Fundamentals of Electrical Drivess - Chapter 3 pps

Symbolic model of transformer with resistive load In this example an excitation voltage u1 is assumed, which in turn corre-sponds to a secondary voltage u2across the load resistance.. Th

THE TRANSFORMER

The aim of this chapter is to introduce the ‘ideal transformer’ (ITF) cept Initially, a single phase version is discussed which forms the basis for atransformer model This model will then be extended to accommodate the so-called ‘magnetizing’ inductance and ‘leakage’ inductance Furthermore, coilresistances will be added to complete the model Finally, a reduced parametermodel will be shown which is fundamental to machine models As in the previ-ous chapters, symbolic and generic models will be used to support the learningprocess and to assist the readers with the development of Simulink/Caspocmodels in the tutorial session at the end of this chapter

con-Phasor analysis remains important as to be able to check the steady-statesolution of the models when connected to a sinusoidal source

The physical model of the transformer shown in figure 3.1 replaces thetoroidal shaped magnetic circuit used earlier The transformer consists of aninner rod and outer tube made up of ideal magnetic material, i.e infinite per-

meability The inner bar and outer tube are each provided with a n2 and n1turn winding respectively The outer n1 winding referred to as the ’primary’

carries a primary current i1 The inner n2 winding is known as the secondary

winding and it carries a current i2 The cross-sectional view shows the layout

of the windings in the unity length inner rod and outer tube together with the

assumed current polarity Furthermore, a flux φ mis shown in figure 3.1 which

is linked with both coils It is assumed at this stage that the total flux in thetransformer is fully linked with both windings In addition, the airgap between

the inner rod and outer tube of the transformer is taken to be infinitely small atthis stage.

The magnetic material of the transformer is, as was mentioned above, sumed to have infinite permeability at this stage, which means that the reluctance

as-R mof the magnetic circuit is in fact zero Consequently, the magnetic potential

u iron core across the iron circuit must be zero given that u iron core = φ m R m,

where φ mrepresents the circuit flux in the core, which is assumed to take on afinite value The fact that the total magnetic potential in the core must be zero

shows us the basic mechanism of the transformer in terms of the interactionbetween primary and secondary current

Let us assume a primary and secondary current as indicated in figure 3.1.Note carefully the direction of current flow in each coil Positive current direc-tion is ‘out of the page’, negative ‘into the page’ The MMF of the two coilsmay be written as

M M F coil 1 = +n1 i1 (3.1a)

M M F coil 2 = −n2 i2 (3.1b)The MMF’s of the two coils are purposely chosen to be in opposition giventhat it is the ‘natural’ current direction as will become apparent shortly Theresultant coil MMF ‘seen’ by the magnetic circuit must be zero given that the

magnetic potential u iron = 0 at present This means that the following MMF

condition holds:

n1i1− n2i2= 0 (3.2)

Equation (3.2) is known as the basic ITF current relationship This expression

basically tells us that a secondary current i2 must correspond with a primary

current i1= n2

n1i2(see equation (3.2))

The second basic equation which exists for the ITF relates to the primary andsecondary flux-linkage values If we assume, for example, that a voltage source

is connected to the primary then a primary flux-linkage ψ1value will be present

This in turn means that the circuit flux φ m will be equal to φ m = ψ1

Figure 3.2 Symbolic model

cur-pointing to the right In some applications it is more convenient to reverse bothcurrent directions, i.e both pointing to the left The ITF model (figure 3.1) isdirectly linked to the symbolic model of figure 3.2 in terms of current polari-ties If we choose the primary current ‘into’ the ITF model then the secondarydirection follows ‘naturally’ (because of the reality that the total MMF must bezero) i.e must come ‘out’ of the secondary side of the model.

The generic diagram of the basic ITF module is linked to the flux-linkageand current relations given by equations (3.5c) and (3.5d) respectively Thegeneric diagram that corresponds with figure 3.2 is given by figure 3.3(a)

It is, as was mentioned earlier, sometimes beneficial to reverse the current

direction in the ITF, which means that i2 becomes an output and i1 an input

Under these circumstances we must also reverse the flux directions, i.e ψ1output, ψ2 input This version of the ITF module, named ‘ITF-Current’ isgiven in figure 3.3(b)

The instantaneous power is given as the product of voltage and current, i.e

u1i1and u2i2 For the ITF model, power into the primary side corresponds to positive power (p in = u1i1) Positive output power for the ITF is defined as

(p out = u2i2) out of the secondary as shown in figure 3.4.

Figure 3.4 Power

conven-tion ITF

3.3 Basic transformer

The ITF module forms the cornerstone for transformer modelling in this bookand is also the stepping stone to the so-called IRTF module used for machineanalysis An example of the transformer connected to a resistive load is shown

in figure 3.5

Figure 3.5 Symbolic model

of transformer with resistive load

In this example an excitation voltage u1 is assumed, which in turn

corre-sponds to a secondary voltage u2across the load resistance The ITF equationset is given by equation (3.6)

The ITF current on the primary side is renamed i 

2and is known as the primaryreferred secondary current It is the current which is ‘seen’ on the primary side,

due to a current i2on the secondary side In this case i1= i 

2as may be observedfrom figure 3.5 The equation set of this transformer must be extended with

the equation u2 = i2R L A generic representation of the symbolic diagramaccording to figure 3.5 is given in figure 3.6

Figure 3.6 Generic model of

transformer with load

The ITF module is shown as a ‘sub-module’, which in fact represents the

generic model according to figure 3.3(a) with the change that the current i1 is

differential represents the secondary voltage which will cause a current i2 in

the load The secondary current leads to a MMF equal to i2n2on the secondary

side which must be countered by an MMF of n1i1 on the primary side giventhat the magnetic reluctance of the transformer is taken to be zero at present

Note that an open-circuited secondary winding (R L = ∞) corresponds to a

zero secondary and a zero primary current value The fluxes are not affected

as these are determined by the primary voltage, time and winding ratio in thiscase (we have assumed that the primary coil is connected to a voltage sourceand the secondary to a load impedance)

Note that a differentiator module is used in the generic model given in

fig-ure 3.6 Differentiators should be avoided where possbile in actual simulations,

given that simulations tend to operate poorly with such modules In most casesthe use of a differentiator module in actual simulations is not required, giventhat we can either implement the differentiator by alternative means or buildmodels that avoid the use of such modules

In electrical machines airgaps are introduced in the magnetic circuit which,

as was made apparent in chapter 2, will significantly increase the total magnetic

circuit reluctance R m Furthermore, the magnetic material used will in realityhave a finite permeability which will further increase the overall magnetic circuitreluctance The transformer according to figure 3.7 has an airgap between theprimary and secondary windings

The circuit flux φ m now needs to cross this airgap twice Consequently, agiven circuit flux will according to Hopkinson’s law correspond to a non-zero

magnetic circuit potential u M in case R m > 0 The required magnetic potential

must be provided by the coil MMF which is connected to the voltage source Inour case we have chosen the primary side for excitation with a voltage sourcewhile the secondary side is connected to, for example, a resistive load The

implication of the above is that an MMF equal to n1i m must be provided via

the primary winding The current i m is known as the ‘magnetizing current’,

which is directly linked with the primary flux-linkage value ψ1and the so-called

magnetizing inductance L m The relationship between these variables is of the

equation (3.7) zero magnetizing current i m.

The presence of a core MMF requires us to modify equation (3.2) given thatthe sum of the coil MMF’s is no longer zero The revised MMF equation isnow of the form

n1i1− n2i2 = n1i m (3.8)which may also be rewritten as:

In expression (3.9) the variable i 

2 is shown, which is the primary referredsecondary current, as introduced in the previous section Note that in the mag-

netically ideal case (where i m = 0) the primary current is given as i1 = i 

2.The ITF equation set according to equation (3.6) remains directly applicable tothe revised transformer model

Figure 3.8 Symbolic model

of transformer with load and

finite L m

The symbolic transformer diagram according to figure 3.5 must be revised toaccommodate the presence of the magnetizing inductance on the primary side

of the ITF The revised symbolic diagram is given in figure 3.8

The complete equation set which is tied to the symbolic transformer modelaccording to figure 3.8 is given as

Figure 3.9 Generic model of

transformer with load and

fi-nite L m

is an ITF sub-module which is in fact of the form given in figure 3.3(a), with

the provision that the current output i1(of the ITF module) is now renamed i 

2,which is known as the primary referred secondary current

3.5 Steady-state analysis

The model representations discussed to date are dynamic, which means thatthey can be used to analyze a range of excitation conditions, which includestransient as well as steady-state Of particular interest is to determine how suchsystems behave when connected to a sinusoidal voltage source Systems, such

as a transformer with resistive loads will (after being connected to the excitationsource) initially display some transient behaviour but will then quickly settledown to their steady-state

As was discussed earlier the steady-state analysis of linear systems connected

to sinusoidal excitation sources is of great importance Firstly, it allows us togain a better understanding of such systems by making use of phasor analysistools Secondly, we can use the outcome of the phasor analysis as a way tocheck the functioning of our dynamic models once they have reached theirsteady-state

3.5.1 Steady-state analysis under load with magnetizing

inductance

In steady-state the primary excitation voltage is of the form u1 = ˆu1cos ωt,

which corresponds to a voltage phasor u1= ˆu1 The supply frequency is equal

to ω = 2πf where f represents the frequency in Hz.

The aim is to use complex number theory together with the equations (3.10)

and (3.6) to analytically calculate the phasors: ψ1, ψ2, i2, i 

2, i1and u2 The linkage phasor is directly found using equation (3.10a) which in phasor form is

flux-given as u1 = jωψ1 The corresponding flux-linkage phasor on the secondary

side of the ITF module is found using (3.6d) which gives ψ2 = n2

n1ψ1 Thesecondary voltage equation (3.10b) gives us the secondary voltage (in phasor

form) u2 = jωψ2, which in turn allows us to calculate the current phasor

according to i2 = R1

L u2 This phasor may also be written in terms of the

primary voltage phasor u1 = ˆu1as

The corresponding primary referred secondary current phasor is found using

equations (3.6d), (3.11) which gives i 

R L The primary current

phasor is found using (3.10c), where the magnetizing current phasor i m is

found using (3.10d) namely i m = L1

m ψ1 The resultant primary current phasormay also be written as

It is instructive to consider equation (3.12) in terms of an equivalent circuitmodel as shown in figure 3.10.

diagram of the transformer with a resistive load R Land magnetizing inductance

L m which corresponds with the given phasor analysis and equivalent circuit(figure 3.10), is shown in figure 3.11

Figure 3.11 Phasor diagram

of transformer with load and

2 is in phase with u2 given that we have a resistive load

Fur-thermore, the primary current is found by adding (in vector form) the phasors i 

2

and i m Note that the primary current will be equal to the magnetizing current

when the load resistance is removed, i.e R L=∞.

The corresponding steady-state time function of, for example, the current i1can be found by using

i1(t) =
i1e jωt

(3.13)

where i1is found using (3.12)

3.6 Three inductance model

The model according to figure 3.7 assumes that all the flux is linked withboth coils In reality this is not the case as may be observed from figure 3.12

This diagram shows two flux contributions φ σ1 and φ σ2, which are known asthe primary and secondary leakage flux components respectively The leakagefluxes physically arise from the fact that not all the flux in one coil is ‘seen’ bythe other

model with finite airgap and leakage

Consequently these components are not linked with both coils and they are

represented by primary and secondary leakage inductances

L σ1 = ψ σ1

L σ2 = ψ σ2

where the primary and secondary leakage flux-linkage values are given by

ψ σ1 = n1φ σ1 and ψ σ2 = n2φ σ2 respectively The total flux-linkage seen bythe primary and secondary is thus equal to

ψ2 = ψ 

The flux φ m which is linked with the primary coil is now renamed ψ m = n1φ m

Similarly the circuit flux φ m which is linked with the secondary coil gives us

the flux-linkage ψ 

m = n2φ m The terminal equations for the transformer in its

current form are given as

The symbolic representation of the transformer according to figure 3.8 must

be extended to include the leakage inductance The revised symbolic model

as given in figure 3.13 shows the leakage inductances The generic model thatcorresponds to figure 3.13 is given in figure 3.14

including algebraic loops

It is further noted that the model is not suited to simulate the no-load situation

(with a passive load), i.e open circuited secondary winding, since current i2is

an output and thus cannot be forced zero by any load

The model in figure 3.14 contains two algebraic loops which may causenumerical problems during the execution of the simulation The term ‘alge-braic loop’ refers to the problem where the output of a simulation model isalgebraically linked to itself For example in the equation

output variable y is not only a function of the input x but also of itself.

In the generic diagram of figure 3.14, the following two loops appear

(iden-tified by the modules in the loop, where Σ is used to indicate a summation

unit):

m, closing

the loop via lower Σ back to L σ1

of ITF to the upper right Σ From here down through L1

σ2, back through

current input of ITF, closing the loop via lower Σ back to L σ1

Execution of this simulation would cause an ‘algebraic loop error’ message inthe simulation The simulation results obtained with the presence of an algebraicloop may be erroneous Hence, such loops should be avoided if possible Inthe example, rewriting equation (3.18) into

y = x G

solves the algebraic loop

The problem with the three inductance model as discussed in section 3.6 lieswith the fact that it is extremely difficult to determine individual values for thetwo leakage inductances in case access to the secondary side of the model isnot possible For transformers this is not an issue (when the winding ratio isknown) but for squirrel cage asynchronous machines, to be discussed at a laterstage, this is certainly the case A triple inductance model of the transformer is

in fact not needed given that its behaviour can be perfectly modelled by a twoinductance based model when all inductances are linear

In this section two such models will be examined together with a third modelwhich makes use of mutual and self inductance parameters

Prior to discussing these new models it is instructive to introduce the so-called

coupling factor κ which is defined as

κ =



1− L short −circuit

Equation 3.20 contains the parameters L open −circuit (the inductance of the first

winding while the second winding is open-circuit) and L short −circuit (the ductance of the first winding while the second winding is shorted) Note thatthe open and short-circuit inductance will be different when the transformer is

in-viewed from the primary or secondary side, however the coupling factor κ will

L s short −circuit = L σ2+ L 

m L  σ1

L 

m + L  σ1

(3.22b)

with L2 = L σ2 + L 

m , L  σ1 = 

The symbolic model given in figure 3.15 is referred to as a primary basedmodel because the two inductances are located on the primary side of the trans-former The new configuration is defined by a new magnetizing inductance

L M , new leakage inductance L σ and ITF transformer ratio k These

parame-ters must be chosen in such a manner as to insure that the new model is identical

Figure 3.15 Two inductance

symbolic transformer model

to the original (figure 3.13) when viewed from either the primary or secondaryside of the transformer It can be shown that the following choice of parameterssatisfies these criteria.

The two inductance model representation is now of the form given byfigure 3.16 The new configuration is again defined by a magnetizing inductance

induc-tance alternative symbolic transformer model