance Drive Systems

Special Topics (2016)1. () [9/23~10/21] 2. Basic Concepts & Magnetic Modeling [Hanselman 2012] () [1]3. Analysis and Simulation of a Gapped Inductor with Inserted PM Using Maxwell () [2]4. Electrical and Mechanical Relationships [Hanselman 2012] () [3] 5. Brushless Motors Fundamentals [Hanselman 2012] () [4] 6. PSIM Simulation of a Single-Phase PWM Inverter () [5] 7. Power Circuit Analysis and Design of a Three-Phase PWM Inverter () [6] 8. SPWM, SVPWM, Carrier-Based SVPWM, and Discontinuous SVPWM () [11] 9. Current Control of a Three-Phase PWM Inverter ()10. PSIM Simulation of a DC Motor Drive () [7] 11. Modeling of SPMSM and IPMSM in a-b-c and d-q Reference Frames () [8] 12. FOC of IPMS and MTPA (Maximum Torque Per Ampere) Control () [?] 13. FOC of SPMSM Using STM32F4 () [?] 14. Sensorless Control of SPMSM and IPMSM Using STM32F4 () [?] 15. Sensorless Control of SPMSM and IPMSM with Sliding Mode Observer () [?] 16. PSIM Simulation of a Current-Controlled BLDC Motor Drive with Hall Sensors () [9] 17. Modeling and Identification of Stator Inductance of an IPMSM () [10] 18. One-Shunt Current Sensing Techniques for Motor Drives () [?] 19. Initial Position Detection of an IPMSM () [?]

6/450

From Power System Design to Power IC Design

2016923

Filename: \Filename: \C01 \\011.ppt

7/450

Contents

An Overview of Electric Drives Keys for Understanding Electrical Machines Basic Physical Laws for Electrical Drives Torque and Back EMF of Electrical Drives Motor-Load Torque-Speed Characteristics Motion Profile of a Motor Drive Control System Electromechanical Energy Conversion Modeling of Electric Drives Motor Losses Motor Sizing and Selection Magnetic Materials for Electrical Machines Permanent Magnets

~ 8/450

An Overview of Electric Drives

~

9/450

Introduction

Why use electro-mechanical energy conversion? Key components of an electrical drive system What characterizes high performance drives? Modes of Operation: Motoring and Braking Notational conventions Use of building blocks to represent equations Magnetic principles Machine sizing principles Tutorials for Chapter

10/450

Why use electro-mechanical energy conversion?

Large power range: Actuators and drives are used in a very wide range of applications from wrist-watch level to machines at the multi megawatt level, i.e. as used in coal mines and the steel industry.

No clutches: Electrical drives are capable of full torque at standstill, hence no clutches are required. Wide speed control range: Electrical drives can provide a very large speed range, usually gearboxes can

be omitted. Clean operation: no oil-spills to be expected. Safe operation: For environments with explosive fumes (pumps in oil-refineries). Immediate use: electric drives can be switched on immediately. Low service requirement: electrical drives do not require regular service as there are very few

components subject to wear, except the bearings. This means that electrical drives have a long life expectancy, typically in excess of twenty years.

Low no-load losses: when a drive is running idle, little power is dissipated since no oil needs to be pumped around to keep it lubricated. Typical efficiency levels for a drive is in the order of 85% in some cases this may be as high as 98%. The higher the efficiency the more costly the drive technology, in terms of initial costs.

Quite Operation: Electric drives produce very little acoustic noise compared to combustion engines. Excellent control ability: electrical drives can be made to conform to precise user requirements. This may,

for example, be in relation to realizing a certain shaft speed or torque level. Four-quadrant operation: Motor- and braking-mode are both possible in forward or reverse direction,

yielding four different quadrants: forward motoring, forward braking, reverse motoring and reverse braking. Positive speed is called forward, reverse indicates negative speed. A machine is in motor mode when energy is transferred from the power source to the shaft i.e. when both torque and speed have the same sign.

11/450

Hierarchical Control Architecture of Motor Drive Control

POSITION

VELOCITY

TORQUE

TorqueController

MVelocityController

PositionController

Torquecommand

Velocitycommand

Positioncommand

FeedbackProcessor

Power Conversion (Current & PWM, Power Flow) Control Sensorless Startup Control (Alignment & Signal Injection) Torque/Field Decoupling (FOC and MTA) Control Servo (Position & Velocity) Control Motion (Interpolation & Ramping, Acc./Dec.) Control

12/450

Architecture of an Electric Motor Drive

Controller

PowerConverter

Motor Load

-50

0

50phase response

10 0 101

10 2 10 3 104 105

frequency(rad/sec)

10 0 10 1 10 2 10 3 10 4 10 510

1

102

103

magnitude response

frequency(rad/sec)Man-Machine Interface PWM Control Vector Control Control Loop Design Power Factor Control

PowerSource

Speed

Torque

Speed

Torque

13/450

Power Flow of a Motor Drive System

Controller

PowerConverter Motor

LoadPowerSource

rad/sec]Nm, ;[Watt mmmmm TtT

tWP

1 HP 746 WattsmP (rad/s) 104.7 6021000 RPM 1000

mP

21 Nm 1 Kg m/sec 10 Kgw cm

Nm 0.1Nm 098.0Kgm/sec 098.0m01.09.8m/secKg 1cm Kgw 1 22

EPSP

command

14/450

Key Components of an Electric Drive System

MotionController

MotorController

Powersource

PWMModulator

Motor Load

SpeedTorque

Sensor signals

Converter

Motor control is an inherent system integration technology based on control theory.

Power FlowSignal Flow

15/450

Key Components of an Electric Drive System

Powersource Motor

Load

SpeedTorque

Converter

Power Flow

PWMModulator

MotorControllerMotion

Controller

Signal Flow

Battery-Powered Bi-Directional Motor DriveBattery

1S 3S 5S

2S 4S 6S

dcVa

bc

VDC = 340~380~420 VDC

1S 3S 5S

2S 4S 6SdcV

ab

c

1Q

2Q

S

VDC = 340~380~420 VVBAT = 42~48~54 V VCAP = 42~200 V

Battery Powered Regenerative Motor Drive for EVs

Super Cap.

VBAT = 42~48~54 V

17/450

Grid Connected PFC Bi-Directional Motor Drive

1S 3S 5S

2S 4S 6SdcV

ab

c

3Q

4Q

R

1Q

2QS

VDC = 340~380~420 VDC

3 kW sensorless IPMSM inverter drive with bi-directional power flow and power factor control.

Measure the MTA performance of the sensorless drive. Measure the efficiency, power factor, and grid current THD as a function of

motor/generator power (%) both in motoring and regenerative modes. Make simulation to get the calculated (, pf, THD) in considerations of RDS(ON) of the

power MOSFET and VF of the power diode and compared with the experimental results. Adjust (VDC, fs) to optimize (, pf, THD).

18/450

3-Phase Back-to-Back Inverter Motor DriveControl of Power Converters and Motor Drives

dcV

Inputconverter

Outputconverter

u1u2u3

N S

to switches

PWM Control Inverter Control DTC Vector Control Sensorless Control Servo Control Auto-Tuning

Power Factor Control

Regenerative Braking Control

DC-Link Voltage Regulation

DC-Link Cap. Minimization

to switches

uud

dC

~

19/450

Matrix Converter as a Power Processor

(a) matrix converter (b) per-phase equivalent circuit

Power Processor

InputsOutputs . . .

.

.

20/450

Matrix Converter Motor Drive

SMPS

IM3~

Auxiliary circuit supply unit(gate-drivers, transducers, control)

ab

c

u v w

Clamp circuit

Inpu

t filt

er

Cclamp

motor

Mat

rix C

onve

rter

Pow

er S

tage

GR1

Gw1

GR2

GU2

GR3

GV3

GS1

GU1

GS2

GV2

GS3

GW3

GT1

GV1

GT2

GW2

GT3

GU3

WVU

RST3-Phase

AC Input

M. Munzer, EconoMacThe first all in one IGBT module for matrix converters, in Proc. Drives and Control Conf., London, U.K., 2001. The Eupec ECONOMAC matrix module (2001)

Popular 3-Phase Voltage Source Converters

adcV b

c

(a) 2-level VSC

(c) T-type 3-level VSC with BB-IGBT (d) T-type 3-level NPC VSC with RB-IGBT (Fuji)

(b) 3-level NPC VSC

1dC

2dC

a

b

c

1T V1T U 1T W

2T V2T U 2T W

3T U

4T U3T V

4T V3T W

4T W

a

o

1dC

2dC

b

c

1dC

2dC

abc

22/450

2-Level vs. 3-Level NPC Inverters

Type 2 Level I-NPC 3L T-NPC 3L T-ANPC

Circuit

Efficiency (%) 97,06 97,48 97,59 97,73

On Noise (%) 100 50 50 50

Filter $ (%) 100 50 50 50

IGBT+Diode $ (%) 100 200 200 150

Total $ (%) 100 99 96 82

Figure Of Merit 1.00 1.01 1.05 1.23

1T

2T

1T

2T

3T

4T

U

N

P

M

1T

2T

3T

4T

M U

C

N

3T 4T 2T

1T

P

GOOD Excellent Highest Efficiency / Cost Ratio (FOM)

23/450

Bi-Directional Bridgeless Totem-Pole PFC Converter with 3L T-Type Active NPC Inverter Motor Drive

1dC

2dC

3Q

4Q

1Q

2Q

M. Schweizer, I. Lizama, T. Friedli, and J. W. Kolar, Comparison of the chip area usage of 2-level and 3-level voltage source converter topologies, IEEE IECON Conf. Proc., November 2010.

T-Type Active NPC Inverter for Motor Drive Single-Phase T-Type Inverter for Bi-Directional Power Factor Control GaN/MOSFET Hybrid PWM Rectifier for Efficiency Enhancement PWM Strategies for T-Type Active NPC Inverter Active Neutral Voltage Balancing Control

24/450

Integration of Power, Motor, and Motion Control

Motor

MCU-BasedDigital Controller

Mechanical Load

Signal Flow

Power Flow

~

Power Converter

25/450

The Complete Family of Electric Motors

AC

Asynchronous Synchronous

Induction

SinglePhase

PolyPhase

CapacitorStart

CapacitorRun

ShadedPole

Cast Rotor

InsertedRotor

WoundRotor

BLDC Sine Hysterisis Step

PSM

WoundField

SurfacePMSM

InteriorPMSM

Reluctance PMDC WoundField

Commutator Homopolar

PermanentMagnet

Hybrid

VariableReluctance

SRM

SynchronousReluctance

Shunt

Compound

Series

Universal

DC

PM AssistedSM

26/450

Torque-Speed Characteristics of Different Motorsmotor DC shunt-

wound motorDC series-wound motor

three-phaseasynchronousmotor

three-phasesynchronousmotor(DC excitation of rotor)

single-phase(universalmotor)

single-phaseasynchronousmotor with condensator

single-phaseasynchronousMotor (Ferraris motor)

circuit diagram

torque-speedcharacteristics

torquecharac-teristicsformani-pulation

different manipu-latedVariables ------normal:

manipu-latedvariables

UA armaturevoltage

IE excitation current

RA armature resistance

U voltage U voltage frequencyR rotor

resistance

frequency U voltageR armature

resistance

Ust manipulatedvoltage

M

EU

EI

AI

AU MEI

I

U

,U ,U

RU

M~1

I

U UM~1 M~1

STU

EU

T

T

T

T

T

T

T

T

EI

AU

AR

T

T

T

T

T

U

R

U

R

UStU

27/450

Block Diagram of a PM DC Servo Motor Drive

dcV

(Full Bridge PWM Amplifier)

(Encoder)

PM DC Servo Motor

EMI Filter

av

1Q

2Q

3Q

4Q5Q

1Q 2Q 3Q 4Q

28/450

AS8446 Programmable PWM DC Motor Driver & Controller (1-Quadrant DC Motor Drive IC)

Block Diagram of a PWM IC for PMDC Motor Drive

dcV

T1

T2

T3

T4

T 1 T 2 T 3 T 4

http://www.eecs.berkeley.edu/~acarlson/ismc.htm

30/450

Block Diagram of a PMSM AC Servo Drive

dcV

T5

T6

T 1 T 2 T 3 T4

T3

T4

T1

T2

T 5 T6

Current sensors

Hall sensors Encoder

EMI Filter

PWM

5Q

PWMAmplifier Motor LOAD

CurrentController

VelocityLoop

Controller

PositionLoop

Controller

MotionController

Current FeedbackVelocity FeedbackPosition Feedback

Torqueestimator

PWMControl

Sensors andSignal Conditioning Unit

PWM for Power Switches ControlCurrent cmd.Torque cmd.Velocity cmd.Position cmd.

Hierarchical Motor Control Architecture

Torque/FieldController

VectorController

32/450

Hierarchical Control Architecture for AC Drives

ServoController

AC/DCConverter

DC/ACConverter

FOC VectorController

Field &Torque

Controller

FluxEstimator

PMSMShaft

sensor

CurrentController

Current Control

Torque ControlField-Weaken Control

Acceleration, Velocity, & Position ControlForce, Motion Profile Control

PWMController

Position & Velocity

estimator

Current &

voltage&

positionfeedback

Feedback Sensor: Hall-Effect SensorEncoder or ResolverVoltage & Current SensorsSensorless

DC-Link

N

S

SN

PWM Control

Sensorless Control

33/450

AC Drive with Field-Oriented Vector Control

for asynchronousmachines

field weakening field controller

speedcontroller

currentcontrollers

_

M3~

Encoder

n* i*q

i*d*

machine

model je

je

je

for synchronousmachines

,

cba ,,

,

cba ,,

i

v

34/450

Typical FOC Sensorless Control of IPMSM

Field WeakeningController

,e ed q

,s sd q

,e ed q

,s sd q

* qei

dei

dev

qev

dei

qei

,e ed q

,s sd q

,a b

dsv

qsv

ai

bi

*dsv

*qsv

DCv

PI-regulator PI-regulator

PI-regulator

Phase transf.

Phase transf.Estimator

Inve

rter

Phase transf.

VT

35/450

Field-Oriented Vector Control (FOC) of AC Motors with Space Vector Pulse-Width Modulation (SVPWM)

Torque Current Command

Field Current Command

,

cba ,,

2-Axis Current Control Scheme in Synchronous Reference Frame

ii

ii

rr

rr

q

d

cossinsincos

PMAC Motor

,

qd ,

,

qd ,SVPWM

ba

a

iii

ii

32

31

PI

PI

3-phaseInverter

dr

r

r

r

r

Park-1 t.

Park t. Clarke t.

ekXi

0.5an

dc

vV

0.5an

dc

vV

0.5an

dc

vV

27T

21T

22T

20T

2sT

1

-11

-11

-1

refV

t

t

t

q

i

idi

qi

bi

ai

v

vdv

qv

*di

*qi

*y

y iK

pK

corK

1z ku ku1

dcV

2 (110)V

1(100)V

11

TT

V

22V

TT

Sensorless Field Oriented Control of PMSM Motors

,

cba ,, ,

qd ,

,

qd ,

3-phase Inverter

PI

PI

PI

InversePark Transform

ParkTransform

ClarkeTransform

Position

Speed ()

i

i

v

v

r

dcV

*di

*qi

dv

qv

di

qi

v

v

SVPWM PMAC Motor

ai

bi

Current Sampling and Reconstruction

m

*m

Position & Speed Estimator

0 0

0 0

1

( )

( )

tan

t

s d

t

s

r

v R i dt

v R i dt

Start-up sequencer

Start-up Controller

t

r

*r

37/450

What Characterizes High Performance Drives?

Advances in motor design and manufacturing technologies Availability of fast and reliable power semiconductor switches for the

power converters Availability of fast microcontroller (MCUs, FPGA, etc.) for (real time)

embedded control Advances in motor control techniques Better simulation packages Better sensors Better materials

Basic Theories for Electric Machines

Electromagnetic Theory

Newtonian Theory

Magnetic Circuits

Force and Torque

Circuit Theory

Torque and Back EMF

Motion Profile

Torque-Speed Characteristics

Magnetic Materials

Electromechanical Energy

Conversion

Motor Loss

Motor Sizing

Modeling

Characterization

4Q Operation

Parameter Identification

Motor Selection

Inductor

Transformer

Simulation

MachineDesign

39/450

References: Introduction to Electrical Drives

[1] Chapter 1 Introduction of Fundamentals of Electrical Drives, Andre Veltman, Duco W.J. Pulle, R.W. de Doncker, Springer, 2007.[2] T.J.E. Miller, Brushless Permanent-Magnet and Reluctance Motor Drives, Clarendon Press, Oxford University Press in Oxford, New

York, 1989. [3] D. C. White and H. H. Woodson, Electromechanical Energy Conversion, Wiley, New York, 1959, Chapters 4 and 7 to 10. [4] G. K. Dubey, Fundamentals of Electrical Drives, Alpha Science International, Ltd, March 30th 2001. [5] S.D. Umans, Fitzgerald & Kingsley's Electric Machinery, 7th Ed, McGraw-Hill Book Company, 2013. [6] Chapter 1 Basic Principles for Electric Machine Analysis, Analysis of Electric Machinery and Drive Systems, P. C. Krause, O.

Wasynczuk, S. D. Sudhoff, and S. Pekarek, IEEE Press and Wiley Inter-Science, 3rd Ed., 2013.

40/450

Power Electronic Systems & Chips Lab., NCTU, Taiwan

Keys for Understanding Electrical Machines

Power Electronic Systems & Chips Lab.

~

Design of Rotating Electrical Machines, 2nd Ed., [Chapter 4: Inductances] Juha Pyrhonen, Tapani Jokinen, Valeria Hrabovcova, October 2013, Wiley.

41/450

Modeling the Stator Inductance of an IPMSM

iv iL

Rviv

Lv

Li

L

R Rv 1L

LR

1s

0( )L t

1( ) ?rL

Assume the rotor produces a sinusoidal flux distribution across the air gap, how to model the stator winding inductance as a function of the rotor position of an interior permanent magnet synchronous motor (IPMSM)?

REF: Chapter 4 Inductances, Design of Rotating Electrical Machines, Juha Pyrhonen, Tapani Jokinen, Valeria Hrabovcova, 2nd Ed., October 2013, Wiley.

1 1( , , ) ?rL I T

Modeling of Synchronous Machine in dq-Frame

q d

PM

PM

PM

s q qi L

d di L

ss

sv

si

r

e

d

axisa

axisb q

axisc

aiavFv

bvbi

Fi

ci

cv

di1q1 fd

d1qi1 r

a

'a

sR e q qL i

dv

dLdi sR e d dL i

qv

qLqi

e PM d s d e q d PM

q e d s q q e PM

v R sL L i sv L R sL i

PMs

3 3( ) ( ) ( )2 2 2 2e d q d q d q d d PM q d q d q

P PT i L L i i i L i L L i i

Electric Equations:

Torque Equations:

mm

Lfe BdtdJTTT

Torque Characteristics of Synchronous Machines

d qL L d qL L

d

q

d

q

dq

d

q

d qL L d qL L0m

3 [( ) ( )]2 2e d d PM q q d d q

PT i L i i i L L

emT erT

0erT 0emT eem rT T eem rT T

Inductance Plays a Key Role in Motor Characteristics

1( ) ?rL

The rotor structure determines the major characteristics of a synchronous machine (SM). For SM with concentrated winding stator, the inductance of the coil of a segmented teeth can be calculated as a function its rotor position if the rotor has an anisotropic structure.

[1] I. A. Viorel, A. Banyai, C. S. Martis, B. Tataranu, and I. Vintiloiu, On the segmented rotor reluctance synchronous motor saliency ratio calculation, Advances in Electrical and Electronic Engineering, vol. 5, vo. 1-2, June, 2011.

[2] B.J. Chalmers and A. Williamson, AC Machines Electromagnetics and Design, Research Studies Press Ltd., John Wiley and sons Inc., 1991.[3] Jong-Bin Im, Wonho Kim, Kwangsoo Kim, Chang-Sung Jin, Jae-Hak Choi, and Ju Lee, Inductance calculation method of synchronous

reluctance motor including iron loss and cross magnetic saturation, IEEE Transactions on Magnetics, vol. 45, no. 6, pp. 2803-2806, 2009.

Design of Rotating Electrical Machines, 2nd Ed., [Chapter 4: Inductances] Juha Pyrhonen, Tapani Jokinen, Valeria Hrabovcova, October 2013, Wiley.

3 [ ( )]2 2e q m q d d q

PT I I I L L Generated electric torque of synchronous machine:

45/450

Basic Physical Laws for Electrical Drives

~ 46/450

Basic Physical Laws for Electrical Drives

Basic Electromagnetic Theory and Magnetic Circuits Modeling of Inductors Transformer Force and Torque Generation in Electric Machines Motion Profile of a Motor Drive Control System Motor-Load Torque-Speed Characteristics Electromechanical Energy Conversion Torque and Back EMF

~

47/450

Power Electronic Systems & Chips Lab., NCTU, Taiwan

Basic Electromagnetic Theory and Magnetic Circuits

Power Electronic Systems & Chips Lab.

~

Chapter 1 Magnetic Circuits and Magnetic Materials, Fitzgerald & Kingsley's Electric Machinery, 7th Ed, S.D. Umans, McGraw-Hill Book Company, 2013.

48/450

1865

18652020(Oliver Heaviside) (Josiah Gibbs)(Heinrich Hertz)1884

Source of Electric Field Changing Electric Field Source of Magnetic Field Changing Magnetic Field

A Students Guide to Maxwells Equations (Daniel Fleisch), , 20101018

0

E

t

BE

0 B0 0 t

EB J

49/450

Basic Notations for Electromagnetism

Electric field strength E [V/m] Magnetic field strength H [A/m] Electric flux density D [C/m2] Magnetic flux density B [Vs/m2], [T] Current density J [A/m2] Electric charge density, dQ/dV [C/m3]

D E

B H

permittivity of free space (Farads/m)

permeability of free space (Henrys/m)

Maxwell's Equations (1860s~1970s)

(Gausss Law for Electric Field)

0

encS

Qnda

E

0S

nda B

0 0 encC Sdd I ndadt

B l E

C S

dd ndadt

E l B

0

E

(Gausss Law for Magnetic Field)

(Faradays Law)

- (The Ampere-Maxwell Law)

t

BE

0 B

0 0 t

EB J

Symbols and Units of Electromagnetic QuantitiesSymbol Field Variable Name MKS Rationalized Units

H Magnetic field intensity A/m

Jf Free current density A/m2

Kf Free surface current density A/m

B Magnetic flux density Wb/m2

M Magnetization density A/m

E Electric field intensity V/m

D Electric displacement C/m2

f Free charge density C/m3

f Free surface charge density C/m2

P Polarization density C/m2

F Force density N/m3

0 Permeability of free space 4 10-7 H/m

0 Permittivity of free space 8.854 10-12 F/m

Summary of Quasi-Static Electromagnetic EquationsDifferential Equations Integral Equations

Magnetic field system (1.1.1) (1.1.20)

(1.1.2) (1.1.21)

(1.1.3) (1.1.22)

(1.1.5) (1.1.23)

Electric field system (1.1.11) (1.1.24)

(1.1.12) (1.1.25)

(1.1.14) (1.1.26)

(1.1.15) (1.1.27)

where E E v B

0 B

0f J

f H J fC Sd d H l J n a0

Sd B n a

0fS d J n a

0C

d E lfS V

d dV D n af fS V

dd dVdt

J n afC S S

dd d ddt

H l J n a D n awhere f f f J J v

H H v D

t

BE

0 E

f D

ff t

J

f t

DH J

C S

dd ddt

E l B n a

53/450

Basic Relations of Electrical and Magnetic Field

Faradays Law

Amperes Law

terminalcharacteristics

Corecharacteristics

( )v t ( ), ( )B t t

( ), ( )H t F t( )i t

Magnetic CircuitsElectrical Circuits54/450

Magnetic Field

Magnetic fields are produced by electric currents, which can be macroscopic currents inwires, or microscopic currents associated with electrons in atomic orbits. The magneticfield B is defined in terms of force on moving charge in the Lorentz force law. Theinteraction of magnetic field with charge leads to many practical applications. Magneticfield sources are essentially dipolar in nature, having a north and south magnetic pole.The SI unit for magnetic field is the Tesla, which can be seen from the magnetic part of theLorentz force law Fmagnetic = qvB to be composed of (Newton x second)/(Coulomb xmeter).A smaller magnetic field unit is the Gauss (1 Tesla = 10,000 Gauss).

55/450

Right-Handed System and Left-Handed System

x

y

z

y

x

z

Right-Handed SystemLeft-Handed System 56/450

Magnetic Field of Current: Right-Handed Rule

The magnetic field lines around a long wire which carries an electric current formconcentric circles around the wire. The direction of the magnetic field isperpendicular to the wire and is in the direction the fingers of your right handwould curl if you wrapped them around the wire with your thumb in the directionof the current.

57/450

Amperes Law

(a) General formulation of Amperes law. (b) Specific example of Amperes law in the case of a winding on a magnetic core

with air gap.

Direction of magnetic field due to currents Amperes Law: Magnetic field along a path

(a) (b) idlH

1i ni

H

1 mean path lengthl

Airgap: gH

1i

1N

1Core: H

g

58/450

Amperes Law

B H H = magnetic field intensity (Ampere-turns/m) = magnetic permeability of material (Wb/A.m, or Henery/m)B = magnetic flux density (Tesla, Weber/m2)

r 0

= permeability of free space

074 10 H / m

r = relative permeability (between 2000-80,000 for ferromagnetic materials)

H l I d

I

Id

enclosenot doescontour if ,0 enclosescontour if ,I

lH

IlB dld

B

59/450

Permeability: Relationship Between B and H

Ampere,s Law H l I d

H = magnetic field intensity (Ampere-turns/m) = magnetic permeability of material (Wb/A.m, or Henry/m)B = magnetic flux density (Tesla, Weber/m2)

r 0

= permeability of free space

074 10 H / m

r = relative permeability (between 2000-6000 for general ferromagnetic materials used in electrical machines)

permeability = = BH

In magnetics, permeability is the ability of a material to conduct flux. The magnitude of thepermeability at a given induction is a measure of the ease with which a core material can bemagnetized to that induction. It is defined as the ratio of the flux density B to the magnetizingforce H. Manufacturers specify permeability in units of Gauss per Oersted (G/Oe).

cgs: = 1 gaussoersted oersted

0410

tesla mks: = 4 henrrymeter

0710

60/450

(Wb), (Tesla)

(SI)Wb

11 11Wb=1Vs

2-2 -1 (m 2 kgs-2A-1) = (Voltsec)

188218951948

CGS

1108[1 Wb = 108 Maxwell]

(Tesla) [1 Tesla = 1 Wb/m2]

1 oersted = 1000/4 ampere/turn = 79.57747154594 ampere/meter 80 A/m

61/450

Magnetic Flux and Flux Density

Bar magnet flux and flux density plot

(a) Flux distribution (b) Flux density distribution

The color scale shown on the right of the flux density plot shows the highest flux density in red.

62/450

Magnetic Circuits

Coil flux and flux density plot

(a) Flux distribution (b) Flux density distribution

You may observe that there is also a C and I outline shown in red in both figures. These are in fact the outlines of a steel structure which in this case has been constructed of air, i.e. the coil does not see this structure at this point of our discussion. This also implies that these are regions for optimum placement of a CI magnetic core for an inductor.

63/450

Flux Distribution of a CI Core

(a) Flux distribution (b) Flux density distribution

fringing effect ()

Hopkinson's law: the magnetic analogy to Ohm's lawMagnetic Circuit Electric Circuit mmf NI () Flux reluctance permeability

V I R 1/, where =resistivity

N

Reluctance

)H :(unit 1-Al

Inductance

2N NLI i

I

l

NI

is the number of turns of the coilis the mean length if the flux is the magnetic motive force (mmf)is the flux across the cross-section area is the flux linkage of the coil

N

I

Nl

mmf

NI

65/450

Magnetic Circuit of a CI Core with an Air Gap

c

C Core

Assume there is no fringing effect.

g

N turns

gg

cc

cc A

l

Reluctance of core

g

gg A

l

0 Reluctance of air-gap

NI

Reluctance of the air gap, this is the dominant factor!

66/450

Flux-linkage and Self Inductance

The flux-linkage () refers to the amount of flux linked with the coil, i.e. = N.

c

C Core Armature

Assume there is no fringing effect.

g

N turns

Flux-linkage N

The inductance is defined as the flux-linkage resulted per ampere.

IN

IL

cgcg

NI

cg

NL

2

67/450

Flux-linkage and Self Inductance

IN

IL

linkage-flux

dtdiLv LLv

Li

The voltage across the inductor (measure with reference direction) is given by Faraday's induction law as

( ) L LL Ld dLi di dLv t L idt dt dt dt

dtdiLtv LL )(

t

LLL dttvLiti

0 ) (1)0()(

If L does not vary with time, is constant,

The integral form of the inductance is:

68/450

Magnetic Circuit of a CI Core

c

g

It can be observed that for a large air gap, there is a significant fringing effect and we need to modify the effective cross sectional area of the air gap. A Carter factor can be used to allow for fringing effect in estimation of the effective air-gap cross sectional area.

g

If cg

0

ggc

lA

If the air gap is very small, the estimated reluctance of air-gap

69/450

Magnetic Saturation

Reluctance change due to saturationFlux-linkage versus current: with saturation effects

mR0

0 2.0

Reluctance

Flux density B(T)

Current i

linear case

Flux-linkage

Saturation of the Inductor in a Forward Converter

Forward converter schematic. Output is 12 V, 5 A, switching at 100 kHz. The reset circuit of the transformer is omitted for simplicity.

47 H, 5A

Primary switch current waveforms for a forward converter with a 15H off-the-shelf drum core inductor: (a) initial operation, (b) after 60 seconds at room temperature ambient, (c) after 3 minutes operation. The inductance is a function of its core temperature.REF: High Frequency Power Inductor Design (Ridley 2007)

Ringing due to the resonance between transformer leakage inductance and the MOSFET junction capacitance.

Increased current spike due to the inductor saturation.

71/450

Simulation and Estimated Calculation of Inductance

Parameters for magnetic C core example

72/450

Tutorial Example 1: Estimation of Inductance

A magnetic core with an air gap is shown in the following figure with key dimensions (in millimeters) as shown. The cross-section of the magnetic core is rotational symmetric. A single n = 1000 turn coil is shown which carries a current of icoil = 5A. The steel used has a permeability of 1060, where 0 represents the permeability in vacuum (air). The model in question was analyzed with a finite element package and gave the results as given in the following table.

5A

Output finite element program

The computer simulation in calculating the inductance is 2.19 H. What is your estimated calculation results?

73/450

Tutorial Example 1: Estimation of Inductance

MMF

0

ggc

lA

Estimated reluctance of air-gap

3

7 3 30

10 10 104 10 2.0 (2 (50 )10 25 10 )2

gg

c

lA

HNLg

17.22

74/450

Tutorial Example 2: E Core

The distance between the I segment, which is also part of the total magnetic circuit and E core is 10mm. A500 turn coil is wound around the center leg of the E core and carries a current of 20A. The depth of bothmagnetic components is taken to be 20mm. Furthermore, the magnetic material is taken to be magneticallyideal. Key dimensions (in mm) are shown in the following figure which relate to the airgaps between the twomagnetic components.

Tutorial Example 3

Magnetic Circuits and Magnetic Materials, Fitzgerald & Kingsley's Electric Machinery, 7th Ed, S.D. Umans, McGraw-Hill Book Company, 2013.

A magnetic circuit containing hard magneticmaterial, a core and plunger of high (assumedinfinite) permeability, and a 100-turn windingwhich will be used to magnetize the hardmagnetic material. The winding will beremoved after the system is magnetized. Theplunger moves in the x direction as indicated,with the result that the air-gap area varies overthe range 2 cm2 Ag 4cm2. Assuming thatthe hard magnetic material is Alnico 5 and thatthe system is initially magnetized with Ag =2cm, (a) find the magnet length lm such thatthe system will operate on a recoil line whichintersects the maximum B-H product point onmagnetization curve for Alnico 5, (b) devise aprocedure for magnetizing the magnet, and (c)calculate the flux density Bg in the air gap asthe plunger moves back and forth and the airgap varies between these two limits.

76/450

Properties of Ferromagnetic Materials

1.4

1.2

1.0

0.8

0.6

0.4

0.2

00 200 400 600 800 1000

H, A-turn/m

B, Wb/m2

B H r 0

Ferromagnetic materials, composed of iron and alloys of iron with cobalt,tungsten, nickel, aluminum, and other metals, are by far the most commonmagnetic materials.

Transformers and electric machines are generally designed so that somesaturation occurs during normal, rated operating conditions.

DC Excitation

i

N

AB

A toroidal coil and the magnetic field inside it.

A is the cross-sectional area

77/450

B-H Curve, Permeability, and Incremental Permeability

Relation between B- and H-fields.

H

Bs

Hs

Linear region

BH

HB

HB

HB

B

H

HHB r 0

Magnetic intensity H, [A-turns/m]

Incremental PermeabilityB The B-H characteristics of acore material is high nonlinear.Depends on its averagecurrent, current ripple,switching frequency, andoperation temperature.

When measuring theinductance of a magneticcircuit, it should first toidentify its operating point.

78/450

B-H Curve of Major Materials

This is because there is a limit to the amount of flux density that can be generated by the core as all thedomains in the iron are perfectly aligned. Any further increase will have no effect on the value of M, andthe point on the graph where the flux density reaches its limit is called Magnetic Saturation also known asSaturation of the Core and in our simple example above the saturation point of the steel curve begins atabout 3000 ampere-turns per meter.

The set of magnetization curves as shown inleft figure represents an example of therelationship between B and H for soft-iron andsteel cores but every type of core material willhave its own set of magnetic hysteresis curves.You may notice that the flux density increasesin proportion to the field strength until itreaches a certain value were it can notincrease any more becoming almost level andconstant as the field strength continues toincrease.

B-H Characteristics of a Magnetic Material

Performance Tradeoffs: saturation Bs, permeability , resistivity (core loss), remanence Br, and coercivity Hc.

sH

Magnetic Flux Density

( )B

SB

rB

CH

CH( )H

= Saturation Flux DensitySB= Remnant Flux DensityrB= Coercive ForceCH

BH Curve(Orange)

max maximum permeability

initial permeabilityi

BH

rB

SB

incremental permeability reversible permeabilityrev

0limrev H Minor Hysteresis Loop

Magnetic FieldIntensity

Normal or MajorHysteresis Loop

Initial Magnetization Curve

Approx. Point of Max. Perm.P

erm

eabi

lity

Flux

Den

sity

Permeability

B-H Curve

Magnetizing ForceH

Approx. OperatingPoint

SaturationB, tesla

80/450

Flux Density or B-Field

Determination of the magnetic field direction via the right-hand in (a) the general caseand (b) a specific example of a current-carrying coil wound on a toroidal core.

(a) (b)

H-fieldCross-sectional area A

HHB r 0

i

iH

N

The total flux pass through the coil with N turns is called flux linkage and named as .

BA

N

N

81/450

Continuity of Flux

A1 A2

A3

1 23

dABA 0surface) (closed dABA

k

k 0

0or 0 321332211 ABABAB

82/450

Magnetic Cores

Ideal Inductor

v N ddt

dN

vdt 1

The above equation shows that the change in flux during a time interval t0-t1 isproportional to the integral of the voltage over the interval, or the volt-seconds appliedto the winding.

Negligible winding resistance Perfect coupling between windings An ideal core

v

i

N

1

0

1)()( 01t

tvdt

Ntt

83/450

Ideal Inductor [Define its Initial Conduction]

(a) Circuit model. (b) -i characteristic (or B-H curve).

(c) v is a step input; (t0) = 0. (d) v = Vm sin t ; (t0) = 0.

(e) v is a square wave; (t0) = -m. (f) v = Vm sin t ; (t0) = -m.

i

N

v

i

0

v

0

v

t0

v

t0

v

84/450

Magnetic Field Strength H of Some Configurations

long, straight wire

Toroidal Coil

Long solenoid

85/450

Inductance of Wound Magnetic Core

magnetic flux per turnwebers (Wb) [1 Wb = 108 Maxwell]

magnetic flux density webers/meter2 (teslas)

flux linkage webers

core cross-sectional area square meters

magnetic field strength ampere-turns/meter

number of turns

coil current ampere

mean length of magnetic flux path meters

permeability henrys/meter (410-7 in perfect vacuum)

inductancehenrys

B A H N i

lm

L

The inductance of a wound magnetic core is directly proportionalto the incremental permeability of the core material, which is theslope of the B-H curve.

v L didt

N ddt

ddt

L N ddi

ddi

BA H Nil m

L N Al

dBdH

N Alm m

2 2

and

N

v N

i

86/450

Inductance of a Core

slope L

(a)

(b)

2NL

lAC

1 lACC oo

1

2

21

2

NCNC

lANL

r

or

ore

1 elL

AL

The inductance L represents the capability of magnetic flux density produced by unit current of a circuit loop.

v

i

A

el

Flux saturation

N

i

87/450

Magnetic Reluctance and Permeance

Reluctance

Mean path length l Cross-sectional area A

Permeability

Al

Ni

d H l N i H l

lNiH

AB

lNiH

Al

Nil

ANi

Magnetic-motive force (mmf) Ni

Permeance

1

N

i

88/450

Inductance of a Toroidal Core (Self Inductance)

Amp (I)

Weber-turns (=N)

Li

Mean path length lCross-sectional

area A

Permeability

For a magnetic circuit that has a linear relationship between and i because of material ofconstant permeability or a dominating air gap, we can define the -i relationship by the self-inductance (or inductance) L as

iN

iL AN i

l

lAN

lANi

iN

iL 2

where =N, the flux linkage, is in weber-turns. Inductance is measured in henrys or weber-turns per amp.

N

i

89/450

Flux Density Distribution of a Toroidal Core

Representing the magnetic vector potential (A), magnetic flux (B), and current density (j) fields around a toroidal inductor of circular cross section. Thicker lines indicate field lines of higher average intensity. Circles in cross section of the core represent B flux coming out of the picture. Plus signs on the other cross section of the core represent B flux going into the picture. Div A = 0 has been assumed.

A toroidal coil and the magnetic field inside it. 90/450

Energy Stored in a Core

Mean path length l Cross-sectional area Ac

Permeability

I

N: number of turns

22 A NL N

l

The energy stored in the core:

tt

L LIdiLiPdtE 02

0 21''

The energy density (energy/volume) is:2 2 2 21

22 2

2 2

0

1 12

12 2

cB

c c

r

LI N A B lA l A l l N

B B

The energy stored in the core:

coreBL VLIE 2

21

Vcore: volume of the core

Chapter 11 Inductance and Magnetic Energy of Introduction to Electricity and Magnetism, MIT 8.02 Course Notes, Sen-Ben Liao, Peter Dourmashkin, and John Belche, Prentice Hall, 2011.

91/450

Typical Energy Density of a Ferrite Core

0

2

2

re

cB

BVE

For a typical ferrite, assuming the relative permeability is about r = 2000, and the saturation flux density Bsat = 0.3 T (3000 G), we get (for most ungapped ferrite cores) a typical power density of

3J/m 9.1710420002

3.02 7

2

0

2

re

cB

BVE

2Newton/A H/m

7

70

104

104

3000G)B 2000,( J/cm 18J/m 18 satr

33 e

c

VE

(Ferrite core)18100 kHz50%(CRM)3.63610

92/450

Inductance of Air-Core Solenoid

H dl N i

Long air-core solenoid Hl Nic

2 2 27 27

0( ) 4 10 10c

c c

N Dd d BA dH N AL N N NAdi di di l l

inductance in henrysTotal number of turnscross-sectional area inside of solenoid coil in square meters ( )diameter of solenoid in meterslength of solenoid in meters

LN A

Dc lc

Dc2 4/

where

Hdl dl dl dl Nil

l

l d

l d

l d

l d

l dc

c

c

c

c

c

c

0

2

2

2 20 0 0

( ) ( ) ( )

cD

cl

clN

iH

2Newton/A H/m

7

70

104

104

93/450

Inductance of a Solenoid

This is a single purpose calculation which gives you the inductance value when you make any change in the parameters.Small inductors for electronics use may be made with air cores. For larger values of inductance and for transformers, iron isused as a core material. The relative permeability of magnetic iron is around 200.This calculation makes use of the long solenoid approximation. It will not give good values for small air-core coils, since theyare not good approximations to a long solenoid.

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/indsol.html94/450

Inductance of a Solenoid

D=10 mm

l=50 mmN=30

WD=1.0 mmWire diameter

I

a

b c

d

I

Id

enclosenot doescontour if ,0 enclosescontour if ,I

lH

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/indsol.html

95/450

HW: Inductance of an Air-Core Solenoid

D=10 mm

l=50 mmN=30

WD=1.0 mmWire diameter

I

a

b c

d

An air-core solenoid with construction parameters as shown above, solve the followingproblems:1. Calculate the ideal equivalent inductance of the air-coil solenoid?2. Compute the equivalent inductance of the air-coil solenoid.3. Make a Maxwell simulation of the flux distribution of the air-core solenoid and compare

the simulated inductance with the the analytical result.

Simple Magnetic Circuits

Analogy between electric and magnetic circuits.

cF

c

F

g c

97/450

Electrical-Magnetic Analogy

Magnetic Circuit Electric Circuitmmf NiFlux reluctance permeability

viR1/, where =resistivity

N

i

98/450

Equivalent Electrical Circuit of a Magnetic Circuit

Reluctance

)H :(unit 1-Al

ANi

1

m

mmk

k iN

0k

k

/:law sOhm'

AlR

iv

m

mk

k vRi :law voltage sKirchhoff'

0 :lawcurrent sKirchhoff' k

ki

Magnetic Electrical

Inductance

2Ni

Ni

L

N

i

99/450

Magnetic Circuits of a Gapped Core

mean flux path in the ferromagnetic material

N1gAirgap: Hg

i1

l1 = mean path length

Core: H1

i1 in

H

(a) (b)

100/450

Modeling of a Simple Magnetic Circuit

IlH dH dl H dl Niia

b

gb

a

Hi : Magnetic field intensity in the ferromagnetic materialHg : Magnetic field intensity in the air gap

magnetic motive force (mmf)(unit: Ampere-turns)

H l H l Nii i g g

mean flux path in theferromagnetic material

v

li

ab

mean flux path in the air gapgl

N

i

101/450

Modeling of a Simple Magnetic Circuit

B H B lB

l Niii

ig

gg

B SA dFlux

The surface integral of flux density is equal to the flux.

If the flux density is uniformly distributed over the cross-sectional area, then

i i iB A g g gB A

The streamlines of the flux density are closed, therefore i g

lA

lA Ni

i

i i

g

g g

ii

ii A

l

gg

gg A

l

Nigigi )(

102/450

Modeling of the Air-Gap

gR

Ni

li

ab

lg

mean flux path in the air gap

mean flux path in theferromagnetic material

cR

In general, cg RR

v N

i

103/450

Inductance of a Slotted Ferrite Core

L NB Ai

N Al

c c c

g

2

0

a

b

~

glv

AC: Cross Section Area

N

i

The shearing of an idealized B-H loop due to an air gap.

Sheared B-H Loop

Normal B-H Loop

H

B [tesla]

104/450

Air-gap Fringing Fields

[1] Colonel Wm. T. McLyman, Fringing Flux and Its Side Effects, AN-115 Kg Magnetics Inc. [2] Colonel Wm. T. McLyman, Chapter 3 Magnetic Cores of Transformer and Inductor Design Handbook, Fourth Edition, CRC

Press, April 26, 2011. [3] W.A. Roshen, Fringing field formulas and winding loss due to an air gap, IEEE Transactions on Magnetics, vol. 43, no. 8, pp.

3387-3394, 2007.

gg

gg A

l

The effect of the fringing fields is to increase the effective cross-section area Ag of the air gap. Fringing flux decreases the total reluctance of the magnetic path and, therefore, increases the inductance by a factor, F, to a value greater than the one calculated.

Reluctance of the air gap:

Fringingfields

Flux lines

Air gap

105/450

Fringing Flux at the Gap

Core

Core Core

Core

Core

Core

Fringing Flux

Gap

Minimum Gap Small Gap Large Gap

gl

The effect of the fringing fields is to increase the effective cross-section area Ag of the air gap. The fringing flux effect is a function of gap dimension, the shape of the pole faces, and the shape, size and location of the winding. Its net effect is to shorten the air gap. Fringing flux decreases the total reluctance of the magnetic path and, therefore, increases the inductance by a factor, F, to a value greater than the one calculated. In most practical applications, this fringing effect can be neglected.

106/450

A Simple Wound-Rotor Synchronous Machine

The magnetic structure of a synchronous machine is shown schematically in the right figure. Assuming that rotor and stator iron have infinite permeability ( ), find the air-gap flux and flux density Bg. For this example I = 10 A, N = 1000 turns, g = 1 cm, and Ag = 2000 cm2.

Calculate the air-gap flux density Bg

2

7

2 2 104 10 0.2

gg

g g

lA

1000 10 0.13 Wbg g

F

0.13 0.65 T0.2g g

BA

Stator Air gap length g

Air gap permeability

0N turns

Pole face,area Ag

Rotor

I

Magnetic fluxlines

107/450

Flux linkage, Inductance, and Energy

Faradays Law When magnetic field varies in time an electric field is produced in space as

determined by Faradays Law:

C S

ddt

E ds B da

( ) d dv t Ndt dt

Line integral of the electric field intensity E around a closed contour C is equal to the time rate of the magnetic flux linking that contour.

Since the winding (and hence the contour C) links the core flux N times then above equation reduces

The induced voltage is usually referred as electromotive force to represent the voltage due to a time-varying flux linkage.

( ) dv tdt

108/450

Direction of EMF

The direction of emf: If the winding terminals were short-circuited a current would flow in such a direction as to oppose the change of flux linkage.

max max( ) sin sinct t A B t

tEtNte coscos)( maxmax

maxmaxmax 2 BNAfNE c

max2 BNAfE crms

e(t) N

109/450

Example: Estimate the Inductance of a Gapped Core

The magnetic circuit of Fig. (a) consists of an N-turn winding on a magnetic core of infinite permeability with two parallel air gaps of lengths g1 and g2 and areas A1 and A2, respectively.Find (a) the inductance of the winding and (b) the flux density Bl in gap 1 when the winding is carrying a current i. Neglect fringing effects at the air gap.

i

N turns

Gap 1

Gap 2

Area A1 Area A2

(a)

2g1g

(b)

Ni

1

1R 2R

2

110/450

Example: Plot the Inductance as a Function of Relative Permeability

The magnetic circuit as shown below has dimensions Ac = Ag = 9 cm2, g = 0.050 cm, lc = 30 cm, and N = 500 tums. With the given magnetic circuit, using MATLAB to plot the inductance as a function of core relative permeability over the range 100 r 100,000.

(b)

(a)

iMagneticflux lines

Air gaplength g

Mean corelength lc

Air gap,permeability 0,Area Ag

Magnetic corepermeability ,Area Ac

Winging,N turns

0 0 71 65432 8 9 10410

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Core relative permeability

Indu

ctan

ce [H

]

111/450

Example: Plot the Inductance as a Function of the Air Gap Length

The magnetic circuit as shown below has dimensions Ac = Ag = 9 cm2, lc = 30 cm, and N = 500 tums. With the given magnetic circuit, r = 70,000. , using MATLAB to plot the inductance as a function of the air gap length over the range 0.01 cm g 0.1 cm.

(a)

iMagneticflux lines

Air gaplength g

Mean corelength lc

Air gap,permeability 0,Area Ag

Magnetic corepermeability ,Area Ac

Winging,N turns

112/450

Example: Magnetic circuit with two windings

The following figure shows a magnetic circuit with an air gap and two windings. In this case note that the mmf acting on the magnetic circuit is given by the total ampere-turns acting on the magnetic circuit (i.e., the net ampere turns of both windings) and that the reference directions for the currents have been chosen to produce flux in the same direction.

2 0 01 1 1 1 1 2 2

c cA AN N i N N ig g

1 2

1i 2i

gN1

turnsN2

turns

Air gap

Magnetic corepermeability ,mean core length lc,cross-sectional area Ac

113/450

Example: Analysis of a Switching Inductor

Calculate the current ripple (peak-to-peak) of the inductor current.

dcV

L

A switching inductor can be used as a fundamental energy storage cell with a switching power converting system. Assume components in the following circuit are all ideal, make an analysis of the given problems. Assume the duty ratio for the MOSFET switch is 20%.

D

S

Li

SiDi

48 V20 kHz5 mH

10

DC

s

VfLR

R

114/450

Recommended Books

A Students Guide to Maxwells Equations (Daniel Fleisch), , 20101018

A Student's Guide to Vectors and Tensors, Daniel A. Fleisch, Cambridge University Press, 1st Ed., November 14, 2011.

Introduction to Electrodynamics, David Griffiths, 4th Ed., Addison-Wesley, October 6, 2012.

115/450

Extended Readings

Electricity and Magnetism, W. N. Cottingham and D. A. Greenwood, Cambridge University Press, 1st Ed., November 29, 1991.

Introduction to Electricity and Magnetism, MIT 8.02 Course Notes, Sen-Ben Liao, Peter Dourmashkin, and John Belche, Prentice Hall, 2011.

116/450

References: Magnetic Components & Circuits

[1] A Students Guide to Maxwells Equations, (Daniel Fleisch), , , 20101018.

[2] Chapter 1: Magnetic Circuits and Magnetic Materials, Fitzgerald & Kingsley's Electric Machinery, S.D. Umans, 7th Ed, McGraw-Hill Book Company, 2013.

[3] Daniel A. Fleisch, A Student's Guide to Vectors and Tensors, Cambridge University Press, 1st Ed., November 14, 2011.[4] Chapter 11 Inductance and Magnetic Energy, Introduction to Electricity and Magnetism, MIT 8.02 Course Notes, Sen-Ben Liao, Peter

Dourmashkin, and John Belche, Prentice Hall, 2011. [5] David Griffiths, Introduction to Electrodynamics, 4th Ed., Addison-Wesley, October 6, 2012. [6] W. N. Cottingham and D. A. Greenwood, Electricity and Magnetism, Cambridge University Press, 1st Ed., November 29, 1991. [7] W.G. Hurley and W.H. Wolfle, Transformers and Inductors for Power Electronics: Theory, Design and Applications, Wiley, 1th Ed.,

April 29, 2013. [8] Colonel Wm. T. McLyman, Transformer and Inductor Design Handbook, CRC Press, 4th Edition, April 26, 2011. [9] S.V. Kulkarni and S.A. Khaparde, Transformer Engineering: Design, Technology, and Diagnostics, CRC Press, 2 Ed., September 6,

2012.[10] Frederick W Grover, Inductance Calculations, Dover Publications, ith Ed., October 22, 2009.[11] G. K. Dubey, Fundamentals of Electrical Drives, Alpha Science International, Ltd, March 30th 2001. [12] Chapter 4 Inductances, Design of Rotating Electrical Machines, Juha Pyrhonen, Tapani Jokinen, Valeria Hrabovcova, 2nd Ed., October

2013, Wiley.

117/450

Power Electronic Systems & Chips Lab., NCTU, Taiwan

Modeling of Inductors

Power Electronic Systems & Chips Lab.

~ 118/450

AC Excitation of Ferromagnetic Materials

H (or F)

B (or )

Hysteresis Loopt

i(t)

a

b

c

d

e

N

i

119/450

Magnetic Domains

Magnetic domains oriented randomly. Magnetic domains lined up in the presenceof an external magnetic field.

magnetic moment (dipole) magnetic domain

120/450

Hysteresis Curves of a Ferromagnetic Core in AC Excitation

H

B

Hysteresis Loop

H

B

Br

-Hc

Residual Flux Density

Coercive Force

Magnetization or B-H Curve

area hysteresis loss

saturation

~

121/450

AC Excitation of a Magnetic Circuit

Ac: cross-section surface area

mean flux path in theferromagnetic material

From Faradays law, the voltage induced in the N-turn winding is

Assume a sinusoidal variation of the core flux (t); thus

(t)= maxsint=Ac Bmax sint

maxmax 2)( BfNANwdtdtv c

where and

amplitude of the flux density

v v fNA Brms c 12

2max max maxmaxmax 2 BfNANv c

( )v t

cl

N

i

122/450

AC Excitation of a Magnetic Circuit

Excitation phenomena. (a) Voltage, flux, and exciting current; (b) corresponding hysteresis loop.

123/450

AC Excitation Phenomena of a Magnetic Circuit

To produce the magnetic field in the core requires current in the excitation winding know as the excitation current iFor the given , we can obtain the corresponding i from the B-H hysteresis loop. Because = BcAc , and i = HcBLc /NThe saturated hystersis loop will result peakly excitation current with sinusoidal flux variation

(a) Voltage, flux, and excitation current; (b) corresponding hysteresis loop.

(a) (b)

t

iii i

i

i

t t

CH

sv

sv

N

i

124/450

Core Saturation Due to Over-Excitation

sv

)()(1)( 011

0

tdttvN

tt

t s

0( ) 0t

N

i

125/450

Core Saturation Due to Over-Excitation

sv

N

i

126/450

Inrush Electric Current Due to Saturation

sv

What you assume

What really happen

N

i

127/450

Typical Waveform of Magnetizing Inrush Current

In practical applications, the winding resistance and losses of the core will decay residual flux and the dc offset due to initial volts-sec integration.

A a soft start procedure can be used to reduce this effect and a balance control loop can be used to eliminate this dc offset in application to inverters.

128/450

Core Saturation Due to Residual Flux

Power transformer inrush current caused by residual flux at switching instant; flux (green), iron core's magnetic characteristics (red) and magnetizing current (blue).

sv

)()(1)( 011

0

tdttvN

tt

t s

0( ) 0t

N

i

129/450

Small-Signal Modeling of Practical Inductors

Ideal impedance model is for a simple linear relationship between frequency and impedance.

Not true across the whole frequency range for real components! For practical capacitors and inductors with nonlinear characteristics, its frequency responses

are only valid for small signal perturbation around its operating point this operating point are generally highly dependent on its dc value, frequency, and temperature.

( )Z j

L

01LC

R

1C

(a) Examples of inductor (b) Equivalent circuit

RAC

C

L

RDC

RC

(c) Frequency response

130/450

Building a Model of a Real Inductor

Ideal inductor Perfectly conducting wire Core of ideal magnetic

material

Practical inductor Real wires have small DC resistance Real wire resistance is frequency dependent Real inductor may saturate Real magnetic materials for inductors are both

frequency and temperature dependent! Parasitic capacitance exists between turns of

the coil, between layers if wound in layers Parasitic lead capacitance

dtdiLv LLv

Li

Li

( )R f ( )L f

C

131/450

Simple Model for Real Inductors

The inductor is modeled as a constant inductance with a series connected resistance (RESR).

As frequency increases, the inductive impedance increases This model does not resonate (no capacitance) There is a corner frequency where the inductive impedance begins

to dominate

ESRR L

132/450

Simple Model for Real Inductors

Example Parameter: L=100 nH, R=2

R

L

( )Z j j L R

LR

Time Constant [sec]

31/2 2dB

RfL

Corner frequency [Hz] 3 92 3.18

2 2 100 10 MHzdB

RfL

Improved Model for Practical Inductors

Parallel resonant circuit Resonant frequency is

R = series resistanceC = parallel capacitance

2( ) 1R j LZ j

j RC LC

R L

C

01LC

1( ) ( ) ||Z j R j Lj C

20 1r

LCQR

1

2Q

2

2

14rR

LC L

In general, R and C are quite small, and the resonant frequency can be approximated to the undamped natural frequency 0:

2L CR 01

r LC

[1] R. L. Boylestad, Introductory Circuit Analysis, 12th Edition, Prentice Hall, 2010.[2] Electromagnetic Compatibility Handbook, Kenneth L. Kaiser, CRC Press, 2005. [3] Cartwright, K., E. Joseph, and E. Kaminsky, Finding the Exact Maximum Impedance Resonant Frequency of a Practical Parallel

Resonant Circuit without Calculus, Technology Interface Internat. J., vol.11, no. 1, Fall/Winter 2010, pp. 26-36.134/450

Improved Model for Practical Inductors

Example Parameter: L=100 nH, R=2 C=10 pF

R

LC

1( ) ( ) ||Z j R j Lj C

135/450

More Complex Models for HF Inductor

Fairly accurate model for SMT chip inductor

ESRR

LstrayC

leadL

leadL

136/450

Simple Electro-Magnetic Circuits

Toroidal Inductance

Block Diagram

( )i t

( )v t

n turns

length ml

i

Lv

Equivalent Circuit

v 1n 1

L

i

0

1( ) ( ) (0)t

i t v t dt iL

137/450

Transient Response of Inductance

dcV

( )v t

If the above PWM voltage is applied to an ideal inductor, what will be the current waveform? What about a practical inductor?

( )i t

( )v t

n turns

length ml

( )v t

( )i t

( )t

0

0

0

v

vT

vTL

T

t

t

tot138/450

Inductor with Resistance

Equivalent circuit of a linear inductor with coil resistance

Block Diagram

i

v LLdvdt

R

v Lv

1 1

Li

( )iRR

139/450

Magnetic Saturation

Amp (I)

Weber-turns (=N)

iL

iv iL

RvLv

Li

L

LR Rv 1L

LR

1s

0( )L t

iv

v Lv1

i

R

( )i

Rv

140/450

Saturation of an Inductor (Biased Incremental Inductance)

Amp (I)

Weber-turns (=N)

( )x

xi I

L Ii

Mean path length l

Permeability

Cross-sectional area Ac

Ni

A practical inductor will saturate as the current is increased. The incremental inductance is defined as the inductance at a specified current

with small signal perturbation, this is equivalent to a linear inductance for currentaround this operating point.

Note: In the given example, the current source as a perturbation source.

xI

141/450

Measuring the Incremental Inductance at Specific Operating Points (1/3)

N

Amp (I)

Weber-turns (=N)

( )x

xi I

L Ii

Practical inductors are nonlinear and its incremental inductance (small-signal inductance) is highly dependent on its operating point, such as its average current, the magnitude of current ripples, the switching frequency, and the core temperature, etc.

The winding inductance of a synchronous machine is nonlinear, especially for an interior PMSM. This characteristics is useful for the detection of its rotor pole position under sensorless control. Devise a scheme to measure the incremental inductance for different operation points (A, B, and C)?

dcV

3S

4S

1S

2S

A

B

C

142/450

Measuring the Incremental Inductance at Specific Operating Points (2/3)

Design an inductor with 0.1 mH, average current from 1 A to 6A, and operating withswitching frequency of 20 kHz. An illustrated design example can be found in [1].

Devise an incremental inductance measurement scheme for operation points of A (0A), B(4A), and C (8A) with a current ripple of 20 kHz, 2 A (peak-to-peak).

Make a simulation study in consideration of the RDS(ON) and the diode forward voltage drop. Make experimental verifications for the proposed scheme.

dcV

3S

4S

1S

2S

0.1mH, IL(pp)=2A, 20 kHz inductor

8 Ohm, 50W, Cement Resistors

oR

L ESRR

ABv

A

B

( )2e

ESR o DS ON

L LR R R R

REF: [1] Inductor Design in Switching Regulators (Technical Bulletin SR-1A, Magnetics).pdf

ABv

s eT

143/450

Compute the Inductance of a Toroidal Ferrite Core

[1] Rosa Ana Salas and Jorge Pleite, Simple procedure to compute the inductance of a toroidal ferrite core from the linear to the saturation regions, Materials, no. 6, pp. 2452-2463, 2013.

60-turn toroidal inductor with the TN23/14/7 ferrite core

TN23/14/7 Ring Core

TN23147-3R1 - FerroxcubeEffective Core Parameters

Permeability as a Function of Frequency of Different Materials

146/450

B and H Magnetic Fields Inside the Toroidal Core

Moduli of the B and H magnetic fields as a function of the distance from the center of the inductor core (x = 30 mm) obtained by 2D (red dashed line) and 3D (black solid line) simulations, for (a,b) I = 0.0057 A (linear region); (c,d) I = 0.16 A (intermediate region); (e,f) I = 3 A (saturation region).

147/450

Winding Inductances of an IPMSM

a-bc

3 cos22 2 2

d q d qL L L LL

S N

S

N

SN

S

N

S N

Rotor pole position ()0 3/2 2/2

1.5 qL

1.5 dL

IPMSM stator coil

a-bcLa

b c

a-bcL

148/450

Modeling of the Stator Winding Inductance

1( ) ?rL

1. Define the stator structure, mechanical dimensions, windingmechanism, and material parameters of the segmented motor.

2. Construct an equivalent circuit for a single segment of the statorteeth and calculate its inductance. Make a Maxwell simulation toverify the calculation.

3. Put the segmented teeth into the stator but without the rotor, makea Maxwell simulation to calculate the inductance of a single statorsegment.

4. Define the rotor structure and material parameters and make aMaxwell simulation to calculate the inductance of a single statorsegment as a function of the rotor pole position.

1 ?L

149/450

Extended Readings

Transformer and Inductor Design Handbook, Colonel Wm. T. McLyman, CRC Press, 4th Edition, April 26, 2011.

Transformers and Inductors for Power Electronics: Theory, Design and ApplicationsW.G. Hurley and W.H. Wolfle, Wiley, 1th Ed., April 29, 2013.

Inductance Calculations, Frederick W Grover, Dover Publications, ith Ed., October 22, 2009.

Handbook of Transformer Design and Applications, William Flanagan, McGraw-Hill Education, 2nd Ed., January 22, 1993.

150/450

References: Magnetic Components & Circuits

[1] A Students Guide to Maxwells Equations, (Daniel Fleisch), , , 20101018.

[2] Chapter 1: Magnetic Circuits and Magnetic Materials, Fitzgerald & Kingsley's Electric Machinery, S.D. Umans, 7th Ed, McGraw-Hill Book Company, 2013.

[3] Daniel A. Fleisch, A Student's Guide to Vectors and Tensors, Cambridge University Press, 1st Ed., November 14, 2011.[4] Chapter 11 Inductance and Magnetic Energy, Introduction to Electricity and Magnetism, MIT 8.02 Course Notes, Sen-Ben Liao, Peter

Dourmashkin, and John Belche, Prentice Hall, 2011. [5] W.G. Hurley and W.H. Wolfle, Transformers and Inductors for Power Electronics: Theory, Design and Applications, Wiley, 1th Ed.,

April 29, 2013. [6] David Griffiths, Introduction to Electrodynamics, 4th Ed., Addison-Wesley, October 6, 2012. [7] William Flanagan, Handbook of Transformer Design and Applications, McGraw-Hill Education, 2nd Ed., January 22, 1993. [8] Colonel Wm. T. McLyman, Transformer and Inductor Design Handbook, CRC Press, 4th Edition, April 26, 2011. [9] S.V. Kulkarni and S.A. Khaparde, Transformer Engineering: Design, Technology, and Diagnostics, CRC Press, 2 Ed., September 6,

2012.[10] Frederick W Grover, Inductance Calculations, Dover Publications, ith Ed., October 22, 2009.[11] G. K. Dubey, Fundamentals of Electrical Drives, Alpha Science International, Ltd, March 30th 2001. [12] Chapter 4 Inductances, Design of Rotating Electrical Machines, Juha Pyrhonen, Tapani Jokinen, Valeria Hrabovcova, 2nd Ed., October

2013, Wiley.

151/450

Power Electronic Systems & Chips Lab., NCTU, Taiwan

Transformer

Power Electronic Systems & Chips Lab.

~

Chap15Transformer Design (Erickson 2001).pdf

152/450

Functions of a Transformer

ac input

supplydc output

voltage

+

v1(t)

i1(t)+

v2(t)

i2(t)n1:n2

+

i3(t)

v3(t)

n3

Isolation Turns Ratio Provide Wide Range Output Multiple Outputs Transformer

Input rectificationand filtering

duty cyclecontrol

controlcircuitry

HighFrequency

switch

PowerTransformer

Output rectificationand filtering

Vref

mosfet orbipolar

TPWM

OSC

Signal Coupling

153/450

Objectives of Transformer

Isolation of input and output ground connections, to meet safety requirements

Extend the voltage conversion range Reduction of transformer size by incorporating high frequency

isolation transformer inside converter Minimization of current and voltage stresses when a large step-up

or step-down conversion ratio is needed use transformer turns ratio

Obtain multiple output voltages via multiple transformer secondary windings and multiple converter secondary circuits

Transformer isolation is required for all circuits operating at a dc input voltage of 60 V DC or more.

154/450

Applications of Transformers

Line Frequency Transformers in Electric Power Distribution Systems

High Frequency Transformers in Power Conversions Systems

LOAD

155/450

Samples for Transformer

Three-Phase 200V, 5kVA60Hz Transformer

Single-Phase 250V, 5kVA20kHz Transformer

10W Flyback Switching Transformer

35W Switching Transformer

20 kVA Switching Transformer

EFD Surface Mount Transformer

ETD Switching Transformer

Example: 10W, 12V Flyback Converter

AN-EVAL3BR4765JG 10W 12V SMPS Evaluation Board with CoolSET F3R ICE3BR4765JG (infineon, Aug 2011)

157/450

Switching Waveform of vds in a Practical Flyback Converter Operating in DCM

n1:n2

ton toff0

Primarycurrent

Seccurrent

Switchingvoltage

T

0

0

Ip

ISW

IS

ID

VceorVds

leakageinductance

spike

t

t

t

(discontinuous)

discontinuous

700V

504V317V

374V (220x120%x1.414)187V (110x120%x1.414)

650V

ON OFF

Clamp Diodeforward recovery

Leak. Inductancedemagnetization

Current flows at the secondary side

Leak. Inductance resonates with drain capacitance

Transformer demagnetized

Prim. Inductance resonates with drain

capacitance

margin

in ONp

p

V TIL

s diodeI I

12in o

nV Vn

inV

Q

inVov

DoC

1T

RV

inV

spikeVdsv

158/450

Ideal Transformer

v p

i p

vs

is

NsN p

Ideal wires (No winding loss)Perfect coupling (No leakage flux)An ideal core (No core loss)No Energy Storage

vv

NN

p

s

p

s

sspp iNiN (b) symbols of a transformer(a) sketch of an ideal transformer

i p is

vsv p

159/450

Dot Convention to Denote the Polarity of a Transformer

i p is

vsv p

symbols of a transformer1. If the primary voltage is positive at the dotted end of the winding with respect to the

undotted end, then the secondary voltage will be positive at the dotted end also.Voltage polarities are the same with respect to the dots on each side of the core.

2. If the primary current of the transformer flows into the dotted end of the primarywinding, the secondary current will flow out of the dotted end of the secondarywinding.

Physical meaning of the dot convention:i

i

produce positive F

produce negative F

i p is

vsv p

Ideal Transformer

For ideal transformer, there is no leakage flux, therefore, the net MMF must be maintain zero at any instant.

An ideal transformer has an infinite magnetizing inductance and therefore, a zero magnetizing current im.

Ideal wires (No winding loss)Perfect coupling (No leakage flux)An ideal core (No core loss)No Energy Storage

161/450

A Simple Transformer Model

Multiple Winding Transformer Equivalent Circuit Model

...)()()(0

...)()()(

332211

3

3

2

2

1

1

tintintinn

tvn

tvn

tv

)(1 ti )(2 ti21 : nn

)(3 ti

3: n

)(1 ti )(2 ti

)(1 tv )(2 tv

21 : nn

)(3 ti

)(3 tv

3: n

)(1 ti

)(tiM

ML

idealtransformer

)(1 tv )(2 tv

)(3 tvMagnetizing Inductance

162/450

Transformer with Magnetizing Inductance

A practical transformer need a non-zero magnetizing current im to buildup the magnetizing field for energy coupling.

A real inductor exhibits saturation, hysteresis, and loss. Modeling of the core material If the secondary winding is disconnected: Only primary winding with core are left; primary winding then behaves as an inductor; the resulting inductor is the magnetizing inductance, referred

to the primary winding. Magnetizing current causes the ratio of winding currents to differ

from the turns ratio

163/450

The Magnetizing Inductance LM

Models magnetization of transformer core material

Appears effectively in parallel with windings

If all secondary windings are disconnected, then primary winding behaves as an inductor, equal to the magnetizing inductance

At dc: magnetizing inductance tends to short-circuit. Transformers cannot pass dc voltages

Transformer saturates when magnetizing current iM is too large

Transformer core B-H characteristic

dttvtB )()( 1

)()( titH M

ML slope

saturation

164/450

Magnetizing Inductance and Leakage Inductance

Lm: magnetizing inductance

Block Diagram Representation of a Transformer with Magnetizing Inductance

165/450

Leakage Inductances

+_

+

_

+_

+

_

1l 2l1v 2v

1v 2v

1i 2i

1i 2i1l 2l

m

m

166/450

Coupling Coefficient and Effective Turns Ratio

effective turns ratio

mutual inductance

primary and secondary self-inductances

1 111 12

2 212 22

v t i tL L dv t i tL L dt

1 2 212

1mp

n n nL LR n

111 1 12

2l

nL L Ln

222 2 12

1l

nL L Ln

2211

eLnL

12

11 22

LkL L

Ideal

coupling coefficient

1i 1lL 2lL

1v 2v

2i1 2:n n

112

2mp

nL Ln

167/450

Total Leakage Inductance

'2lL

'21total, lll LLL

2

2

2

1'2 ll LN

NL

1i 2i

1v 2v

1i 2i

1v 2v

mi

mL 1e

1 mi i1lL 2lL

1 mi i1R 1lL

2e

1N 2N

1N 2Nmi

mL

168/450

Excitation of a Real Transformer

When an ac power source is connected to a real transformer, a current flows in itsprimary winding, even when the secondary winding is open-circuited. Thisexcitation current is the current required to produce flux in a real ferromagnetic core.

excitation current = magnetization current + core-loss current

magnetization current im : the current required to produce the flux in the transformer core

core-loss current ih+e : the current required to make up for core losses.

i i iex m h e

v p vs

is

NsN p

pi

Transformer core B-H characteristic

dttvtB )()( 1

)()( titH M

ML slope

saturation

169/450

Magnetization Current in a Real Transformer

If the leakage flux can be neglected:

If the primary voltage is:

then

dttvN PP)(1

[wb] tN

VdttVN P

MM

P

sincos1

Volts] [cos tVv MP

With a given flux excitation, the magnetization current can bederived from the magnetization curve of the transformer core.

Transformer core B-H characteristic

dttvtB )()( 1

)()( titH M

ML slope

saturation

170/450

Magnetization Curve in a Real Transformer

(a) The magnetization curve of thetransformer core;

, Wb

(a)

(t) andvp(t)

( ) sint VN

tMp

t F = N i

(b)

F, A turns.

(b) the magnetization current caused bythe flux in the transformer core.

mi

pv

171/450

Notes on Magnetization Current

1. The magnetization current in the transformer is not sinusoidal. Thehigher-frequency components in the magnetization current are dueto magnetic saturation in the transformer core.

2. Once the peak flux reaches the saturation point in the core, a smallincrease in peak flux requires a very large increase in the peakmagnetization current.

3. The fundamental component of the magnetization current lags thevoltage applied to the core by 90o.

4. The higher-frequency components in the magnetization current canbe quite large compared to the fundamental component. In general,the further a transformer core is driven into saturation, the larger theharmonic component will become.

172/450

Note on Core Loss Current

1.Core-loss current is greatest as the flux passes through zero.2.The core-loss current is nonlinear because of the nonlinear effects of

hysteresis.3. The fundamental component of the core-loss current is in phase with

the voltage applied to the core.

The core-loss current in a transformer.

t

h ei h ei

173/450

Excitation of a Real Transformer

excitation current = magnetization current + core-loss current

sNN p

(a) Excited transformer with no load. (b) Excitation current

pv

exipi si

pv sv t iCH

(c) B-H characteristics of the core

174/450

Equivalent Circuit of a Practical Transformer

Major considerations in the construction of the transformer model

Copper Losses (proportional to current square)Eddy Current Losses (proportional to voltage square)Hysteresis Current Losses (proportional to excitation frequency)Leakage Flux (represented by self-inductance)Magnetization Flux (represented by magnetization inductance)

idealtransformer

magnetization inductance (reactance)core-loss resistance

primarywinding

resistance

primarywinding

self-inductance

Rc

1i

1v

mi

mL

1 mi i1R 1lL 2i

2v1e

2R2lL

2e

1N 2N

175/450

Secondary Referred to Primary

Rc

Rc

'2lL ' 2lR

'21total, lll LLL

2

2

2

1'2 ll LN

NL

2

2

2

1'2 RN

NR

'21total RRR

1i 2i

1v 2v

1i 2i

1v 2v

mi

mL 1e

1 mi i

mi

1R 1lL 2R2lL

mL

1 mi i1R 1lL

2e

1N 2N

1N 2N

176/450

Energy Storage in a Transformer

An ideal transformer stores no energyall energy is transferred instantaneously from input to output.

In practice, all transformers do store some undesired energy. Leakage inductance represents energy stored in the non-magnetic regions between

windings, caused by imperfect flux coupling. In the equivalent electrical circuit, leakage inductance is in series with the windings, and the stored energy is proportional to load current squared.

Mutual inductance (magnetizing inductance) represents energy stored in the finite permeability of the magnetic core and in small gaps where the core halves come together. In the equivalent circuit, mutual inductance appears in parallel with the windings. The energy stored is a function of the volt-seconds per turn applied to the windings and is independent of load current.

177/450

Transformer Core Construction

Core-type provides larger winding window and even flux density around the core. However, suffers with higher leakage flux.

Shell-type provides lower leakage flux and the magnetic flux has a closed path around the coils, this has the advantage of decreasing core losses and increasing overall efficiency.

Modeling of HF Transformer

(a) A simple 1:1 transformer designed for a 100 kHz, 60 W forward converter.

(c) Equivalent circuit model used for 2-winding transformer.

(b) Winding layout.

Different types of laminated cores

179/450

Transformers for Asymmetrical and Symmetrical Switching Converters

Asymmetrical Converter Symmetrical Converter

2Bs

B

H

Bssymmetricalconverters

asymmetricalconverters

symmetricalconverters

forwardconverter

flybackconverter

AvailableFlux swing

A

BCDN2

N1

v1

i1

v2

i2

Ll2Ll1

Lm

A

B

C

D

180/450

Induction Machine as a Rotary Transformer

V s

Is Rs jXsl

I m

Rc jXm E s

+

_

+

_

(a)

Er

Ir Rr

+

_jsXrl

(b)

Equivalent Circuit of an Induction Machine

n

equivalent circuit of the rotor

a s

as'

b s'

bs

c s

c s'

br

b r' cr

c r'

a r

a r'

A

C

B

S1

S2

S3

S4

S5

S6

3-PhasePowerSupply

oVdc U VW

asi

bsi

csi

asv

bsv csv

arv

brv crv

ari

bri

cri

181/450

Sinusoidal Distribution of Stator Windings

The details of the slot shape are not shown for clarity

sinmwinding density (Turns/Radian)

0

Idealized induction machine illustrating sinusoidal distribution of one phase winding

0

g

Inductances of AC Machinesabcs s abcs abcsr p v i

( )

as abs acs

abcs s abs bs bcs abcs

acs bcs cs

L L LL L LL L L

i

, , ,

( ) , , ,

, , ,

as ar as br as cr

abcs r bs ar bs br bs cr abcr

cs ar cs br cs cr

L L LL L LL L L

i

, , ,

( ) , , ,

, , ,

ar as ar bs ar cs

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-axisar

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Magnetic axes of a three phase induction machine

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Extended Readings

Transformer Engineering: Design, Technology, and Diagnostics, S.V. Kulkarni and S.A. Khaparde, CRC Press, 2 Ed., September 6, 2012.

Transformer Design Principles: With Applications to Core-Form Power Transformers, Robert M. Del Vecchio, Bertrand Poulin, Pierre T. Feghali, Dilipkumar M. Shah and Rajendra Ahuja, CRC Press; 2 edition (June 2, 2010)

Transformers & Induction Machines, M. V. Bakshi and U. A. BakshiTechnical Publications, 2009

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References: Magnetic Components & Circuits

[1] A Students Guide to Maxwells Equations, (Daniel Fleisch), , , 20101018.

[2] Chapter 1: Magnetic Circuits and Magnetic Materials, Fitzgerald & Kingsley's Electric Machinery, S.D. Umans, 7th Ed, McGraw-Hill Book Company, 2013.

[3] Daniel A. Fleisch, A Student's Guide to Vectors and Tensors, Cambridge University Press, 1st Ed., November 14, 2011.[4] Chapter 11 Inductance and Magnetic Energy, Introduction to Electricity and Magnetism, MIT 8.02 Course Notes, Sen-Ben Liao, Peter

Dourmashkin, and John Belche, Prentice Hall, 2011. [5] W.G. Hurley and W.H. Wolfle, Transformers and Inductors for Power Electronics: Theory, Design and Applications, Wiley, 1th Ed.,

April 29, 2013. [6] David Griffiths, Introduction to Electrodynamics, 4th Ed., Addison-Wesley, October 6, 2012. [7] Colonel Wm. T. McLyman, Transformer and Inductor Design Handbook, CRC Press, 4th Edition, April 26, 2011. [8] S.V. Kulkarni and S.A. Khaparde, Transformer Engineering: Design, Technology, and Diagnostics, CRC Press, 2 Ed., September 6, 2012.[9] Frederick W Grover, Inductance Calculations, Dover Publications, ith Ed., October 22, 2009.[10] G. K. Dubey, Fundamentals of Electrical Drives, Alpha Science International, Ltd, March 30th 2001. [11] Chapter 4 Inductances, Design of Rotating Electrical Machines, Juha Pyrhonen, Tapani Jokinen, Valeria Hrabovcova, 2nd Ed., October

2013, Wiley.

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Power Electronic Systems & Chips Lab., NCTU, Taiwan

Force and Torque Generation in Electric Machines

Power Electr