【交流驅動】03：Magnetic Circuits

Power Electronic Systems & Chips Lab., NCTU, Taiwan

Magnetic Circuits

電力電子系統與晶片實驗室 Power Electronic Systems & Chips Lab. 交通大學 • 電機控制工程研究所 Chapter 1 Magnetic Circuits and Magnetic Materials, Fitzgerald & Kingsley's Electric Machinery, 7th Ed, S.D. Umans, McGraw-Hill Book Company, 2013.

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台灣新竹‧交通大學‧電機控制工程研究所‧電力電子實驗室~鄒應嶼 教授

Modeling the Stator Inductance of an IPMSM L (t0 ) R

vi

vR

vi

1 s

iL L

vL

1 L

iL

vR RL

L1 ( r )  ?

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.

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Modeling of Synchronous Machine in dq-Frame 

q

s

is

b  axis

id Ld



q

iq Lq d

λs

ib

vb i1q

PM

s

1q



λPM

e



a' r

1d

i1d

fd

vF

vs

d

ia

iF

va

a  axis

vc

r 

PM

id

Electric Equations:

a

ic

Rs

vd  Rs  sLd e Lq id   sPM  v     L i     vd  R sL q e d s q     q  ePM 

c  axis

Ld

e Lqiq

iq

sPM

Torque Equations: 3P 3P d iq  (Ld  Lq )id iq   (id Ld  PM )iq  (Ld  Lq )id iq  Te  22 22

Rs

Lq e Ld id

vq

ePM

Te  Tf  TL  J

dm  Bm dt

Torque Characteristics of Synchronous Machines q

q q

q

d

d

d d

Ld  Lq

Ld  Lq

Ter  0

Tem  Ter

Te 

Ld  Lq

Ld  Lq

m  0

Tem  0

Tem  Ter

3P [(id Ld  PM )iq  iq id ( Ld  Lq )] 22

Tem

Ter

Inductance Plays a Key Role in Motor Characteristics L1 ( r )  ?

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. Generated electric torque of synchronous machine:

Te 

3P [ I q m  I q I d ( Ld  Lq )] 22

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

Basic Notations for Electromagnetism 電場強度

Electric field strength

E

[V/m]

磁場強度

Magnetic field strength

H

[A/m]

電力線密度

Electric flux density

D

[C/m2]

磁力線密度

Magnetic flux density

B

[Vs/m2], [T]

電流密度

Current density

J

[A/m2]

電荷密度

Electric charge density, dQ/dV

ρ

[C/m3]

D  E

 電容率 permittivity of free space (Farads/m)

B  H

 磁導率 permeability of free space (Henrys/m) 6/91

1865年～電磁理論誕生～馬克士威電磁方程式 1865年英國物理學家詹姆斯·馬克士威以先前的電磁研究成果為基礎，經 由有系統地整理和綜合提出了由20個等式和20個變量組成的電磁理論。 現今電磁理論的數學形式分別由英國的奧利弗·黑維塞(Oliver Heaviside) 、 美 國 的 約 西 亞 · 吉 布 斯 (Josiah Gibbs) 、 與 德 國 的 海 因 里 希 ‧ 赫 茲 (Heinrich Hertz)於1884年前後以向量形式重新表達成目前的四組方程式。

   



Source of Electric Field   E  0 Changing Electric Field Source of Magnetic Field   B  0 Changing Magnetic Field

B t E     B  0  J   0  t  

E  

電磁學天堂祕笈：輕鬆解析最實用的馬克士威方程式 A Student’s Guide to Maxwell’s Equations 作者：夫雷胥 (Daniel Fleisch), 譯者：鄭以禎 出版社：天下文化, 出版日期：2010年10月18日

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Maxwell's Equations (1860s~1970s) 現今使用的馬克士威的方程式包含四個方程式組，它們分別描述了靜電場 、靜磁場、感應電場、與感應磁場的關係，其積分與微分形式表示如下：

 電場的高斯定律 (Gauss’s Law for Electric Field) Q enc  ˆ nda E     E  

0

S

0

 法拉第定律 (Faraday’s Law)



C

E  dl  

d dt



S

ˆ B  nda

E  

B t

 磁場的高斯定律 (Gauss’s Law for Magnetic Field)



S

ˆ B  nda 0

B  0

 安陪-馬克士威定律 (The Ampere-Maxwell Law)

d  B l      d I 0  enc 0  C dt 

 E   ˆ E  nda       B J   S 0 0   t  

Symbols and Units of Electromagnetic Quantities

Summary of Quasi-Static Electromagnetic Equations

Electromechanical Dynamics (MIT Course Notes)

s r

is

ir

s  is Ls  Lsr ( )ir r  Lsr ( )is  ir Lr dL ( ) Te  is ir sr d

Te



Electromechanical Dynamics, Discrete Systems (Part 1), Herbert H. Woodson and James R. Melcher, Wiley, 1st Ed., January 15, 1968. REF: Electromechanical Dynamics - Part 1 Discrete Systems (Woodson & Melcher, MIT 1968)

Basic Relations of Electrical and Magnetic Field

v(t )

Faraday’s Law

terminal characteristics

B (t ),  (t ) Core characteristics

H (t ), F (t )

i (t ) Ampere’s Law

Electrical Circuits

Magnetic Circuits 12/91

Magnetic Field

Magnetic fields are produced by electric currents, which can be macroscopic currents in wires, or microscopic currents associated with electrons in atomic orbits. The magnetic field B is defined in terms of force on moving charge in the Lorentz force law. The interaction of magnetic field with charge leads to many practical applications. Magnetic field 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 the Lorentz force law Fmagnetic = qvB to be composed of (Newton x second)/(Coulomb x meter). A smaller magnetic field unit is the Gauss (1 Tesla = 10,000 Gauss). 13/91

Right-Handed System and Left-Handed System z

z x

y y

Left-Handed System

x

Right-Handed System

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Magnetic Field of Current: Right-Handed Rule

The magnetic field lines around a long wire which carries an electric current form concentric circles around the wire. The direction of the magnetic field is perpendicular to the wire and is in the direction the fingers of your right hand would curl if you wrapped them around the wire with your thumb in the direction of the current. 15/91

Ampere’s Law

(a)

H

 dl 



i

(b)

(a) General formulation of Ampere’s law. (b) Specific example of Ampere’s law in the case of a winding on a magnetic core with air gap.

 Direction of magnetic field due to currents  Ampere’s Law: Magnetic field along a path

16/91

Ampere’s Law

 H  dl  I

 B  dl  I

B dl

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)  = permeability of free space

 0  4   10 7 H / m

r

I, if contour encloses I

 H  dl  0, if contour does not enclose I

  0

r = relative permeability (between 2000-80,000 for ferromagnetic materials) 17/91

Permeability: Relationship Between B and H Ampere,s Law

 H  dl  I

permeability =  =

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

B H

r 

 0

 = permeability of free space  0  4   10  7 H / m

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

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

cgs:

 gauss 

tesla

 0 = 1   10 4   oersted  oersted

mks:

 henrry   meter 

 0 = 4  10 7 

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磁通量單位：韋伯 (Wb), 磁通量密度單位：特斯拉(Tesla) 磁通量的國際制(SI)單位，紀念德國物理學家韋伯而命名。簡稱韋﹐用Wb表示。 韋伯定義如下﹕令通過單匝線圈的磁通量在 1秒鐘內均勻地減小到零。如果在該線 圈中激發產生的感應電動勢為 1 伏特，則原來通過該線圈的磁通量為 1 韋伯。即 1Wb=1V．s。 韋伯是國際單位制的導出單位﹐用基本單位表示的關係式為： 米2‧千克‧秒-2 ‧安培-1 (m 2 ‧kg‧s-2‧A-1) = 伏秒 (Volt‧sec) 1882 年西門子在英國科學進展協會上第一次提出以『韋伯』作為磁通量單位， 1895年得到英國科學進展協會承認，1948年得到國際計量大會的承認。

韋伯和 CGS電磁系中的磁通量單位馬克斯威之間的換算關係為﹕ 1韋伯相當於108馬克斯威。[1 Wb = 108 Maxwell]

磁通量密度的單位是特斯拉(Tesla)。 [1 Tesla = 1 Wb/m2] 1 oersted = 1000/4 ampere/turn = 79.57747154594 ampere/meter  80 A/m 19/91

Properties of Ferromagnetic Materials



B, Wb/m2

B

 A is the cross-sectional area A

B  r0H

1.4 1.2

i

1.0 0.8

N

0.6 0.4 0.2

DC Excitation

0

0

200

400 600 800 H, A-turn/m

1000

A toroidal coil and the magnetic field inside it.

 Ferromagnetic materials, composed of iron and alloys of iron with cobalt, tungsten, nickel, aluminum, and other metals, are by far the most common magnetic materials.  Transformers and electric machines are generally designed so that some saturation occurs during normal, rated operating conditions.

20/91

B-H Curve, Permeability, and Incremental Permeability B  H   r  0 H

Relation between B- and H-fields. Incremental Permeability

B

H Bs

B

  B



H

B H

B B  H H H

Linear region Hs

Magnetic intensity H, [A-turns/m]

 The B-H characteristics of a core material is high nonlinear. Depends on its average current, current ripple, switching frequency, and operation temperature.  When measuring the inductance of a magnetic circuit, it should first to identify its operating point.

21/91

B-H Curve of Major Materials The set of magnetization curves as shown in left figure represents an example of the relationship between B and H for soft-iron and steel cores but every type of core material will have its own set of magnetic hysteresis curves. You may notice that the flux density increases in proportion to the field strength until it reaches a certain value were it can not increase any more becoming almost level and constant as the field strength continues to increase.

This is because there is a limit to the amount of flux density that can be generated by the core as all the domains in the iron are perfectly aligned. Any further increase will have no effect on the value of M, and the point on the graph where the flux density reaches its limit is called Magnetic Saturation also known as Saturation of the Core and in our simple example above the saturation point of the steel curve begins at about 3000 ampere-turns per meter. 22/91

B-H Characteristics of a Magnetic Material



Hs

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

Flux Density or B-Field B  H   r  0 H Cross-sectional area A H-field

H

i

  BA

N

i

  N (a)

(b)

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

The total flux pass through the coil with N turns is called flux linkage and named as .   N  24/91

Continuity of Flux   

   B dA

A(closed surface)

A

B dA  0

A2

A1

1

2

3

A3

1  2  3  0

B1 A1  B2 A2  B3 A3  0 or



k

0

k

25/91

Magnetic Cores Ideal Inductor i

 Negligible winding resistance  Perfect coupling between windings

v



N

 An ideal core

v  N

d dt

d 

1 vdt N

 ( t1 )   ( t 0 ) 

1 N



t1

t0

vdt

The above equation shows that the change in flux during a time interval t0-t1 is proportional to the integral of the voltage over the interval, or the volt-seconds applied to the winding. 26/91

Ideal Inductor [Define its Initial Conduction] 



i

v

i

N

(a) Circuit model.

(b) -i characteristic (or B-H curve).

v

v



0



t

0

(c) v is a step input; (t0) = 0.

(d) v = Vm sin t ; (t0) = 0. v

v



 0

t

(e) v is a square wave; (t0) = -m.

0

(f) v = Vm sin t ; (t0) = -m.

27/91

Magnetic Field Strength H of Some Configurations long, straight wire

Toroidal Coil

Long solenoid

28/91

Inductance of Wound Magnetic Core 

  magnetic flux per turnwebers (Wb) [1 Wb = 108 Maxwell]

i

v

B  magnetic flux density webers/meter2 (teslas)   flux linkage webers

N

A  core cross-sectional area square meters H  magnetic field strength ampere-turns/meter N  number of turns

d di d  N  dt dt dt d d  L  N di di

v  L

i  coil current ampere l m  mean length of magnetic flux path meters   permeability henrys/meter (4×10-7 in perfect vacuum)

  N   BA

and

L  inductancehenrys

Ni H  lm

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

N 2 A dB N 2 A L   l m dH lm

29/91

Inductance of a Core i

A

N

v

N2  A L   r  o le



le

 r  o  C1  N 2

L  N2

 r  C  N 2

L  le1

(a)

Flux saturation



slope L

(b)

i

L A

C1 

A l

C   o  C1   o 

A l

The inductance L represents the capability of magnetic flux density produced by unit current of a circuit loop. 30/91

Magnetic Reluctance and Permeance Cross-sectional area A

Mean path length l

i

 H  dl  H 

N

H  Permeability 

  Magnetic-motive force (mmf)

  Ni l A

Reluctance



Permeance

1  

Hl  Ni Ni l

Ni    B  l A

ANi Ni     l  l     A   



 Ni    31/91

Inductance of a Toroidal Core (Self Inductance) Cross-sectional area A

Mean path length l

Weber-turns (=N)

i N L

 i Amp (I)

Permeability 

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

L

 i



N i



 

ANi  l

L

 i



N ANi A   N 2 i l l

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

32/91

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. 33/91

A toroidal coil and the magnetic field inside it.

Energy Stored in a Core N: number of turns Mean path length l

Cross-sectional area Ac

The energy stored in the core: t

E L   Pdt  0



t

0

Li ' di ' 

1 2 LI 2

The energy density (energy/volume) is:

I

LI 2 1 1   N 2 Ac  B   Ac l Ac l 2  l 1 B2 B2   2  2r 0 1 2

Permeability 

A N2 LN    l 2

  B 2l 2   2 2    N 

The energy stored in the core:

EL 

1 2 LI   BVcore 2

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. 34/91

Typical Energy Density of a Ferrite Core Ec B2 B   2r 0 Ve

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 0 .3 2 Ec B2 B     17 . 9 J/m 3 7 2  r  0 2  2000  4  10 Ve 0  4  10 7 H/m  4  10 Newton/A 7

2

Ec  18 J/m 3  18 μJ/cm Ve

(  r  2000, B sat  3000G)

3

一般典型的鐵氧體鐵心 (Ferrite core)的儲能密度約為每立方公分18微 焦爾。假設開關頻率為100 kHz，最大開關責任比為50%，工作於臨界 導通模式(CRM)，則其可處理之平均功率約為3.6瓦。以36瓦的反馳式 電源供應器為例，其鐵心體積約為10立方公分。

35/91

Inductance of Air-Core Solenoid  H  dl  N i Dc

lc



lc

0

Hdl  

Long air-core solenoid

lc  d

lc

( 0 )dl  

Hl c  Ni

2 lc  d

lc  d

( 0 )dl  

2 lc  2 d

2 lc  d

( 0 )dl  Ni

H N  i lc

d d ( BA) dH 4 107 N 2 A N 2 Dc2 2 N  0 NA   107 LN di di di lc lc

0  4  10 7 H/m  4  10 7 Newton/A 2

where L  inductance in henrys N  Total number of turns A  cross-sectional area inside of solenoid coil in square meters (   D c2 / 4 ) Dc  diameter of solenoid in meters

l c  length of solenoid in meters 36/91

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 is used 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 they are not good approximations to a long solenoid. http://hyperphysics.phy-astr.gsu.edu/hbase/electric/indsol.html 37/91

Inductance of a Solenoid D=10 mm

I b

c

l=50 mm

N=30

WD=1.0 mm

a

d

Wire diameter

 I , if contour encloses I does not enclose I

 H  d l   0, if contour http://hyperphysics.phy-astr.gsu.edu/hbase/electric/indsol.html

38/91

HW: Inductance of an Air-Core Solenoid D=10 mm

I b

c

l=50 mm

N=30

a

WD=1.0 mm

d

Wire diameter

An air-core solenoid with construction parameters as shown above, solve the following problems: 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. 39/91

Simple Magnetic Circuits



c

F

 

F c

 g  c

Analogy between electric and magnetic circuits.

Electrical-Magnetic Analogy Magnetic Circuit mmf Ni Flux  reluctance  permeability 

Electric Circuit v i R 1/, where =resistivity





i



N



41/91

Equivalent Electrical Circuit of a Magnetic Circuit 

Reluctance



i





N



Inductance   

Magnetic Ni





1 A



N N 2  L   i i

Electrical Ohm' s law :

l v  R  A/ i

  k   N mim

Kirchhoff' s voltage law : i  Rk   v m



Kirchhoff's current law :  ik  0

k

k

k

m

k

0

l (unit : H -1 ) A

m

k

42/91

Magnetic Circuits of a Gapped Core

l1 = mean path length i1

in i1

Airgap: Hg N1

g

H Core: H1

(a)

(b)

mean flux path in the ferromagnetic material 43/91

Modeling of a Simple Magnetic Circuit

i v

mean flux path in the air gap

lg b a

N

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

li mean flux path in the ferromagnetic material

 H  dl  

b a

I

a

H i dl   H g dl  N i b

Hi : Magnetic field intensity in the ferromagnetic material Hg : Magnetic field intensity in the air gap

H i l i  H g l g  Ni

44/91

Modeling of a Simple Magnetic Circuit Bi

B  H

i

li 

Bg

g

l g  Ni

Flux

 

B A

 dS

The surface integral of flux density is equal to the flux. If the flux density is uniformly distributed over the cross-sectional area, then

 g  B g Ag

 i  Bi Ai

The streamlines of the flux density are closed, therefore  i   g lg li    Ni  i Ai  g Ag li  i Ai

i 

g 

 i    g    (  i   g )  Ni

lg  g Ag

45/91

Modeling of the Air-Gap mean flux path in the air gap



lg i

v

Rg N

li

b a

mean flux path in the ferromagnetic material

Ni Rc

In general,

Rg  Rc

46/91

Inductance of a Slotted Ferrite Core AC: Cross Section Area i

v

N

lg

b a

NBc Ac N 2  0 Ac L  i lg The shearing of an idealized B-H loop due to an air gap. 台灣新竹‧交通大學‧電機控制工程研究所‧電力電子實驗室~鄒應嶼 教授

47/91

Air-gap Fringing Fields Reluctance of the air gap:

g 

lg

 g Ag

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.

[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.

48/91

Fringing Flux at the Gap

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. 49/91

A Simple Wound-Rotor Synchronous Machine Calculate the air-gap flux density Bg

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.

2  lg

2  10 2 g    g Ag 4  10 7  0.2



F 1000  10   0.13 Wb g g

Bg 

 Ag



0.13  0.65 T 0.2

50/91

Flux linkage, Inductance, and Energy Faraday’s Law  When magnetic field varies in time an electric field is produced in space as determined by Faraday’s Law:

 E  ds   C

d B  da  dt S

v(t ) 

d 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.

v(t ) 

d d N dt dt

51/91

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.

 (t )  max sin t  Ac Bmax sin t

e(t )   Nmax cos t  Emax cos t e(t)

N

Emax   Nmax  2 f NAc Bmax

Erms  2 f NAc Bmax

52/91

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.

53/91

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.

(a)

(b)

54/91

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)

55/91

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.

 0 Ac  g

1  N1  N12 

  0 Ac  i N N 1 1 2   g

  i2 

56/91

Example: Analysis of a Switching Inductor 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%. iL L

R

Vdc

iS S

D iD

VDC  48 V f s  20 kHz L  5 mH R  10 

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

57/91

Recommended Books

電磁學天堂祕笈：輕鬆解析最實用的馬克士威方程式 A Student’s Guide to Maxwell’s Equations 作者：夫雷胥 (Daniel Fleisch), 譯者：鄭以禎 出版社：天下文化, 出版日期：2010年10月18日

A Student's Guide to Vectors and Tensors, Daniel A. Fleisch, Cambridge University Press, 1st Ed., November 14, 2011. Introduction to Electricity and Magnetism, MIT 8.02 Course Notes, Sen-Ben Liao, Peter Dourmashkin, and John Belche, Prentice Hall, 2011.

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

58/91

References: Magnetic Circuits [1] [2] [3] [4]

G. K. Dubey, Fundamentals of Electrical Drives, Alpha Science International, Ltd, March 30th 2001. Chapter 1: Magnetic Circuits and Magnetic Materials, Fitzgerald & Kingsley's Electric Machinery, S.D. Umans, 7th Ed, McGraw-Hill Book Company, 2013. 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. Chapter 4 Inductances, Design of Rotating Electrical Machines, Juha Pyrhonen, Tapani Jokinen, Valeria Hrabovcova, 2nd Ed., October 2013, Wiley.

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

59/91

Power Electronic Systems & Chips Lab., NCTU, Taiwan

Modeling of Practical Inductors 電力電子系統與晶片實驗室 Power Electronic Systems & Chips Lab.

交通大學 • 電機控制工程研究所

台灣新竹‧交通大學‧電機控制工程研究所‧電力電子實驗室~鄒應嶼 教授

60/91

Modeling of Practical Inductors Ideal impedance model is for a simple linear relationship between frequency and impedance. Z ( j )

RAC

L

RDC RC

1 C

C R

L

0  (a) Examples of inductor

1 LC



(c) Frequency response

(b) Equivalent circuit

 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.

61/91

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

iL iL

vL

vL

diL dt

L( f )

R( f ) C

62/91

Simple Model for Real Inductors RESR

L

 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

63/91

Simple Model for Real Inductors Example Parameter: L=100 nH, R=2 

L R Z ( j )  j L  R

 f 3dB

L R

Time Constant [sec]

1/  R Corner frequency [Hz]   2 2  L

f3dB 

R 2   3.18 MHz 2  L 2 100109 64/91

Improved Model for Practical Inductors L

R

 Parallel resonant circuit  Resonant frequency is r  0 1   2

C

L 1  Q  C 0  2Q R

R = series resistance C = parallel capacitance

1 Z ( j )  ( R  j L ) || jC Z ( j ) 

R  j L 1  j RC   2 LC

1 LC

r 

1 R2  LC 4 L2

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

L  CR 2

r  0 

1 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.

Improved Model for Practical Inductors Example Parameter: L=100 nH, R=2 C=10 pF

L C R

Z ( j )  ( R  j L) ||

1 jC

66/91

More Complex Models for HF Inductor Fairly accurate model for SMT chip inductor

67/91

Simple Electro-Magnetic Circuits

Equivalent Circuit

Toroidal Inductance Block Diagram 68/91

Transient Response of Inductance

Vdc

u (t )

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

69/91

Inductor with Resistance

Block Diagram Equivalent circuit of a linear inductor with coil resistance

70/91

Magnetic Saturation 

Weber-turns (=N)

L

 i Amp (I)

71/91

AC Excitation of Ferromagnetic Materials 

B (or  )

i

b c

N a

H (or F)

i(t) e d

t

Hysteresis Loop

72/91

Magnetic Domains

magnetic moment (dipole)

magnetic domain

Magnetic domains oriented randomly.

Magnetic domains lined up in the presence of an external magnetic field. 73/91

Hysteresis Curves of a Ferromagnetic Core in AC Excitation

B

B

Magnetization or B-H Curve

Residual Flux Density Br Coercive Force

saturation -Hc

H

H

area  hysteresis loss

Hysteresis Loop

台灣新竹‧交通大學‧電機控制工程研究所‧電力電子實驗室~鄒應嶼 教授

74/91

AC Excitation of a Magnetic Circuit  Ac: cross-section surface area

i N

v(t)

mean flux path in the lc ferromagnetic material

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

(t)= maxsint=Ac Bmax sint

amplitude of the flux density

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

v(t)  where

d  Nw  max  2  fNA c B max dt

v rms 

1 v max  2fNAc Bmax 2

and

v max   N  max  2  fNA c B max 75/91

AC Excitation of a Magnetic Circuit

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

76/91

AC Excitation Phenomena of a Magnetic Circuit  i

vs



vs



 

 



 i

N i

t i 

t 

i

t

i

(a)

i HC

(b)

(a) Voltage, flux, and excitation current; (b) corresponding hysteresis loop. To produce the magnetic field in the core requires current in the excitation winding know as the excitation current i For the given , we can obtain the corresponding i from the B-H hysteresis loop. Because  = BcAc , and i = HcBLc /N The saturated hystersis loop will result peakly excitation current with sinusoidal flux variation 77/91

Core Saturation Due to Over-Excitation  i

N

vs

 ( t1 ) 

1 N



t1

t0

v s ( t ) dt   ( t0 )

 ( t0 )  0

78/91

Core Saturation Due to Over-Excitation  i N

vs

79/91

Core Saturation Due to Residual Flux  i

vs

N

 ( t1 ) 

1 N



t1

t0

v s ( t ) dt   ( t0 )

 ( t0 )  0 Power transformer inrush current caused by residual flux at switching instant; flux (green), iron core's magnetic characteristics (red) and magnetizing current (blue).

80/91

Inrush Electric Current in a Transformer 

What really happen

i

vs

N

What you assume

81/91

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. 82/91

Saturation of an Inductor (Incremental Inductance) Weber-turns (=N)

Cross-sectional area Ac

Mean path length l

L(I x ) 

N

i

 i

iIx

Amp (I)

Permeability 

I

x  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 current around this operating point.  Note: In the given example, the current source as a perturbation source. 83/91

Measuring the Incremental Inductance at Specific Operating Points (1/3) Weber-turns (=N) S1

Vdc

B

N S2

C

S3

L(I x ) 

S4

 i

iIx

Amp (I)

A

 Practical inductors are nonlinear and its incremental inductance (smallsignal 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)? 84/91

Measuring the Incremental Inductance at Specific Operating Points (2/3) 0.1mH, IL(pp)=2A, 20 kHz inductor

S1

S3

8 Ohm, 50W, Cement Resistors

e 

A

v AB

Vdc S2

S4

Ro B

v AB

L

L L  R R ESR  Ro  2 R DS ( ON )

Ts   e

RESR

 Design an inductor with 0.1 mH, average current from 1 A to 6A, and operating with switching 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. REF: [1] Inductor Design in Switching Regulators (Technical Bulletin SR-1A, Magnetics).pdf

85/91

Compute the Inductance of a Toroidal Ferrite Core

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

TN23/14/7 Ring 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.

86/91

TN23147-3R1 - Ferroxcube Effective Core Parameters

Permeability as a Function of Frequency of Different Materials

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). 89/91

Winding Inductances of an IPMSM a

La-bc 1.5Lq

1.5Ld

La-bc b

IPMSM stator coil

N

N S

La-bc 

3/2 2 Rotor pole position ()

S

N S



/2

0

c

 3   Ld  Lq   Ld  Lq   cos 2   2 2 2  

S

S

N

N

90/91

Mid-Term Report (April 15, 2016) Modeling of the Stator Winding Inductance

L1  ?

L1 ( r )  ?

1. Define the stator structure, mechanical dimensions, winding mechanism, and material parameters of the segmented motor. 2. Construct an equivalent circuit for a single segment of the stator teeth and calculate its inductance. Make a Maxwell simulation to verify the calculation. 3. Put the segmented teeth into the stator but without the rotor, make a Maxwell simulation to calculate the inductance of a single stator segment. 4. Define the rotor structure and material parameters and make a Maxwell simulation to calculate the inductance of a single stator segment as a function of the rotor pole position. 91/91

UNITS

FOR MAGNETIC

PROPERTIES

Quantity

Symbol

Gaussian & cgs emu a

Conversion factor, C b

SI & rationalized mks C

Magnetic flux density, magnetic induction

B

gauss (0) d

10- 4

tesla (T), Wb/m 2

maxwell (Mx), G·cm 2

Magnetic flux

weber (Wb), volt second (V·s)

Magnetic potential difference, ma~netomotive force

U,F

gilbert (Gb)

Magnetic field strength, ,magnetizing force

H

oersted (Oe),e Gb/cm

A/m!

(Volume) magnetization g

M

emu/cm 3 h

A/m

(Volume) magnetization

477M

G

A/m

Magnetic polarization, intensity of magnetization

1,1

emu/cm 3

477 X 10- 4

T, Wb/m 2i

(Mass) magnetization

(F,

emu/g

1 477 X 10- 7

A.m 2/kg Wb·m/kg

Magnetic moment

m

emu, erg/G

10- 3

A.m 2, joule per tesla (l/T)

Magnetic dipole moment

j

emu, erg/G

477 X 10- 10

Wb·m'

M

10/477

dimensionless henry per meter (H/m), Wb/(A.m)

dimensionless, emu/cm 3

(V 01 ume) susceptibility

ampere (A)

(Mass) susceptibility

Xp, K p

cm 3/g, emu/g

477 X 10- 3 (477)2 X 10- 10

m3/kg H.m 2/kg

(Molar) susceptibility

Xmo}, K mo]

cm 3/mol, emu/mol

477 X 10- 6 (477)2 X 10- l3

m3/mol H·m 2/mol

Permeability

1J-

dimensionless

477 X 10- 7

H/m, Wb/(A·m)

Relative permeability j

1J-r

not defined

(Volume) energy density, energy product k

W

erg/cm 3

10- l

J/m 3

Demagnetization factor

D,N

dimensionless

1/477

dimensionless

dimensionless

a. Gaussian units and cgs emu are the same for magnetic properties. The defining relation is B =H + 477M. b. Multiply a number in Gaussian units by C to convert it to SI (e.g., 1 G X 10- 4 T /G = 10- 4 T). c. SI (Systeme International d'Unites) has been adopted by the National Bureau of Standards. Where two conversion factors are given, the upper one is recognized under, or consistent with, SI and is based on the definition B = 1J-o(H + M), where 1J-o = 477 X 10- 7 H/m. The lower one is not recognized under SI and is based on the definition B =1J-oll +J, where the symbol I is often used in place of J. d. 1 gauss = 105 gamma (1'). e. Both oersted and gauss are expressed as em -1I2. g 1l2.S -1 in terms of base units. /. A/m was often expressed as "ampere-turn per meter" when used for magnetic field strength. g. Magnetic moment per unit volume. h. The designation "emu" is not a unit. i. Recognized under SI, even though based on the definition B = 1J-oll +J. See footnote c. j. 1J-r = 1J-/1J-o = 1 + X, all in SI. 1J-r is equal to Gaussian 1J-. k. B·H and 1J-oM·H have SI units J/m 3; M·H and B·H /477 have Gaussian units erg/cm 3. R. B. Goldfarb and F. R. Fickett, U.S. De~artment of Commerce, National Bureau of Standards, Boulder, Colorado 80303, March 1985 NBS Special Publication 696 For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402

CHAPTER Fitzgerald & Kingsley's Electric Machinery,

1

Stephen Umans, McGraw-Hill Education, 7th Ed., Jan. 28, 2013.

agnetic Circuits and agnetic Materials he objective of this book is to study the devices used in the interconversion of electric and mechanical energy. Emphasis is placed on electromagnetic rotating machinery, by means of which the bulk of this energy conversion takes place. However, the techniques developed are generally applicable to a wide range of additional devices including linear machines, actuators, and sensors. Although not an electromechanical-energy-conversion device, the transformer is an important component of the overall energy-conversion process and is discussed in Chapter 2. As with the majority of electromechanical-energy-conversion devices discussed in this book, magnetically coupled windings are at the heart of transformer performance. Hence, the techniques developed for transformer analysis form the basis for the ensuing discussion of electric machinery. Practically all transformers and electric machinery use ferro-magnetic material for shaping and directing the magnetic fields which act as the medium for transferring and converting energy. Permanent-magnet materials are also widely used in electric machinery. Without these materials, practical implementations of most familiar electromechanical-energy-conversion devices would not be possible. The ability to analyze and describe systems containing these materials is essential for designing and understanding these devices. This chapter will develop some basic tools for the analysis of magnetic field systems and will provide a brief introduction to the properties of practical magnetic materials. In Chapter 2, these techniques will be applied to the analysis of transformers. In later chapters they will be used in the analysis of rotating machinery. In this book it is assumed that the reader has basic knowledge of magnetic and electric field theory such as is found in a basic physics course for engineering students. Some readers may have had a course on electromagnetic field theory based on Maxwell's equations, but an in-depth understanding of Maxwell's equations is not a prerequisite for mastery of the material of this book. The techniques of magneticcircuit analysis which provide algebraic approximations to exact field-theory solutions

T

1

2

CHAPTER 1

Magnetic Circuits and Magnetic Materials

are widely used in the study of electromechanical-energy-conversion devices and form the basis for most of the analyses presented here.

1.1 The complete, detailed solution for magnetic fields in most situations of practical engineering interest involves the solution of Maxwell's equations and requires a set of constitutive relationships to describe material properties. Although in practice exact solutions are often unattainable, various simplifying assumptions permit the attainment of useful engineering solutions. 1 We begin with the assumption that, for the systems treated in this book, the frequencies and sizes involved are such that the displacement-current term in Maxwell's equations can be neglected. This term accounts for magnetic fields being produced in space by time-varying electric fields and is associated with electromagnetic radiation. Neglecting this term results in the magneto-quasi-static form of the relevant Maxwell's equations which relate magnetic fields to the currents which produce them. (1.1)

i

B-da=O

(1.2)

Equation 1.1, frequently referred to as Ampere's Law, states that the line integral of the tangential component of the magnetic field intensity H around a closed contour C is equal to the total current passing through any surface S linking that contour. From Eq. 1.1 we see that the source of His the current density J. Eq. 1.2, frequently referred to as Gauss' Law for magnetic fields, states that magnetic flux density B is conserved, i.e., that no net flux enters or leaves a closed surface (this is equivalent to saying that there exist no monopolar sources of magnetic fields). From these equations we see that the magnetic field quantities can be determined solely from the instantaneous values of the source currents and hence that time variations of the magnetic fields follow directly from time variations of the sources. A second simplifying assumption involves the concept of a magnetic circuit. It is extremely difficult to obtain the general solution for the magnetic field intensity Hand the magnetic flux density B in a structure of complex geometry. However, in many practical applications, including the analysis of many types of electric machines, a thtee-dimensional field problem can often be approximated by what is essentially 1 Computer-based numerical solutions based upon the finite-element method form the basis for a number of commercial programs and have become indispensable tools for analysis and design. Such tools are typically best used to refine initial analyses based upon analytical techniques such as are found in this book. Because such techniques contribute little to a fundamental understanding of the principles and basic performance of electric machines, they are not discussed in this book.

1.1

Introduction to Magnetic Circuits

Mean core length lc Cross-sectional areaAc Magnetic core permeability f1,

Figure 1.1 Simple magnetic circuit. )._ is the winding flux linkage as defined in Section 1.2.

a one-dimensional circuit equivalent, yielding solutions of acceptable engineering accuracy. A magnetic circuit consists of a structure composed for the most part of highpermeability magnetic material. 2 The presence of high-permeability material tends to cause magnetic flux to be confined to the paths defined by the structure, much as currents are confined to the conductors of an electric circuit. Use of this concept of the magnetic circuit is illustrated in this section and will be seen to apply quite well to many situations in this book. 3 A simple example of a magnetic circuit is shown in Fig. 1.1. The core is assumed to be composed of magnetic material whose magnetic permeability fJ., is much greater than that of the surrounding air (JJ., >> tJvo) where /Jvo = 4n x 1o- 7 Him is the magnetic permeability of free space. The core is of uniform cross section and is excited by a winding of N turns carrying a current of i amperes. This winding produces a magnetic field in the core, as shown in the figure. Because of the high permeability of the magnetic core, an exact solution would show that the magnetiC flux is confined almost entirely to the core, with the field lines following the path defined by the core, and that the flux density is essentially uniform over a cross section because the cross-sectional area is uniform. The magnetic field can be visualized in terms of flux lines which form closed loops interlinked with the winding. As applied to the magnetic circuit of Fig. 1.1, the source of the magnetic field in the core is the ampere-turn product N i. In magnetic circuit terminology N i is the magnetonwtive force (mmf) F acting on the magnetic circuit. Although Fig. 1.1 shows only a single winding, transformers and most rotating machines typically have at least two windings, and N i must be replaced by the algebraic sum of the ampere-turns of all the windings. 2

In its simplest definition, magnetic permeability can be thought of as the ratio of the magnitude of the magnetic flux density B to the magnetic field intensity H. 3 For a more extensive treatment of magnetic circuits see A.E. Fitzgerald, D.E. Higgenbotham, and A. Grabel, Basic Electrical Engineering, 5th ed., McGraw-Hill, 1981, chap. 13; also E.E. Staff, M.I.T., Magnetic Circuits and Transformers, M.I.T. Press, 1965, chaps. 1 to 3.

3

4

CHAPTER 1

Magnetic Circuits and Magnetic Materials

The net magnetic flux ¢ crossing a surface S is the surface integral of the normal component of B; thus ¢=

1

B·da

(1.3)

In SI units, the unit of¢ is the weber (Wb). Equation 1.2 states that the net magnetic flux entering or leaving a closed surface (equal to the surface integral of B over that closed surface) is zero. This is equivalent to saying that all the flux which enters the surface enclosing a volume must leave that volume over some other portion of that surface because magnetic flux lines form closed loops. Because little flux "leaks" out the sides of the magnetic circuit of Fig. 1.1, this result shows that the net flux is the same through each cross section of the core. For a magnetic circuit of this type, it is common to assume that the magnetic flux density (and correspondingly the magnetic field intensity) is uniform across the cross section and throughout the core. In this case Eq. 1.3 reduces to the simple scalar equation (1.4)

where c/Je =core flux Be = core flux density Ae = core cross-sectional area

From Eq. 1.1, the relationship between the mmf acting on a magnetic circuit and the magnetic field intensity in that circuit is. 4 (1.5) The core dimensions are such that the path length of any flux line is close to the mean core length le. As a result, the line integral of Eq. 1.5 becomes simply the scalar product Hele of the magnitude of H and the mean flux path length le. Thus, the relationship between the mmf and the magnetic field intensity can be written in magnetic circuit terminology as (1.6) where He is average magnitude of H in the core. The direction of He in the core can be found from the right-hand rule, which can be stated in two equivalent ways. (1) Imagine a current-carrying conductor held in the right hand with the thumb pointing in the direction of current flow; the fingers then point in the direction of the magnetic field created by that current. (2) Equivalently, if the coil in Fig. 1.1 is grasped in the right hand (figuratively speaking) with the fingers 4

In general, the mmf drop across any segment of a magnetic circuit can be calculated as portion of the magnetic circuit.

JHdl over that

1.1

Introduction to Magnetic Circuits

pointing in the direction of the current, the thumb will point in the direction of the magnetic fields. The relationship between the magnetic field intensity H and the magnetic flux density B is a property of the material in which the field exists. It is common to assume a linear relationship; thus (1.7)

where f.1, is the material's magnetic permeability. In SI units, His measured in units of amperes per meter, B is in webers per square meter, also known as teslas (T), and JL is in webers per ampere-turn-meter, or equivalently henrys per 1neter. In SI units the permeability of free space is /Lo = 4Jr X 1o- 7 henrys per meter. The permeability of linear magnetic material can be expressed in terms of its relative permeability JLr, its value relative to that of free space; JL = /Lr/LO· Typical values of /Lr range from 2,000 to 80,000 for materials used in transformers and rotating machines. The characteristics of ferromagnetic materials are described in Sections 1.3 and 1.4. For the present we assume that J.l,r is a known constant, although it actually varies appreciably with the magnitude of the magnetic flux density. Transformers are wound on closed cores like that of Fig. 1.1. However, energy conversion devices which incorporate a moving element must have air gaps in their magnetic circuits. A magnetic circuit with an air gap is shown in Fig. 1.2. When the air-gap length g is much smaller than the dimensions of the adjacent core faces, the core flux c/Jc will follow the path defined by the core and the air gap and the techniques of magnetic-circuit analysis can be used. If the air-gap length becomes excessively large, the flux will be observed to "leak out" of the sides of the air gap and the techniques of magnetic-circuit analysis will no longer be strictly applicable. Thus, provided the air-gap length g is sufficiently small, the configuration of Fig. 1.2 can be analyzed as a magnetic circuit with two series components both carrying the same flux¢: a magnetic core of permeability f.l,, cross-sectional area Ac and mean length lc, and an air gap of permeability fLo, cross-sectional area Ag and length g. In the core (1.8)

+

Mean core length lc

i ---+-

+--Air gap, permeability fl-o, AreaAg Magnetic core permeability fl-, AreaAc

Figure 1.2 Magnetic circuit with air gap.

5

6

CHAPTER 1

Magnetic Circuits and Magnetic Materials

and in the air gap ¢

(1.9)

Ba=Ac

1:>

Application of Eq. 1.5 to this magnetic circuit yields

+ Hgg

:F = Hclc

(1.10)

and using the linear B-H relationship of Eq. 1.7 gives

B + ___! g

(1.11)

JLo

Here the :F = Ni is the mmf applied to the magnetic circuit. From Eq. 1.10 we see that a portion of the mmf, Fe = Hclc, is required to produce magnetic field in the core while the remainder, :Fg = Hgg produces magnetic field in the air gap. For practical magnetic materials (as is discussed in Sections 1.3 and 1.4), Be and He are not simply related by a known constant permeability JL as described by Eq. 1.7. In fact, Be is often a nonlinear, multi-valued function of He. Thus, although Eq. 1.10 continues to hold, it does not lead directly to a simple expression relating the mmf and the flux densities, such as that of Eq. 1.11. Instead the specifics of the nonlinear Be-He relation must be used, either graphically or analytically. However, in many cases, the concept of constant material permeability gives results of acceptable engineering accuracy and is frequently used. From Eqs. 1.8 and 1.9, Eq. 1.11 can be rewritten in terms of the flux ¢cas

:F = ¢ (__!_:__ J-LAc

+

_g_) JLoAg

(1.12)

The terms that multiply the flux in this equation are known as the reluctance (R) of the core and air gap, respectively, (1.13) g

Ra=-1:>

JLoAg

(1.14)

and thus (1.15) "Finally, Eq. 1.15 can be inverted to solve for the flux

:F

¢=---

Rc+Rg

(1.16)

or

:F

¢ = ---;---

+-g!LoAg

(1.17)

1.1

I

Introduction to Magnetic Circuits

~

---+-

nc

Rl +

+

v

:F

ng

R2

I---v-

¢=

- (R 1 +R2)

:F ('Rc + 'Rg) (b)

(a)

Figure 1.3 Analogy between electric and magnetic circuits. (a) Electric circuit. (b) Magnetic circuit.

In general, for any magnetic circuit of total reluctance Rtob the flux can be found as

¢

= _!__

(1.18)

Rtot

The term which multiplies the mmf is known as the permeance P and is the inverse of the reluctance; thus, for example, the total permeance of a magnetic circuit is Ptot

=

1

(1.19)

rn /'\-tot

Note that Eqs. 1.15 and 1.16 are analogous to the relationships between the current and voltage in an electric circuit. This analogy is illustrated in Fig. 1.3. Figure 1.3a shows an electric circuit in which a voltage V drives a current I through resistors R1 and R2 • Figure 1.3b shows the schematic equivalent representation of the magnet~c circuit of Fig. 1.2 . Here we see that the mmf :F (analogous to voltage in the electric circuit) drives a flux ¢ (analogous to the current in the electric circuit) through the combination of the reluctances of the core Rc and the air gap Rg. This analogy between the solution of electric and magnetic circuits can often be exploited to produce simple solutions for the fluxes in magnetic circuits of considerable complexity. The fraction of the mmf required to drive flux through each portion of the magnetic circuit, commonly referred to as the mnif drop across that portion of the magnetic circuit, varies in proportion to its reluctance (directly analogous to the voltage drop across a resistive element in an electric circuit). Consider the magnetic circuit of Fig. 1.2. From Eq. 1.13 we see that high material permeability can result in low core reluctance, which can often be made much smaller than that of the air gap; i.e., for (fl,Ac/lc) >> (f.loAg/ g), Rc