Tài liệu Fundamentals of electromechanical energy conversion docx

Converters with linear motion are called linear electricmachines, while those that rely on rotating motion are called rotating electric machines.. Once when the conductor moves, accordin

FUNDAMENTALS OF ELECTROMECHANICAL ENERGY

CONVERSION

Electromechanical energy conversion is achievable in a number of ways Thesepossibilities rely on different fundamental laws of electrical engineering As the only methodthat has importance on the large scale is electromechanical conversion achieved by means ofelectromagnetic converters, this section is fully devoted to the analysis of basic principlesinvolved in electromagnetic electromechanical conversion

Electromechanical energy conversion is achieved by devices that are usually calledelectric machines In principle, laws of electromagnetics can be used to design converters withthe linear and with the rotary motion Converters with linear motion are called linear electricmachines, while those that rely on rotating motion are called rotating electric machines Vastmajority of existing machinery belong to the category of rotating electric machines Theseinclude all the machines used to generate the electricity, as well as the most of the machinesused in industry to perform some useful work while converting electric into mechanicalenergy Linear machines are used relatively rare for somewhat specialised applications It isfor this reason that only rotating electric machines will be dealt with here Prefix ‘rotating’will be omitted and the converters will be called simply electric machines, implying thatdevices under consideration are characterised with rotational movement

Operating principles of electric machines involve two basic laws of electromagnetism,namely the law of the electromagnetic induction (Faraday’s law) and the law of force creation

in an electromagnetic field (Bio-Savart’s law)

Consider the situation shown in Fig 1 A conductor is connected to an electric source

and it carries current I It is placed in the magnetic field of certain flux density B (which is of

course a vector; hence the arrow above the symbol in Fig 1) Interaction of the flux densityand the conductor current leads to the creation of an electromagnetic force

B

l

I

where l is the conductor length This electromagnetic force will cause the movement of the

conductor, which will start travelling at certain linear speedv to the left The electromagnetic

force will be balanced by another, mechanical force that acts in the opposite direction (to theright) The equilibrium will be established when the two forces are mutually equal and theconductor will then travel at a constant speed Note that the magnitude of force in (1) willsimply beF e = IlB, since the angle between the conductor and the flux density is 90 degrees.

Once when the conductor moves, according to the law of electromagnetic induction anelectromotive force will be induced in the electric circuit,

( )v B l

The magnitude of this emf is simplye =vBl, since the angle between the speed vector and flux

density vector is 90 degrees, while the angle between the conductor length vector and thevector product is zero degrees

A process of electromechanical conversion is established in this way The energy will

be converted from electrical to mechanical and the process is calledmotoring In order for the

motoring to happen it is necessary to: i) establish flux density, using permanent magnets forexample; ii) create an electric circuit, that is connected to a voltage source and is placed in theflux density This will lead to establishment of the electromagnetic force, which causes linearmovement of the conductor This movement is counterbalanced by the applied mechanical

force (not shown in Fig 1) and the equilibrium is established when the conductor travels at aconstant speed Under this condition the electromagnetic force and the mechanical force aremutually equal, but act in the opposite directions.

Consider next Fig 2, where the same conductor is placed in the same flux density.However, the conductor is now not connected to the electric source; instead, the electriccircuit is closed by using, say, an external resistance The conductor is now dragged throughthe flux density using mechanical force at certain speed and this is the origin of the movement

in this case The sequence of events now reverses An electromotive force, given with (2), is

at first induced in the conductor Since the circuit is closed, a current starts flowing.Interaction of the current and the flux density causes creation of the electromagnetic force.This force again acts in the opposite direction to the mechanical force and the equilibrium isestablished when the two forces are equal but act in the opposite direction Note that in thiscase the source of motion is the supplied mechanical energy The mechanical energy is nowconverted into electrical energy and the process is calledgeneration.

Fig 2 – Illustration of generation.

It is important to note here that the process of electromechanical energy conversion isreversible This means that either electric energy can be converted into mechanical energy, ormechanical energy can be converted into electric energy, by means of the same physicalassembly Note as well that both the expression for electromagnetic force acting on aconductor and the expression for induced electromotive force due to relative movement ofconductor with respect to flux density, which are vectorial, reduce to very simple expressionsdue to the relative position of flux density vector, conductor and speed of motion This isexactly the situation that is encountered in electric machines Therefore equations (1) and (2),which contain scalar and vectorial multiplications, reduce to a very simple form of F e = IlB

ande = vBl.

Nothing changes in principle when rotational movement is under consideration instead

of the linear movement Table I gives the analogy between the linear and the rotationalmovement Creation of torque in the case of rotational movement is illustrated in Fig 3

Table I – Analogy between linear and rotational movement.

Speed Source of motion Road travelled Power Linear motion Linear, v [m/s] Force, [N] Linear, s [m] F v

Rotational motion Angular, ω [rad/s] Torque, [Nm] Angle, θ [rad] Tω

Suppose once more that there is a certain flux density, in which a structure is placed.This structure can rotate and is of radius r Assume that there are two conductors placed on

the structure, 180 degrees apart, as shown in Fig 3, and let these two conductors carry current

in designated (mutually opposite) directions An electromagnetic force, F e = IlB, is created on

each of the two conductors However, one of these two forces acts to the left, while the other

one acts to the right (due to opposite directions of the current flow in the two conductors).Now, a torque is created on each of the two conductors, that equals the product of the forceand the radius However, since forces act in opposite directions at opposite sides of thestructure, the torques will both act in anticlockwise direction, initiating the rotation of thestructure in anticlockwise direction The total electromagnetic torque will in general be thesum of all the individual torques acting on individual conductors.

Fig 3 – Torque creation in the rotating structure.

Every electric machine consists of ferromagnetic iron cores and windings mounted onthe iron cores, these elements being of essential importance for electromechanical conversion

An electric machine consists of a stationary element, called stator, and a rotating element(such as the one in Fig 3) called rotor The winding is placed in slots of the stationary statorand/or in slots of rotational rotor The winding consists of an appropriate number of turns Aturn is composed of two conductors which are placed in such a way that the inducedelectromotive forces in them sum up The current therefore flows in the opposite direction, asillustrated in Fig 3

As already noted and explained, the operation of electric machines relies on Faraday'slaw of electromagnetic induction and on Bio-Savar's law of electromagnetic force (torque).One important point to note is that the induced emf will be described with (2) only if thecurrent in the system is pure constant DC current A more general expression for the inducedemf says that, if the total flux through the electric circuit is changed, an electromotive force isinduced,

θω

θθψ

ψ

d

dL i dt

di

L

e

dt d d dL i dt di L dt dL i dt

The first term in this expression will exist only in circuits with AC currents and it is called

transformer emf The second term is what corresponds to (2) and it is the induced emf due to

the movement of a conductor in certain flux density It is called rotational emf Note that,

according to (3), a rotational emf will be induced only if the inductance of an electromagneticstructure is a function of the rotor position θ This may sound awkward but will be clarifiedlater on In deriving (3) the use was made of the correlation between the angle travelled by therotor and its speed of rotation,

= ωdt

that reduces for a constant speed of rotation toθ = ωt Chain differentiation rule was applied

as well The total flux of the winding is called flux linkage and is denoted with ψ in (3) Itdepends on the flux seen by each conductorΦ and on the number of turns N Flux linkage is

ψ = NΦ.

Electromotive force in an electric machine is induced either due to rotation of awinding in the flux density, or due to rotation of the flux density with respect to a stationarywinding Change of flux linkage can be caused either by mechanical motion or by change ofcurrent in time This is reflected in (3) and will be elaborated in detail later on

Let us further clarify the two operating regimes of electric machines, generating andmotoring Generating is discussed first Due to the action of the prime mover (which delivers

mechanical energy to the machine’s shaft) rotational part of the machine is forced to rotate(Fig 4) Consequently, the speed of rotation is constant (n = const.) and T e = T PM Voltage atmachine terminals and induced emf differ because of the voltage drop in the winding; forgenerating induced emf is greater than terminal voltage (in the sense of rms values in ACmachines, i.e in the sense of average values in DC machines) Note that in generationdirection of the speed of rotation coincides with the direction of the mechanical (primemover’s) torque, while the electromagnetic torque of the machine opposes motion

During motoring (Fig 4) created electromagnetic torque, which is a consequence ofelectric energy delivered to the machine, acts as the source of motion, i.e it causes the rotorrotation In this case the direction of speed and the electromagnetic torque coincide, while themechanical torque (that is now load torque) acts against the direction of rotation Once morethe speed of rotation is constant (n = const.) and T e = T L During motoring induced emf hasthe opposite polarity since it balances the applied voltage It is therefore usually calledcounter-electromotive force The terminal voltage is greater than the counter-emf in motoring

Fig 4 – Torque and speed directions in generation (left) and motoring (right).

In what follows a generalised electromechanical converter is discussed at first Theanalysis is valid for any type of electric machine; the only constraint is that there is only onedegree of freedom for mechanical motion (i.e rotor can rotate along one axis only)

Efficiency of an electric machine is defined in the same way as for any other device,

as ratio of the output to input power

=

loss out loss loss

out

out in

out

P P

P P

P

P P

mech loss Fe Cu out

m Fe

One important point to note is that the nature of the input and output power depends

on the role of the machine In motoring the input power is electrical, while the output power ismechanical In generation it is the other way round, the input power is mechanical while theoutput power is electrical (in generation, there may be some windings that take electricalpower as well, while some other windings generate electrical power) It has to be rememberedthat the rated power of the machine (power for which the machine has been designed), which

is always given on the nameplate of the machine, is the output power Hence, in generation

the known rated power (always identified further on with an index n) is the output electrical

power, while in motoring it is the output mechanical power

Since the role of the input and the output power is dependent on the function that themachine performs, the two cases are treated separately In what follows lower case symbolsare used for all the quantities, meaning that instantaneous time-domain variables are under

consideration The idea behind the subsequent development is to develop a generalmathematical model that is valid for any rotating electric machine It is for this reason that thenumber of windings is not specified Instead, it is taken as being equal to n, where this is an

arbitrary number The electric machine is for the time being a black box There are two doorsthat enable access to the machine, electrical door and mechanical door The power can beeither delivered to the machine, or taken away from the machine, through these two doors.Electromechanical conversion takes place inside the box and the converted power isp c Fig 5illustrates power flows inside an electric machine for motoring and generating Apart fromthese two doors there are two windows that are unwanted outputs only These windows areoutputs for winding losses and mechanical losses, which are inevitably created within amachine, and which represent lost power Note that the iron (or core) loss is omitted from thisrepresentation The reason is that it is of electromagnetic nature and it does not take place inthe windings The existence of the iron loss can be accounted for at a later stage, in anapproximate manner, as it is done in transformer theory Both doors can be either inputs oroutputs (depending on whether the machine operates as a motor or as a generator), whilewindows are outputs only Normally, one door will be the input while the other door will bethe output, although in generation some windings make consume electrical energy, while

other windings are generating it (as shown in Fig 5) The role of doors is thus reversible, asthe machine can operate both as a generator and as a motor.

Electrical input

output power

Copper losses in windings

Mechanical loss

Converted power

Electromagnetic energy storage

Mechanical energy storage

Electrical output

power

Copper losses in windings

Mechanical loss

Converted power

Electromagnetic energy storage

Mechanical energy storage

Small electrical

input power

Fig 5 – Power flow in an electric machine for motoring and generation, respectively.

As can be seen from Fig 5, apart from input and output power and losses, there aretwo internal storages of energy inside the machine The first one is the stored electromagneticenergy, while the second one is the stored mechanical energy Stored mechanical energy is theenergy stored in rotating masses (kinetic energy) and it is in every aspect analogous to theenergy stored under linear movement (which is 2

2

1

mv

W mech = , where m is the mass of the

body) Rotating bodies are characterised with so-called inertia (that is function of the mass

and dimensions) J [kgm2], while instead of the linear velocity one uses angular velocity.Hence

i Li

2

2

2

2 12

1

1

2 2 1 2

12 2 2 2 2

1

2

12

1

2

1

i L

L i L i

(10b)

Taking indexe for electrical power and index m for mechanical power in Fig 5, one can write

the following power balance equations:

Motoring:

m m mech

loss

c

c e Cu

e

p dt

dW p

p

p dt

dW

p

p

++

=

++

c

c m mech

loss

m

p p dt

dW

p

p

p dt

dW p

p

−++

=

++

(12)

Note that storages are energies, as defined in (9)-(10) Powers are time derivatives of energiesand this is taken into account in formulation of (11)-(12) In generation some windings make

take the power (p e2 ), while other winding actually generate the power (p e1)

Equations (11)-(12) enable formulation of the converted power that is defined as

m

e

c t

in terms of other known powers and derivation of the equation for motion of rotating masses

in terms of known parameters and inputs of the machine This is a tedious procedure for the

generalised n-winding converter and most of the derivations will be therefore omitted Only

the starting equations and the final equations are presented in the next sub-section It is to benoted that all the powers, as well as all the other variables (currents, flux linkages) weredenoted with lower-case letters in this section These are instantaneous time domainquantities, and the same approach is used in the following sub-section This enables creation

of a general mathematical model, in terms of time-domain instantaneous quantities, that isvalid for all possible existing types of electric machines with rotational movement

Each of the n windings of the machine is a piece of wire Hence each winding can be

characterised with its resistance and inductance In addition, there are mutual inductancesbetween any two windings An induced emf appears in general in each winding Hence thevoltage equilibrium equation for one particular winding can be written as

n in i

i

i

i i i

i

i L i

L i

L

i

L

dt d i R e

i

R

v

++

++

2 12 1

ψ

ψ

(14)

There is one flux linkage equation and one voltage equation for each of the n windings It is

convenient to use further on matrix notation to express these and other equations, since matrix

notation will enable substitution of n equations with a single matrix equation Hence for all the n windings one has (matrices are underlined):

ù

êêêêêêêê

ë

é

=

nn n

n n

n n n

L L

L L

L L

L L

L L

L L

L L

L L

3 33

32 31

2 23

2 21

1 13

12 1

û

ù

êêêêêêêê

ë

é

=

n i

i i i

i

3 2 1

úúúúúúúú

û

ù

êêêêêêêê

ψ

3 2 1

(16b)

Note that in any electrical machine L ij = L ji

Input electrical power in motoring is

v i i v i

v

i

v

p e = 1+ 22+ + n n = T (17)Note that current matrix in (17) has to be transposed to satisfy the rules of matrixmultiplication In generation the output power is

v i i v i

v i v i

v

i

v

p e = 1+ 22+ + k k − k+1k+1− − n n = T (18)

where winding 1…k generate electricity, while windings k+1…n consume electric energy.

Current vector in (18) has positive currents for the windings that generate and negativecurrents for the windings that consume electric power

Winding losses can be expressed as

i R i i R i

i L i L i

L i L i L i

L i

n n

n e

2 23 1 1 3

13 2 1 12 2 2

2 2 2

1

=

+++

+++

++

++

(20)

Current sign in voltage equation (15) is such that the current is positive when it flowsinto the winding Hence in generation all the windings that generate will have negativecurrents since the current flow will be in the opposite direction from assumed positive currentflow

Mechanical power and mechanical loss are governed with

ω

ωω

dt d

Stored mechanical energy remains to be given with

is considered, the converted power is found to be

i d

L d i

generation

motoringω

ω

ωω

k dt

in nature One then arrives at the equation of mechanical motion in the frequently used form

generationmotoring

motoring0

The equations presented in this sub-section fully describe any rotational electricalmachine, in terms of the instantaneous time-domain variables The full mathematical model issummarised in the following sub-section

2.4 Summary of the mathematical model

Any rotating electric machine, regardless of the actual structure of the stator and rotor andregardless of the number of windings, is completely described with the following set ofequations:

=

+

generation0

motoring0

,

e PM

e L m

m

e

t T

t T t

k dt

d J

t

i d

L d

û

ù

êêêêêêêê

ë

é

=

nn n

n n

n n n

L L

L L

L L

L L

L L

L L

L L

L L

3 33

32 31

2 23

2 21

1 13

12 1

û

ù

êêêêêêêê

ë

é

=

n i

i i i

i

3 2 1

úúúúúúúú

û

ù

êêêêêêêê

ψ

3 2 1

(29)

Equations (28)-(29) constitute the mathematical model of a generalised n-winding

electromechanical energy converter Note once more that all the variables (voltages, currents,flux linkages, electromagnetic torque, speed of rotation) are instantaneous time-domainquantities Note as well that voltage equation is valid for current flowing into the winding.Hence in generation some of the currents will be negative since they will be flowing out of themachine

Equation (25) shows that power will be converted if and only if the machine rotates Thismeans that at zero speed converted power is always zero Further, one can see thatelectromagnetic torque can exist at zero speed (a machine will always start from standstill andthe torque at zero speed is called starting torque) In order for an electromagnetic torque toexist it is necessary that at least some windings carry current and that at least someinductances of the machine are functions of the rotor angular position Note that unless this

condition is satisfied, torque will be zero The issue of dependence of machine’s inductances

on angular position of the rotor will be discussed later

Although an electromagnetic torque will exist if appropriate currents flow in themachine and there are inductances that depend on the rotor position, this is still not sufficient

to realise useful electromechanical energy conversion Assume that the torque of ahypothetical electric machine varies as a sine function of time, with the period equal to theperiod of rotation The instantaneous torque does exist But, it is positive in the first half-cycleand negative in the second half-cycle The average torque is zero and hence the averageconverted power will be zero even if the machine runs at a constant speed The machine will

do motoring in the first half-cycle and generating in the second half-cycle, with a net zeroconverted power over one cycle Thus it follows that, if useful electromechanical conversion

is to take place, average torque of the machine must differ from zero Average

electromagnetic torque T e can only exist if the certain correlation between stator current(voltage) frequency, rotor current (voltage) frequency and the frequency of rotation is

satisfied It can be shown that T ewill be of nonzero value if and only if

r

ω

where indices s and r identify stator and rotor angular frequency Note that DC case is

encompassed by (30) Note as well that, according to (30), it is not possible to realise usefulelectromechanical energy conversion if both stator and rotor windings are supplied with DCcurrents In such a case an average torque can only exist at zero speed But converted powerequals zero at zero speed

On the basis of (30) is it is now possible to classify the most commonly used electricmachines into three categories:

1 Synchronous machines: rotor frequency is zero Hence frequency of rotation

equals stator frequency

2 Induction machines: both stator and rotor windings carry AC currents Rotor speed

is related with the two angular frequencies as ω =ωs−ωr

3 DC machines: stator frequency is zero Hence frequency of rotation must equal

frequency of rotor winding currents

Note that even when (30) is satisfied, this still does not mean that the electromagnetic torque

is constant However, if the machine’s torque varies in time, then it follows from the equation

of the mechanical motion that the speed will constantly vary although mechanical torquemight be perfectly constant It is therefore necessary to provide such arrangements inelectrical machines that not only (30) is satisfied but in addition

Consider as an example a rotating electric machine with one winding on stator and onewinding on rotor General mathematical model (28)