Switched Reluctance Motor Modelling

8/17/2019 Switched Reluctance Motor Modelling

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SWITCHED RELUCTANCE MOTOR MODELLING

3.1Mechanical Dynamics

A simplified physical model of an SR machine is shown in Figure 3.1. The variables

Fi!"e 3.1 Free-body diagram of mechanical subsystem

in this rotational system are

−θ , angular displacement in radians rad!

−ω, angular velocity in radians per second rad"s!

−T p , tor#ue in phase p in $ewton-meters $m!

al1 of which are functions of time. Angular displacement is measured from a reference position.

%hase tor#ue is the tor#ue produced by one phase& and tor#ue production is identical in al1

phases. The total instantaneous tor#ue produced by a machine is the sum of the tor#ues produced

in each phase& i.e.&

T total=∑ p=1

m

T p (3.1)

where m is the total number of phases.


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'onsider the setup in Figure 3.1. The rotating mass of inertia J is sub(ected to the

phase tor#ueT p

and a load tor#ueT l . )etting B denote the rotational friction

coefficient* we have from $ewton+s second law

J ώ=−Bω+∑ p=1

m

T p−T l(3.2)

,e thus arrive at the state model

θ́=ω ώ=1

J (∑ p=1m

T p−Bω−T l)(3.3)

3.1.1 Mane#ic Cha"ac#e"is#ics

The total flu in the stator pole is called flu-linage&Ψ

& which varies with rotor

position and phase current* thus we write Ψ (θ , i) . /t increases as the rotor pole starts to

overlap with the stator pole and approaches the aligned position. )arger magnetic flu-linage is

created as the phase current is increased. At low current levels& flu-linage is approimately a

linear function of phase current. 0owever& beyond a certain level of current& magnetic saturation

occurs in the material& thus yielding a nonlinear relationship between flu-linage and phase

current. This critical current level is called the saturation current. The characteristic curve of the

flu linage of one phase with respect to the phase current is called the magnetiation curve. The

magnetiation curves for different rotor angles from unaligned to aligned position are shown in

Figure 3.2. The curveC u at the bottom is the flu curve in the


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Fi!"e 3.$ agnetiation curve

unaligned position& while C a at the top is the flu curve in the aligned position. As one can

see& the flu path is more susceptible to saturation in the aligned position. /nductance of an

unsaturable inductor is defined as L=ψ / i

. 4nlie the constant inductance of typical

solenoids& phase inductance of an SR machine varies with its rotor position and phase current.

5elow the saturation current& however& the inductance is a function only of rotor position because

the flu-linage is approimately a linear function of current& i.e.&

ψ (θ , i )= L (θ )i (unsaturated ) (3.4)

The inductance in the unsaturated region is inversely proportional to the magnetic

reluctance& and it achieves a maimum in the aligned position and a minimum in the unaligned

position. /n the saturated region& because the flu-linage is a nonlinear function of the phase

current& a new epression of inductance& called incremental inductance& is given by

l (θ , i )=∂ Ψ (θ , i)

∂i

(3.5)

/n Figure 3.2& one can thin of the incremental inductance& l(θ , i) & as the slope of the

magnetiation curve whenθ

is held constant. The incremental inductance in the saturated

region does not vary as much as the inductance in the unsaturated region when the rotor rotates


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from the unaligned position to the aligned position under a constant current. Figure 3.3 shows

phase inductance in both cases as a function of rotor position.

Fi!"e 3.3 6ariation of phase inductance with respect to rotor position

/t is also important to note that when ad(acent phases are ecited at the same time& there

eists a coupled inductance called mutual-inductance. 0ence& in general& phase inductance of an

SR machine is the sum of self-inductance and mutual-inductance.

0owever& mutual-inductance is neglected in this research because it is typically very

small compared to self-inductance.

3.1.$ De"i%a#i&n &' T&"(!e E(!a#i&n

The tor#ue produced during motoring operation is the tor#ue re#uired to rotate the rotor.

The instantaneous tor#ue can be viewed as the mechanical wor done by the motor resulting in a

displacement of the rotor angle. The most general epression for tor#ue produced by one phase

at any rotor position is given by

T (θ , i )=∂W c(θ ,i)

∂θ (3.6)

whereW C is referred to as the coenergy. /n the first approimation& the coenergy can be

thought of as the amount of mechanical energy converted from the electrical energy. Assuming

that a constant current&i1 & is applied during the rotation from θ=0 to θ=θ1 & the


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coenergy is the area below the rnagnetiation curve& shown in Figure 3.7. The shaded area&

W f & is the energy stored in the magnetic field& and the rectangular area&Ψ 1× i1 & is the total

input energy. The coenergy can be epressed as the definite integral

W c (θ1 ,i 1 )=∫0

i1

Ψ (θ1, i )dt (3.7)

Since the magnetiation curve changes as the rotor rotates& the instantaneous tor#ue can

be visualied graphically in Figure 3.8. 0ere&∆ W m , is the mechanical wor done at a

constant current as the rotor moves through an infinitesimal displacement& ∆θ .

Therefore& the instantaneous tor#ue is∆W m/∆θ . 9uring such a displacement& there

are echanges of energy among the coenergy& the stored field energy& and the power supply. The

constraint of constant current ensures that& during the displacement& the mechanical wor done is

eactly e#ual to the change in coenergy& as shown in the following :22;.

The energy echanged with the power supply during the displacement from A to B in

Figure 3.8 is given by the change of total input energy corresponding to the displacement∆θ

that is&

∆ W total= ABCD (3.8)

where ABCD denotes the area delimited by the vertices& A , B , C ,and D . The change in

stored field energy is

∆ W f =BC −AD (3.9)

where& e.g.& BC

denotes the area delimited by the curveB

and the verteC

.


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¿1

2 i2 dL(θ)

dθ (3.12)

Figure 3.= shows the basis of tor#ue production in the unsaturated region with an idealied linear

variation of phase inductance over one rotor pole-pitch. The aligned

Fi!"e 3.* Tor#ue production in the unsaturated region

and unaligned positions are denoted by A

and!

, respectively. As the stator and rotor

poles start to overlap& the inductance increases from the unaligned value& L

min eventually

reaching the fully aligned value& Lma" . /n the figure& the tor#ue is shown for a constant current

in one phase. $otice that the sign of the tor#ue depends on the sign of the change in inductance

with respect to the rotor angle& i.e.&dL(θ)dθ

# Therefore& in order to produce only positive

tor#ue& the phase windings must be energied only during the rising inductance period. 0ence&

this period is called the effective tor#ue one and is approimated to the lesser pole-arc of the

overlapping stator and rotor poles. For the continuous tor#ue production with al1 the phases in an

?"< SR machine& consider Figure 3.?.


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,e number stator poles as shown& in a clocwise fashion. )et θ denote the rotor

angle& reference value being when a reference rotor pole is vertically up. The range of B is

(−$, $) . /n addition& we have four phase angles& θ p( p=1,%, 4) . The range of these

angles is [−& 6 , & 6 ] .

Fi!"e 3.+ %hases and their range

These angles denote distances between stator and rotor poles& as follows. The first angle&

θ1 , denotes the distance from stator pole 1 to the nearest rotor pole. That is& the graph of

θ1

as a function ofθ

is as follows>

Fi!"e 3., %hase angleθ1 as a function of θ ranging from

−& 6 ¿

&

6


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)iewise&θ2

is the distance from stator pole 2 to the nearest rotor pole. The graph ofθ2

as

a function ofθ

is the previous graph shifted to the right by&

12 that is&

θ2=θ1(θ− & 12 )(3.13)

And so on forθ3 and

θ4 .

The relationships between θ and the phase angles are shown in Figure 3.1@. The shaded

area represents the phase ecitation period for producing positive tor#ue in each phase. The

process is controlled by switching the supply voltage on at the turn on angle&θon

and

switching off at the turn-off angle&θoff . /n general& the turn-on and turn-off angles are chosen

to maimie the performance of an SR machine& such as efficiency and tor#ue density. 0owever&

the most important principle in choosingθon and

θoff is to produce always nonero tor#ue.

$otice that the effective tor#ue one in one phase overlaps with ad(acent phases. 0ence&

by choosing properθon and

θoff continuous tor#ue is produced. The control aspect of turn-

on and turn-off angles is described in the net chapter in detail. The operation in the saturated

region is similar to that in the unsaturated region ecept that the output tor#ue is larger and has a

distorted shape. The purpose of operating an SR machine in the saturated region is to increase the

tor#ue density& i.e.& tor#ue per unit volume. As the phase current is increased to the saturated

region& the coenergy taes up a larger portion of the total input energy. 0ence& with the same sie

of machine& larger tor#ue can be produced. 5ecause the tor#ue density of unsaturable machines

is very low& they are not of practical interest.

3.$C&n#"&l M&-es


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SR drives are controlled by synchroniing the energiation of the motor phases with the

rotor position. Figure 3.11 illustrates the basic strategy.

As #uation 3.12! suggests& positive or motoring! tor#ue is produced when the motor

inductance is rising as the shaft angle is increasing&dl

dθ>0

. Thus& the desired operation is to

have current in the SR winding during this period of time. Similarly& a negative or braing!

tor#ue is produced by supplying the SR winding with current whiledl

dθ


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tae advantage of all potential tor#ue output from a given phase. A simple and effective

commutation scheme is depicted in Figure 3.12.

/n the plot of Figure 3.12& the dashed line shows the tor#ue that would be generated by

phase A& should constant current flow through the phase winding during an entire electrical cycle

of the SR. ,ith the idealied current waveforms of the figure& the resulting net tor#ue from the

motor is shown by the solid line. The turn-on and turn-off angles coincide with the region where

maimum tor#ue is obtained for the given amount of phase current.

This commutation se#uence tends to optimie efficiency. 0ere& a dwell angle of 12@

electrical degrees is used& which is the minimum dwell angle that can be used for a three-phase

SR& without regions of ero tor#ue. Bf interest to note from Figure 3.12 is that constant current

results in non-constant tor#ue. This application& although not covered in this report& is well suited

for 9S% implementation.

Fi!"e 3.1$ 'ommutation of a 3-%hase SR


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Figure 3.12 illustrates the effect that the choice of commutation angles can have upon the

SR performance. #ually important is the magnitude of the current that flows in the winding.

'ommonly& the phase current is sensed and controlled in a closed-loop manner& and as seen in the

voltage curve of Figure 3.11& the control is typically implemented using %, techni#ues. SR

control is often described in terms of Dlow-speedE and Dhigh-speedE regimes. )ow-speed

operation is typically characteried by the ability to arbitrarily control the current to any desired

value. Figure 3.11 illustrates waveforms typical of low-speed SR operation. As the motors

speed increases& it becomes increasingly difficult to regulate the current because of a

combination of the bac F effects and a reduced amount of time for the commutation

interval. ventually a speed is reached where the phase conducts remains on! during the entire

commutation interval. This mode of operation& depicted by Figure 3.13& is called the single-pulse

mode.

Fi!"e 3.13 Single-%ulse ode C otoring& 0igh Speed

,hen this occurs& the motor speed can be increased by increasing the conduction period

a greater dwell angle! or by advancing the firing angles& or by a combination of both. 5y

ad(usting the turn-on and turn-off angles so that the phase commutation begins sooner& we gain

the advantage of producing current in the winding while the inductance is low& and also of

having additional time to reduce the current in the winding before the rotor reaches the negative

tor#ue region. 'ontrol of the firing angles can be accomplished a number of ways& and is based

on the type of position feedbac available and the optimiation goal of the control. ,hen


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position information is more precisely nown& a more sophisticated approach can be used. Bne

approach is to continuously vary the turn-on angle with a fied dwell.

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Switched Reluctance Motor Modelling