Electrical Machines , Vol. 1

Con tents

Pr eface

11

Introduction 1-1 Bas ic Defin itions 13 1-2 Con version of El ectric Energy by the Tr an sforme r 15 El ectromechanical Energy Con versio n by an 1-3 Electrical Machin e 18 1-4 Functiona l Classifi cation of Elect romagnetic En er gy Conve r t ing Devices 24

13

1 Chapter

1

1-1 1-2 Chapter

2

2-1 2-2

2-3 2-4 2-5

2-6 2-7 2-8

Chapter

3 3-1 3-2 3-3

Transformers

An Outline of 'I'ransiormers P urpose, Applications, Ratings 27 Const ru ction of a Transformer 31 Elec tromagne tic Processes in the Transformer a t No-Load Th e No-Load Condition 43 Voltage Equations 45 Vari ations in EMF with Time. An EMF Eq uation 46 Th e Magnetization Curve of the Transforme r Th e No-Load Cur ren t W aveform 49 T ra nsformer Eq ua tio ns a t No-Lo ad in Compl ex Form .50 No-Loa d Losses 52 The Effect of the Core Loss on t he T ransformer 's Pe rform ance at No-Loa d 53

27

43

47

El ectromagnetic Pro cesses in the Transformer on Load 56 The Magnetic Field in a T ran sform er on Load. Th e MMF Equati on. Th e Leakage Inductance of th e Windings 56 Voltage E qua ti ons of the Tr an sform er Windings 60 Transferring t he Secondar y Quan tities to t he Primary Sid e 6? . . .

6

Contents 3-4 3-5 3-6 3-7 3-8

Chapter

4 4-1 4-2 4-3 4-4

The Phasor Diagram of a Transformer 65 The Equivalent Circuit of the Transformer 68 The Per-Unit Notation 69 The Effect of Load Variations on the Transformer 72 Energy Conversion in a Loaded Transformer 75 Transformation of Three-Phase Currents and Voltages 79 Methods of Three-Phase Transformation. Winding Connections 79 A Three-Phase Transformer on a Balanced Load 83 Phase Displacement Reference Numbers 84 The Behaviour of a Three-Phase Transformer During Magnetic Field Formation 89

Chapter

5 5-1 5-2

Measurement of Transformer Quantities The Open-Circuit (No-Load) Test 99 The Short-Circuit Test 102

Chapter

6 6-1

Transformer Performance on Load 106 Simplified Transformer Equations and Equivalent Circuit for 11» 1 0 106 Transformer Voltage Regulation 107 Variations in Transformer Efficiency on Load 111

6-2 6-3 Chapter

7

7-1

7-2

Chapter

8 8-1 8-2

Chapter

9 9-1 9-2

99

Tap Changing Off-Load Tap Changing 113 On-Load Tap Changing 114

113

Calculation of Transformer Parameters No-Load (Open-Circuit) Current and Mutual Impedance 117 Short-Circuit Impedance 119

117

Relationship Between Transformer Quantities and Dimensions Variations in the Voltage, Current, Power and Mass of a Transformer with Size 121 Transformer Losses and Parameters as Functions of Size 123

121

125

Chapter 10 10-1 10-2

Multiwinding Transformers. Autotransformers Multiwinding Transformers 125 Autotransformers 133

Chapter 11 11-1 11-2

Transformers in Parallel 138 Use of Transformers in Parallel 138 Procedure for Bringing Transformers in for Parallel Operation 139 Circulating Currents due to a Difference in Transformation Ratio 141 Load Sharing Between Transformers in Parallel 14:/

11-3

tH

7

Contents Chapter 12 12-1 12-2 12-3 12-4 12-5 12-6 12-7

Three-Phase Transformers Under Unbalanced Load 145 Causes of Load Unbalance 145 Transformation of Unbalanced Currents 146 Magnetic Fluxes and EMFs under Unbalanced Load Conditions 151 Dissymmetry of the Primary Phase Voltages under Unbalanced Load 154 Dissymmetry of the Secondary Voltages under Unbalanced Load 156 Measurement of the ZPS Secondary Impedance 160 Single- and Two-Phase Unbalanced Loads 161

Chapter 13 13-1 13-2

Transients in Transformers Transients at Switch-On 164 Transients on a Short-Circuit Across the Secondary Terminals 167

164

Chapter 14 14-1 14-2

Overvoltage Transients in Transformers Causes of Overvoltages 171 The Differential Equation for the Initial Voltage Distribution in the Transformer Winding 172 Voltage Distribution over the Winding and Its Equalization 175

171

Special-Purpose Transformers General 177 Three-Phase Transformation with Two Transformers 177 Frequency-Conversion Transformers 178 Variable-Voltage Transformers 179 Arc Welding Transformers 180 Insulation Testing Transformers 181 Peaking Transformers 182 Instrument Transformers 182

177

Heating and Cooling of Transformers Temperature Limits for Transformer Parts under Steady-State and Transient Conditions 184 Transformer Cooling Systems 186

184

14-3 Chapter 15 15-1 15-2 15-3 15-4 15-5 15-6 15-7 15-8 Chapter 16 16-1 16-2 Chapter 17 17-1 17-2 17-3

2 Chapter

18

189 Transformers of Soviet Manufacture USSR State Standards Covering Transformers 189 Type Designations of Soviet-made Transformers 190 Some of Transformer Applications 191 A general theory of electromechanical energy conversion by electrical machines

E1eclromechanical ~a.chjn!lS

Processes

in

Electrical

Contents

8 18-1 18-2 Chapter 19 19-1 19-2 19-3 19-4 19-5 Chapter 20 20-1 20-2 20-3 20-4

Chapter

21 21-1 21-2

Chapter 22 22-1 22-2 22-3 22-4 22-5 22-6 22-7 22-8 Chapter 23 23-1 23-2 23-3 23-4

Classification of Electrical Machines 192 Mathematical Description of Electromechanical Energy Conversion by Electrical Machines 195 Production of a Periodically Varying Magnetic Field in Electrical Machines 201 A Necessary Condition for Electromechanical Energy Conversion 201 The Cylindrical (Drum) Heteropolar Winding 202 The Toroidal Heteropolar Winding 206 The Ring Winding and a Claw-Shaped Core 206 The Homopolar Ring Winding and a Toothed Core 206 Basic Machine Designs Modifications in Design 207 Machines with One Winding on the Stator and One Winding on the Rotor 211 Machines with One Winding on the Stator and Toothed Rotor and Stator Cores (Reluctance Machines) 214 Machines with Two Windings on the Stator and Toothed Cores for the Stator and Rotor (Inductor Machines) 218 Conditions for Unidirectional Energy Conversion by Electrical Machines The Single-Winding Machine 227 Two-Winding Machines 230

207

227

Windings for A. C. Machines 235 Introductory Notes 235 The Structure of a Polyphase Two-Layer Winding 235 Connection of Coils in a Lap Winding. The Number of Paths and Turns per Phase 240 Coil Connection in the Wave Winding 244 The Selection of a Winding Type and Winding Characteristics 246 A Two-Pole Model of a Winding. Electrical Angles between Winding Elements 247 Two-Layer, Fractional-Slot Windings 250 Field Windings 255 Calculation I Zn l

1 2 < 1 2 ,n and the transformer is underloaded . When I Z I is infinity, which occur s when the transformer is disconnected from the io V, e, ~

\

1, ~o

-

@

r-r--r-«

hI

iz=O

I

~

F ig . 2-1 Single-p hase tw o-windin g transformer on no-l oad

receiving line on the secon dary side (the secondary is opencircuited), the secondary current falls to zero . In th e circumstances , the transformer supplies no-load current, which is why this state is calle d the no-load (open-circuit) condition,

Ch. 2 Proc esses in Transformer at No-Load

45

The electromagnetic pro cesses occurring in a transformer at no load ar e far simpler than they are und er load, with 10 > 0, so their study can best be begun with the no-load c~ndition.

Consider the electrom agnet ic proce sses at no-load in the single-phase two-winding t r ansf ormer shown in sketch form in Fig . 2-1. This is a core-t ype transformer whose primary and secondary windings are shown for conv eni ence located on different limbs. (The ac tual arrangement of the windings on a core-type transformer has been described in Sec . '1-3, see Fig. 'I-Sa.) 2.2

Voltage Equations

The supply voltage VI im pressed on the primary winding gives rise in it to an alternating current i o , called the noload curr ent . This current produces two fluxes , namely the mu tual (usefu l) magnetic f lux which has its path wholly within the core of a very high permeabili ty, ~tr ~ '1, and links all the turns WI an d W 2 of the primary and secondary windings, and als o the leakage flux which links onl y the primary turns . If we find the mutual magnetic flux cD at an y section of the dosed magnetic circuit , we sha ll be able to find the mutual flux linkage with the primary winding 1f on = w1 cD and with the secondary winding "If 02 1 = w2 cI) The leakage flux has its path completed through nonmagnetic materials (air gaps, insulation) with a permeability equal to that of free space , ~to, and substant ially smaller than that of t he magnetic core . Therefore, the leakage flux linkage with the primary winding at no-load , 1f a O, is a small fraction of the mutual flux linkage with the primary, "If on (Fig . 2-'1 ). Th e periodicall y varying mutual and leakage fluxes induce electromotive forces in the windings with which they link. For t he posi tive directions of currents, voltages, emfs and magnetic lines of force shown in Fig . 2-'1, the primary emf oj mutual induction is e1 = - W I dCV/d t = - d"lf on /dt (2-2) whereas the secondary emj oj mutual induction is ez = - w z dcD/dt = - d"lfOZl/dt (2-3)

46

Part One. Transformers

and the leakage primary emf is e ao = -elcDao/elt

~ e1

(2-4)

Interpreting VI as an emf impressed on the winding from the supply line, we may write Kirchhoff's voltage equation VI

+ e + e ao =

R 1i o

l

(2-5)

where R 1 is the resistance of the primary winding. The no-load voltage across the secondary is the same as the emf induced in it V 2 = e2 2-3

l I

Variations in EMF with Time. An EMF Equation

1

For all power transformers (and for most microtransformers) ,II we may neglect in Eq. (2-5) both the voltage drop across R 1 . and the leakage emf e cu I

I u.i, I ~ I el I I e ao I -e; I ell I'l---l----+--'.--'lf-----,I-z-f{

and deem, with sufficient accuracy , that the primary emf of mutual induction is in antiphase with the primary voltage (Fig. 2-2):

el

=

-VI =

=

-EI,~

V1 ,m cos rot cos rot (2-6) -

Fig. 2-2 Time variations in voltages, emfs and magnetic flux of a transformer

It follows from Eq. (2-6) that the emf of mutual induction . varies with time harmonically, and its peak (rms) value does not differ from the peak (rms) value of the voltage

E 1 ,m = VI,m,

(E I = VI)

(2-7)

From a comparison of Eqs. (2-2) and (2-3), we may conclude that the ratio of e2 and l is time-invariant. This ratio is called the transformation, or turns, ratio

e

e21el

= E 2 , mlEl,m = E 2/E1 =

W 2/Wl

=

1221

(2-8)

47

Ch. 2 P rocesses in Transformer at No-Loa d

On the basis of E qs . (2-6) an d (2-8), we may arg ue that e2 varies likewise harmonically and is in phase with el . We may express the magnet ic flux (D in terms of el by integrating t he differential equation (2-2) subject to Eq . (2-6): t

t

. ~ E (1)= - - J e 1 d t= \' cos or dz 1

WI

WI

U

"

0

(2-9)

- Q)msin ro t where

(D m = E1 ,m/Wlro (2-'10) is the peak va lue of the magnetic fl ux . Using Eq . (2-10), we can der ive an equati on giv ing the rms va lue of e1 from the given peak magnetic flux or flux linkage E 1 = E 1 •n/ j/2 = row1cDm/ 1I 2 = ro'P'o11 ,m/ V 2 or E 1 = (2n /

11 2) j W1(I)m

. (2-11)

Accordingly, the rms va lu e of e2 is E 2 = rowlI)m/

11 2 = rolfo:d ,n,l 11 2

or (2-12)

Referring to the plot of Fig. 2-2, the magnetic fl ux l ags behind VI by 90° (it is said to be in qu adrature lagging with the primary voltage) , and leads e1 and e 2 by 90° (it is said to be in qua drature leading with the two emfs). 2-4

The Magnetization Curve of the Transformer

The thick ness and m aterial of the lami nations for a transformer core are always chosen accord ing to t he frequency of the magnetizing current , so as to keep eddy currents t o a minimum . The instantaneous magnetic flu x may th en be determined from t he inst ant aneous pr imary mm f, iOw 1, at no-l oa d. The r esultant r el at ionship between t he in st ant aneous va lues of the two quantities, cD = j (i o), is i dent ical

Part One. Transform ers

48

to that obtained with d .c., when eddy currents are nonexi stent. Graphically , the nonlinear re lationship betwee n t he flux cD in the core an d the direct curren t i o in the primary winding is dep icted by what is called the cl.c. magnetization curve (or characteristic) of a tran sformer. It can be construc ted on the basis of Ampere's circuital la w in in t egral form . On aligning the loop enc losing t he current in all the primar y turns, iOw 1, with a line of for ce of t he mutual magnetic flux in the core, Ampere's circuital law ma y be written

I

iowt =

I

~ Hzdl

The procedure yielding the cir culation of the H vector is as follows: (1) Assign a desired va lue to the magnet ic flux in th e core . (2) Bre ak up the core into n portions of length I" each , such that within each portion the act ive iron cross-sectional are a A" and the permeabili ty r em ai n constan t for the specifi ed m agnetic flux . (3) Calculate the magnetic induction within each port ion, B" = it is inevit able that R 12 ~ X 1 2 . Fina lly , we may write

n, = X ~ /R12 ' x, = X 1 2 , R o « R 12 (2-29) The quantity X 0 retains the name of the mutual inductive reactance of the primar y winding. R o is a fictitious res istance the loss across which at l ois equa l to th p 1',(>1'0. loss of the t ransformer P ear e = I~R o

.

As is seen from the equivalent circuit in Fig. 2-7b, the . products Rol o and jXol o are, res pectively, the active and reactive components of the primary voltage

T\ .

If 12

Fig. 2-8 Ph asor dia gram of a transform er on no-loa d

F ig . 2-9 Impedance of the equi " valent circuit and no-lo ad cur ren t 1 0 as functions of TTL

The relation hetween the pr imary voltage Tll and the no-load current 1 0 . . Tll = - E 1 = Z oI o (2-30) is illustrated b y the phasor diagram in Fig'. 2-8 which, with P ear e = 0, R 12 = 00, and lo a = 0, is the same as that shown in Fig. 2-6 . Because the magnetic circuit of a transformer is nonlinear , the no-loa d current 1 0 rises at a faster rate t ha n VI' so R o and X 0 depend substan tially on 111 (Fi g. 2-9):

.

X;

=

X 12

'"

Vl /l o

56

Part One. Transformers

and

R o ,.... (V)I o)2 In contrast, as the primary voltage is va ried, R 1 2 remains practically unchanged, because the core loss is pro portional to the square of the magnetic induction, Eq . (2-21), or the primary voltage, Eq . (2-20).

3

Electromagn etic Processes in the Transformer on Load

3-1

The Magnetic Field in a Transformer on load. The MMF Equation. The leakage Inductance of the Windings

When a transformer is operating on load, i ts secondary is traversed by a current

The load current gives rise to a change in the primary current. Proportionate changes also occur in the magnetic flux and the secondary voltage , aud thera is an increase

i,

if Ii.I ~f ~ p/ -

£1!

~f

~2

'--

=

!tZtt62 Z - Pz

Fig. 3-1 Single-phase, two-winding transformer

Oil

load

in the power lost. For a proper estimate of these changes in a transformer on load , it is essential above all to examine its magnetic field and to deve lop voltage equations for its pr imary and secondary windings.

Ch, 3 Processes in Transformer on Load

57

Figure 3-'1 shows a single-phase , two-winding transform er whose second ar y is conne cted acros s a load impedance Z. Assuming that all the relevant electric and magnetic quantities are var ying harmonically , we may write them in complex notation . In doing so, it is important to remember that the instantaneous value of a harmonic quantity is t o be construed as the re al par t of t he respective complex amplitud e multiplied by exp (jwt) i

= Re [V 2 j exp (jwt)]

[V 2Vexp (jwt)] e = Re [V2Eexp (jwt) J

v = Re

= Re [cP m exp (jwt)J lJf = Re [O/m exp (jw t)J

(1)

The adopted positive directions of the abo ve quantities are shown in Fig . 3-'1. Positive directions for II and 1 2 ar e chosen such that they set up a positive mutual magnetic flux . Posit ive directions for t he voltages and emfs across the win dings are the same as for the res pective curren ts . Positive diractions on load are chosen the sam e as for oper ation at no loa d. When a transform er is operating on lo ad, its magnetic flu x is established by the prim ar y current II traversing the primary winding and by the second ar y current 1 2 traversing the secondary winding . To simplify the matters , this magnet ic flux can be visualized as a superposit ion of two fluxes, namely the mutual (or m agn etizing) flux and the leakage flu x . Th e grea ter proportion of the flux linking the windings is the mutual flux whi ch h as all of its pa th within the core and com pletely encloses the wind ings fr om both sides. The mutual flux cD (Fig. 3-'1) is the same at an y section of the core ; it s linkage with t he prim ary is WlcPnll and with the secon dary , w 2 cD m . Und er Amp ere 's circuital law, the magnetic intensity du e to mu tual induction is t he sum of the prim ary and seconda ry mmfs

58

Part One. Transformers

Since t he mutual magnetic induction and the mutual flux are connected to the field int ensity in a well -defined manner (see Chap . 2), we may arg ue that the mutua l flux is established by the sum of t he primary and secondary mmfs. This sum may be vis ua lized as the mmf due to some current i o traversing the primary winding (3-1) ilWl + i 2w2 = iOw l Therefore, the current given by i o = (ilWl

+ i 2w2)/ W l

may be called the mag netizing current, and Eq. (3-1), an mmj equation .

The non linear effects taking pl ace in the transformer core as it undergoes cycles of magnetization by the current i o may be accounted for as in t he case of no-load operation. The nonsinusoidal current i o may be rep laced by an equivalent sinusoidal magneti zing current the rms va lue of which is

=v

10 IBa + I ar and whose active component l oa is related to the core losses. Then we may write the mmf equation in complex notation as

.

I lwl ,;

"'!

I

+

.

.

I 2w 2

=

I owl

(3-2)

In our further discussion, the term "magnetizing current" will refer to the equivalent sinusoidal magnetizing current 1 0 , Now we are in a position to present the primary mmf i l w 1 as a sum of iOw l and (illVl - iolVl) = -i 2lV z which ba lances the secondary mmf i zlV 2 , and the magnetic flux in operation on load as a sum of t hree fluxes, namely: (a) the mutual magnetic flux and the leakage flux with flux li nk age 1Jf aO' set up by the primary mmf iolV l (Fig . 3-2a); (b) the leak age flux established by the mutually ba lancing mmfs, namely (illV l - iolVl) = - i zlV z on the primary side and i zlV 2 on the secondary side (Fig . 3-2b). Referring to Figure 3-2, it is seen that the lines of the leak age flux have their path comp leted through nonmagnetic (air, oil, etc .) gaps alb l and a zb2 comparable in leng th with the portions of the lines accommodated within the core (bla l and bza z) . These lines link either the primary turns (1Jf o i and 1Jf ao), or the secondary turns (1Jf az).

59

Ch. 3 Processes in Transformer on Load

The lines of the leakage flux in a transformer may be divided into two groups-those linking only the primary turn s and giving rise to the flux linkage 1Jf 00 due t o i o and / i,

a,

e,

iz=O

,---

iowrt

'I{,.

b, ~

iri o

J

(i,-i,)

w, t

a,

a2

'Pa,

'f;;z

b,

bz

1

iz=-(i,-ioJw,(W2

~

r-r-r-

'---

~ li,""w·l

~

(6)

(a)

Fig. 3-2 Magnetic flux on load as the sum of (a) mutual flux and (b) leakage flux

1Ya1 due to (i1 -

i o) , and those linking only the secondary turns and giving rise to the flux linkage 1Jf 02' To appraise the relationship between the flux linkages and the currents in the windings, we shall develop an equat ion by Ampere's circuital law for, say, a closed line of the leak age flux linking t he primary winding as shown in Fig . 3-2b:

~ H dl =

:r

b1

«i

«i

b1

Jr n, dI + ~I\ H eor e dl = (ii- i o)

Wi

Let us write the magnetic field in the nonmagnetic region, H 0 ' and the magnetic field in the core , H eore , in terms of the respe ctive induction and permeability: Ho

H eoro ~[. a. e ore

= = =

B oht o

B eore/~t a , eo re ~t r . eore ~t o

~r,eore ~ '1

Therefore , the leakage field ill the core is negligibly small lIeore

=

B eorehta ,eore =

0

60

Part One. Transformers

The total current is equal to the magnetic potential difference across the nonmagnetic gap bl

bl

:0 ) n, dI

) n, dl = al

(ii - io) Wi

=

0

It follows from the foregoing that 1.J.! o i is proportional to (i 1 - io). The same holds for 1.J.! co and 1.J.! cr2 and their respective currents i o and i 2 • Therefore, the leakage inductances of the windings L cr1 = Wcr1/ ( i 1 - io) t.., = 1.J.! cr2/i2 (3-3) L cro = 1.J.! cro/ i o are constant for a gi ven transformer and solely depend 011 the wid th of nonmagnetic gaps and the number of turns in the windings (see Sec . 8-2) . With a high degree of accuracy, the total leakage fl ux linkage with the primary winding may be written Wcrl = 1.J.! co

+ 1.J.! o o = i

Lcroi o

+ L cr1 (i 1-io) ~ L cr1i1

(3-4)

because in operation on load i 1 ~ i o, and we may neglect wh atever diff erence there may be between L cro and L cri and deem t hat L cr o ~ L cr1. By analogy with the mutual inductance [see Eq. (2-16)J, t he leakage inductances may be expressed in terms of the respective permeances, A cr i and A cr 2: L cr1

=

W~AUl'

L cr 2

= w;A U2

or in terms of perm eance coefficients L cr2

= f10 Wi Acrl = ~LoW~Acr2

AU!

= A u!hlo

L cr1

wher e

Acr2 =

3-2

A cr2/ f1 0

(3-5)

(3-6)

V o ltage Equ ations of the Transfor mer Windings

The emf induced in each of the transformer windings can conveniently be presented as the sum of the mutual emf E 1 (or E 2 ) and of t he leak age emf E U 1 (or E cr2).

61

Ch. 3 Processes in Trans form er on Load

The mutual flux shown in Fig . 3-2a does not differ from that in a transformer OIl no-load (see Fig . 2-1) . Th erefore, the mutual emf ma y he expressed in terms of the mutual flux in precisely the same manner as at no-load . Given a certain E I , the magnetizing current i o must be the same as at no-load, provided that E I and CD are the same in either case . Therefore, 1 0 and E I can he conn ected by an equation of the form (3-7)

where

Zo

=

Ro

+ jX o

Using the turns ratio, n 21 = w2 /w ll we can wri te the mutual emf on the secondary side as

.

=

.

=

.

(3-8) The primary and secondary leakage emfs, e crl and e cr2 , are induced by the leakage flux linkages "If o i and 'P" cr2, respectively, proportional to the primary and secondary currents : e crl = - d'P" crl/dt = -L crl dil/dt (3-9) e cr2 = -d'P" cr2/dt = - L cr 2 di 2 /dt - E2

- n 21E I

n 21Z0IO

Using complex notation and differentiating by analogy with Eq . (2-19), we get and, similarly (3-10)

Here, (3-1'1)

are called the leakage inductive reactance of the primary and second ary, respectively. As is seen from Eq. (3-'10), the leakage emfs ar e in quadrature lagging with the associated currents. Now that we have defined the primary and secondary emfs of a loaded transformer and recalling that all the quantities involved vary harmonically*, we ma y write Kirchhoff's

* The nonsiuusoidal magnetizing curren t is replaced b y an equivalent sinusoi dal cur rent .

II

62

Part One. Transformers

voltage equati ons for t he prim ar y and secon dar y windings ill complex form as

.

111

.. . + u, + E = RIll . . . . E 2 + E 0 2 = R 2l 2 + 11

(3-12)

01

2

where R, and R 2 are the resistances of the primar y and secondary windings, respe ctively, including add ition al losses due to altern at ing current (see Sec. 31-2). In writing Eqs. (3-12), positive directions were chosen as shown in Fig. 3-1. The volt age 111 is the supply emf impressed on the winding from an external source. The . . voltage 11 2 = Z l2 is the voltage drop across the l oad on the secondary side with an impedance of value Z = = R jX . Expressing the leakage emfs in (3-12) in terms of the respective leakage induct ive reactances and currents (3-10), we may re-write t he voltage equations as follows:

+

.

(3-13) . . 11 2 = E 2 - l2 Z2 where Zl = R, jX l and Z2 = R 2 jX 2 are the complex imped anc es of the primary and secondary windings, respectively .

+

3-3

+

Transferring the Secondary Quantities to the Primary Side

The performance analysis of a transformer can greatly be simplifi ed, if we transfer the qu an tities associated with the secondary to the primary winding. This technique consists in tha t the real t ransformer having in t he general case different numb ers of primary and secondary turn s, W I and W 2, is rep la ced by an equivalent transformer in which the secondary ha s the same number of turns as the primary, w~ = WI (see Fig . 3-3). The qu an tities associated with the equivalent secondar y ha ving WI turns are said to be transferred (or referred) to th e primary winding or side. They are expressed in terms of the original secondary quantities adjusted in value by a suita bl e factor so that transfer of secondary quantities to the primary side wi ll leave the magnetic fie ld, and the power fluxes PI' P 2, and Q2 unaltered . The procedure is as follows .

o..

63

Ch. 0 Processes in Trans former on Load

('1) To leave the ma gnetic flux (I) unaltered , we must retain the secondary mmf unchanged, that is

.

I~ Wl =

.

I zw z

whe nce

j~ = j ZWZ/Wl

(3-14)

Here and elsewhere , the prime on a secon dary quantity in di cates that it has been tr ansferre d to the prim ar y side . (2) With (I) ke pt constant, t he emf is proportion al t o the t urns number. Therefore, the emf acro ss the secondary 4>

iI

II

t?/!'f/!

~,

"it;z

'. ~

Pf -

-----

jE! It;,

z.~ tv;

- Pz

Fig. 3-3 Transformer of Fig . 3-1 with i ts secondary transferred to = WI the primary,

w;

winding transferre d t o the primary side will in crease W 1/W2 times: E~ = E Zw1/w Z (3-15) (3) To keep unchanged the values of P z an d Qz drawn by the load on t he secon dar y side, its R an d X mu st be repl aced by th ose t ra nsferred t o the primar y side: P 2 = RI~ = R'I~2 Q•... = XI 22 = X ' I'22 Using Eq. (3-14), we get R' = R (W 1/WZ)2 X ' = X (w1h v z)2 Therefo re, (3-16) We can see th at the secon dary impedance can be transferre d to the prim ary side, adjuste d in value by t he turn s ratio squared.

Part One. Transformers

64

The secondary voltage call likewise he transferred to th e primary side, adjusted in value by the turns r a tio

t

(3-17) V; = Z ' j~ = Z (w]/w z)z zWz/w] = VZw]/w z The secondary impedance Zz, its resistive component Hz and its inductive component X z can be transferred to the primary side in about the sam e manner: Z; = H; jX; = Zz (w]/wz) Z H; = Hz (w]/wz)Z (3-18) = (w]/w z)Z As a result, the secondary voltage equation takes the form E~ = Ez (w]/w z) = VZw]/w z Zz (w]/wz)Z jzw z/w j or

+

X;

X;

+

E2'

V'2 + z.i: 2 2

(3-19) Because the primary and secondary windings have the same number of turns, the transferred (or referred) secondary emf is the sam e as the primary emf : =

.

"

E; = E 2w]/w Z = E] The mmf equation for a transformer with its secondary parameters transferred to the pr imary side is ex tended to include the secondary mmf expressed in t erms of the secondary current referred to the primary wind ing

.

.

+

.

I]w] I~w] . lOw] Dividing the above equation through by equation of transformer currents

.

.

WI

give s th e

.

I ] + I ; = 10 (3-20) which has the same physical meaning as the mmf equation (2-30) . With a suffi cient ly heavy load, when the primary current markedly exceeds the magnetizing current, I] ~ 10' th e current equation can approximately be written as

.

.

I] = -I; = - I zwz/w] or

l II

I] /I z = wz/w] (3-21) As is seen, given a heavy load, the referred secon rlary cur rent, I;, does not differ from the primary current , I ].

Ch. 3 Processes in Transformer on Load

3.4

65

The Phasor Diagram of a Transformer

The voltage and current phasor diagram of a transformer is a graphical interpretation of the equations describing the performance of the transformer. These equations includ e - the winding voltage equations

.

111

.

.

+ ZIII

-E I

=

-E~ =

(3-22a)

-i1~

-EI =

-the load voltage equation

.

-11'2

= Z'

+ Z~ (-j~) .

(-1') 2

(3-22b)

(3-22c)

-the mu tual emf equation

- EI

-E~

=

=

z.),

(3-22d)

-the current equation

i,

=

i, -

i;

(3-22e)

Using a ph asor diagram constructed to a cer t ain definite scale, we can determine the voltages, emfs and currents of a transformer on load . The sequence in which a phasor diagram is constructed depends on which quantities are specified to define the operation of the t ransformer and which quantities are to be determined . Suppose that we know the secondary current 1 2 and the load impedance Z = R jX (for an inductive load , X > 0; and for a capacitive load, X < 0). We set out to find the . secondary voltage 112 , the primary emf E I , the magnetizing . . current 1 0 , the primary current II' and the primary voltage VI' Th e phasor diagram is usually constructed for the transformer with its secondary quantities referred to the primary side. Therefore , the first step is to determine the secondary quantities referred to the primary side (that is, adjusted in value by the turns ratio or the t urn s ratio squared). The referred secondary current is

+

I~ = ;;- OI6D

.

I2

(W~/W l)

Part One. Transformers

and t he referred imped ances ar e Z'

=

Z~ =

Z (WI/W2) 2 = R'

Z 2 (WI/W 2)2 = R~

+ jX'

+ jX~

The ph asor diagram is m ade more compact if the complex quantities referred to the primary side are t aken with a minus sign , -1~ an d The first to be pl otted (see

- r;

Fig. 3-4 Pha sor dia gram of a tra nsformer operating int o a resist iveinductive loa d (CP2> 0, X > 0)

Fig. 3:.. 4) should be -1 2 which ma y be drawn in an ar bitrary direction , say along t he positive axis of the complex t ime pl ane and on the sca le adopted for currents. Th en , using t he load voltage equation, we find the referred secondary . voltage: - V~. This voltage has an activ ~ component, R' ( - 1~), an d a re act ive component , j X' ( - 1~ ), wh ich are laid off t o the adopt ed sca le . The active comp-onent is la id off in the direction of -1~ , whereas the reac ti ve com ponent leads -1~ by 90 if the load is inductive and X > O. The actual second ary volta ge is found by Eq . (3-17): 0

,

V2

=

Vi (W2/Wl)

Ch. 3 Proc esses in Transformer on Load

Then we find graphically the mutual emf -E 1 and compute the magnetizing current 10 = E 1/ V Rfi

67

-E'

2

+ X~

and the ph ase angle CjJo =

arctan

(XolR o)

Now j 0 can be laid off on the phasor diagram . The mutual flux cD can be found from Eq . (3-7) and laid off on a scale of its own (the flux is in quadrature lagging with -E1) . The primary current II is deduced from t he current equation . The primary voltage VI is found graphically in a similar way. The construction thus obtained also gives the phase

-ii

R'(-ii)

Fig . 3-5 Ph asor diagra m of a tr ansformer operating into a resi stivecapaciti ve load (rr2 < 0, X < 0)

shift (P2 between t he secondary voltage and current, and the phase shift (PI between the corresponding primary quantities. With a resistive-inductive load , both the primary and the secondary currents lag beh ind the respective voltages in phase, so CjJl and CjJ2 are taken to be positive: CjJl > 0 and CjJ2> 0 (see Fig. 3-4). The ph asor diagram for a resistive-capacitive load is plotted in Fig. 3-5. As is seen, the secondary current leads the voltage by an angle (P2 (CjJ2 < 0). If the load is predominantly capacitive (see Fig . 3-5), the primary current like5*

68

Pa rt One. Transformers

wise leads the voltage by an angle (PI < O. If the capacitive component is less pronounced, the primary current may even lag behind the voltage . 3-5

The Equivalent Circuit of the Transformer

If we treat a single-phase, two-winding transformer as a two-port, the equivalent circuit stems from Eqs. (3-22a) through (3-22d), where the secondary quantities are trans-

ferred to the primary side . Given VI' the circuit equivalent to a given transformer must draw from the supply line the same primary current i, as the transformer itself. In order to identify the configuration of this equivalent circuit, we must express the primary voltage in terms of the primary current. To begin with, we shall express i, in terms of E1 and the circuit parameters

Hence, 11 l/Zo+l /(Z~+Z)

Substituting the above expression into the voltage equation gives

Vi =

j1 Z1 -E 1= i, [Z1 +-

1/Z

o

+ 1I ~Z~ +Z/)

] =

j1 Z eQ

(3-23)

It is seen from Eq. (3-23) that the transformer equivalent circuit drawing a primary current II must have an equivalent impedance given by ZeQ = Z1

+- 1/Zo +1I\Z~+Z/)

This impedance is presented by the circuit in Fig . 3-6 where ZI is shown connected in series with a parallel combination of Zo and (Z; Z'). A detailed analysis would show that the individual arms of the equivalent circuit carry the same currents as the

+-

Ch. 3 Proc esses in Transformer on Load

69

windings of the transformer in whi ch the secondary quantit ies are tra nsferred to the primary sid e . Also, the current s

Fig . 3-6 E quivalent circ uit of a t ransformer

ent ering the nod es of the circ uit and it s loop voltages satisfy the basic tra nsformer equat ions. 3-6

The Per-Unit Notation

E lectrical quantities (such as currents and voltages) and circuit paramet ers (reactances an d resistances) can be expressed each as a fraction of an arbitrarily ch osen base or reference quantity , thereby gi ving per-un it quantities. The per-uni t notation sim plifies the equat ions describing t r ansformer performance. It also simplifies a chec k on the design data an d result s , because the per-unit qu an titi es of different t r ansformers differ much less t ha n the same quantit ies expressed in absolut e units . The base quantities usually chosen for the primary side of t ransformers are : -the rated phase primary voltage, VI, R -the rated phase primary current, I I. H - the rated impedance presented by t he t r ansformer t o th e supply line, I Z1, n I = VI , nlII, n (3-24) -the power r ating of t he t r ansformer

8 1, H = VI, RII , R in the case of a single-phase transform er , and 81, H = 3V I , RII , R for a three-phase tr ansformer ,

70

Part One. Transformers

The base quantities usually chosen for the secondary side are: -the rated phase secondary voltage, V z, R = VI, R (WZIWl); -the rated phase secondary current, l z , R = 11, R (wl/w z); -the impedance presented to the line on the secondary side (at V z, Rand i ; R) I z; R I = V z , RlI z, R =

I z; R I (wZlwl)Z

(3-25)

-the base power on the primary side

S2, R = Sl, R To obtain a per-unit quantity on the primary side, its absolute value is divided by an appropriate base quantity taken in the same units

VI: l

=

Vl/V l , H

1,!:l = lIllI, R

I Z:I:O I = I z, III z; R I I Z*l I = I z, III z; R I P:I: l = Pl/S], R = Vi:l1,r.l cos

(3-26) (PI

where an asterisk stands for per unit. Sometimes, this index may be omitted, if the use of the per-unit notation is referred to in the text. The power equation in per-unit quantities is equally applicable to single- and three-phase transformers. The quantities associated with the secondary winding of a transformer can be expressed as per-unit quantities in anyone of two ways. For example, we may divide a given secondary quantity taken in absolute units by the corresponding secondary quantity taken as the base. Alternatively the secondary quantity may first be referred to the primary side by adjusting it in value by the turns ratio or the turns ratio squared, as the case may be, and the result may then be divided by the adopted base quantity associated with the primary side:

V 21V2 , H = V~fl1l, R 1,1:2 = 1 21I 2 , H = I~lIl, R (3-27) IZ:,:21 = I Z21/1 Z2,R I = IZ~ I/IZl,R I P*2 = P 2/S 2, H = P~/S l, R = V:J: 21,r. z cos crz Vr.z

=

For obvious reasons, the secondary quantities expressed on the per-unit basis carr y no referring index, .

71

Ch. 3 Processes in Transformer on Load

Anyone transformer equation may be written in per-unit notation . . To this end, it must be div ided through by the corresponding base quant it y . As an exam ple, let us do this for Eq. (3-13) which gives t he primar y voltage

or

.

.

V:I: I = -E:':I

. + Z:I:lI,1:l

(3-28)

For the current equation, we obtain

.

or

1*1

.

+ 1*2 =

.

(3-29)

1*0

As is seen , the per- unit equations are written in about the same way as those in absolute quantities, except that they have no indexes to show transferring to the primary side. Per-unit quantities are also helpful in expressing the parameters and quantities involved in equivalent circuits , and in constructing phasor diagrams . The per-u nit parameters and losses of a transformer va ry within a ralative ly narrow range of va lues and depend ma in ly on it s power rating . Let us establis h the relations between some of the perunit quantities . Among other things, we will find that the mutual inductive reactance varies inversely as the no-load current:

X:I: O = I Z:I:O I = I z, II I ZI, R I = (TTl, RII 0) (II, RIV I , R) = I I, RlI o

(3-30)

The resistance during magnetization can be expressed in terms of the no-load current and the core losses (the no-load or open-circuit losses) as

n., = =

Roll z; R I = P coreII, R/3I~Vl, (Pcore/3VI. RII, R) (II, RlI o)2

~ P:~, corel I~:Q

R

(3-31)

Part One. Transformers

Finally , the winding resistances are equal to the copper losses

R :I: 1 = RIll z; R I = 3RII~, R/3T1 1, RI 1 , R = PCu, /8 1, R = P:"cu, I R:1: 2 = R~/I Zl, =

R

(3-32)

I = 3R~I~, R/3VI , nIl , Ii

Pc«, 2/ 81, R = P:1: ClI ,

2

Using the above relations and data sheet values, the range of values for the basic per-unit quantities of three-phase power transformers rated from 25 to 500 000 kVA can readily be defined. Transformers with higher ratings have lower resistances and higher inductive reactances: I,~o =

+

P:':l, CU X*l

0.03 to 0.003

= P*o = 0.005 to 0.000 6

P:I', core P:1' 2, ClI

=

X:':2

=

P:I:, cu = 0.025 to 0.0025 0.03 to 0.07 (3-33) I Z:I:O I = X:I,o = 33 to 330 R:I: I = R u = 0.012 5 to 0.001 25 R,~o~= 5.5 to 65 As is seen from the above figures, as the power is changed by a factor of 20 000, the per-unit quantities change not more than ten-fold (in fact, X*l and X:I: 2 only change by a factor of 2). As can readily be checked , the same parameters expressed in absolute units will change by a factor of many hundred thousand . 3-7

=

The Effect of Load Variations on the Transformer

In a transformer, the primary and secondary windings are coupled by a mutual flux . Therefore, any change in load impedance (the impedance on the secondary side), with the primary voltage held constant, leads not only to a change in the secondary current, but also to a change in the magnetic flux, the magnetizing current, the primary current, and the secondary voltage. After the transients associated with a load change die out, the transformer settles down to a new steady state in which the electric and magnetic circuits are at equilibrium . In other words, the currents in the windings and the magnetic flux in the core take on va lues which again

Ch, 3 Proc ess es in Transformer on Load

73

satisfy the conditions of equilibrium for its electric circuits defined by the voltage equations, (3-13) or (3-19), and for its magnetic circuit define d by t he current equ ation (3-20) supplemented by the emf equations (3-7) and (3-8). A change in the secondary current immediately brings about a change in the peak magnetic flux cD m and the primary emf E l it induces. The state of equilibri um that exist ed on the primary side prior to that change and with which was associa ted a certa in definite pr imary cur rent is ups et, and a current is induced in the primary in accord with Eq. (3-13)

i,

=

n\ -

(-E\)] /Zl

The primary emf and the primary current keep varying un til t he magnetizing current (with the new value of 1 2 ) and the . . corresponding emf, -E l = ZoI o, build up enough for a st ead y-state current to appear in the primar y winding. Considering together the equations written earlier, the primar y current (Fig. 3-7) ma y be written

t. = l\. R/(Zo + Zl) . . =

10 • NL - I~Zo /(Z o

where

.

1 0 • NL

j~Zo/(Zo

+ Zl)

.

=

+ Zr)

VI, R/(Zo

(3-34)

+ Zl )

is the ma gnetizing current at no-load . Because Zl ~ Zo, with a sufficiently large load we have 1 0 • NL ~ 1

and

.

II

=

.

-I~

The magnetic flu x varies directly wi th E, which is in t urn a function of t he magnitude and phase of the primary current

..

-E l

or, in per-unit

=

.

VI. R - ZlI l

..

.

-E*l = V*l . R - Z*l!,l:l (3-35) At no-load , when I I = 1 0 • NL ~ 0, the emf and the flux are equal to the prim ar y voltage taken as unity

eon =

CD n = cD ~/cD ~ l R =

V'!'l, R

= 1

Part One. Transformers

74

At rated load (I*1 = 1*1, R = 1), the emf and fl ux change in significantly in comparison with their no-load' values. . . Ev en when t he phase of I I , R is such that ZlI1 is in the same or oppo sit e direction with VI , R, the emf is

where Z','1 = 0.03 to 0. 07 (see Eqs. 3-33). Thu s , even in t he wor st loading case, with the load rising fr om zero t o its full value (see Fi g. 3-7), t he emf an d flux change by as li t tle as Z,!:1 X tOO = 3 % 7% Given other ph ases for 1, an d 1 2 , the changes in the emf an d flux are still more insignificant . Referring to Figs. 3-4 and 3-5, t he emf decre ases in the case of a resistive-ind ucti ve load an d m ay in crease if t he lo ad is resistive-ca paci t ive ~====~~§~';I2~ and the ph ase sh ift is cl ose I 0; dashed line , resistive-capa citive load , CJl:i < 0; Io x=Io.NL

i o = - E1 /Z o

.

. .

= (VI - Z l I 1)/Z o

.

=1 0 ,

NL -

I 1Z 1 /Z o (3-3G)

In a linear approx ima t ion , this cur re nt varies in the same m an ner as t he pr imar y emf. If we in clu de t he nonlinear beh aviour of t he m agne ti c circuit whi ch causes Zo to vary as well , this cha nge bec omes more pr onounced . The effect of nonlin earity m ay be account ed for by using t he ma gnetizat ion curve, (J) = f (I o) , shown in Fig. 2-9. Plots of II , E 1 , (D , and 1 0 as fun ctions of 1 2 for inductive and capacitive load s ar e sh own in Fi g. 3-7,

75

Ch. 3 Pr ocesses in Transforme r on Load

3.8

Energy Conversion in a Loaded Transformer

The energy fed int o t he pr imary winding of a transformer from a supply li ne is customarily treated as t he sum of two parts. One pa rt is delivered to load and is partly lost in the transformer it self. The average time rate of this unidirectional flow of energy is called the active power drawn by the primary winding from t he supply line. For a single-phase transformer, it is given by PI

=

VIII cos

CPI =

VI lla

=

VIall

(3-37)

where I l a = I I cos CPI is the active current VI a = VI COS CPI is the active voltage The act ive power is taken as positi ve, PI > 0, if CPI li es anywhere between _90 0 and +90 0 (electrical) . The other part of in put energy is spent to establish magnet ic fie lds in t he transformer it self * and also electric an d magnetic fields in the load. The direction of this energy is changed twice every cycle , so the res pective power averaged over a cycle is zero . The transfer of energy between the supply line and a field (electric or magnetic) is described in terms of the peak inst ant aneous power , called the reactive power. The reactive power drawn by t he pri mary winding of a single-phase transformer from the sup ply li ne is given by QI

=

VIII sin

CPI =

VII l r

=

VlrI

I

(3-38)

where I l r = I I sin CPI is the rms value of reactive current Vir = VI sin CPI is t he rms va lue of react ive voltage The reactive power is assumed to be posit ive, QI > 0, if the reactive current is lagging beh ind the voltage, < CPI (PI> - :n , which cor resp onds to a resistive-capacitive load . Consider the conversion of active power in a transformer. Let us write the active component of the primary voltage, VIa = VI COS (PI, as the sum of projections of Eland the

°

°

* The energy associated with the electric field wi thin th e transformer is usually ne~lecte~,

Part One. Transformers

76

voltage drop R]I] (see the phasor diagram in Fig. 3-8a) VIa = V] cos cp] = E] cos 11J] R]I] and the active power p] supplied to the primary winding by a supply line (its direction is shown in Fig. 3-9 by an arrow) as the sum of two components p] = (11] cos (p]) I] = (E] cos 1h) I] (R]I]) I] (3-39) The term fiR 1 = P C ll , ] is the copper loss in the primary winding, that is, the power lost as heat dissipated in the primary turns (see the arrows in Fig. 3-9). Referring to Fig . 3-8b, the active component of the primary current, I] cos 1IJ], is shown as the sum of the active components of the magnetizing current locos CPo and of the secondary current I~ cos 1P2' Therefore, the term (E 1 cos 1P]) I] may likewise he written as the sum of two components: E]I] cos 1p] = E2I~ cos 'ljJ2 E]I O cos CPo

+

+

+

=

P e~

+ P eor e

(3-40)

The term P em = E]I~ cos 11J2 is called electromagnetic power. It is transferred inductively from the primary to the secondary winding. The flow of electromagnetic power crosses the channel between the two windings (Fig. 3.9). The term E]I o cos (Po = E]I oa = P eor e represents core loss in the transformer. Referring to Fig. 3-8c, the active component of the primary emf, E] cos 1P2' can be expressed in terms of the active component of the secondary voltage, V~ cos (P2' and resistive voltage drop, R~I~. Hence, we may write P em = (E] cos 1P2) I~ = (V~ cos (P2) I~ = P2 P CU, 2

+

+ (R~I~) I~

(3-41) Some of the electromagnetic power is expended to make up for the copper loss in the secondary winding, P CU,2

=

I~2R~

The remainder,

P 2 = V~I~ cos CP2 is transferred to the load conductively (see Fig. 3-9). The active power input to a transformer is p] = P CU, ] P rpm =

P CU,2

+ P eor e + P em + P~

(3-42)

77

Ch. 3 Processes in 'transforme r on Load

.

'r

- E,=-Ez

(a) (a)

Fig. 3-8 Phasor diagrams of a transforme r oper ating into a resist ive inductive load (see Fig . 3-4) ,

\

\ I I

If

V,

'\

7

P~qf ) -, /

( k-. .....

\

,11

\ >JlT", tl.ll ~

It. ~ "VI

r:rm

~

r-,

Pea;'

( ) -r-

Va

Pe

-r-

*=-

PC

'-F

Vi,e

F ig. 4-1 Three-phase transform at ion by a bank of si ng le-phase transformers

c

A

8

,L

I

T

.L,

.1

PA-

3 -:

-

rf>;y-

/

/ / --- -

/

\

\

\

~

I'-

\--

"""'~- -- -~--

/.

Tank

(6)

rp

(a)

(b)

Fig. !i-13 Third -h armoni c flux es in various core des igns

higher than that seen by the fun dament al fluxes traver sin g a closed path within the core. In determining the fund amental an d third-harmoni c te rms, we have to invoke different magnetization characteristics . For the third-harmonic flux, this is the lin eal' magnetizing characteristic, cDs = is (is). For the fund amental flux, th is is the nonlinear magnetization cha racterist ic, cD 1 = II (i o), derived for the sinusoidal flu x upon replacing i o with (i o - is) which gives rise t o the mmf associat ed with the fundamental flux (Fig. 4.13b)* .

* This is t r ue, if we consi der the fun da me nta l and t h ird-ha rmo nic terms on ly .

- ,

'

.

,

I

.

. '

,

.

.

,

Gil, 4 Transfo rm ation of 3-Pllase Curr ents and Voltages

Now we shall examine the waveformsof magnetizing -currents, flux es and voltages asso ci ated with the va ri ouswin ding connections an d core designs, assuming t hat at no -l oad the transformer is energized from the HV side . 1. A three-pha se hank of single-phase tr an sf ormer s . M Y connection. With the supply voltage impressed on the delta-connected HV side , the phase voltage is the same as t he sinusoidal line voltage . Therefore, all t he single-phase transformers in the bank are connected t o carry a sinusoidal voltage , an d t hey are magnetized in the same manner as an individual single-phase t r ansforme r is magnetiz ed with a sinusoidal volt age (see Sec. 2-5) . In other words, the flux varies sinusoidally and the magnetizing phase current, nonsinusoidally. The m agnetiz in g current has t he waveshape shown in Fig . 2-4. The line conductors carry harmonic currents whose ord er is no t a multiple of three (esp ecially , t he fu n damental term i OI ,lI n e) ' Their rms values are V:3 tim es the rms values of t he phase quantities l

o1,llne =

VS l o1

[see Eq . (4-6)1. The t riplen h armonics (especi ally i o3 ) t ra verse a closed path within the delta, and ar e no t present in the line con duc to rs (see Fig. 4-12). Bec aus e t he ph ase fluxes con t ain sol ely the fun damental t erms (

fr