Generalized Analysis of Brushless Doubly-Fed Induction ...profdoc.um.ac.ir/articles/a/1080132.pdf · is ignored, the time derivative of magnetizing inductances will yield incorrect

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Abstract— Substantial advantages of brushless doubly-fed induction machine (BDFIM) bring up it as an interesting alternative for conventional solutions. In order to better analyse of BDFIM performance, it is required to employ a more precise model of BDFIM. In this paper, an approach, which completely relies on dynamic modelling in arbitrary reference frame, is proposed to consider both iron loss and main flux saturation effects in two-axis model of BDFIM. In this regard, the dynamic and static inductances are derived by using the magnetizing flux curve. If the cross-saturation is ignored, the time derivative of magnetizing inductances will yield incorrect simulation results. Therefore, to increase the accuracy, the proposed model contains terms which describe the cross-saturation effects. Due to the close relation between torque and power, the torque expression is also achieved in the presence of iron loss and magnetic saturation using expression of electrical input power. The simulation and experimental results are finally given to show the performance of the proposed model.

Index Terms—Brushless doubly-fed induction machine, iron loss, magnetic saturation, two-axis model.

NOMENCLATURE

, 𝐼 , 𝜆 Voltage, current, flux vectors

𝑇𝑒 Electromagnetic torque

𝑃 Power

𝑅 Winding resistance

𝐿𝑙 Leakage inductance

𝐿𝑚 Magnetizing inductance

𝐿𝑀𝑝𝐷 𝐷-axis component of power winding (PW)

magnetizing inductance

𝐿𝑀𝑝𝑄 𝑄-axis component of PW magnetizing inductance

𝐿𝑀𝑐𝐷 𝐷-axis component of control winding (CW)

magnetizing inductance

𝐿𝑀𝑐𝑄 𝑄-axis component of CW magnetizing inductance

𝐿𝑝𝐷𝑄 Cross-coupling inductance between 𝐷- and 𝑄-

axes of PW

𝐿𝑐𝐷𝑄 Cross-coupling inductance between 𝐷- and 𝑄-

axes of CW

𝐿𝑝𝐷 𝐷- axis component of PW self-inductance

𝐿𝑝𝑄 𝑄- axis component of PW self-inductance

𝐿𝑐𝐷 𝐷- axis component of CW self-inductance

𝐿𝑐𝑄 𝑄- axis component of CW self-inductance

𝑝 Number of pole pair

𝑁𝑟 Number of rotor loops (nests)

𝑓 Frequency

𝜔𝑟 Synchronous rotor speed

𝜔𝑎 Arbitrary angular speed

𝜔𝑛 Natural synchronous speed

𝑠 Derivative operator

Subscripts

𝑝 , 𝑐, 𝑟 PW, CW, rotor

𝐷 , 𝑄 Arbitrary frame axis

𝑑 , 𝑞 Stationary frame axis

𝑚 Magnetizing

𝑐𝑢 Copper loss

𝑖𝑟𝑜𝑛 Iron loss

𝑠𝑙𝑙 Stray load loss

𝑓&𝑤 Friction and windage losses

I. INTRODUCTION

ynamic modelling is necessary for analyzing the

performance of brushless doubly-fed induction machine

(BDFIM); moreover, it is helpful to realize the control

techniques. The coupled-circuit model was first suggested by

Wallace et al. [1] which makes it possible to evaluate the

dynamic and the steady-state behavior of BDFIM. In order to

replace the position-dependent inductances with speed-

dependent ones, an approach was proposed in [2], which

transformed the coupled-circuit model into the two-axis form.

Considering the BDFIM as two interconnected induction

machines (IMs), an equivalent circuit was introduced in [3]

which is valid for all modes of operation. To reduce the order

of this model, a simplified model has been developed in [4],

excluding the iron loss effect. This model is a much better

approximation to represent the behavior of BDFIM compared

to the core model.

For the purpose of understanding the performance of BDFIM,

it is required to derive its precise dynamic model considering

the iron loss and magnetic saturation effect. They are possible

sources to deteriorate the electrical machine performance when

the close-loop control strategies are employed [5]. The

combined impact of flux saturation and iron loss has been

studied on operation of vector controlled IM by Levi et al. [6].

It has been shown detuned operation necessarily will occur if

the iron loss and parameter variations are neglected.

A novel modelling method has been introduced for BDFIM

in [7] based on magnetic equivalent circuit (MEC). The

operating characteristics are evaluated using the proposed

model considering saturation. The impact of magnetic

saturation and iron loss on the performance prediction of

BDFIM have been studied in [8] using time-stepping finite

element analysis. By providing the steady-state conditions, the

proposed finite-element model can be used for optimization of

BDFIM design. In [9], the steady-state model of the cascade

doubly-fed induction machine (CDFIM) is presented that

accounts for the magnetic saturation and iron loss. Using this

model, a general performance study of the CDFIM has been

Generalized Analysis of Brushless Doubly-Fed Induction Machine Taking Magnetic Saturation and Iron Loss into Account

Hamidreza Mosaddegh Hesar, Mohammad Ali Salahmanesh, Hossein Abootorabi Zarchi

D

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carried out. A dynamic saturated model is proposed for dual-

stator brushless doubly-fed induction generator (DS-BDFIG)

by Zeng et al. [10] and it is confirmed through experimental

tests which the proposed model is more accurate than the model

without saturation effect. In this research work, the measured

magnetizing inductances are updated in each cycle from fitting

formulas and then placed in the electrical equations of DS-

BDFIG.

According to literature review, although few research works

have been published to date on modelling of types of the

brushless doubly-fed machines taking the both iron loss and

flux saturation into account, this is the first time that they are

considered in electric equivalent circuit (EEC) model of

BDFIM. This paper attempts to fill this void by achieving the

following purposes:

1) The two-axis model is a simple but precise mathematical

method that is useful for studying the electrical machines.

Meanwhile, the arbitrary reference frame provides a direct

means to achieve the voltage equations in all other reference

frames. Accordingly, the dynamic model of the BDFIM is

proposed considering both PW and CW sides iron loss

resistances in the arbitrary reference frame. Furthermore, by

introducing the full nonlinear BDFIM model, the cross-

saturation effect is also considered. If the cross-saturation is

ignored, the time derivative of magnetizing inductances will

yield incorrect simulation results. Generally, there are two

approaches to consider the effect of saturation in the electrical

machines model. In first which is known as simplified model,

the saturation effect is taken into account by substituting a

nonlinear function of magnetizing inductance into the voltage

equations of the unsaturated model [10]. The second approach

is a full nonlinear model in which the time variation of

magnetizing inductance leads to different voltage equations in

comparison with the unsaturated model. Therefore, new terms

arise as dynamic cross-saturation effects in the full nonlinear

model which are neglected in the simplified form. The full

nonlinear model is obviously better than the simplified model

due to the benefits such as high accuracy [11, 12] and fast

response [13].

2) The torque expression is derived in the presence of iron

loss and magnetic saturation using expression of electrical input

power. This power includes the power that is being converted

to mechanical energy, the power that is stored in the field of

machine, and the power for losses. It should be noted that the

various expressions for the electromagnetic torque will not be

derived for the saturated machine, since saturation does not

introduce new terms into the torque expression. Of course, as a

result of saturation, the saturation-dependent machine

parameters which are present in the different expressions for the

torque, e.g. the magnetizing inductance, or the rotor self-

inductance or the stator self-inductance, will be different to

their unsaturated values and are variables which depend on the

machine currents.

3) Due to the complex structure of BDFIM, the iron loss

resistance cannot be determined using the conventional

approaches proposed for standard IM. In order to identify the

BDFIM iron loss resistance, an approach is presented in which

the speed-dependent iron loss resistances are obtained through

some effective experiments. The acceptable accuracy of this

method is validated by comparing the calculated iron loss

resistances of BDFIM to the measured data.

This paper is organized as follows; in Section II the

dynamic modelling principle of BDFIM is introduced which

includes the magnetic saturation and iron loss modelling, as

well as derivation of two-axis model of BDFIM in the arbitrary

reference frame. In Section III the performance of proposed

model is confirmed by simulation and experiments. Section IV

contains the concluding remarks. The torque expression

achieved using active input power is also given in Appendix.

II. MODELLING OF BDFIM

A. Two-Axis Electrical Equations

The two-axis equations of an ideal BDFIM in the arbitrary

reference frame are: pQampDpDlppDppD dtddtdiLiRV (1)

pDampQpQlppQppQ dtddtdiLiRV (2)

cQrramcDcDlccDccD NdtddtdiLiRV (3)

cDrramcQcQlccQccQ NdtddtdiLiRV (4)

rQrpa

mcDmpDrDlrrDrrD

p

dtddtddtdiLiRV

(5)

rDrpa

mcQmpQrQlrrQrrQ

p

dtddtddtdiLiRV

(6)

mpDpDlppD iL , mpQpQlppQ iL (7)

mcDcDlccD iL , mcQcQlccQ iL (8)

mcDmpDrDlrrD iL , mcQmpQrQlrrQ iL (9)

where 𝜆𝑚𝑝𝐷,𝑄 = 𝐿𝑚𝑝𝑖𝑚𝑝𝐷,𝑄 and 𝜆𝑚𝑐𝐷,𝑄 = 𝐿𝑚𝑐𝑖𝑚𝑐𝐷,𝑄 are the two-

axis components of power winding (PW) and control winding

(CW) magnetizing flux linkages.

The orthogonal 𝐷 − 𝑄 axis equivalent circuit of the BDFIM

obtaining from (1)-(9) has been depicted in Fig. 1. This circuit

is valid for all operation modes of BDFIM including

synchronous mode. In this model, the magnetizing inductances

are constant and the iron loss resistances has not been included.

Later in this paper, the BDFIM model is developed taking into

consideration the iron loss and magnetic saturation.

Fig. 1. The 𝐷 − 𝑄 axis dynamic model of BDFIM in arbitrary reference frame

B. BDFIM Model Including Magnetic Saturation

In this section, the effect of iron saturation on the electrical

equations are discussed. As a result of saturation, the

magnetizing inductances are not constant, and vary with

saturation. The variation of the magnetizing inductances will be

incorporated into the voltage equations. In the two-axis model

cRlrLpDi lpL

lcLpRrR

rDi

pDv

pQa rQrpa p )( cQrra N )( cDi

cDv

mpDi mcDi

cRlrLlpL

lcLpRrR

pQv

pQi pDa

mpQi rQi

rDrpa p )(

mcQi

cDrra N )(

cQv

cQi

mpL

mpL

mcL

mcL





introduced in (1)-(6), the saturation effect is neglected and

parameters such as magnetizing inductances, stator and rotor

self-inductances are considered constant. Tacking the magnetic

saturation into account, the voltage equations considerably

differ from those obtained in the ideal form. In order to derive

the dynamic electrical equations, we start with the PW-side.

There is similar procedure for the CW. It is required to compute

the first time derivative of 𝐷- and 𝑄-axis components of PW

magnetizing flux as: dtdLidtdiLdtd mpmpDmpDmpmpD (10)

dtdLidtdiLdtd mpmpQmpQmpmpQ (11)

In the saturation conditions, the first time derivative of

magnetizing inductance can be calculated as follows [14]:

dtidiLLdtdL mpmpmppmp

(12)

where 𝐿𝑝 and 𝐿𝑚𝑝 are called dynamic and static the time variant

inductances of PW, respectively. These inductances are defined

as follows:

mp

mpp

id

dL

, mp

mpmp

iL

(13)

In (12), the first time derivative of magnetizing current

amplitude (|𝑖 𝑚𝑝|) is also obtained as:

dtdidtdidtid mpQpmpDpmp sincos

(14)

where angle 𝜇𝑝 is the angular displacement of 𝑖 𝑚𝑝 with respect

to the direct axis of arbitrary reference frame. By substituting

(12) into (10) and (11) and through straightforward

computation, the first time derivative of 𝐷 and 𝑄 components

of PW magnetizing flux can be rewritten as: dtdiLdtdiLdtd mpQpDQmpDMpDmpD (15)

dtdiLdtdiLdtd mpDpDQmpQMpQmpQ (16)

In a similar way, we have for CW: dtdiLdtdiLdtd mcQcDQmcDMcDmcD (17)

dtdiLdtdiLdtd mcDcDQmcQMcQmcQ (18)

where 𝐿𝑀𝑝𝐷, 𝐿𝑀𝑝𝑄, 𝐿𝑀𝑐𝐷, 𝐿𝑀𝑐𝑄, 𝐿𝑝𝐷𝑄, and 𝐿𝑐𝐷𝑄 are achieved as:

pmpppMpD LLL 22 sincos (19)

pmpppMpQ LLL 22 cossin (20)

cmcccMcD LLL 22 sincos (21)

cmcccMcQ LLL 22 cossin (22)

ppmpppDQ LLL cossin (23)

ccmcccDQ LLL cossin (24)

As observed, (19)-(24) depend on both static and dynamic

inductances. Under linear magnetic conditions, the static

inductances are equal to the dynamic ones. As a result, it can be

simply attained 𝐿𝑀𝑝𝐷 = 𝐿𝑀𝑝𝑄 = 𝐿𝑚𝑝, 𝐿𝑀𝑐𝐷 = 𝐿𝑀𝑐𝑄 = 𝐿𝑚𝑐 ,

and 𝐿𝑝𝐷𝑄 = 𝐿𝑐𝐷𝑄 = 0 .

According to above equations, the two-axis saturated model

of BDFIM is obtained in arbitrary reference frame without rotor

equations as follows:

rQmpapQmplparQpQpDQ

rDMpDpDpDpDppD

iLiLLdtdidtdiL

dtdiLdtdiLiRV

(25)

rDmpapDmplparDpDpDQ

rQMpQpQpQpQppQ

iLiLLdtdidtdiL

dtdiLdtdiLiRV

(26)

rQmcrra

cQmclcrrarQcQcDQ

rDMcDcDcDcDccD

iLN

iLLNdtdidtdiL

dtdiLdtdiLiRV

.

(27)

rDmcrra

cDmclcrrarDcDcDQ

rQMcQcQcQcQccQ

iLN

iLLNdtdidtdiL

dtdiLdtdiLiRV

.

(28)

where 𝐿𝑝𝐷,𝑄 = 𝐿𝑙𝑝 + 𝐿𝑀𝑝𝐷,𝑄, and 𝐿𝑐𝐷,𝑄 = 𝐿𝑙𝑐 + 𝐿𝑀𝑐𝐷,𝑄.

After derivation of PW and CW voltage equations in

saturation condition, the same procedure is applied to achieve

the rotor voltage equations. The first time derivative of 𝐷 and

𝑄 components of rotor flux are obtained as:

dtdLidtdiL

dtdLidtdiLdtdiLdtd

mcmcDmcDmc

mpmpDmpDmprDlrrD

(29)

dtdLidtdiL

dtdLidtdiLdtdiLdtd

mcmcQmcQmc

mpmpQmpQmprQlrrQ

(30)

According to the described process from (10)-(18), (29) and

(30) are restated as:

dtdiLdtdiL

dtdiLdtdiLdtdiLdtd

mcQcDQmcDMcD

mpQpDQmpDMpDrDlrrD

(31)

dtdiLdtdiL

dtdiLdtdiLdtdiLdtd

mcDcDQmcQMcQ

mpDpDQmpQMpQrQlrrQ

(32)

In the next step, the rotor voltage equations are derived by

substituting (31) and (32) into (5) and (6) as:

cQmcpQmprpa

rQmcmplrrparQcQcDQ

rDcDMcDrQpQpDQ

rDpDMpDrDlrrDr

iLiLp

iLLLpdtdidtdiL

dtdidtdiLdtdidtdiL

dtdidtdiLdtdiLiR

.

.

0

(33)

cDmcpDmprpa

rDmcmplrrparDcDcDQ

rQcQMcQrDpDpDQ

rQpQMpQrQlrrQr

iLiLp

iLLLpdtdidtdiL

dtdidtdiLdtdidtdiL

dtdidtdiLdtdiLiR

.

.

0

(34)

The two-axis model of the BDFIM will attain in the arbitrary

reference frame if equations (33) and (34) are combined with

equations (25)-(28), as the matrix form (35) (see top of next

page). As observed in this matrix, not only all windings are

coupled owing to the cross-saturation, but also all inductances

are modified in saturated model. In order to get the magnetizing

curves, the experimental procedure described in [10] is fulfilled

and the results are used to find the best fit of PW and CW

magnetizing curves. The approximation of these curves are

obtained as an exponential function: 208928.1.27009.0

.9551414.0880384.0 mpimp e

(36)

56675.1.18937486.0.8291694.0879062.0 mcimc e (37)

Fig. 2 shows the magnetizing test results (star points) and the

fitted curves (solid lines). Considering (13), the static

inductances of PW and CW are obtained from (36) and (37),

respectively, as follows:

mpi

mp ieL mp

208928.1.27009.0.9551414.0880384.0 (38)

mci

mc ieL mc

56675.1.18937.0.8291694.0879062.0 (39)





TrQrDcQcDpQpD

McQMpQlrrmcmplrrpa

cDQpDQMcQmcrpacDQMpQmprpapDQ

mcmplrrpa

cDQpDQMcDMpDlrrmcrpacDQMcDmprpapDQMpD

McQmcrracDQcQcmclcrracDQ

mcrracDQMcDmclcrracDQcDc

MpQmpapDQpQpmplpapDQ

mpapDQMpDmplpapDQpDp

TrQrDcQcDpQpD

iiiiii

sLLLRLLLp

sLLsLLpsLsLLpsL

LLLp

sLLsLLLRLpsLsLLpsLsL

sLLNsLsLRLLNsL

LNsLsLLLNsLsLR

sLLsLsLRLLsL

LsLsLLLsLsLR

VVVVVV

.00

00

00

00

(35)

The first order derivatives of (36) and (37) are also given the

dynamic inductances corresponding to PW and CW: 208928.1.27009.0208928.0 ..3118723.0 mpi

mpp eiL

(40)

56675.1.18937.056675.0 ..246016.0 mcimcc eiL

(41)

The dynamic and static inductances are illustrated in Fig. 3,

in terms of the magnetizing currents. It should be noted that 𝐿𝑚𝑝

(𝐿𝑚𝑐) is chord (static) slop and 𝐿𝑝 (𝐿𝑐) is tangent slop of PW

(CW) magnetizing curve. The tangent-slop inductance is not

zero under saturated conditions, in contrast to linear magnetic

condition in which it is equal to chord-slope inductance.

(a)

(b)

Fig. 2. Magnetizing curves of BDFIM; (a) PW, (b) CW

(a)

(b)

Fig. 3. Static and dynamic inductance curves; (a) PW, (b) CW

C. Full Nonlinear Model of BDFIM with Incorporating Iron Losses

The rotor speed of BDFIM is not close to the synchronous

speeds of stator windings. Hence, the rotor frequency is high

and the rotor iron loss is significant. Moreover, the existence of

two rotating flux densities in the stator increases the stator

hysteresis loss [15]. Therefore, the iron loss of BDFIM is higher

than the conventional IM with the same capacity. The complete

model of BDFIM is now derived including the iron losses. In

the stationary reference frame, the iron loss resistance can be

simply paralleled with the magnetizing branch in the dynamic

equivalent circuit [16]. Since we have previously defined

𝜆𝑚𝑝 = 𝐿𝑚𝑝𝑖𝑚𝑝, we can write KVL (Kirchhoff’s voltage law) as

follows:

dtiLdiRdtdiR smpmp

siipmp

siip

pp

(42)

where 𝑅𝑖𝑝 is the PW-side iron loss resistance. The “s”

superscript is to emphasize that the frame is stationary. In order

to convert any frame to another frame a general expression is: jexx 12

(43)

where 𝑥 1 and 𝑥 2 are space vectors in old and new frames,

respectively, and 𝜃 is difference between new and old frame

angles (𝜃 = 𝜃2 − 𝜃1). In a particular frame conversion, the

stationary reference frame is converted to an arbitrary rotating

frame. Hence, 𝜃1 = 0 and 𝜃2 = 𝜃𝑎, therefore 𝜃 = 𝜃𝑎. In this

regard, (42) is rewritten as follows:

dtdLei

iLejdtideL

dteiLdeiR

mpj

mp

mpmpj

ampj

mp

jmpmp

jiip

a

aa

aap

(44)

Considering (12), we get the following:

dtidiLLi

iLjdtidLiR

mpmpmppmp

mpmpampmpiip p

.

(45)

The D-axis component of (45) is determined as:

dtdidtdiiLLi

iLdtdiLiR

mpQpmpDpmpmppmpD

mpQmpampDmpiip pD

.sin.cos.

(46)

By knowing 𝑖𝑚𝑝𝐷 = |𝑖 𝑚𝑝|. cos 𝜇𝑝 and after some

mathematical manipulations, (46) is rewritten as follows:

mpQmpampQpDQmpDMpDiip iLdtdiLdtdiLiRpD

.. (47)

2 4 6 8 10 120

0.25

0.5

0.75

1

PW Magnetizing Current (A)

P

W M

ag

ne

tizin

g

Flu

x L

ink

ag

e (

Wb

)

Fitting Curve

Experiment

2 4 6 8 100

0.25

0.5

0.75

1

CW Magnetizing Current (A)

C

W M

ag

ne

tizin

g

Flu

x L

ink

ag

e (

Wb

)

0 2 4 6 8 10 120

0.25

0.5

0.75

1


PW

Ma

gn

eti

zin

g F

lux

(W

b)

Fitting Curve

Experiment

2 4 6 8 10 120

0.05

0.1

0.15

0.2


PW

In

du

cta

nc

e (

H)

Static Ind.

Dynamic Ind.

2 4 6 8 100

0.05

0.1

0.15

0.2

CW Magnetizing Current (A)

CW

In

du

cta

nc

e (

H)

Static Ind.

Dynamic Ind.





The same procedure proposed for obtaining (47) is now

carried out to derive the following equations:

mpDmpampDpDQmpQMpQiip iLdtdiLdtdiLiRpQ

.. (48)

mcQmcrramcQcDQmcDMcDieqi iLNdtdiLdtdiLiRcD

.., (49)

mcDmcrramcDcDQmcQMcQieqi iLNdtdiLdtdiLiRcQ

.., (50)

where 𝑅𝑖,𝑒𝑞 is the CW-side iron loss resistance. Expressions of

(47) – (50) are then considered in the proposed equivalent

circuit. It should be noted that 𝑅𝑖,𝑒𝑞 is introduced for equivalent

resistance of CW iron loss (𝑃𝑖𝑟𝑜𝑛𝐶𝑊 ) and rotor iron loss (𝑃𝑖𝑟𝑜𝑛

𝑟 ).

These power loss components are obtained from power diagram

of BDFIM as:

sllwfrcu

PWiron

CWcu

PWculoadno

riron

CWiron

PPPPPPP

PP

&

(51)

where 𝑃𝑛𝑜−𝑙𝑜𝑎𝑑 is total no-load input power. According to (51),

the calculation process of resistance of CW-side iron loss (sum

of the CW and rotor iron losses) consists of four steps:

Step 1) The no-load power is calculated at the synchronous

mode for rotor speed varied in the range of ±30%

around the natural synchronous speed.

Step 2) Through DC measurements and considering skin effect

and temperature, the PW and the CW copper losses are

determined.

Step 3) The PW iron loss is obtained by proposed approach in

[17] for conventional IM. This step is fulfilled in

simple induction mode.

Step 4) The stray load loss, the friction and windage losses,

and the rotor copper loss are calculated according to

[18], [19], and [20], respectively.

Since the PW is fed by a constant frequency supply, we have

a constant resistance at the PW side. At the rated voltage and

frequency, the PW-side iron loss resistance (𝑅𝑖𝑝) is nearly 1025

Ω. The speed-dependent resistance of CW-side iron loss (𝑅𝑖,𝑒𝑞)

is achieved by supplying the PW through an autotransformer.

In order to keep the CW magnitude flux at a specific level, the

CW is controlled using 𝑉 𝑓⁄ (voltage-versus-frequency) control

approach. The CW-side iron loss has been reported in Table I

for the speed range of 350-650 rpm. The finite element analysis

(FEA) is also presented to verify the experimental results.

According to the proposed results in Table I, values of 𝑅𝑖,𝑒𝑞 are

therefore tabulated in Table II. The experimental and FEA

results have a close agreement which confirm the accuracy of

iron loss calculation algorithm. TABLE I

CW-SIDE IRON LOSS (EXPERIMENTAL AND FEA RESULTS) Rotor speed (rpm)

350 400 450 500 550 600 650

𝑷𝒊𝒓

𝒐𝒏

𝑪𝑾

+𝑷

𝒊𝒓𝒐𝒏

𝒓 (𝑾

)

Exp. 64 60 53 36 51 52.5 53.9

FEM 63 59.3 52.1 35.9 50.01 51.07 53.1

Relative error

between Exp.

and FEM (%)

1.58 1.18 1.72 0.27 1.98 2.8 1.5

TABLE II CW-SIDE IRON LOSS RESISTANCE OBTAINED BY EXPERIMENT AND FEA

Rotor speed (rpm)

350 400 450 500 550 600 650

𝑹𝒊,𝒆𝒒 (𝛀

) Exp. 506.3 540 611.3 900 635.3 617.1 604.5

FEM 514.3 546.4 621.9 902.5 647.7 634.4 610.2

Relative error

between Exp.

and FEM (%)

1.55 1.17 1.7 0.28 1.91 2.72 0.93

Based on the mentioned points in section B and C, the 𝐷 − 𝑄

full saturated model of BDFIM in arbitrary reference frame is

finally derived tacking iron loss into account. In this model

which is illustrated in Fig. 4, the orthogonal 𝐷 − 𝑄 axes are

coupled through cross-coupling inductances (𝐿𝑝𝐷𝑄 and 𝐿𝑐𝐷𝑄).

It is interesting to note that the state-space model will not

comprise clear terms that express the mutual influence of direct

axis on quadrature one and vice versa if the flux linkage are

selected as state variables.

Fig. 4. Two-axis full saturated model of BDFIM including iron loss effect in arbitrary reference frame

cRlrL

eqiR ,

pDi lpLlcLpR

rR

rDi

.pDv

pQa

MpDL

rQrpa p )(

McDL

. mcQrra N )(

cQrra N )( cDi

cDv

pDii mpDi mcDicDii

ipR

cRlrL

eqiR ,

lpLlcLpR

rR

.

mpDa .

ipR

pDQL cDQL

pQv

pQi pDa

pQii

mpQa

mpQi

MpQL

rQi

rDrpa p )(

mcQicQii

mcDrra N )(

cDrra N )(

cQv

cQi

McQL


https://www.powerthesaurus.org/it_is_worth_noting/synonyms




III. RESULTS AND DISCUSSION

The accuracy of proposed dynamic model is confirmed by

means of simulation and experiment. The main parameters of

BDFIM are shown in Table III. Due to the complex structure of

machine, most of its parameters cannot be determined using the

no-load and locked-rotor tests, unlike the standard IM. The

machine parameters except the iron loss resistances were

therefore extracted from D132s prototype BDFIM using the

procedure described in [3]. Accordingly, the parameters were

obtained from experimentally determined torque-speed

characteristics. Finding the best fit to experimental data was

done using non-linear least-squares optimization. The stator

winding resistances are also obtained from DC measurements.

The laboratory setup illustrated in Fig. 5 consists of: a voltage

source inverter with corresponding driver board, a sensor board

and a TMS320F28335 signal processor board designed with

Texas Instrument Co. The stator phase currents are measured

using Hall-effect current sensors and the line-to-line voltages

are detected by voltage sensors. To measure the torque, a

separated excitation DC generator with an external rheostat in

the armature terminal as a load is connected to the shaft of the

BDFIM. The DC generator specifications are given in Table IV.

TABLE III D132 PROTOTYPE BDFIM PARAMETERS

Symbol Parameter Value

𝑝𝑝/𝑝𝑐 PW/CW pole-pairs 2/4

𝑉𝑝 PW rated voltage (rms) 180 (V)

𝑉𝑐 CW rated voltage (rms) 180 (V)

𝐼𝑝 PW rated current (rms) 10 (A)

𝐼𝑐 CW rated current (rms) 4.5 (A)

𝑇𝑒 Rated torque 20 (N.m)

𝑅𝑝 PW resistance 1.3012 (Ω)

𝑅𝑐 CW resistance 3.7171 (Ω)

𝑅𝑟 Rotor resistance 1.1237 (Ω)

𝐿𝑙𝑝 PW leakage inductance 0.0047 (H)

𝐿𝑙𝑐 CW leakage inductance 0.0053 (H)

𝐿𝑙𝑟 Rotor leakage inductance 0.0206 (H)

TABLE IV DC GENERATOR SPECIFICATIONS

Symbol Parameter Value

𝑃𝐷𝐶 Power 4.8 (kW)

𝑉𝐷𝐶 Rated voltage 230 (V)

𝐼𝐷𝐶 Rated current 21 (A)

𝜔𝑟 Rotor speed 157 (𝑟𝑎𝑑 𝑠⁄ )

Fig. 5. Experimental setup

(a)

(b)

(c)

(d)

Fig. 6. BDFIM variables during free acceleration; (a) Rotor speed, (b) Electromagnetic torque, (c) PW current, (d) CW current.

To assess the dynamic performance of proposed model, the

machine variables including rotor speed, torque, as well as PW

and CW currents (Phase A) are obtained during free

acceleration from stall. If the IGBTs is turned off in the high-

DC Link

Capacitors

3 phase Inverter &

IGBT Driver Board

Voltage

Sensors

TMS

Board

Analog FilterCurrent

Sensors

Resistive

load

BDFIMDC

Generator

Encoder

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

150

300

450

600

time (s)

Ro

tor

sp

ee

d (

rpm

)

0 0.1 0.2 0.3 0.4 0.5-15

-10

-5

0

5

10

15

time (s)

CW

cu

rren

t (A

)

Without iron loss and

magnetic saturation

Proposed model

Experimental

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

7

14

21

28

35

time (s)

To

ruq

e (

N.m

)

0 0.1 0.2 0.3 0.4 0.5-15

-10

-5

0

5

10

15

time (s)

CW

cu

rren

t (A

)


magnetic saturation

Proposed model

Experimental

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

-10

0

10

20

time (s)

PW

cu

rre

nt

(A)

0.25 0.26 0.27 0.28 0.29 0.3-8

-4

0

4

8

0 0.1 0.2 0.3 0.4 0.5-15

-10

-5

0

5

10

15

time (s)

CW

cu

rren

t (A

)


magnetic saturation

Proposed model

Experimental

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

-7.5

0

7.5

15

time (s)

CW

cu

rre

nt

(A)

0 0.1 0.2 0.3 0.4 0.5-15

-10

-5

0

5

10

15

time (s)

CW

cu

rren

t (A

)


magnetic saturation

Proposed model

Experimental

0.2 0.21 0.22 0.23 0.24 0.25-6

-3

0

3

6





side (or low-side) of machine-side inverter (MSI), the CW will

be short circuit. In this condition, the BDFIM operates similar

to the self-cascaded machine and it accelerates from zero to

natural synchronous speed. According to Fig. 6, the simulation

and experimental results verify the accuracy of the proposed

model. Hence, it can be understood that the BDFIM dynamic

model including iron loss and magnetic saturation is

sufficiently accurate for purpose of control.

For a specific operating point (magnetizing current), the PW

magnetizing flux and the static and dynamic inductances are

experimentally obtained (Figs. 2a and 3a). In order to more

assess the proposed dynamic model, among the experimental

voltage levels supplied to the PW, the voltages with amplitude

of 120 V, 200 V and 240 V are selected to apply to the PW in

the cascade mode of operation in the simulation. The accuracy

of values obtained for PW flux linkage, static inductance, and

dynamic inductance are validated through the PW magnetizing

curve (Fig. 2a) and PW inductances curve (Fig. 3a). As

observed in Fig. 7, for a same PW voltage, these variables

which are derived from proposed model are nearly equal to

those attained from no-load test.

For additional study, a three-phase terminal fault is applied

in steady-state by setting the power supply voltage to zero at

𝑡 = 0.5 𝑠. The system configuration for three-phase fault is

demonstrated in Fig. 8. In this scenario, the load torque is 50%

of rated torque and the machine is operating at synchronous

mode. The stepping of the terminal voltage to zero at the time

of fault causes transients in the both PW and CW currents.

These transients are damped before the fault is removed and

thereafter the machine currents equal to zero. After 25 cycles

(grid frequency ꞊ 50 Hz) the fault is cleared by closing the

circuit breaker and the source voltage is reapplied to the

BDFIM. Therefore, the machine reestablishes its original

operating condition. As shown in Fig. 9, the simulation results

of the proposed model are very close to the experimental ones.

The error between theoretical and experimental tests is due to

the stray load loss, and the mechanical losses which are not

considered in the BDFIM modelling.

(a)

(b)

Fig. 7. Analysis of saturation effect for three levels of PW voltage (Cascade mode); (a) PW voltage and magnetizing flux linkage, (b) Static and dynamic inductances of PW.

Fig. 8. System configuration for three-phase voltage fault (Synchronous mode).

(a)

(b)

Fig. 9. PW and CW currents during a terminal fault (Phase A); (a) PW current, (b) CW current. (𝑓𝑃𝑊 = 50 𝐻𝑧 and 𝑓𝐶𝑊 = 20 𝐻𝑧).

The accuracy of proposed model is also evaluated under

loading condition. In this regard, the dynamic behavior of

BDFIM during step load change is demonstrated in Fig. 10. As

observed, the load torque is first stepped up to 50% of rated

torque. The load torque is then stepped down and the BDFIM

returns to its initial operating condition. A little disagreement is

observed between the proposed model and experimental results.

We change the load torque standardly with connecting/

disconnecting the load resistor instead of connecting/

disconnecting the DC generator field supply which results in

the sluggish response.

It is worth noting that the transient behavior during the

terminal fault and loading has not been shown in Figs. 9 and 10.

0 0.2 0.4 0.6 0.8 1-400

-200

0

200

400

time (s)

PW

Mag

neti

zin

g f

lux l

inkag

e (

Wb

)

0 0.5 1 1.5 2

-200

-100

0

100

200

time (s)

PW

Vo

lta

ge

(V

)

0

0.5

1

PW

flu

x l

inkag

e (

Wb

)

1.5

0.621 Wb 0.755 Wb0.372 Wb

1

0

-1

-1.5

Voltage

Flux linkage

0 0.2 0.4 0.6 0.8 10

0.1

0.2

time (s)

PW

In

du

cta

nc

e (

H)

Static Ind.

Dynamic Ind.

0.172 H

0.112 H

0.144 H

0.061 H

0.181 H

0.191 H

BDFIM

AC

DC

DC

AC

Grid

Con

trol

Win

din

g

Machine-Side

Inverter

Grid-Side

Inverter

Frequency Converter

Power Winding

DC

Gen.

Circuit

Breaker

Auto-

transformer

0 0.25 0.5 0.75 1 1.25 1.5-20

-10

0

10

20

time (s)

PW

cu

rre

nt

(A)

0.25 0.26 0.27 0.28 0.29 0.3-10

-5

0

5

10

0 0.1 0.2 0.3 0.4 0.5-15

-10

-5

0

5

10

15

time (s)

CW

cu

rren

t (A

)


magnetic saturation

Proposed model

Experimental

0 0.25 0.5 0.75 1 1.25 1.5-20

-10

0

10

20

time (s)

CW

cu

rre

nt

(A)

0 0.1 0.2 0.3 0.4 0.5-15

-10

-5

0

5

10

15

time (s)

CW

cu

rren

t (A

)


magnetic saturation

Proposed model

Experimental

0.5 0.52 0.54 0.56 0.58 0.6-15

-10

-5

0

5


https://www.powerthesaurus.org/it_is_worth_noting_that/synonyms




Fig. 10. Dynamic performance of BDFIM during step change in load torque (Phase A); (a) PW current, (b) CW current. (𝑓𝑃𝑊 = 50 𝐻𝑧 and 𝑓𝐶𝑊 = 20 𝐻𝑧).

(a)

(b)

Fig. 11. Comparison of two magnetic saturation models during starting

transient, (a) 𝑑 − 𝑞 axes PW currents, (b) 𝑑 − 𝑞 axes CW currents

Comparison of two magnetic saturation models of BDFIM

has been given in Fig. 11. One of the models accounts for

dynamic cross-saturation effects (full nonlinear model),

whereas the other neglects them (simplified model). It is

evident that the full nonlinear model is more accurate than the

simplified one ignoring the cross-coupling inductances. The

difference between the models is visible through showing the

two-axis stator currents in stationary reference frame during

free acceleration.

It is worthwhile to provide a comparison of the proposed

model and existing methods in the literature for modeling the

some types of brushless doubly-fed machines. In this regard,

the most significant features of them have been summarized in

Table V.

IV. CONCLUSION

The two-axis theory of the three-phase electrical machines is

well-developed and it was used as a starting point for

development of the model of BDFIM tacking the iron loss and

saturation into account. In this regard, the two-axis full

saturated model of BDFIM was derived in arbitrary reference

frame including iron loss which covers the cross-saturation

effect. The experimental and simulation results for cascade and

synchronous modes were shown the accuracy of dynamic

model during free acceleration, three-phase fault, and step

change in the load torque. Moreover, an experimental process

was presented to determine the fundamental iron loss of

BDFIM and the equivalent iron loss resistances were obtained

from these data. According to the results, the proposed dynamic

model can be used to implement the various control strategies

of the BDFIM as a possible substitute for adjustable speed

drives.

APPENDIX

DERIVATION OF TORQUE EXPRESSION

The active input power can be written for the BDFIM in space

0 0.5 1 1.5 2 2.5 3-10

-5

0

5

10

PW

cu

rre

nt

(A)

time (s)

1.19 1.21 1.48 1.25-10

-5

0

5

10

Without iron loss and magnetic saturation Proposed model Experimental

0 0.5 1 1.5 2 2.5 3-5

0

5

time (s)

CW

cu

rre

nt

(A)

2.8 2.825 2.85 2.875 2.9

-3

0

3

(a)

(b)

0.2 0.21 0.22 0.23 0.24

-10

-5

0

5

10

0.25 0.26 0.27 0.28 0.29 0.3-8

-4

0

4

80 0.05 0.1 0.15 0.2 0.25 0.3 0.35

-10

0

10

20

time (s)

PW

cu

rre

nt

(A)

0.25 0.3 0.35 0.4 0.45

-5

0

5

time (s)

CW

cu

rren

t (A

)

Simplified model

Proposed model

(Full nonlinear model)

Experimental

0.25 0.26 0.27 0.28 0.29 0.3-8

-4

0

4

8

0.08 0.09 0.1 0.11 0.12-10

-5

0

5

10

15

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-20

-10

0

10

20

time (s)

CW

cu

rre

nt

(A)

0.25 0.3 0.35 0.4 0.45

-5

0

5

time (s)

CW

cu

rren

t (A

)

Simplified model

Proposed model

(Full nonlinear model)

Experimental





TABLE V

COMPARISON OF MODELS PROPOSED IN THE LITERATURE

Ref. Type of

machine

Has iron loss

been considered?

Has magnetic saturation

been considered?

Type of

model

Dynamic or

steady-state

modelling?

Model accounting for

magnetic saturation

[1] BDFIM No No EEC1 Dynamic -

[2] BDFIM No No EEC Dynamic -

[3] BDFIM No No EEC Steady-state -

[7] BDFIM No Yes MEC2 - -

[8] BDFIM Yes Yes MEC - -

[9] CDFIM Yes Yes EEC Steady-state Simplified model

[10] DS-BDFIG No Yes EEC Dynamic Simplified model

[20] BDFIM Yes No EEC Steady-state -

Proposed model BDFIM Yes Yes EEC Dynamic Full nonlinear model

1 Electric equivalent circuit (EEC) 2 Magnetic equivalent circuit (MEC)

vector form as:

ccpp IVIVP

Re2

33 (52)

This active input power includes the power for losses, the

power that is stored in the three-phase field of the machine, and

the power that is being converted to mechanical energy. The

power stored in the field is constant in steady state (there is a

constant amplitude three-phase flux density waveform in the

air-gap of the machine), and therefore there is no energy stored

in this, after the initial transient. On the other hand, the

individual phases have reactive power transferred to and from

the machine. These reactive powers add instantaneously to be

zero over the three phases of the machine.

Applying KVL to loop between magnetizing and iron loss

branches at the PW (CW) side, we have respectively:

pDpQpQpDp impQimpDai

mpQi

mpDiip iii

dt

di

dt

dIR ._...

2

(53)

cDcQcQcDc imcQimcDrrai

mcQi

mcDieqi iiNi

dt

di

dt

dIR ._....

2,

(54)

In a similar way, KVL is written for the rotor loop:

0._....

..2

rDrQrQrDrrarQrQ

lrrDrD

lr

rQmcQmpQ

rDmcDmpD

rr

iiNidt

diLi

dt

diL

idt

d

dt

di

dt

d

dt

dIR

(55)

Substituting (53)-(55) into (52) and after a small amount of

manipulation, the expression of active input power is derived as

follows:

rDrQrQrDrpa

imcDimcQrracDcQcQcDrra

impDimpQapDpQpQpDa

rQrQ

lrrDrD

lrmcQmcQ

mcDmcD

cQcQ

lc

cDcD

lcmpQmpQ

mpDmpD

pQpQ

lppDpD

lp

rrieqiiipccpp

iip

iiNiiN

iiii

idt

diLi

dt

diLi

dt

di

dt

di

dt

diL

idt

diLi

dt

di

dt

di

dt

diLi

dt

diL

IRIRIRIRIRP

cQcD

pQpD

cp

...

......

....

22,

222

23

(56)

This expression includes following components:

Resistive losses ꞊

22

,

222

2

3rrieqiiipccpp IRIRIRIRIR

cp

(57)

Changes in Field energy ꞊

rQrQ

lrrDrD

lr

mcQmcQ

mcDmcD

cQcQ

lccDcD

lc

mpQmpQ

mpDmpD

pQpQ

lppDpD

lp

idt

diLi

dt

diL

idt

di

dt

di

dt

diLi

dt

diL

idt

di

dt

di

dt

diLi

dt

diL

2

3

(58)

Power converted to mechanical energy ꞊ rDrQrQrDp

imcDimcQrcDcQcQcDrr

iip

iiNiiNcQcD

..

....

(59)

The electromagnetic torque is therefore obtained from

rotational power (𝑃𝑟𝑜𝑡) as:

** .Im

2

3.Im

2

3rmcmccrmpmpp

r

rote IILpIILp

PT

(60)

REFERENCES

[1] A. K. Wallace, R. Spee, and H. K. Lauw, “Dynamic Modelling of

Brushless Doubly-Fed Machines,” in proc. IEEE Ind. Appl. Society Annu. Meeting, San Diego, CA, USA, 1989, pp. 329-334.

[2] R. Li, A. K. Wallace, and R. Spee, “Two-Axis Model Development of

Cage-Rotor Brushless Doubly-Fed Machines,” IEEE Trans. Energy Convers., vol. 6, no. 3, pp. 453–460, Sep. 1991.

[3] P. C. Roberts, R. A. McMahon, P. J. Tavner, J. M. Maciejowski and T. J.

Flack, “Equivalent Circuit for the Brushless Doubly-Fed Machine (BDFM) including Parameter Estimation and Experimental Verification,”

IEE Elect. Power Appl., vol. 152, no. 4, pp. 933-942, July 2005.

[4] S. Tohidi, “Analysis and Simplified Modelling of Brushless Doubly-Fed Induction Machine in Synchronous Mode of Operation”, IET Elect.

Power Appl., vol. 10, no. 2, pp. 110-116, Feb. 2016.

[5] E. Levi, and V. Vuckovic, “Rotor Flux Computation in Saturated Field-Oriented Induction Machines,” Electric Machines & Power Systems, vol.

21, no. 6, pp. 741-754, Oct. 1992.

[6] M. Sokola, and E. Levi, “Combined Impact of Iron Loss and Main Flux Saturation on Operation of Vector Controlled Induction Machines,” in

proc. Int. Conf. Power Electron. & Variable Speed Drives, Nottingham,

UK, 1996, pp. 36-41. [7] H. Gorginpour, H. Oraee and R. A. McMahon, “A Novel Modeling

Approach for Design Studies of Brushless Doubly-Fed Induction

Generator Based on Magnetic Equivalent Circuit,” IEEE Trans. Energy Convers., vol. 28, no. 4, pp. 902-912, Dec. 2013.


https://ieeexplore.ieee.org/author/37276997700






[8] A. C. Ferreira, and S. Williamson, “Time-Stepping Finite-Element

Analysis of Brushless Doubly-Fed Machine Taking Iron Loss and

Saturation into Account,” IEEE Trans. Ind. Appl., vol. 35, no. 3, pp. 583-

588, May-June 1999.

[9] D. Basic, J. G. Zhu, and G. Boardman, “Modeling and Steady-State Performance Analysis of a Brushless Doubly-Fed Twin Stator Induction

Generator,” in Proc. Australasian Power Eng. Conf., Melbourne,

Australia, 2002, pp. 1–6. [10] Y. Zeng, M. Cheng, X. Wei, and L. Xu, “Dynamic Modeling and

Performance Analysis with Iron Saturation for Dual-Stator Brushless

Doubly-Fed Induction Generator,” IEEE Trans. Energy Convers., vol. 35, no. 1, pp. 260-270, March 2020.

[11] A. Accetta, F. Alonge, M. Cirrincione, M. Pucci, and A. Sferlazza, “Feedback Linearizing Control of Induction Motor Considering

Magnetic Saturation Effects,” IEEE Trans. Ind. Appl., vol. 52, no. 6, pp.

4843–4854, Nov. 2016. [12] M. Pucci, “State-Space Space-Vector Model of the Induction Motor

Including Magnetic Saturation and Iron Losses,” IEEE Trans. Ind. Appl.,

vol. 55, no. 4, pp. 3453-3468, July-Aug. 2019. [13] O. Kiselychnyk, M. Bodson, and J. Wang, “Comparison of Two Magnetic

Saturation Models of Induction Machines and Experimental Validation,”

IEEE Trans. Ind. Electron., vol. 64, no. 1, pp. 81–90, Jan. 2017. [14] P. Vas, “Generalized Analysis of Saturated AC Machines,” Electrical

Engineering, vol. 64, no. 1, pp. 57–62, Jan. 1981.

[15] H. Gorginpour, H. Oraee, and E. Abdi, “Calculation of Core and Stray Load Losses in Brushless Doubly-Fed Induction Generators,” IEEE

Trans. Ind. Electron., vol. 61, no. 7, pp. 3167-3177, July 2014.

[16] E. Levi, M. Sokola, A. Boglietti, and M. Pastorelli, “Iron Loss in Rotor-Flux-Oriented Induction Machines: Identification, Assessment of

Detuning, and Compensation,” IEEE Trans. Power Electron., vol. 11, no.

5, pp. 698-709, Sep. 1996. [17] Test Procedure for Poly-phase Induction Motors and Generators, IEEE

Standard 112, 2017.

[18] Rotating Electrical Machines-Part2-1: Standard Methods for Determining Losses and Efficiency from Tests (Excluding Machines for Traction

Vehicles)), IEC 60034, 2014.

[19] H. Mosaddegh Hesar, H. A. Zarchi, and G. A. Markadeh, “Modeling and Dynamic Performance Analysis of Brushless Doubly Fed Induction

Machine Considering Iron Loss”, IEEE Trans. Energy Convers., vol. 35,

no. 1, pp. 193-202, March 2020. [20] M. Yousefian, H. Abootorabi Zarchi, and H. Gorginpour, “Modified

Steady-State Modelling of Brushless Doubly-Fed Induction Generator

Taking Core Loss Components into Account”, IET Elect. Power Appl., vol. 13, no. 9, pp. 1402-1412, Sep. 2019.

Hamidreza Mosaddegh Hesar received the

B.Sc. and M.Sc. degrees from Ferdowsi

University of Mashhad, Iran, in 2011 and

2014, respectively, from Ferdowsi University of Mashhad, Iran, where he is currently

working toward the Ph.D. degree.

His current interests and activities include control of high-performance drives, nonlinear

control, and modelling of electrical machines.

Mohammad Ali Salahmanesh received the

B.Sc. degree in electrical engineering from the Sajjad University of Technology, Iran, in 2017,

and the M.Sc. degree from Ferdowsi

University of Mashhad, Iran, in 2019 (both with honors).

His research interests and activities include control of electric drives, nonlinear control, modeling of electrical machines and power electronics.

Hossein Abootorabi Zarchi received the M.S. and Ph.D. degrees from the Isfahan University of Technology, Isfahan, Iran, in 2004 and 2010, respectively. He was a Visiting Ph.D. Student with the Control and Automation Group, Denmark Technical University, Denmark, from May 2009 to February 2010. He is currently an Assistant Professor in the Department of Electrical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran.

His research interests include electrical machines, applied nonlinear control in electrical drives, and renewable energies.


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Generalized Analysis of Brushless Doubly-Fed Induction ...profdoc.um.ac.ir/articles/a/1080132.pdf · is ignored, the time derivative of magnetizing inductances will yield incorrect