System Normalization and Iron Saturation Based on Generalized Coupled Circuits Analysis as

Fundamentals for Electric Machines Modeling Course Rene Wamkeue, Senior Member, IEEE Nneme Nneme Léandre, Fouad Slaoui-Hasnaoui École de Génie, UQAT Départ. G. électrique, ENSET École de Génie, UQAT Rouyn-Noranda, QC, Canada Douala, Cameroun Rouyn-Noranda, QC, Canada rene.wamkeue@uqat.ca leandren@gmail.com fouad.slaoui-hasnaoui@uqat.ca Abstract— This paper describes the use of a suitable approach to teaching generalized magnetically coupled electric circuits as an introduction chapter to the electric machines modeling and simulation course for power engineering students. The teaching methodology focuses on some common concepts and fundamentals of electrical machine theory such as machine inductances (self, leakage and mutual), equivalent circuits, magnetic circuits, iron saturation, reciprocal per unit system, state modeling and simulation, so that the modeling approach of each type of classical machine can easily be deduced from the general theory established by the chapter on magnetically coupled electric circuits.

Keywords— Magnetically Coupled circuits; iron saturation; per unit system normalization, state space model, application and simulations

I. INTRODUCTION Magnetically coupled electric circuits are central to the operation of transformers and electrical machines, as specified by Prof. Krause et al. [1]. A good understanding of the theory underlying magnetically coupled circuits greatly simplifies both the teaching and the learning of transformers and electrical machines [2]-[4]. Therefore, for power engineering students, special attention should be paid to teaching magnetically coupled circuits, with an emphasis on their electrical machine structure, as the first chapter of the electrical machines course where transient modeling and simulations are included. The modeling of the equivalent electric circuits of electrical machines is based on magnetically coupled circuits. The modeling and simulation of electrical machines are great challenges for power system engineering students around the world. Currently, in many universities around the world, the magnetically coupled circuits are only taught in the ‘electric circuits’, which is a common undergraduate course for first year electrical engineering programs [2]-[4]. Although transient behaviors are included in ‘electric circuits’ course, the study of electric machines for undergraduate power engineering students is most of time still limited to the steady state performances [5]. Furthermore, there are not sufficient pedagogical links between the courses ‘electric circuits’ and many undergraduate textbooks treating electric machines which in general don’t content magnetically coupled circuits [5]-[6]. Very few authors have realized the importance to insert the magnetically coupled circuits as the first chapter of their research textbook [1], [7]-[8]. However, even though the magnetically coupled circuits developed in [1], [7] included the saturation phenomenon and simulation analysis, their electrical models given in block-diagram form, are not flexible to adapt to the rotating machines study and control purposes. As shown in the present work, the course on magnetically coupled circuits offers interesting opportunities to introduce

main fundamentals of the electrical machines modeling and simulation. In this paper, generalized concepts that could be applied to all electrical machines analysis such as the state model, electrical equivalent circuits, per-unit system normalization, iron saturation, numerical simulations are introduced in a course on magnetically coupled circuits to facilitate the understanding and interest of future power engineering students commencing the study and analysis of electrical machines. Repeating these concepts over time when analyzing every type of electrical machine (redundancy principle) will increase the student’s understanding of the pedagogical objectives to be achieved. II. EQUATIONS OF GENERALIZED MAGNETICALLY COUPLED

CIRCUITS Magnetically coupled electric circuits with an adjustable number of coils are shown in Fig.1. The flux equations for n coils are given in (1) and reorganized in (2) while the voltage equations are defined in (3) [1]:

( )( )

( )

1

2

1 1 1 1 1 2 2

2 2 2 1 1 2 2

1 1 2 2

( )

( )

( )n

f m n n

f m n n

n n f n m n n

N N N i N i N i

N N N i N i N i

N N N i N i N i

λ φ

λ φ

λ φ

= + ℜ + + +

= + ℜ + + +

= + ℜ + + +

(1)

( )( )

( )

1

2

1 11 12 1 1 1 12 2

2 21 22 2 21 1 2 2

1 2 1 1 n

n m

n m

n n n nn n m n n

L l i L i

L i L l i

L i L l i

λ λ λ λ

λ λ λ λ

λ λ λ λ

= + + + = + + +

= + + + = + + +

= + + = + + +

(2)

1 21 1 1 11 12 1

1 22 2 2 21 22 2

1 21 2

nn

nn

nn n n n n nn

di di die ri L L Ldt dt dtdi di die r i L L Ldt dt dt

di di die r i L L Ldt dt dt

= + + + +

= + + + +

= + + + +

(3)

with:

978-1-4673-5261-1/13/$31.00 ©2013 IEEE

2i

i

mim i

i

NL Ni

φ= =

ℜ; i j

ij ji

N NL L= =

ℜ (

ifi i

i

l Ni

φ=

; i im f

ii i i mi i

L N N Li i

φ φ= + =

The following relationships are derived bvariables and parameters of the coils:

Fig.1. Coupled electric circuit with adjustable numbe

III. EQUIVALENT CIRCUIT REFERRED FRO The main objective when modeling an electricamagnetically coupled circuits for the study of eleis to derive flux and voltage equations such thcircuit without a galvanic link as illustrated drawn. The Kirchhoff mesh and nodal laws applied to the equivalent circuit model obtainedthe solution of the physical magnetically couorder to achieve this goal, flux and voltage equawill be modified so that the current, flux and vcoil can be observed or referred from coil 1.‘referred circuit’ obtained by creating fictitiouparameters allows the galvanic link of Fig.2between coils and, thus creates their electricasuch a context, the ampere-tours ( j jN i ) in a coil j can be mathematically replaced by the fampere-tours ( 1 jN i′ ) so that the current in the c

its equivalent ( )1j j ji N N i′ = referred from c

1( )e t

1i

1mL

1l1r2l

2mL

12 21L L M= =

1me2me

Fig. 2 Equivalent circuit of two magnetically coupled circuconnection (with a galvanic link)

( )( )

(

1 1 1

1 1 1

1 1 1

1 1 1 2 3

2 1 2 2 3

1 2 3

m m m

m m m

n m m m n

l L i L i L i

L i l L i L i

L i L i L i l

λ

λ

λ

′ ′= + + + + +

′ ′ ′ ′= + + + + +

′ ′ ′ ′= + + + + +

( i j≠ ) (4)

im il+ (5)

between various

er of windings

OM COIL 1

al system such as ectrical machines

hat the equivalent in Fig.2 can be can therefore be

d for analysis and upled circuits. In ations (2) and (3)

voltage of a given The concept of us variables and

2 to be removed al connection. In given secondary

fictitious primary coil j is given by

coil 1.

2i

2( )e t

2r

uits without electrical

)

1

1

1

m n

m n

m n

L i

L i

L i

′+

′+

′

(6)

11 1 1 2 2 2;d de ri e r i

dt dλ λ′ ′ ′= + = +

where:

21 1

21 1 1 1

; ;

; ;;

i

ij i j i

i i i i i i i

m m i i i

a N N i j e a

a r a r lL L a L Lλ λ

′= ≠ =

′ ′ ′= = == = =

Fig.3. Electrical equivalent circuit for an coupled coils referred (observed) from windi According to the energy conservatiofictitious circuit and in the actual se

j j j je i e i=′ ′ ). Introducing the abvoltage referred from coil 1 in (2) can be organized as given in (6) ansecondary j both in motor mode (fand 2 absorb power from externcircuit of the generalized magneticreferred from coil 1 (in the motor important to observe that Fig.1 equivalent circuit of a variable relwith adjustable damper windinginduction machine. The magnetiz

called the mutual inductance since L IV. NORMALIZATION OF GENERALIZ

UNIT SYSTEM

A. Definition of Base Values Normalization schemes are frequmodeling and simulation. The pernormalization of electric machine clear to students. The great advscheme is that the system independently from which coil thmagnetically coupled circuits suctransformers are defined by the racircuits (i): apparent power iNScurrent iNI (A), and the rated p

where nf (Hz) is the nominal

winding voltage 1( )e t (coil 1)network. In single-phase circuit anstudy, the base quantities for a gTable 1. From this table is obsdistinct coils i and j: ib jb jE E I=

2 ; ; nn n n

de r idt dtλ λ′ ′′ ′ ′= +

(7)

1

1 1

21

1

; 1

; 1; 1

i j i i i

i i

m i

a e i i a i

a l iL a i

′ = ≠

= ≠≠

(8)

adjustable number of magnetically

ing #1 ( primary)

on principle, the power in the condary should be the same (bove fictitious current and

and (3), the latter equations nd (7) with the primary and for example, in Fig.2, coils 1

nal sources). The equivalent cally coupled circuit of Fig.1 mode) is given in Fig.3. It’s is quite close to electrical

luctant synchronous machine gs or multi-rotor windings zing inductance

imL is also

im ij ijL a L= .

ZED COUPLED CIRCUITS: PER

uently used in power systems r-unit systems applied for the equations are not more often

vantage of the normalization equations are the same

ey are referred. The classical ch as electrical machines and ated values of their individual

N (VA), voltage iNE (V),

pulsation 2n nfω π= (rad/s);

frequency of the primary

) supplied by the external nalysis, as it’s the case in this given coil (i) are defined in erved that for two arbitrary

1jb ib ij jiI a a= = thus:

1 1 2 2 3 3

1 2 3

b b b b b b nb nb

b b b nb

E I E I E I E IS S S S

= = = == = = =

(9)

The above relationship (9) is fundamental and general in electrical machines analysis when system normalization is concerned. It states that: in the per-unit normalization, the VA bases of coupled circuits should be equal.

TABLE I: BASE VARIABLES OF COIL #i Quantity Base units Dimen.

Power ib iNS S= [VA] [ei]

Voltage ib iNE E= [V] [e]

Current ibib iN

ib

SI IE

= =

[A] [i]

Impedance ibib

ib

EZI

= [ Ω ]

[z]

Pulsation 2b n nfω ω π= = [rad/s] [1/t]

Time 1

bb

Tω

=

[s] [t]

Flux ibib ib b

b

EE Tλω

= =

[Wb-turns] [et]

Inductance ibib

ib

LIλ=

[H] [et/i]

B. Normalization of Voltage Equations

For given base voltages and currents 1 1,b bE I , 2 2,b bE I , ,

,nb nbE I for circuits of Fig.1, the voltages equations (3) of generalized coupled circuits can be organized as expressed in (10) where the index (u) indicates a per unit quantity:

1 21 1 1 1 1 11 1 12 2

1 22 2 2 2 2 21 1 22 2

1 21 1 2 2

u ub u b u b b b b

b b

u ub u b u b b b b

b b

u unb nu n nb nu b n b b n b

b b

di diE e r I i L I L Idt dt

di diE e r I i L I L Idt dt

di diE e r I i L I L Idt dt

ω ωω ω

ω ωω ω

ω ωω ω

= + + +

= + + +

= + + +

(10)

with:

;i i ii b ii iiiu iiu

ib ib ib ib ib ib ib ib

ii ii b iiiiu iiu

ib ib ib

r r L L Lr LE I Z E I I LL L XL XL Z Z

ωλ

ω

= = = = =

= = = = (11)

According to relationships (11), the reciprocity of the mutual inductances should be preserved in (12) (since ij jiL L= ). Another important result obtained from the per-unit system

normalization is that mutual inductances should be equal as expressed in (13) (derived from (12)).

;( )

;( )

i

i

j

j

m bij jb ij ibb b ij m u

ib ib ib

m bji ib ji jbb b ji m u

jb jb jb

LL I L Ia L i j

E E ZLL I L I

a L i jE E Z

ωω ω

ωω ω

= = = ≠

= = = ≠ (12)

1 2 nm u m u m u muL L L L= = = = (13)

C. Normalized Equations The voltage equations (3) in (SI system) can be organized as given in (14) in the reciprocal per-unit system (pu) without index (u) and minuscule symbols for leakage inductances. The voltage equations (14) are formatted in the matrix form (15), where nJ in (16) is an n order matrix with all entries equal 1.

1 1 21 1 1 1

1 2 22 2 2 2

1

m mb b b

m mb b b

n nn n n m m n

b b b

di di die ri l L Ldt dt dtdi di die r i L l L

dt dt dt

di di die r i L L ldt dt dt

ω ω ω

ω ω ω

ω ω ω

= + + + +

= + + + +

= + + + +

(14)

( )1 1 ( )( ) ( ) ( )

b b

d d ie t Ri t Ri t Ldt dtλ

ω ω= + = +

(15)

With:

[ ]( )( )

1 2

1 2

1 2

T

n

n

m m n m m n

g g g g

R diag r r r

L diag l L l L l L L J

=

=

= − − − +

(16)

V. SATURATION MODEL OF GENERALIZED COUPLED CIRCUITS In order to introduce the saturation phenomenon in previous coupled circuits, neglecting the voltage drop in leakage inductance 1l and resistance 1r , the coil #1 voltage in (14) yields:

( ) ( )1 2 101

1 1nm m

b b

d i i i d ie L L

dt dtω ω+ + +

= = (17)

Assuming a coil #1 sinusoidal input voltage and current, (17) can be written into complex form (18):

( )1 10 10

1m b m m m

b

E jL I jL I jL Iωω

= = = (18)

In (18) the magnitude of the magnetic current mI only depends

on the magnitude of terminal voltage 1E and not on the state of

the circuit (loaded or not). Consequently, if 1E is maintained

constant, the unloaded ( 2 3 0ni i i= = = ) magnetizing 10icurrent will remain the same even if other windings are loaded. Thus:

10 10 1 2m m nI I i i i i i= ⇒ = = + + + (19)

10 10

10m m

m

E EX LI I

= = = (20)

Using the no-load magnetizing characteristic 10 10( )E f I=obtained experimentally (no load is connected to secondary coils of the coupled circuits), the magnetic saturation curve

10( )mL f i= given in Fig.4 can be drawn point per point from (20).

0

0mL

( )m mL I

10 1 2m ni i i i i= = + + +

(pu))

(pu)

linear zone (unsaturated)

Fig.4. 10( )mL f i= characteristic of the unloaded coupled circuits (in pu)

The curve in Fig.4 can be easily modeled as polynomial function given by (21) where parameters 2α and 4α are computed from curve fitting using polyfit function of Matlab program and the characteristic 10( )mL f i= . 0mL is the unsaturated magnetic inductance of the circuit which is obtained from no-load characteristic of Fig.4.

( )2 410 100 2 4

10

1m m m mm

E EL L I II I

α α= = = + + (pu) (21)

1( )e t

1i

10 1 2i i i= −

mL

1l1r2r 2l

2 ( )e t

2i

( )me t

Fig.5. Equivalent circuit of two magnetically coupled circuits (transformer or per phase equivalent circuit of wound-rotor induction motor)

1( )e t

1i

10 1 2i i i= − +

mL

1l1r2r 2l

2 ( )e t

2i

( )me t

Fig.6. Equivalent circuit of two magnetically coupled circuits (AC generators)

It’s to note that the magnetizing current 10i depends on the structure of the coupled circuits under study as shown by examples of two magnetically coupled coils of Figs.5 and 6 where it takes the following expressions respectively:

10 1 2i i i= − ; 10 1 2i i i= − + (23)

VI. SATURATED STATE SPACE MODELING OF MAGNETICALLY

COUPLED CIRCUITS A. State Space Model of Magnetically Coupled Circuits Let us consider a dynamic linear system defined by the differential equation (24); the so-called standard form of the state space model of a linear dynamic system, a well-known control theory suggested for undergraduate engineering students in [9], can be summarized as given in (25). The electrical matrix equation (26) derived from (15) can be organized in standard state form (27):

( )( ) ( ( ), ( ), )

dx tx t f x t u t t

dt= =

(24)

( ) ( ) ( )

( ) ( ) ( )

x t Ax t Bu t

y t Cx t Du t

= +

= + (25)

1

1

( ) ( )

( ) ( )1 ( )

b

i t t

e t R t

LdL

dt

λ

λλ

ω

−

−

=

= +

⎧⎪⎨⎪⎩

(26)

( ) ( )

( )

1

1

( ( )) ( ) ( )

( ) ( )

b b n

d t RL t I e tdt

i t L t

λ ω λ ω

λ

−

−

= − +

=

(27)

with:

[ ]

1 1,

1 2

; ; ;

;b b n n n

T

n

A RL B I C L D

i i i i u e

ω ω− −= − = = = Ο

= =

(28)

Where ( )tλ is the state vector, ( )e t the input control variable and A, B, C and D, the state matrix, the state control matrix, the state output matrix and the state output control matrix respectively. B. Saturated State Model of Magnetically Coupled Circuits

Leakages inductances in (14) 1;i i nl = due to magnetic losses in the air gap are not concerned with the saturation phenomenon. The magnetic saturation can be taken into account in the state model (27) at each instant kt by adapting the corresponding

value ( )m kL t of the magnetic inductance computed from (21) in

matrices A and C (28). The implemented process to account for saturation is the following:

1. Obtain the characteristic 10 10( )E f i=

2. Obtain the curve 10( )mL f i= of Fig.4

3. Obtain the reactance 0mL from 10( )mL f i=

4. Compute parameters 2α and 4α usi

polyfit of Matlab program and mL f=5. Define a time vector 0 Fkt t⎡ ⎤⎣ ⎦ usin

1n− steps of simulation. 6. Set 0k = ; 0m mL L= and compute m

using circuit parameters and steady

0 0( )tλ λ= 7. Simulate the state system (27) usin

(.)lsim of Matlab control toolbox interval [ ]1k kt t + .

8. Set 1k = and use currents obtainedcompute ( )m ki t given by (19) and mL

9. Compute state matrices A, B, C 10. Repeat step 7 until

Fk kt t=

VII. EXAMPLE OF NUMERICAL APPLICATION U In order to provide students with a sample applicsimulations were performed on a 60-Hz, 110-V330-VA single-phase transformer with two sec(Fig.7) whose parameters are listed in Table IItests were performed. First, the two secondaryshort-circuited and a 50-V DC voltage wasprimary. In the second test, the DC voltage w60-Hz 110-V AC voltage. Per unit inductancmatrices computed form base values of Tablerespectively:

TABLE II: PARAMETERS OF THE TRANSF

1mL 0.09 H 2 3m mL L= 22

1l 0.01H 1N 100

2 3ll = 1r

2.5 mH

10 Ohms

2 3N N=

2 3r r=

500

2.5

TABLE III: BASE VARIABLES OF COIVariable Base

Voltage 1 1b nE E=

Current 11 1

1

bb n

b

SI IE

= =

Power 1 1 1b n nS E I= 33

Impedance 11

1

bb

b

EZI

= 3

Pulsation 2b n fω ω π= = 377

Flux 11

nb

b

Eλω

= W

Inductance 11

1

bb

b

LIλ= 0.

ing the function

10( )f i data

ng n values for

matrices A, B, C state conditions

ng the function in one time step

d from step 7 to ( )kt from (21)

USING MATLAB

cation, numerical V/55-V/55-V, and condary windings I. Two different y windings were s applied to the

was replaced by a ce and resistance es III and IV are

FORMER 2.5 mH

00 turns

0 turns

Ohms

IL #1 Value 110 V

3 A

30 VA

36.7 Ω

7 rad/s

0.292 Wb-turn

093 H

Fig.7. Electrical equivalent circuit of three w

TABLE IV: BASE VARIABLEVariable Base

Power 2 2 2b n nS E I=Voltage 2 2b nE E=

Current 2 2b nI I= =

Impedance 22

2

bb

b

EZI

=

Pulsation 2b nω ω= =

Flux 22

nb

b

Eλω

=

Inductance 22

2

bb

b

LIλ=

;

1.03 0.93 0.930.93 1.03 0.930.93 0.93 1.03

RL = =

⎛ ⎞ ⎛⎜ ⎟ ⎜⎜ ⎟ ⎜⎜ ⎟ ⎝⎝ ⎠

Fig.8 The no-load magnetizing characteristic

Fig.9 The no-load magnetizing characteristic

In both cases, simulation of the curmagnetizing current of the transform

windings transformer

S OF COILS #2AND #3 e Value

n 330 VA

55 V

2

2

b

b

SE

6 A

9.167 Ω

2 fπ 377 rad/s

0.146 Wb-turn

0.0243 H

0.272 0 00 0.272 00 0 0.272

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

(31)

c of the transformer

c 10( )mL f i= of the transformer

rrent of each winding and the mer was requested. The

no-load saturation characteristic of the transformFig.8 while the curve 10( )mL f i= is plotted inof the secondary windings are set to zero (shorfirst test. The primary simulated current is showthe magnetizing current and the secondary currFigs.11 and 12 respectively. It is to be notsecondary windings are the same. After the sudden applied voltage, the secondary current besignal). For the second test, the sinusoidalsinusoidal currents as shown in Figs.13-15. Tcurrent is given by 10 1 2 3i i i i= − − . The impactphenomenon is observed on various figures.

Fig.10. Current of the transformer primary following a 50-primary voltage with the secondary windings in short-circuit

Fig.11 Magnetizing current following a 50-V step change owith the secondary windings in short-circuit

Fig.12 Current of the transformer secondary following a 50primary voltage with the secondary windings in short-circuit

Fig.13 Current of the transformer primary following a 110-of the primary winding with the secondary windings in short

mer is provided in n Fig.9. Voltages rt-circuit) for the

wn in Fig.10 while rents are given in ted that the two peak due to the ecomes zero (DC l voltage yields The magnetizing t of the saturation

-V step change of the t

of the primary voltage

-V step change of the t

-V sinusoidal voltage t-circuit

Fig.14 Magnetizing current following a 110-winding with the secondary windings in shor

Fig. 15 Secondary windings current of thsinusoidal voltage of the primary winding wcircuit.

VIII. CONCL

A suitable approach to teaching mapower engineering undergraduate paper in a more general approach. that the understanding and teachinsimulation of electrical machines special attention is given to the teafor magnetically coupled circuits, machine theory can be easily introdon per-unit system normalization power system theory and not oftenAll fundamentals raised are commoand can be easily well adapted to theOnly simulation results are providedcarry out experimental setup toframework study.

REFERENC

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McGraw-Hill, 1998 [6] S.A. Nasar, Handbook of Electric Mach

USA, 1990 [7] Chee-Mun Ong,, Dynamic Simulatio

Matlab/Simulink, Prentice Hall, 1997, [8] P. M Anderson, A. A. Fouad, Power Sys

edition, IEEE Power Engineering Societ[9] T. Martinez-Marin, “State Space Formu

IEEE Transactions on Education, vol.532010

V sinusoidal voltage of the primary rt-circuit

he transformer following a 110-V

with the secondary windings in short-

LUSION

agnetically coupled circuits to students is proposed in this The work attempts to show

ng of dynamic modeling and can be greatly improved if

aching methodology adopted where various concepts in

duced. The paper emphasizes concept frequently used in

n well explained to students. on to classical AC machines eir modeling and simulation. d in this work. It is planned to o more assess the present

CES

hoff, Analysis of Electrical 95 cuits, 7th Edition, John Wiley and

ctric Circuits, 6th Edition

Engineering Circuits Analysis,

nery Fundamentals, Third Edition,

hines, McGRAW-HILL, New York,

ons of Electric Machinery using

stem Control and Stability, second ty, IEEE PRESS, 2002 ulation for Circuits Analysis’’ 3, No.3, pp. 490-503, August,