Electric Machines

Energetic Systems in Offshore Industry 1

ELECTRIC MACHINES


The transformer

• Is a quadripole working in steady regime (AC) or pulse regime which transform the effective values of the electric current or the electric voltage by using the electromagnetic induction phenomenon. • Is also an electromagnetic static machine.• It contains a core whereon are placed 2 windings (primary with N1 turns and secondary with N2 turns, crossed by electric currents i1(t) and i2(t) and having between the terminals the voltages u1(t) and u2(t).• Transformers working in steady regime are classified as follows:- without ferromagnetic core (with air), used at high frequencies when the ferromagnetic core losses could become too high decreasing too much its working efficiency. The windings are differential coupled through a mutual inductance. This type is used in radiotechnics (by electromagnetic coupling between different amplifier stages), for galvanic insulation of the circuits (between the power part and the command part) and for pulses synchronization needed in order to command semiconductor components.- with ferromagnetic core, used at low frequency and medium frequency. The windings are placed on a ferromagnetic core and thus ensure a tight magnetic coupling, so a convenient transmission efficiency. Because the ferromagnetic core has a non-linear characteristics, the device has also a non-linear behaviour.


Examples of different cores of the transformers


Transformer without magnetic core (with air)

This device is characterized by 2 equations concerning the voltages in the primary winding and respectively in the secondary winding, see the figure bellow:

dt

tdiL

dt

tdiLtiRtu

dt

tdiL

dt

tdiLtiRtu

222

112222

212

111111

Zs is the charge impedance.

This equations system could also be written in the simplified complex domain:


Z1 = R1 + jωL11 – the impedance of the primary winding, Z2 = R2 + jωL22 – the impedance of the secondary winding ZM = jωL12 – the impedance of the differential coupling

221

211

2

1

2

2

1

122

2111

22

2212

2111

0

IZIZ

IZIZ

U

UZ

ZZ

I

I

IZIZZ

IZIZU

IZU

IZIZU

IZIZU

M

M

M

s

Ms

M

s

M

M

The ratio of the primary electric current and the secondary electric current and the ratio of the similar voltages depend on charge impedance, Zs, which is the most important disadvantage of this transformer. Taking into consideration the equations above, it could be drawn the specific phasor diagramme of this transformer. It is suppose to know the charge impedance, its voltage and the parameters of the primary winding and of the magnetic coupling. The final result of the diagramme is the voltage U1, considering the voltage U2 as phase origin.


222222112

212111111

ILjIRUILj

ILjILjIRU


The active power balance of these transformers is:

2

222

2

111

2

222112

2

222

2112

2

222

2

2222222

2

112112

2

111

2112

2

111

2

11111

2112

12

22

11

sin

sincos*

sin

sincos*

*

*2

1

IRPIRP

PIRIILIRP

jIILjILjIRjQPSIU

PIRIILIRP

jIILjILjIRIUS

IUS

eIIIIeII

eIIjj

j

The last equation is the active power balance and shows that the active power provided by the power source to the transformer is equal with the sum of the losses power on the primary and secondary resistances of the windings and the power transfered to the charge impedance connected on the secondary winding, P2.


The transformer with ferromagnetic coil

The layout of the windings on the ferromagnetic core has two purposes:-to achieve a tight magnetic coupling between the primary and secondary

windings so that the most part of the field lines crossing the turns of the primary winding will cross also the turns of the secondary winding. In such way the dispersion magnetic flux which determine the losses is minimum.

-to achieve a minimum of the magnetomotive force of the windings.


The transformer with ferromagnetic coilThe primary winding is supplied with a sinusoidal voltage, so will be crossed by a sinusoidal current which will produce a magnetic field and consequently a magnetic flux through the primary winding having N1 turns. This magnetic flux, time variable, will close its field lines through the air and the turns of the secondary winding. Because of the electromagnetic induction law, an induced electromotive voltage e2 will be produced in the secondary winding which will produce an electric current i2 through the charge impedance Z2, so a drop voltage u2 between its terminals.This transformer is characterized by 3 equations, the third one resulting from the balance of the magnetomotive forces. Notations: Ψ1 – total magnetic flux corresponding to the primary winding, Ψ2 – total magnetic flux corresponding to the secondary winding, φ1 - magnetic flux on one spire of the primary winding, φ2 - magnetic flux on one spire of the secondary winding, φd1 – dispersion magnetic flux on one spire of the primary winding, φd2 – dispersion magnetic flux on one spire of the secondary winding, Ld1, Ld2 – dispersion inductances defining dispersion reactances Xd1, Xd2; φ0= φ12+ φ21 – magnetic flux on one wire that closes exclusively through the core (the main magnetic flux); i0 – idle running current; Fmm – magnetomotive force; Rm – magnetic reluctance; n – transformation ratio.


For a convenient efficiency of the transformer the idle electric current must be minimum. This supposes a minimum magnetic reluctance, so a core with important values of the magnetic permeability and a section as large as possible. At the same time the construction of the transformer must ensure minimum values of the dispersion inductances and of the winding resistances (made by copper). The section of the conductors must be important and the balance equation of the magnetomotive forces becomes:

nN

N

u

un

N

N

i

iiNiN

i

iNiNiN

2

1

2

1

2

1

1

22211

10

2211101

0

Taking into consideration the working equations of the transformer with ferromagnetic coil and the ratio values U2’, R2’ and Ld2’ , it is obtained a final system with three equations. Based on these equations it could be constructed the phasor diagram of the transformer with ferromagnetic coil:


dt

diL

dt

dN

dt

ddt

diL

dt

dN

dt

d

iLN

iLNNNN

NNN

iNiNiNdt

diRu

dt

diRu

iNiNiNdt

de

dt

de

iRue

iRue

d

d

dd

dd

d

d

22

02

2

11

01

1

2222

1111

2202222

1101111

2211101

2222

1111

2211101

22

11

2222

1111


'

'''''

';';'

'1

''

2110

2222201

0111111

2

2

1

2

122

2

1

2

1222

2

1

212

1

2110

1

2211101

012

2

1

2

122

2

1

2

122

2

1

2

1

0222222

0111111

III

ILjIRUNj

NjILjIRU

LN

N

N

NLR

N

N

N

NRUU

N

N

IIIN

NII

NINININ

NjIN

N

N

NLjI

N

N

N

NRU

N

N

N

NNjILjIRU

NjILjIRU

d

d

dd

dd

d


The phasor diagram of the transformer with ferromagnetic coil


Three-phase circuits

The phase is a circuit composed by generator, connection conductors and receiver, working in sinusoidal steady regime. Because the sinusoidal signals could be easier obtained when several phases are associated, there are achieved poly-phase signals characterized by the same frequency of all signals. If more signals have the same frequency, the same amplitude and equal phase shifts, then they form a symmetrical polyphase system. The most used example is the three-phase system:

333

222

111

sin)(

sin)(

sin)(

tAta

tAta

tAta

m

m

m

a) If Am1 = Am2 = Am3 = Am şi γ1 - γ2 = γ2 – γ3 = γ3 - γ1 =2π/3, then the signals a1(t), a2(t), a3(t) form a direct symmetric three-phase system:

)3

4(

3

)3

2(

2

1

133

12

11

1

1

1

3

4sin)(

3

2sin)(

sin)(

j

m

j

m

jm

m

m

m

eAA

eAA

eAA

tAta

tAta

tAta


00 321321 aaaAAA

0

1

2

3

2

12

3

2

1

32

3

63

3

42

3

2

aaa

ea

jea

jea

j

j

j

13

4

3

123

2

2

1

1

1

1

AaeaeA

AaeaeA

eaA

jj

jj

j


b) If Am1 = Am2 = Am3 = Am şi γ1 – γ2 = γ2 – γ3 = γ3 – γ1 = - 2π/3, then the signals a1 (t), a2 (t), a3 (t) form an inverse symmetrical three-phase system and the phases succeed in counterclockwise:

00 321321 aaaAAA

c) If Am1 = Am2 = Am3 = Am and γ1 – γ2 = γ2 – γ3 = γ3 – γ1 = 2π, then the signals a1 (t), a2 (t), a3 (t) form a homopolar symmetrical three-phase system:

03 1321

)4()2(321

111

AAAA

eAeAeAAAA jm

jm

jm


Connexion of the polyphase nets. Signal systems in these nets

They are defined the following three-phase systems of current and voltages:-if1, if2, if3 – phase currents crossing each impedance of the circuit, from generator to receiver;-u1, u2, u3 – phase voltages to receivers, produced by the phase currents;- u1’, u2’, u3’ – phase voltages at generators.The phase currents of the circuit and the phase voltages between the terminals are associated to the rules of the generators. If the phases are different, then must be used 6 conductors with similar cross-section. If they are connected together the begining of the phase of the generator and the end of the phase of the receiver, there are obtained two null points, called N and N’. It appears that the 3 back wires can be replaced by one single conductor with reduced section or the conductor can be even eliminated. A such connexion is called star connexion, with or without neutral conductor.


If there is no neutral conductor between N and N’, then INN’ =0 and Il1+ Il2 + Il3 =0.They could be defined the lines voltages u12, u23 şi u31. Their sum is equal to zero:

1331

3223

2112

uuu

uuu

uuu

If the system of the phase voltages is direct symmetric and the receiver is symmetric, Z 1=Z2 =Z3 =Z, then the following expressions could be written:

j

j

f

j

f

j

fl

fl

ZeZ

eUUaU

eUUaU

eUUUU

II

UU

ZZZZ

UaUUaUUU

6

5

1231

212

223

612112

321

1312

211

3

3

3'','',''


The line currents and voltages, the phase currents and voltages compose three-phase symmetric systems. The system of the line voltages is phase shift with π / 6 before the system of the phase voltages. For the calculation of the instantaneous power of the system there are used the voltages and the currents of the system as following:

cos3)(

3

4sin

3

2sin

sin

,

3

4sin

3

2sin

sin

332211321

3

2

1

3

2

1

IUiuiuiutptptptp

tIi

tIi

tIi

tUu

tUu

tUu

um

um

um

um

um

um

It could be observed that the instantaneous power is time independent, the system is balanced and U and I are the phase signals. The active power and the reactive power take the following values:

llfff

llfff

llfff

fff

fff

fff

IUIUSS

IUIUQQ

IUIUPP

IUS

IUQIUP

333

sin3sin33

cos3cos33sincos


llff

llff

llfffff

ll

j

fll

j

fl

fl

fl

lll

IUIUS

IUIUQ

IUIUPIUP

IaI

eIIaI

eII

ii

uu

uuu

III

33

sin3sin3

cos3cos3cos

3

3

3

0

0

13

6

5

12

2

61

312312

321


An induced electromotive voltage could be obtained by rotating a conductor turn (spire) in a homogeneous magnetic field. The turn is crossed by the magnetic flux with the maximum value Φm, obtained when the magnetic field lines are perpendicular to the plane of the coil:

tEete

dt

de

t

t

mm

mm

sinsin

coscos


Synchronous machine

Industrial synchronous generators working in AC are based on the principle of the induced electromotive voltage (described on the previous slide). Their structure is different and is composed by:-a rotor R having placed on its poles the inductive winding, supplied in DC; - a stator S having inserted in its gaps the induced winding which could be monophase or three-phase. When the rotor is rotating, being mechanically driven by a motor, and the inductive winding is supplied, in the turns of the induced winding is produced by electromagnetic induction an electromotive voltage with a periodic sinusoidal evolution.


Synchronous machine

The electromotive voltage has its period when the rotor is rotating with the angle equal to the double of the polar step. If the rotor has p pole pairs, then for a complete rotation of the rotor the electromotive voltage covers p periods. If the speed of the rotor is n [rot/sec], then the frequency of the electromotive voltage is f = n p showing that its value is directly proportional with the speed of the generator. For this reason the name of this machine is “AC synchronous generator“.


The rotating magnetic field

The magnetic rotating fields could be produced by three-phase systems of the electric currents and underlying the operation of the rotating electric machines. These machines have been made firstly by G. Ferrari (1884) and the first polyphase engine (a two-phase one), based on the rotating magnetic field, was achieved by N. Tesla in 1888. These magnetic fields could be classified: - with rotating vector of the magnetic induction; - with rotating radial division.

The theorem of Leblanc: any alternative magnetic field is resulted from superposition of 2 circular rotating fields having the amplitude Bm /2 and contrary rotating with angular velocity ω.

Circular rotating magnetic fields could be ensured by 2 identical coils placed in perpendicular planes with a common axis and crossed by electrical currents with the same frequency but with a quadrature phase shift (for example the monophase machines with auxiliary phase).


The rotating magnetic field

dt

dt

B

Bttgtg

B

tBBB

tBBB

tIi

tIi

tBB

tBB

y

x

m

my

mx

m

m

m

m

;

cos

sin

)2

sin(

sin

cos

sin

2

1

2

1

2

1

In these conditions it is produced an equivalent field with constant amplitude and angular velocity equal to the angular frequency through the coil, which is rotating clockwise and increases the α angle. If the sense of the electric current through one of the coils is inverse, then the angular velocity becomes –ω and the equivalent rotating field changes its rotation in counterclockwise.


Rotating magnetic field with rotating vector of the magnetic induction

The end effect is neglected and it is assumed that the coils are placed close enough to origin (the point 0).

3033

2022

1011

13

12

11

3

4sin

3

2sin

sin

inkB

inkB

inkB

tIi

tIi

tIi

im

im

im


k is a coefficient of proportionality. Each of the magnetic fields having the amplitudes B1, B2 or B3 is an alternating magnetic field. We are interested by the equivalent value obtained in origin.

2332

321321

321

13

12

11

2

3

6cos

6cos

23cos

3cos

3

4sin

3

2sin

sin

BBBBB

BBBBBBB

BBBB

tkIB

tkIB

tkIB

y

x

im

im

im

1

22

1

11 2

3

cos2

3

sin2

3

2

3

iy

x

myx

imy

imx

tB

Barctg

kIBBB

tkIB

tkIBB


By superposing the magnetic fields in the origin 0 is obtained a rotating magnetic field with an amplitude of 50 % bigger than the amplitude of each alternative magnetic field, which uniformly rotates with the angular velocity ω of the electric currents. The extremity of the magnetic induction B describes a circle into x0y plan. Consequently it was obtained a circular rotating magnetic field. If the three-phase currents system is non-symmetrical or the electric currents through the coils are not phase shift with 2π / 3, then the rotating magnetic field is elliptic. If into the symmetrical system there is inversed the order of the coils (coil 2 changed with coil 3), then in origin point 0 it is produced a magnetic induction B’, characterized by:

'

'

'

' BB

BB

BB

yy

xx


In this way is produced a rotating magnetic field which keeps its amplitude and the angular velocity, but with opposite sense.

If for the same symmetrical system is maintained the order of the coils and is changed the sense of the electrical current through the coil 1, then is produced an elliptic rotating field having the magnetic induction B”:

49

''''

cos2

3''

sin2

1

3'' 222

2

1

1my

x

imyy

imx

x IkBB

tkIBB

tkIB

B


Rotating magnetic field with rotating radial division The rotating electric machines are built from 2 coaxial ferromagnetic cylinders separated by an air gap. The exterior cylinder, called stator, is fixed and the interior one, called rotor, is moving. If in the stator slots is placed a winding crossed by an electric current and composed by insulated conductor bars, having connections with the frontal parts of the stator, then the surface bounded by the circuit of the winding will be crossed through a magnetic field having the field lines which close through the stator, the air gap and the rotor. The areas where the field lines enter and leave the stator could be considered as poles (N and S) of the stator magnetic field. Where inside the slots is placed one winding, then we obtain p = 1. If inside the slots are placed two windings with a phase shift of π / 2, then we obtain a magnetic field with four poles (p = 2). Because of the important value of the magnetic permeabilities for the stator and the rotor, the field lines into the air gap are perpedicular to this one. In this case the magnetic field is circular. When is scrolled through the air gap, the magnetic field B takes minimum or maximum value on the neutral axis, which is the axis passing through the middle of the polar step. This parameter is periodic with Tp.

pT

Tp

Tp

pT

pT

p

pp

2

2

212


In order to obtain then spatial distribution of the magnetic induction B they are conducted the circumferences of the stator and of the rotor. An uniform distribution of conductors involves trapezoidal spatial distributions of magnetomotive force and also of the magnetic induction. The lines of the magnetic field are closing around the conductors passed by the electric currents and are symmetrically distributed from the middle of each side of the coil width. The field lines have opposite sense from each side of the neutral axis. Taking in consideration infinite values of the magnetic permeabilities of the stator and of the rotor and applying the Ampere's theorem, it is obtained the magnetic induction into a point placed to the distance x from the neutral axis (where δ is the length of the air gap, N – number of the turns of the coil and b – the width of the coil):

b

xiNB

b

xiNBldB xxSll

2

2 000

For the points placed on the side of the coil x = b and the magnetic induction becomes:

20

NiB

If τ / 2=Tp / τ, then bp = 2π / 3, the third harmonics and the multiples of 3 are missing from the spectrum of the magnetic induction. Its spatial division is close from a sinusoidal signal.


pBB sin0

im

imim

tpBB

tBBtIi

sinsin

sinsin 0

where α is the spatial angular variable and β – the initial phase. If the magnetic induction is produced by a sinusoidal current, then it will have also a sinusoidal time evolution:


The time evolution of the magnetic induction corresponds to an alternative radial field. The analytical expression of the magnetic induction is similar to a stationary wave having the antinodes at the poles position and the nodes at the position of the poles’ axis. There is considered that into the stator slots are placed 3 identical windings having phase shifted with 2π / 3 and crossed by a three-phase system of electric currents. The values of the magnetic inductions produced by the three-phase system of currents are:

3

4sin

3

4sin

3

2sin

3

2sin

sinsin

13

12

11

ptBB

ptBB

ptBB

im

im

im


1

321

cos2

3,

,

im ptBtB

BBBtB

The resultant value of the magnetic induction is not anymore a stationary wave, bu a progressive wave, spatial distributed after a sinusoidal law. This wave is moving into the air gap with a constant speed which could be calculated with formula:

dt

d

pdpdtcstpt i

01 0.

By superposing the three radial alternative magnetic fields is obtained a radial magnetic field with rotating spatial distribution. It rotates with the angular velocity Ω0 called synchronous speed. This field could be obtained inside the air gap of a three-phase electric machine with 2 poles, being provided by three identical windings placed inside the slots of the stator, spatially shifted with 2π / 3p and crossed by a symmetrical three-phase system of the electric currents.


Three-phase asynchronous induction machine

For the study of the spatial distribution of the magnetic induction B it is considered the circumferences development of the stator and of the rotor. The stator winding is the inductor of the machine and is supplied from a three-phase symmetric net. The rotor winding is the induced of the machine. The three-phase symmetric system of the electric currents produces into the air gap a magnetic field with a radial rotating distribution. This magnetic field rotates with respect to the conductor with the angular velocity Ω0. Electromotive forces are induced by electromagnetic induction into the winding of the rotor, placed in a rotating magnetic field of the stator. The electromotive forces produce electric currents providing alternative magnetic fields, because the stator winding is usually in shortcircuit. The result of these magnetic fields into the air gap is a rotating magnetic field corresponding to the rotor. There is a tendency to keep invariable the values of the magnetic fluxes through the rotor windings surfaces.But because the rotor has only one degree of freedom it will result its rotation in the same direction with the same angular velocity as the rotating field of the stator.


Three-phase asynchronous induction machine

In this way could be explained the aperance of the shaft torque and thus the operation of this machine as a motor. In fact, due to losses and the load, the rotor rotates at a rotational speed less than the synchronous value:

0

The relative difference of the two speeds is assessed using a parameter called synchronous motor slip:

%;1,0;0

0

sss


When the machine is starting the motor slip is s =1. When the electric machine reaches synchronism, Ω = Ω0 and s = 0. In this situation the electromagnetic induction phenomenon is stopped and the shaft torque is zero. At angular velocities lower than the synchronous value, inside the rotor winding are induced electric currents having the next angular frequency:

ssppr 00

0' sp

s

pr

r

The rotational speed of the rotating magnetic field relative to the rotor is:

To determine the angular velocity of the field relative to the stator must be taken into account the relative velocity between the stator and rotor, Ω:

0000 1' sssrrTherefore the rotating magnetic fields of the stator and of the rotor of an asynchronous three-phase induction machine are rotating in the same direction and with the same angular velocity, independently of the angular velocity of the rotor. Their common velocity is that one corresponding to the synchronism of the machine .


If the stator winding is supplied from a three-phase net and the rotor is mechanically driven in meaning of the stator rotating field at a speed higher than synchronous value (Ω > Ω0), then the electric machine behaves as an asynchronous generator with the motor slip s <0. If the stator winding is supplied from a three-phase net and the rotor is mechanically driven in opposite meaning of the stator rotating field (sign Ω = - sign Ω0), then the electric machine behaves as an asynchronous brake with the motor slip s >1. In three-phase circuits containing rotating electrical machines, such as three-phase induction machine, direct impedance is different from the reverse impedance. For nominal working regime when is applied a direct symmetric system of voltages to the stator windings, the magnetic rotating field of the stator and the rotor are rotating in the same sense. The rotation velocity of the rotor is usually smaller with 1,5 ÷ 4 % comparative with the rotating magnetic field of the stator, because of the motor slip. The motor behaves differently if it keeps the same sense as in previous case (being mechanically engaged by another motor, for example) and the stator windings are supplied from an inverse symmetrical system of voltages. In this case the rotating magnetic field of the stator will rotate with the same angular velocity as in previous case, but the sense of rotation is opposite.


Consequently the relative velocity of the rotating magnetic field of the stator relative to the rotor will be almost the double of the angular velocity of the same magnetic field relative to the stator and much bigger than the relative velocity relative to the rotor in nominal conditions of functioning (when the stator windings are supplied from a direct symmetric system of voltages).Because of this the amplitudes of the induced currents in the rotor winding will be much higher. Taking into consideration the principle of electromagnetic inertia, these currents will produce magnetic fields which will decrease the rotating magnetic field of the stator in a more important manner that in any nominal working regime. This contributes to reduce the electromotive force induced by the rotating magnetic field of the stator and consequently, considering Ohm's law, (e+u=ri), will result an increase in the amplitudes of the stator currents. At equal values of the amplitude and of the frequency of the symmetrical direct and reverse applied voltage of the stator winding and at the same angular velocity, the direction of rotation of the rotor in both cases being the same, the electric currents of the reverse succession appears to be more intense than the one of direct succession. Consequently the opposite motor impedance of the currents reverse system is less than the opposite impedance of the currents direct system: Zi < Zd.


The mechanical characteristics of the asynchronous machine

These are: the velocity evolution function of the useful power, n (P2), the slip evolution function of the useful power, s (P2), the shaft torque evolution function of the useful power, M (P2), the yield evolution function of the useful power, η (P2), the power factor evolution function of the useful power, cos φ1 (P2), the absorbed electric current evolution function of the useful power, I1 (P2), the absorbed primary power evolution function of the useful power, P1 (P2), established in normal conditions. The supply voltage and the frequency of the supply voltage are constant (U1 = ct. = U1N , f1 = ct. = f1N). The additional resistance of the rotor winding is zero, R2s = 0, in the case of the asynchronous motors with wound rotor motor. In the next figure all these characteristics are shown, in the case of an asynchronous motor with squirrel cage (nominal power of 50 kW). The dependence of n = f (P2) has a similar shape with n = f (M), (represented in the previous figure only the portion corresponding to the slid range of 0 ÷ sk).The analysis of this dependence shows a little decrease of the speed for an increasing load, so this dependence is rigid.The shape of the energy efficiency characteristic is similar to the characteristic of the transformer. The nominal values of this efficiency are from 0.85 to 0.9 for average power machines. These values are slightly lower than in the case of the transformer because the ventilation losses, the mechanic losses and other supplementary losses.

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The mechanical characteristics of the asynchronous machine

Power factor characteristic, cos φ1 = f(P2), has a similar feature to the yield allure.Because asynchronous motor absorbs reactive power for magnetization, the power factor is always below unit.The operating characteristics of the induction motor can be determined either by direct charging, changing the charge between 0 ÷ 1,2 · PN, or by a synthetic method. The first of these, although involving high energy consumption, it is more accurate.For direct charging it is assumed that are known mechanical losses, pmec, the ventilation, v, and iron losses, pFe, separated by an idle test and also the measured resistance R1 at warm.It is measured for each charge the load consumption P1, the electric current I1 and the motor slip s and is applied the formula:

111

11cos

IUm

P


The processing of the experimental data suppose the following steps:-Subtracting from the power P1 of the power losses from the stator winding and the iron losses, it is obtained the electromagnetic power PM and consequently the mechanic torque M. - Because the sliding parameter s is measured, the lossed from the rotor winding could be calculated with formula pJ2 = s PM.- Substracting from PM the losses termspJ2 and pmec+v it is obtained the useful power P2 and the efficiency of the transformer:

Repeating the calculations for different loads, it is experimental determined the working characteristics of the machine.

1

2

P

P

Electric Machines

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