an introduction to electrical machines - unipv · pdf filetransformer: static electrical...

An Introduction to Electrical Machines

P. Di Barba, University of Pavia, Italy Academic year 2011-2012

Contents

Transformer • 1. An overview of the device • 2. Principle of operation of a single-phase transformer • 3. Power (three-phase) transformer, signal transformer, autotransformer

Rotating electrical machines (REM) • 1. Ampère law, Faraday principle, Lorentz force • 2. An overview of REM • 3. Rotating magnetic field

Asynchronous machine: induction motor Synchronous machine: alternator DC machine: commutator motor, dynamo Permanent magnet motor Stepping motor

Transformer: static electrical machine operating in AC regime. It converts electric power into electric power, by varying the power factors (V,I) and keeping the power (approximately) constant.

1 : primary circuit (power in) 2 : secondary circuit (power out)

Single-phase

Three-phase

Applications Power transformer • Production systems operate at a reduced voltage in order to decrease dielectric material size • Transmission systems operate at a low current in order to decrease conducting material size • User systems operate at a low voltage in order to increase safety Signal (and power) transformer • Adapts the voltage of two systems to be connected • Adapts the impedance of two systems to be connected • Couples two systems, keeping them electrically disconnected

Step up transformer: increases V and decreases I Step down transformer: decreases V and increases I Power range: from 1 VA to 1000 MVA Voltage range: from 1 V to 400 kV

Basically, a transformer is composed of: • a magnetic circuit; • (at least) two magnetically-coupled electric circuits.

1 2

yoke

lim

b

2 1

low-voltage terminal

high-voltage terminal lamination package (core)

A>25 kVA forced cooling

A<25 kVA natural cooling

Single-phase power transformer (25 Hz < f < 400 Hz)

windings

a b c

1 2

Three-phase power transformer (f=50 Hz) : three pairs of windings

Signal transformer single phase 20 Hz < f < 20 kHz

open magnetic circuit (f>20 kHz, core made of Fe2O3)

closed magnetic circuit

Three primary windings 1a,1b,1c and three secondary windings 2a,2b,2c Used connections: star-delta, delta-delta, delta-star.

Operating principle (ideal case) Model assumptions: infinite core permeability, zero losses in the magnetic circuit and in the two electric circuits.

1: N1 turns

2: N2 turns

221121

21

2

112

IVIVAA

Ik

1I

N

Nk,V

k

1V

Transformer equations

1 1I

2

2I

Z

F

Transformer laws (ideal model) I No-load operation (I2=0) Let a sinusoidal voltage source v1(t) supply winding 1 Then, an infinitesimal magnetizing current I10 flows in winding 1 and gives rise to a magnetomotive force (mmf) Hopkinson law: magnetic flux

infinitesimal mmf infinitesimal reluctance non-infinitesimal sinusoidal flux

F101IN

Transformer laws (ideal model) II Flux F links both windings 1 and 2 Faraday-Lenz law: induced emf KVL v1=-e1 , v2=-e2 In phasor notation: In terms of RMS values, it turns out to be: transformer voltage law

dt

dNe

F 11

dt

dNe

F 22

kN

N

V

V

2

1

2

1

F 11 NjV F 22 NjV

F

2

1 10I

1

2

0EV

0EV

22

11

Kirchhoff voltage law

Terminal marking: currents entering the marked terminals originate like fluxes in the magnetic core.

I10 magnetizing current

Primary circuit: supplied by V1

Secondary circuit: open circuited

Transformer laws (ideal model) III On-load operation Current I2 is delivered by winding 2 to load Z mmf N2I2 originates a secondary flux F2 opposing the primary flux F1 = FM , but cannot vary due to the applied primary voltage V1

fN

VM

1

1

44.4F

Transformer laws (ideal model) IV Consequently, voltage source V1 is forced to deliver an additional current I1 to winding 1 in order to compensate the flux mismatch: N1I1-N2I2=N1I10=0 It turns out to be: transformer current law Finally, the complex power conservation holds:

kN

N

I

I 1

1

2

2

1

2211 IVIV

Vector diagram (Kapp diagram)

A resistive-inductive load is assumed

A more realistic model - 1

In a real transformer, power losses are not zero: P0=GV2 active power in the magnetic core (eddy current and hysteresis) Pc=RI2 active power in the electric circuits due to the Joule effect

Moreover, the finite magnetic permeability of the core causes a non-zero magnetizing power: Q0=BV2 reactive power in the magnetic circuit of main flux (inside the core) QC=XI2 reactive power in the magnetic circuit of stray flux (outside the core)

P0 and Q0 dominate in the no-load operation (measured in the open-circuit test)

Pc and Qc dominate in the on-load operation (measured in the short-circuit test)


As a consequence, in a real transformer: V1 is different from –E1 due to voltage drops caused by R and X of the windings F is not constant with respect to E1 because E1 varies with I1 for a given V1

V1 is different from kV2 due to on-load voltage drops I1 is different from I2/k due to no-load primary current A1 is different from A2 due to active and reactive power losses

Transformer equations based on the ideal model are valid as a first approximation only


stray flux main flux

power loss (eddy currents, hysteresis)

power loss (Joule effect)

Z R jX

Y G jB

21

2221

21

2221

121

212

BVXIQQ

GVRIPP

VYIk

1I

IZVk

1V


Primary winding admittance

Secondary winding impedance


Modified Kapp diagram

2

11

kZ

VI

Impedance adaption - 1

Problem: how to modify the value of an impedance Z to be connected at a terminal pair A-B ?

Impedance adaption - 2

Zk

V

k

1

Z

1

k

V

k

1

Z

V

k

1II

2

11221

Signal transformer: frequency response - 1

Xm Xc

Signal transformer: frequency response - 2

senjIIjXRVIjXRVVVk

22222201 cos)(1

V V20 V2

Voltage drop - 1

sincos 22220 XIRIVVV

Voltage drop - 2

Insulation transformer - 1

Measurement

Safety



Damping

kN

NN

V

V

2

12

2

1

k

1

NN

N

I

I

12

2

2

1

AAIVIVA 222111

212121 )()( VIIIVVAd

Auto-transformer

1A

Ad

Rotating Electrical Machine Basically, it consists of: • an electric circuit (inductor) originating the magnetic field; • a magnetic circuit concentrating the field lines; • an electric circuit (armature) experiencing electrodynamical force or electromotive force.

bi

Ampère law

e

Faraday law

transformer effect

b(t) e(t), i(t)

Faraday law

evb

(motional effect)

Fib

Lorentz force

stator

air-gap

rotor

inductor circuit

armature circuit

r

Bd

ld

i

B

n

Magnetic field of a circular coil

nr2

iB 0

1B

2B

j axis

r axis

B

1

2 Rotating magnetic field (two-phase) I

tr

Int

r

IB

cos

2cos

2

01

01

tsenr

Ijntsen

r

Int

r

IB

222cos

2

02

02

02

1B

2B

j axis

r axis

B

1

2

Rotating magnetic field (two-phase) II

tjer

Itsenjt

r

IBBB

2cos

2

0021

1

r axis

B

j axis

2

3

tjer

IB

22

3 0

Rotating magnetic field (three-phase): Ferraris field

10

1 cos2

ntr

IB

20

2

3

2cos

2nt

r

IB

30

3

3

4cos

2nt

r

IB

Three-phase asynchronous motor (induction motor)

graphic symbol

As a motor, it converts electric power into mechanical power

three-phase inductor circuit

ventilating fan

laminated stator and rotor

shaft

air gap

W

Squirrel-cage rotor Wound rotor

slip-ring commutator (single-phase)

end ring

bar

Induction motor: operating principle

• Inductor terminals are supplied by a three-phase symmetric voltage V1

• A three-phase balanced current I1 is delivered to inductor windings. • An induction field B1 rotating at speed N0 = 120 f p-1 is originated (f current frequency, p number of poles, speed in rpm). • In stator and rotor windings, subject to a sinusoidal magnetic flux F, electromotive forces E1=2.22 m1 f F and E2=2.22 m2 f F are induced, respectively. F max value of flux, E1,2 rms value of electromotive force m1,2 number of conductors in a winding

E2 three-phase balanced current I2 in the rotor windings

Magnetic effect of I2 : • a synchronous rotating field B2 (speed N0) is originated, which tends to decrease the inductor rotating field B1

• B1 depends on the applied voltage V1 and cannot vary • voltage source V1 is forced to deliver an additional current I3 to the inductor, such that rotating field B3 due to I3 compensates field B2

Mechanical effect of I2 : the axially-directed conductors placed in the radially-directed field B1

experience tangentially-directed forces Ft

The rotor experiences a torque C=2RFt: if unconstrained, it rotates.

Having defined the slip factor s=(N0-N)/N0

as the speed of the rotating field with respect to the rotor, the frequency of E2 and I2 is f2=sf

If the opposing torque is equal to zero then, the running torque CM=0 and f ≠ 0 I2=0 E2=0 f2=0 s=0 N=N0 the rotor is synchronous wrt the rotating field

If the opposing torque is different from zero then, the running torque CM≠0 and f ≠ 0 I2 ≠ 0 E2 ≠ 0 f2 ≠ 0 0<s<1 N<N0 the rotor is asynchronous wrt the rotating field (the rotor follows the field)

Torque-speed curve

Torque C is proportional to both flux F1 of the rotating field and real part of the armature current I2cos f2 (active component):

Current I2 and power factor cos f2 depend on emf E2 and armature circuit impedance:

On the other hand, electromotive force E2 depends on both relative rotor speed N0-N and flux F1

It turns out to be:

221 cosIkC F

2

2

2

22 LsRZ resistive-inductive impedance R2+jsL2

102 ' F NNkE

2

12

2

2

0

2

20

20

2

02

11

2

2

2

21 , VCNCC

LNNRN

RNNNk

Z

R

Z

EkC

FF

N N0 0

CR

CS

C

operating point

CM

Torque-speed curve

p

f120N0 synchronous speed (rpm)

self-starting Cs = k1F1

2 N0R2/(R22+2L2

2)

mechanical power Pm = CN electric power Pe = √3 VI cosf

Max torque CM = k1F1

2 N0/(2L2) when N = N0(1-R2/L2)

N

C

1

2

N

C

1

2

N

C

1

2

Speed control

increasing R2 (wound rotor only)

decreasing f decreasing V

Synchronous generator

turbine alternator

Three-phase output

Alternator structure

Three-phase armature circuit

DC inductor circuit (two-pole) In low-power alternators it can be replaced by a permanent magnet.

As a generator, it converts mechanical power into electric power.

• The inductor circuit is supplied by a DC current IE

• A magnetic flux FR sinusoidally distributed in the air-gap is originated

• A speed W is applied to the rotor shaft

• In the armature circuit a three-phase symmetric electromotive force

E = kWFR at an angular frequency = Wp/2 is induced • If the armature circuit is on load, a three-phase (balanced) current I is delivered, so providing an active power P = 3EI cos f

• An equal amount of mechanical power must be supplied to the shaft

• Current I generates a synchronous rotating field (speed ), which originates a flux FS opposing flux FR

• In order to keep E constant, excitation current IE must be increased

Alternator operating principle

I I

2

pW

Two-pole DC winding Four-pole DC winding

number of poles electric frequency

rotor speed

DC machine

graphic symbol

From electric to mechanical power: motor From mechanical to electric power: dynamo

ventilating fan

laminated stator and rotor

shaft sliding brushes with commutator

armature

inductor

salient pole

air gap

DC machine - Principle of operation I

MOTOR Inductor circuit is supplied by DC voltage VE

Current IE is originated Magnetic field B, and so flux F, take place Armature circuit is supplied by DC current IA Armature conductors, carrying axial current IA and placed in radial field B, are subject to tangential force F Main effect F C N Pm = CN Secondary effect Due to the rotor motion, in the armature conductors an AC emf is induced, which is subsequently rectified by the multi-sector commutator. Therefore, a back emf E = kNF appears at the commutator terminals. It turns out to be: E = V (V voltage across armature terminals) Pe = EIA = CN = Pm (power balance)

DC machine - Principle of operation II

DYNAMO Inductor circuit is supplied by DC voltage VE Current IE is originated Magnetic field B, and so flux F, take place Rotor is given speed N, externally applied to the shaft In the armature conductors, an AC emf is induced At the commutator terminals a rectified emf E = kNF is available Main effect If armature terminals are connected to a resistive load E IA Pe = EIA

Secondary effect Armature conductors, carrying axial current IA and placed in radial field B, are subject to tangential force F opposing the motion F C N Pm = CN To maintain the motion, power Pm must be supplied to the shaft (Pm = Pe) .

+

B

A

b

a

B

A

b

a

t t1

a b

t2

t

VAB

t

VAB

Multi-sector commutator: principle of operation

As a result, the original AC signal is mechanically transformed into a DC signal, and viceversa.

with two conductors/sectors

with several conductors/sectors

The polarity of each sector is reversed when crossing the A-B line joining the brushes (orthogonal to the pole axis).

Each armature conductor is electrically connected to a sector.

The voltage VAB between brushes exhibits always the same polarity.

MVE

I

Series excitation (motors for traction) When Cres varies, both N and C vary in such a way that CN const (constant power motor)

MVE

IE IA

VA

Independent excitation

MV

IE IA

Derived excitation

When Cres varies, excitation current IE varies N const (constant speed motor)

DC motor excitation schemes

F kNE

AA

R

EV'kI'kC

FF

EIFF

Equivalent circuit of the DC motor (with independent excitation)

k voltage constant

k’ torque constant

Induced emf (rectified)

Excitation flux

inductor circuit

armature circuit (Ra resistance of armature conductor)

Torque

F F(IE )

F

k

VN0

AS

R

V'kC F

Torque-speed curve (independent-excitation motor)

self-starting

NCkNVR

kC

A

FF

'

Speed control

Ideal diode: current-voltage curve

AC to DC conversion - 1

iR i1 i2

R RiR

Transformer + single phase rectifier


Single-phase rectifier (full-wave): principle of operation


A rectified voltage vR is originated: non-zero average value

Wound synchronous motor

Excitation

windings

Permanent magnet synchronous motor

Magnets

Magnets

SPM IPM

Synchronous motor: torque-speed curve

Reference signal Rotor position sensor (Hall probe)

feedback Control system

Three-phase inverter

Brushless DC motor: principle of operation

Stepping motor: principle of operation

Permanent-magnet stepping motor

Variable-reluctance stepping motor

an introduction to electrical machines - unipv · pdf filetransformer: static electrical...

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