Introduction to the Space Vector Modeling of the

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Introduction to the Space Vector Modeling of the Brushless Doubly Fed Reluctance Machine a

R. E. BETZ & M. G. JOVANOVIĆ

b

a

School of Electrical Engineering and Computer Science, University of Newcastle, Callaghan, NSW, 2308, Australia b

School of Engineering, University of Northumbria, Newcastle upon Tyne, United Kingdom Published online: 21 Jun 2010.

To cite this article: R. E. BETZ & M. G. JOVANOVIĆ (2003): Introduction to the Space Vector Modeling of the Brushless Doubly Fed Reluctance Machine, Electric Power Components and Systems, 31:8, 729-755 To link to this article: http://dx.doi.org/10.1080/15325000390219785

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EMP 31(8) #5897

Electric Power Components and Systems, 31:729–755, 2003 c Taylor & Francis Inc. Copyright ISSN: 1532-5008 print/1532-5016 online DOI: 10.1080/15325000390219785

Introduction to the Space Vector Modeling of the Brushless Doubly Fed Reluctance Machine

Downloaded by [York University Libraries] at 02:06 15 June 2013

R. E. BETZ School of Electrical Engineering and Computer Science University of Newcastle Callaghan, NSW, 2308 Australia

´ M. G. JOVANOVIC School of Engineering University of Northumbria Newcastle upon Tyne, United Kingdom The brushless doubly fed reluctance machine (BDFRM) is the least known of a group of electrical machines that include the classic cascaded induction machine, the brushless doubly-fed induction machine (BDFIM), and the doublefed slip ring induction machine (DFSRIM). Since its initial development some 30 years ago, the BDFRM has been largely ignored because of the performance limitations imposed by the reluctance rotor design. However, improvements in reluctance rotors, a by-product of the development of the synchronous reluctance machine, have resulted in renewed interest in the BDFRM. This together with the promise of higher efficiency and simpler control compared to the BDFIM means that further investigation of the BDFRM is warranted. This paper is designed to be a starting point for research into the BDFRM. It develops the fundamental modelling equations that are required to carry out research into its dynamics and control. The approach is partly tutorial in nature as it presents some “well-known” analysis techniques that are scattered throughout the literature on the machine. It develops from first principles the space vector model of the machine, which is then used to derive the steadystate BDFRM equations. Standard sinusoidal spatial variation and linearity assumptions are used throughout the analysis. Where relevant, the physical concepts behind the machine’s operation are emphasized. In addition to the full dynamic model of the machine, the paper also introduces a per-unit system that forms the basis for machine-independent performance expressions. Keywords brushless doubly fed machine, reluctance machine, electric machine control, space vector modeling

1. Introduction The brushless doubly fed reluctance machine (BDFRM) is a close cousin of the brushless doubly fed induction machine (BDFIM). The latter has been extensively Manuscript received in final form on 6 June 2002. Address correspondence to R. E. Betz. E-mail: [email protected]

729


730

R. E. Betz and M. G. Jovanović

researched in the literature over the last 20 years. However, the reluctance machine version has not received the same attention. The BDFRM was first proposed in [1]. A machine was built and tested, but the performance achieved was relatively poor compared to that of the BDFIM. This was mainly due to the poor saliency of the rotor used, which was constructed from an induction machine rotor with sections machined out of it. Consequently interest in the machine waned until the early 1990s. In the meantime considerable work was occurring on the design of the synchronous reluctance machine (Syncrel) rotor [2–4]. These rotor designs are able to yield much larger saliency ratios, and the radially laminated versions are simple, robust, and economical to build. With such rotors, the BDFRM’s performance could be competitive with the BDFIM. Furthermore the cageless structure and consequent lower rotor losses in the BDFRM mean that it is more efficient than the BDFIM, when both machines are driven by an inverter [5].1 In the early 1990s papers began to appear that included the modeling of the machine. For example, [6] proposed a steady-state model based on using a number of approximations. However, up to this time the most comprehensive papers published on the transient model of the machine were [7, 8]. These papers developed the dq model of the machine, which was shown to be similar to that for the wound rotor induction machine. Many of the modeling details were omitted in these papers, and the connection to the space vector model was not developed. The purpose of this paper is to provide a first principles-based rigorous development of the dynamic equations for the BDFRM. It should be emphasized that the approach is theoretical in nature and utilizes the standard assumptions from induction machine modeling that the windings have a sinusoidal spatial variation and linearity of the magnetic material.2 The process of developing these expressions is very useful in gaining an in-depth understanding of the operation of this complex machine. The paper extends the work in [7, 8], with the development of a space vector–based dynamic model. In addition it considers modeling issues associated with the machine inductances and from the space vector model develops a per-unit (PU) system that forms the basis of the machine-independent performance expressions. The paper is organized as follows. Section 2 outlines the physical structure of the machine and gives a brief explanation of its basic operational principles. Section 3 then derives the space vector electrical equations. Section 4 develops the torque expressions and presents them in a number of different forms. Section 5 is concerned with the development of all the power expressions for the BDFRM. Section 6 derives the PU bases for the machine, and a set of PU performance equations are developed using them. Finally the Appendix develops the expressions for the inductances using two different models for the machine air gap.

2. Preliminaries Figure 1 is a conceptual diagram of the BDFRM. The machine is constructed with two conventional sinusoidally distributed windings wound onto the same induction 1 This is not the case when the machines are operated in dead-online asynchronous mode. 2 The equations developed have limitations because of the ideal modeling.


Brushless Doubly Fed Reluctance Machine

731

Figure 1. Conceptual diagram of the brushless doubly fed reluctance machine.

machine stator.3 The two windings have different numbers of pole pairs, so if the machine had a round rotor, then ideally there should be no coupling between them. One of the two windings of the machine is known as the primary winding, this name referring to the fact that it is connected to the grid supply. The other winding is called the secondary winding and in a modern BDFRM drive system would be inverter-fed. The rotor of the machine is shown in Figure 1 as a crude “dumbbell”-type rotor. This would be appropriate for high-speed applications, but the more likely choice would be a modern radially laminated multiple flux barrier Syncrel rotor. Similar to the Syncrel, the rotor saliency is the key to satisfactory operation of the BDFRM. As stated previously, the normal situation is that there would be not coupling between the two windings. However, the presence of the variable reluctance path for the flux in the machine essentially modulates the stator mmf waveforms, resulting in the formation of corresponding flux density harmonics, which can link to the opposite winding. The generation of these harmonic fluxes can be more clearly understood if we cast the problem into a slightly more formal framework. First, let us state the assumptions that are being made in the analysis: 1. The iron of the machine is infinitely permeable. 2. The stator windings are ideal spatially sinusoidally distributed windings. 3. The stator currents are ideal three-phase temporal currents. In the following simplified discussion we shall also assume that the inverse machine air gap function can be modeled as [10] g −1 (θ, θrm ) = m + n cos pr (θ − θrm ),

(1)

3 It is possible to wind the machine so that the two windings not only share the same magnetic circuit, but also the same copper. These are known as parallel path or mixed-pole windings [1, 9].

732


where the constants are m ≥ n > 0 and pr the number of rotor poles, θrm the rotor mechanical angle with reference to the primary “A” phase axis, θ the mechanical angle around the periphery of the machine with respect to the “A” phase axis, 1/m the mean air gap length. Remark 1. The above equation means that


• The maximum air gap is gmax = 1/(m − n) • The minimum air gap is gmin = 1/(m + n). If we assume that the primary and secondary windings are excited by threephase sinusoidal currents of angular frequencies ωp and ωs , respectively, then it is well known that the resultant mmfs per air gap are Fp (θ) = Fpm cos(ωp t − pp θ),

(2)

Fs (θ) = Fsm cos(ωs t − α − ps θ),

(3)

where: pp,s the pole pairs of the windings, np,s the effective turns per pole per phase, Ip,s the peak phase currents, F(p,s)m =

3 np,s Ip,s . 2

Now that we have the air gap function and the mmf produced by the windings, we can use B(θ, θrm ) = µ0 g −1 (θ, θrm )F (θ)

(4)

to calculate the resultant flux density. Let us consider the primary winding. Substituting Equations (1) and (2) into Equation (4) we can write the expression for Bp (θ, t) as n cos[(ωp − pr ωrm )t − (pp − pr )θ] Bp (θ, t) = µ0 mFpm cos(ωp t − pp θ) + 2m n (5) + cos[(ωp + pr ωrm )t − (pp + pr )θ] , 2m where θrm = ωrm t and ωrm the rotor angular velocity in mech. rad/s. Remark 2. Equation (5) is similar to that one obtains in amplitude modulation in communications systems. In fact, one can say that the inverse rotor air gap function effectively “modulates” the sinusoidal mmf of the windings, producing a fundamental flux density waveform (equivalent to the carrier component in AM modulation) plus two sidebands.


733

Remark 3. One very interesting and important aspect of Equation (5) is that the “modulation” occurs in both a temporal and spatial sense. For example, the temporal frequency of the primary has been transformed from ωp to ωp ±pr ωrm . The spatial frequency, which is equivalent to the pole pair number, has also been converted by the process—the effective pole pairs of the sidebands are pp ± pr .


In a similar manner to the primary, one can carry out the same procedure to evaluate the secondary resultant flux density waveform: n Bs (θ, t) = µ0 mFsm cos(ωs t − α − ps θ) + [cos((ωs − pr ωrm )t + (pr − ps )θ − α) 2m + cos((ωs + pr ωrm )t − (pr + ps )θ − α)] . (6) The secondary, similar to the primary, has a fundamental and two temporal and spatial sidebands. Examination of Equations (5) and (6) indicates that a judicious choice of the rotor pole number will result in one of the sidebands of one winding being of the same temporal and spatial frequency as the other winding. Consider, for example, the first of the primary sideband terms. Written mathematically the previous assertion is equivalent to (ωp − pr ωrm )t = ωs t,

(7)

(pp − pr )θ = ps θ,

(8)

which leads to the following conditions: ωrm =

ωp − ωs , pr

p r = pp − p s , ∴ ωrm =

ωp − ωs . pp − ps

(9) (10) (11)

Because cosine is an even function, we can also consider the negative version of the arguments on the left-hand side of Equations (7) and (8): −ωp + pr ωrm = ωs ,

(12)

−pp + pr = ps ,

(13)

ωp + ωs , pr

(14)

which results in ωrm =

p r = pp + p s , ∴ ωrm =

ωp + ωs . pp + ps

(15) (16)

This process can be repeated with the second primary harmonic component. One ends up with the same rotor pole number expression as for the first harmonic

734


component, but of opposite sign. Theoretically speaking, this is correct since the air gap function (1) is defined for pr both positive and negative. Of course, from a practical viewpoint, only the situations corresponding to pr > 0 are of interest. The other point to note about the expressions for this harmonic is that the ωrm terms turn out to be identical to those of the first primary harmonic case.


Remark 4. Note that for any particular pr value, only one of the harmonics couples to the complementary winding. The remaining harmonic contributes to the leakage flux of the machine, as does the fundamental component of the winding fluxes. This relatively weak coupling between the windings, and consequent high leakage, result in relatively poor torque per unit volume for the BDFRM [11]. The same results can be obtained by performing a similar analysis for the secondary-to-primary winding terms. Summary 1. There are two choices for the rotational speed and pole numbers for both the primary-to-secondary, and secondary-to-primary couplings: pr = pp + ps and ωrm =

ωp + ωs , pr

pr = |pp − ps | and ωrm =

ωp − ωs . pr

(17) (18)

Remark 5. The relationships in Equations (17) and (18) are required for magnetic coupling to exist between the primary and secondary winding. In other words, the number of rotor poles has to obey Equation (17) to produce field harmonics of the correct spatial frequency (i.e., to effectively pole change the fundamental mmfs of the two windings). Furthermore, the rotor has to be rotating at the specific mechanical velocity defined by Equation (18) for the harmonic flux density distribution to be rotating at the speed of the fundamental mmf of the other winding. Remark 6. The above analysis does not indicate that torque will be developed by the machine. It is possible to show (see the Appendix) that only the mutual inductances between the windings are rotor position dependent, the other inductances all being constant. As the rotor moves there is a change in the co-energy of the system, and consequently torque can be produced. This assertion will be proved in Section 3. Remark 7. It should be noted that the rotor angular velocity relationships of Equations (17) and (18) also apply to the steady-state operation of the classic cascaded connection of two wound rotor induction machines with pp and ps pole pairs.

3. Space Vector Electrical Equations The starting point for the development of dynamic equations for any electrical machine is Faraday’s law. The equations for the primary and secondary windings


735

of the BDFRM in a stationary reference frame (denoted with the sub-subscript s) are v ps = Rp ips +

dλps , dt

(19)

v ss = Rs iss +

dλss . dt

(20)


In order to flesh out these equations we need to find the expressions for the flux linkage vectors. The self and mutual inductances of the BDFRM, using two different modeling approaches, are derived in the Appendix. The derivation of the flux expressions can be made more elegant if we rewrite the primary-to-secondary inductance expressions of Equations (137)–(139) as LAa = LBc = LCb =

1 Lpsmax (ejθr + e−jθr ), 2

(21)

LAb = LCc = LBa =

1 Lpsmax (a2 ejθr + ae−jθr ), 2

(22)

LAc = LCa = LBb =

1 Lpsmax (aejθr + a2 e−jθr ), 2

(23)

where the phases of the primary and secondary windings are denoted as “ABC” 2π and “abc,” respectively, a = ej 3 and θr = pr θrm . Expanding the expressions for the flux linkages in matrix form we get         λA LAA LAB LAC iA LAa LAb LAc ia λB  = LBA LBB LBC  iB  + LBa LBb LBc   ib  , (24) λC LCA LCB LCC iC LCa LCb LCc ic         λa Laa Lab Lac ia LaA LaB LaC iA  λb  =  Lba Lbb Lbc   ib  +  LbA LbB LbC  iB  . (25) λc Lca Lcb Lcc ic LcA LcB LcC iC If we substitute for the inductances using Equations (21)–(23) and (133)–(136) in the Appendix we can manipulate Equations (24) and (25) to give   1 1 − Lmp − Lmp Lmp + Llp 2 2       1 1  [λABC ] =  − Lmp Lmp + Llp − Lmp   [iABC ] + [LABC−abc ] [iabc ] , 2   2   1 1 − Lmp − Lmp Lmp + Llp 2 2   1 1 − Lms − Lms Lms + Lls 2 2       1 1  [λabc ] =  L L − L + L − ms ls ms  [iabc ] + [Labc−ABC ] [iABC ] ,  2 ms 2     1 1 − Lms − Lms Lms + Lls 2 2

(26)

(27)

736 where

R. E. Betz and M. G. Jovanović   1 a2 a 1 [LABC−abc ] = [Labc−ABC ] = Lpsmax a2 a 1  ejθr 2 a 1 a2   1 a a2 1 + Lpsmax  a a2 1  e−jθr . 2 a2 1 a

(28)


Expanding one of these terms, say, λA (since the other flux linkage expressions follow the same pattern), 1 1 λA = (Lmp + Llp )iA − Lmp iB − Lmp iC 2 2

1 + Lpsmax (ia + a2 ib + aic )ejθr + (ia + aib + a2 ic )e−jθr . 2

(29)

If we assume no zero sequence currents can flow (i.e., a star-connected winding with an isolated neutral), then iA = −iB − iC , and the first term of Equation (29) becomes 1 1 3 (30) (Lmp + Llp )iA − Lmp iB − Lmp iC = Lmp + Llp iA . 2 2 2 Now considering part of the second term of Equation (29) one can see the following space vector relationships: (ia + a2 ib + aic ) ejθr + (ia + aib + a2 ic ) e−jθr ,

(31)

3 2 iss

3 ∗ 2 iss

which allows Equation (29) to be written more succinctly as

1 λA = Lp iA + Lps i∗ss ejθr + iss e−jθr , 2

(32)

where 3 Lmp + Llp , 2 3 = Lpsmax 2

Lp = Lps

(33) (34)

are the constant three-phase inductances expressed in terms of the inductances defined in the Appendix. Via a similar procedure one can derive the expressions for λB and λC :

1 λB = Lp iB + Lps a2 i∗ss ejθr + a iss e−jθr , 2

1 λC = Lp iC + Lps a i∗ss ejθr + a2 iss e−jθr . 2

(35) (36)

Remark 8. Equations (32), (35), and (36) each have the two flux terms— one is related to the self-produced flux (e.g., Lp iA,B,C ), and the other, containing


737

the complex variables, represents the mutual flux components. Note, however, that only the mutual flux terms participate in the electro-mechanical energy conversion, since only these terms depend on the rotor position (via θr ). We are now in a position to form the primary flux space vector: λps =

2 (λA + aλB + a2 λC ). 3

(37)

Substituting Equations (32), (35), and (36) into Equation (37), and simplifying allows us to write the stationary frame expression for the primary flux:


λps = Lp ips + Lps i∗ss ejθr .

(38)

Using a similar derivation process we can obtain the expression for the secondary flux space vector: λss = Ls iss + Lps i∗ps ejθr .

(39)

Remark 9. Equations (38) and (39) clearly contain a term associated with the self-produced flux (i.e., Lp ips and Ls iss ), and another term associated with the mutual flux (that is being coupled to the winding from the other winding due to the previously mentioned rotor mixing action). Notice that Lps determines the degree of coupling and is therefore crucial in the determination of the machine’s performance. One can see from Equation (142) that Lpsmax , and hence Lps via Equation (34), can be improved by modifying the n parameter of Equation (1). The same applies to the amplitudes of the two flux density sidebands, as indicated by Equations (5) and (6). It should be noted that there is a trade-off with the n parameter: while the torque-producing harmonic component is increased, the other component, which is effectively a leakage, is also proportionally higher. Remark 10. The i∗ss ejθr and i∗ps ejθr terms in Equations (38) and (39) are little difficult to interpret. At first sight one may suspect that these expressions are in the form of a reference frame conversion. However, they are not: all of Equations (38) and (39) are in a stationary reference frame, therefore the ejθr terms do not carry out a spatial frame conversion, but instead a translation from the primary to secondary frequency (and vice versa). This frequency transformation results from the rotor modulating influence on the stator mmf’s considered previously. Remark 11. The real effect of the i∗ss ejθr and i∗ps ejθr terms can be seen by simplifying the two expressions as follows. Expanding i∗ss ejθr under the assumption that Equation (17) holds, we can write i∗ss ejθr = is e−jθs ej(θp +θs ) = is ejθp = ipss ,

(40)

where the meanings of the current angles are self-evident from Figure 2 and θr = θp +θs due to Equation (17). Therefore the above term is a secondary current vector referred to the primary winding side (but in a frequency, not turns, ratio sense) and is rotating at the same angular velocity as the primary counterpart (i.e., at ωp ) with respect to the stator. We can do the same with the other expression, to give i∗ps ejθr = ip e−jθp ej(θp +θs ) = ip ejθs = isps .

(41)


738


Figure 2. Reference frames and current vectors used in the BDFRM equations (dq frame is stationary and aligned with the primary A-phase axis).

Using the simplifications of Remark 11, we can write Equations (38) and (39) as λps = Lp ips + Lps ipss ,

(42)

λss = Ls iss + Lps isps .

(43)

Substituting Equations (42) and (43) and taking the appropriate derivatives, we can now rearrange Equations (19) and (20) into a slightly expanded form: dλps v ps = Rp ips + + jωp λps , (44) dt θp const dλss v ss = Rs iss + + jωs λss . (45) dt θs const Now that we have the stationary frame flux and voltage expressions, the next step is to convert them to an arbitrary dp qp reference frame (Figure 2). Let us start with Equation (42). Using the conventional frame conversion formulae [10], and assuming that the angle of the primary reference frame is θ, then we can write λpr = λps e−jθ ,

(46)

ipr = ips e−jθ ,

(47)

˜is = ips e−jθ . r s

(48)


739

Substituting these into Equation (42) and simplifying, we get the same form of expression, but in the dp qp frame: λpr = Lp ipr + Lps˜isr .

(49)

Considering the last term of the equation, and using Equation (40), we can manipulate Equation (48) as ˜is = i∗s ej(θr −θ) = [is e−j(θr −θ) ]∗ . r s s

(50)


Examination of this expression suggests that a reference frame at the angle θr − θ may be a good choice for the secondary equations. If this is the case, then ˜isr = i∗sr . Using this frame one can refer Equation (43) to the ds qs frame (Figure 2). The primary and secondary flux linkage space vectors can finally be expressed as λpr = Lp ipr + Lps i∗sr ,

(51)

λsr = Ls isr + Lps i∗pr .

(52)

Remark 12. The previous equations are in two different reference frames: Equation (51) is in dp qp frame (at angle θ) and Equation (52) in ds qs frame (at angle θr −θ). The last terms in both the equations represented the currents reflected from the other winding via mutual coupling. It is very interesting that these “mapped” vectors are actually the complex conjugates of their originals. Using the above flux linkage expressions, we are now in a position to derive the corresponding rotating frame voltage equations for the machine. Applying the stationary-to-primary reference frame transformation we can write d λpr ejθ dt dλpr = Rp ipr ejθ + ejθ + jωλpr ejθ . dt

v pr ejθ = Rp ipr ejθ +

(53) (54)

In a similar fashion one can get the secondary voltage equation with respect to the ds qs frame. Summary 2.

The arbitrary rotating frame form of the voltage equations are dλpr + jωλpr , dt dλ + sr + j(ωr − ω)λsr . dt

v pr = Rp ipr +

(55)

v sr = Rs isr

(56)

Remark 13. Up to this point the dp qp reference frame has been rotating at an arbitrary angular velocity of ω radian/sec. If the frame is chosen to be rotating at ω = ωp , then the primary equations are obviously in the ωp frame, and the secondary ones are in the ωr − ωp = ωs frame. Therefore this selection moves each of the equations into their “natural” reference frames where the respective vector components appear as DC in steady state. Another interesting reference frame is to let θ = 0, that is, a stationary dq reference frame for the primary equation, and the rotor reference frame (at θr

740


angle) for the secondary equation. This form of the equations is particularly useful for the development of the steady state model. In this case ω = 0 and Equation (55) becomes identical to Equation (19). Clearly under this condition the primary flux vector is λpr = λps = λp ejωp t . Substituting this into Equation (19), taking the derivative, and setting the time rate of change terms to zero, one can write the steady-state voltage equation:


v ps = Rp ips + jωp λps .

(57)

One can repeat the process for the secondary equation, realizing that for this situation λss = λs ejωs t and λsr = λss e−jθr . Differentiating as indicated in Equation (56) (remembering that dλs /dt = 0 in steady state and dθr /dt = ωr ) gives v sr = Rs isr + jωs λsr .

(58)

In order to allow us to derive a steady-state model for the BDFRM that is similar to that of other more conventional machines, we need to be able to connect the primary and secondary equivalent circuits together. However, the frequencies in either circuit are different. The use of a generalized slip makes it possible for a frequency referral operation to be carried out. Consider the standard expression for the slip of a 2p-pole induction machine: s=1−

pωrm . ωe

(59)

If pole and angular velocity relationships (17) are substituted into Equation (59), where ωe = ωp and p = pr , we get the slip definition for a BDFRM: s=1−

ωp + ωs ωs =− . ωp ωp

(60)

Substituting Equation (51) into Equations (57) and (52), and Equation (60) into Equation (58) yields the steady-state equations with reference to the ωp frequency: v ps = Rp ips + jωp (Lp − Lps )ips + jωp Lps (ips + i∗sr ), v ∗sr s

=

Rs ∗ i + jωp (Ls − Lps )i∗sr + jωp Lps (i∗sr + ips ). s sr

(61) (62)

The corresponding equivalent circuit is shown in Figure 3. Remark 14. We can define im = ips + i∗sr , which can be considered to be the magnetizing current. Remark 15. Figure 3 has the same form as the equivalent circuit for the doubly fed slip ring induction machine. If the secondary is shorted, then the model is that of a traditional squirrel cage induction machine. This reinforces the close connection between the modeling of the BDFRM and the induction machine. It also indicates that the BDFRM can function as an induction machine in case of inverter failure.


741


Figure 3. The steady-state equivalent circuit of the BDFRM.

4. Torque Expressions The complete development of the BDFRM torque expressions is too long to be included in this paper. Therefore only the general outline shall be given with relevant observations highlighted (for more details refer to [12]). The torque for an electro-magnetic system can be determined by using the change of field energy Ef e (or co-energy) with rotor rotation, that is, Te = dEf e /dθrm [10]. This basic relation can be used to develop the following well-known matrix torque expression [13]: 1 pr [i]T [G][i], 2

Te =

(63)

where [i] is a vector of machine instantaneous currents, [G] = dθdr [Lij ], and [Lij ] is a matrix of machine inductances. Specifically, for the BDFRM the current vectors are

(64) [i]T = [iA , iB , iC , ia , ib , ic ] = [iABC ]T , [iabc ]T , and the matrix of inductance derivatives (where i = A,B,C,a,b,c; j = A,B,C,a,b,c) becomes   d [0] [LABC−abc ]  dθr   . [G] =  (65)    d  [Labc−ABC ] [0] dθr Remark 16. The diagonal block matrices of Equation (65) are zero because the corresponding elements represent the phase inductances of the windings, which are constant; they are not a function of θr as follows from Equations (133)–(136) in the Appendix. If one substitutes Equations (64) and (65) into Equation (63), bearing in mind that [LABC−abc ] = [Labc−ABC ], one gets T

Te = pr [iABC ]

d [LABC−abc ] [iabc ] . dθr

(66)

742


The next step in the derivation is to substitute Equation (28) into Equation (66) and then differentiate. After considerable manipulation, it is possible to show that   ∗ jθ iss e r − iss e−jθr 3 d   (67) [LABC−abc ] [iabc ] = j Lpsmax a2 i∗ss ejθr − a iss e−jθr  . 4 dθr ∗ jθr 2 −jθr a iss e − a iss e Using this expression in Equation (66), and given Equation (34), one obtains the electromagnetic torque in terms of the stationary frame variables:


3 Te = j pr Lps i∗ps i∗ss ejθr − ips iss e−jθr . 4

(68)

Equation (68) can be converted into the rotating reference frame (i.e., primary quantities to the dp qp frame, and the secondary ones to the ds qs frame). The result is

3 Te = j pr Lps i∗pr i∗sr − ipr isr . 4

(69)

This equation can be further simplified by using the flux linkage expressions (51) and (52) to give

3 Te = j pr λpr i∗pr − λ∗pr ipr 4

(70)

3 pr [λpd ipq − λpq ipd ] . 2

(71)

or in rectangular form Te =

Remark 17. Equation (71) for the BDFRM is exactly the same form as the torque expression for the conventional 2pr -pole induction machine. This fact again indicates that the two machines are closely related despite a fundamentally different physical operating principle. Remark 18. Equation (71) contains only primary winding currents. It is possible to get a variety of different torque expressions for the machine, but the ones that contain the secondary winding current values are the best suited for the development of controllers for the BDFRM (since an inverter is usually connected to this winding). From a control perspective, and considering the nature of the BDFRM operation, one of the most useful forms of the torque expression is when the torque is expressed in terms of the primary winding flux and the secondary winding currents. The reason for this is that the primary flux of the machine is essentially fixed (since the primary winding is fed with a constant voltage and frequency grid supply), whereas the secondary currents can be controlled via an inverter. Expanding i∗pr from Equation (52) and substituting this into Equation (70), we can derive the following relationship: Te =

3 Lps pr [λpd isq + λpq isd ] . 2 Lp

(72)


743

This equation can be simplified by arbitrarily aligning the dp qp frame with the primary flux vector. This means that there is no λpq component by definition, that is, λpd = λp , and therefore Equation (72) becomes Te =

3 Lps λp isq . pr 2 Lp

(73)

Remark 19. Equation (73) is important as it shows that the torque and flux in the BDFRM are independent. This is a significant advantage of the BDFRM, since decoupled control of these parameters is possible, similarly to a DC machine. The flux vector, λp , is fixed by the primary winding grid connection, while the isq current is controllable via the secondary winding.


5. Power Expressions The other set of important expressions for any machine are those for power. The instantaneous input power expression for the BDFRM is S 3φ = P3φ + jQ3φ =

3 v pr i∗pr + v sr i∗sr . 2

(74)

The real power component of Equation (74) includes the copper losses, rate of change of energy stored in the field, and mechanical power output. Substituting Equations (55) and (56) into Equation (74) allows us to write dλpr dλ 3 + jωi∗pr λpr + Rs i2sr + i∗sr sr + j(ωr − ω)i∗sr λsr . P3φ = Rp i2pr + i∗pr 2 dt dt (75) To simplify the expression we shall arbitrarily put ω = 0 (since power is reference frame invariant), which makes the expression dλpr 3 2 ∗ 2 ∗ dλsr ∗ + jωr isr λsr . + Rs isr + isr P3φ = Rp ipr + ipr (76) dt 2 dt Equation (76) can be broken down into individual components as follows: 3 (Rp i2pr + Rs i2sr ), (77) 2 dλpr dλ 3 Changes in field energy = i∗pr + i∗sr sr , (78) 2 dt dt 3 Rotational power = jωr i∗sr λsr . (79) 2 The last term is the only one that involves the rotation of the machine shaft, therefore it must be the rotational power. This can be expressed as Resistive losses =

3 {jωr (λsd + jλsq )(isd − jisq )} 2 3 = pr ωrm (λsd isq − λsq isd ). 2

Prot =

(80) (81)

Remark 20. Equation (81) is totally in terms of secondary current variables. Dividing by ωrm gives a resultant torque expression that is the dual of Equation (71).

744


We shall now consider the primary and secondary components of the power separately. In the following equations we shall assume that the voltages are in steady state—that is, we are using Equations (57) and (58). The primary apparent power is therefore 3 3 v i∗ = (Rp i2p + jωp Lp i2p + jωp Lps i∗sr ips ), (82) 2 2 ps ps where ip and is are the magnitudes of the primary and secondary current vectors.


Sp =

Remark 21. The development of the power expressions under steady-state conditions may seem to be limiting. However, notice that Equation (81), which is derived using the rotational power and ignoring the stored energy, is of the same form as the torque developed using the field energy approach. It is possible to derive Equation (81) from Equation (71). Therefore, the rotational power component of the power expression is present regardless of the transient field storage of the machine. Consequently Equation (81) is not only valid in steady state, but also instantaneously. Equation (82) can be developed further by substituting the following current relationships (Figure 2):  ipr = ip ejαp = ip ej(θp −θ)      j(θ+αp ) jθp  = ip e ips = ip e . (83)  isr = is ejαs = is ej(θs +θ−θr )      iss = is ejθs = is ej(θr −θ+αs ) If we let θ = ω = 0, then ipr = ips ⇒ θp = αp . Therefore, Equation (82) becomes after a little manipulation

3 Sp = Rp i2p + ωp Lps is ip sin(αs + αp ) + j ωp Lp i2p + ωp Lps is ip cos(αs + αp ) . 2 (84) One can obviously break Equation (84) into real and imaginary components as follows:

3 Rp i2p + ωp Lps is ip sin(αs + αp ) , (85) Pp = 2

3 ωp Lp i2p + ωp Lps is ip cos(αs + αp ) . (86) Qp = 2 Remark 22. As noted in Remark 21, even though the power expressions are in steady state the rotational power expression is valid transiently. Therefore, Equation (85) is the primary winding contribution to the machine real power (omitting any real power that is being stored as field energy). In a similar fashion, one may obtain the secondary power expressions for the BDFRM:

3 Ps = (87) Rs i2s + ωs Lps is ip sin(αs + αp ) , 2

3 ωs Ls i2s + ωs Lps is ip cos(αs + αp ) . (88) Qs = 2


745

The connection between the real powers and the torque produced by the machine can be shown by expanding Equation (69) using Equation (83) to obtain 3 pr Lps ip is sin(αs + αp ). 2

Te =

(89)

If one combines Equations (85) and (87) and subtracts the copper loss components, then one gets the total mechanical power output for the machine:


PT = Prot =

3 (ωs + ωp ) Lps ip is sin(αs + αp ). 2

(90)

As indicated, this expression is equivalent to Equation (81) for the machine rotational power. Since ωrm = ωr /pr = (ωp +ωs )/pr , then comparing with Equation (89) it can be easily deduced that PT = Te ωrm as expected. Similarly, if we ignore resistive losses, the following power relationships for each individual winding are also immediately obvious: ωp Te , pr ωs ωs Te = Pp = −sPp , Ps = pr ωp

Pp =

(91) (92)

where s is the generalized slip previously defined in Equation (60). Remark 23. Equation (92) clearly shows the slip power recovery property of the machine. The power required from the secondary winding inverter is directly proportional to ωs , that is, the slip s. Therefore, supplying DC to the secondary (i.e., ωs = 0) results in the inverter having to supply zero power (except for the resistive losses), regardless of the torque produced by the machine (which is rotating at synchronous speed ωsyn = ωp /pr ). In general, if the machine is to be operated in a narrow range around ωsyn (as is the case in pump-type applications, for example), a fractionally rated inverter can be used. Remark 24. From Equation (92) it can be also seen that for ωs ≥ 0 then Ps ≥ 0. But if ωs < 0 (i.e., the phase sequence of the secondary is reversed), then obviously Ps < 0. Under this condition the machine is operating below synchronous speed. In this mode the BDFRM is inefficient as power is flowing from the primary into the machine, and then some of it is being regenerated into the supply by the inverter. This circulation of energy incurs losses in the process.

6. Per-Unit Bases In order to develop machine-independent performance equations for the BDFRM a per-unit (PU) system is required. The following development is slightly unusual because it assumes that a full set of nameplate parameters for the machine are not available (which is the case for machines that are only in the conceptual design phase). Therefore, there is a degree of arbitrariness in the selection of the PU bases. However, some of the base values to choose are fairly obvious—the primary voltage, the primary frequency, and the primary self-flux (which is related to the previous two bases).

746


The first step in developing the PU bases is to consider the PU torque. In order to do this we will manipulate the torque expression (72) into an appropriate form. Starting from Equation (51), we can write it in rectangular form as follows: λpr = λpd + jλpq = Lp ipd + Lps isd + j(Lp ipq − Lps isq ).

(93)

Assuming that the primary reference frame is aligned with the primary flux vector, then we can say that the imaginary component of Equation (93) is zero. Therefore the flux with the primary flux alignment condition is


λpr = λp = λpd = Lp ipd + Lps isd .

(94)

Given that λpq = 0, the following relationship between the primary self and mutual inductances and the q axis current components can be established: isq is sin αs Lp = = . ipq ip sin αp Lps

(95)

Using the current angle definitions in Figure 2 together with Equations (94), (95), and (72), we can write Te =

L2ps 3 pr Lp (i2p sin 2αp + 2 i2s sin 2αs ). 4 Lp

(96)

Remark 25. This equation shows some similarity with the torque expression for the synchronous reluctance machine, due to the presence of the sin 2α terms. Applying an approach similar to that above, the primary and secondary current magnitudes can be expressed in terms of the primary flux and current angles: ip =

λp sin αs , Lp sin(αp + αs )

(97)

is =

λp sin αp . Lps sin(αp + αs )

(98)

Substituting Equations (97) and (98) into Equation (96) gives Te =

3 λ2p sin αs sin αp pr . 2 Lp sin(αs + αp )

(99)

Allowing αs = αp = π/4 defines Te as the base torque:4 TB =

3 λ2p pr . 4 Lp

(100)

Note that pr the number of the rotor poles. 4 The choice of αs = αp = π/4 is to some extent arbitrary. However, it does allow a simplification of Equation (96).


747

As mentioned at the beginning of this section three obvious base variables are5 vB base voltage, equal to the grid voltage,

(101)

ωB the base frequency, equal to the grid frequency, v p vB r λB = (the base flux). ωB ωB

(102) (103)

In a straightforward manner the remainder of the bases can be derived as


PB

2ωB TB , pr

iB

λB , Lp

LB Lp .

(104)

Given the above definitions it is possible to develop the following normalized expressions for torque, currents and fluxes of the BDFRM: Tn =

2 sin αs sin αp 2 = isn sin αs , sin(αp + αs ) ζ

(105)

ipn =

sin αs , sin(αp + αs )

(106)

isn =

ζ sin αp , sin(αp + αs )

(107)

λpn = 1,

1 λsn = Lsn isn + i∗pn . ζ

(108)

The PU real and reactive power expressions are derived from Equations (85)– (88): Ppn =

1 sin αp sin αs 1 isn ipn sin(αs + αp ) = = Tn , ζ sin(αs + αp ) 2

Qpn = i2pn + =

ipn isn cos(αs + αp ) , ζ

sin αs [sin αs + sin αp cos(αs + αp )], sin2 (αs + αp )

=1−

isdn sin αs cos αp = , ζ sin(αs + αp )

1 ωsn ωsn isn ipn sin(αs + αp ) = Tn , ζ 2 isn ipn cos(αs + αp ) 2 , = ωsn Lsn isn + ζ sin αp ωsn sin αp = + sin αs cos(αs + αp ) , 2 kps sin2 (αs + αp )

(109) (110) (111) (112)

Psn =

(113)

Qsn

(114) (115)

5 In the λB expression below we are implicitly assuming that ωp λp >> Rp ip , which is normally the case as ωp is a constant at line frequency.

748


where ζ = Lp /Lps and kps = Lps / a transformer.

Lp Ls is the coupling coefficient defined as with


Remark 26. For a good machine design the ζ parameter should be as close d to unity as possible. A link between ζ and the traditional saliency ratio, ξ = L Lq , which characterizes the rotor design for the synchronous reluctance machine, can be established. See [11] for details. The PU bases and the consequent PU performance equations developed above offer a significant simplification of the machine expressions in some cases. For example, Equation (113) explicitly shows the connection between the applied secondary frequency and the power required from the inverter. Similarly, Equation (109) demonstrates that the primary power is only related to the torque required, and not the frequency (as it should since this is fixed).

7. Conclusion The BDFRM is a very unusual and interesting electrical machine. In order to design control strategies and understand its complex dynamics it is necessary to have the full set of dynamic equations. This paper has developed, from first principles, the full space vector dynamic model for the BDFRM using the standard sinusoidal assumptions used when modeling induction machines. The steady-state equations were subsequently derived from this. The modeling similarities of the BDFRM and the DFSRIM were emphasized where appropriate. The torque expressions for the BDFRM were derived and subsequently used to obtain the power expressions for the machine. Finally, a per-unit normalization of the dynamic and performance equations was carried out. In the Appendix two techniques for the development of the inductance expressions for the machine’s parameters are outlined. The derivation approaches shown in this paper should serve as an excellent introduction to the analysis of the BDFRM, and should be very invaluable to researchers wishing to carry out more advanced study and control of this machine.

Appendix: Machine Inductances One of the crucial parts of modelling the BDFRM is the calculation of the self and mutual inductances of the windings. Most of the existing techniques involve a large degree of approximation in order to keep the analysis mathematically tractable. We shall present two approximations for the development of the relevant inductances in this appendix.

A.1. Sinusoidal Air Gap Model The first approximation implies the modelling of the air gap as a sinusoidal variation using Equation (1), which we shall rewrite here for convenience: g −1 (θ, θr ) = m + n cos pr (θ − θrm ), where m ≥ n ≥ 0.

(116)


749

The technique to be used for the evaluation of the inductances is the winding function technique [7, 11, 12], which can be succinctly stated in the following expression: 2π g −1 (θ, θrm )Ni (θ)Nj (θ)dθ, (117) Lij = µ0 rl 0

where the two winding functions are defined as


give

Ni = ni cos(pi θ − αi ),

(118)

Nj = nj cos(pj θ − αj ).

(119)

One can substitute Equations (116), (118), and (119) into Equation (117) to Lij = µ0 ni nj rl

m 2

0

m + 2 n + 4 n + 4 n + 4 +

2π

n 4

cos((pi + pj )θ − αi − αj )dθ 2π

0

2π

0

2π

0

2π

0

0

2π

cos((pi − pj )θ − αi + αj )dθ

cos((pi + pj + pr )θ − θr − αi − αj )dθ cos((pi + pj − pr )θ + θr − αi − αj )dθ cos((pi − pj + pr )θ − θr − αi + αj )dθ cos((pi − pj − pr )θ + θr − αi + αj )dθ

(120)

where pr the number of rotor poles pi ; pj pole pairs of the phase windings i and j, respectively ni ; nj effective turns per pole per phase for windings i and j, respectively θ arbitrary mechanical angle around the stator circumference θr the electrical angle of the rotor high-permeance axis (d-axis) = pr θrm r radius of the machine to the mean of the air gap l the axial length of the machine αi ; αj spatial phase of the windings i and j, respectively m; n some constants dependent on the machine design parameters. Equation (120) is completely general and would allow one to develop inductance expressions for a variety of different machine types by choosing appropriate values for m and n.

750

R. E. Betz and M. G. Jovanović The following nomenclature is used throughout the rest of the analysis:

Primary windings: A, B, C Secondary windings: a, b, c Poles: pp the pole pairs of the primary windings ps the pole pairs of the secondary windings pr pp + ps the rotor poles


Primary winding function: NA (θ) = np cos pp θ, 2π , NB (θ) = np cos pp θ − 3 2π , NC (θ) = np cos pp θ + 3

(121) (122) (123)

Secondary winding function: Na (θ) = ns cos ps θ, 2π , Nb (θ) = ns cos ps θ − 3 2π , Nc (θ) = ns cos ps θ + 3

(124) (125) (126)

where np and ns are the turns per pole per phase for the primary and secondary windings, respectively. Note also that for the BDFRM pp = ps . Let us initially consider primary-to-primary winding inductances. LAA Inductance (αi = αj = 0) In this particular case we are calculating the self-inductance of the primary winding A-phase. Hence αi = αj = 0 and pi = pj = pp . Substituting for this situation into Equation (120), one can write the following: m 2π m 2π n 2π 2 cos(2pp θ)dθ + dθ + cos((3pp + ps )θ − θr )dθ LAA = µ0 np rl 2 2 0 4

0

0 +

+

n 4

n 4

=0

2π

0

0

n cos((pp − ps )θ + θr )dθ + 4 =0

2π

cos((−pp − ps )θ + θr )dθ =0

0

=0

2π

cos((pp + ps )θ − θr )dθ =0

! + Llp .

(127)


751

Therefore we have ∴ LAA = µ0 n2p mπrl + Llp .

(128)

The other phases’ self-inductances follow a similar pattern for the primary and the secondary winding. LAB Inductance (αi = 0, αj = 2π/3)


We can evaluate the mutual inductance between phases on the same winding by following a similar process to that above, yielding 1 LAB = − µ0 n2p mπrl. 2

(129)

Now look at the primary-to-secondary winding mutual inductances. LAa Inductance (αi = αj = 0) Note that the “A” primary phase, and the “a” secondary phase axes are assumed to be co-linear (this is the case if the two windings are accommodated in the same slots). Again using Equation (120) with the appropriate substitutions (pi = pp and pj = ps ) we can write LAa = µ0 np ns rl

m 2

n + 4

n + 4

2π

0

(pr θ)dθ +

2π

0

m 2

2π

0

cos((pp − ps )θ)dθ =0

n cos((4pp θ − θr )dθ + 4

2π

0

cos θr dθ

=0

0

2π

n cos(2pp θ − θr )dθ + 4 =0

∴ LAa =

0

2π

! cos(−2ps θ + θr )dθ

1 µ0 np ns nπrl cos θr . 2

,

(130)

=0

(131)

LAb Inductance (αi = 0, αj = 2π/3) The same procedure can be carried out for the LAb mutual inductance evaluation, giving 1 2π . (132) LAb = µ0 np ns nπrl cos θr − 2 3 One can continue on with the rest of the inductance expressions in a similar manner. However, a pattern emerges that makes this unnecessary. One only has to

752


consider the integral terms in Equation (120), where the coefficient of the θ term in the cos expression goes to zero. All the other terms integrate to zero. The result of this process is summarized below. Summary 3. The BDFRM inductances evaluated using the sinusoidal air gap assumption are Self inductances: LAA = LBB = LCC = Lmp + Llp ,


Laa = Lbb = Lcc = Lms + Lls .

(133) (134)

Primary-to-primary inductances: 1 LAB = LBC = LCA = − Lmp . 2

(135)

Secondary-to-secondary inductances: 1 Lab = Lbc = Lca = − Lms . 2

(136)

Primary-to-secondary inductances: LAa = LBc = LCb = Lpsmax cos θr , 2π , LAb = LCc = LBa = Lpsmax cos θr − 3 2π LAc = LCa = LBb = Lpsmax cos θr + , 3

(137) (138) (139)

where Lmp = µ0 n2p mπrl,

(140)

Lms = µ0 n2s mπrl,

(141)

Lpsmax =

1 µ0 np ns nπrl, 2

(142)

Llp the primary leakage inductance, Lls the secondary leakage inductance. Remark 27. Notice that the self-inductance and primary-to-primary and secondary-to-secondary inductance expressions do not contain any θr terms, which means that their values do not change with the rotor position. Therefore these inductances cannot play any part in the production of torque. On the other hand, though, the primary to secondary inductances do contain θr terms, therefore they do participate in the generation of torque.


753


Figure 4. Inverse air gap function for a four-pole salient rotor (pr = 4).

A.2. Twin Air Gap Model The sinusoidal air gap model of the previous section is simpler than the twin air gap model. However, this simplicity has the price that the saliency ratio range for the rotors that can be represented by it is limited [11]. This is especially a problem when comparing the performance of the BDFRM with other reluctance machines. A more realistic model is the twin air gap model [7]. The modeling approach used here, however, is more general than that in [7] as the maximum air gap length has been assumed finite, that is, the minimum inductance value (along the rotor q-axis) is not zero. The corresponding inverse air gap plot is shown in Figure 4. This function can be written mathematically as  Λd = Λmax for     π π    θrm + (2k − y) ≤ θ ≤ θrm + (2k + y)   pr pr  , (143) g −1 (θ, θrm ) = and    Λq = Λmin for     π π   θrm + (2k + y) ≤ θ ≤ θrm + (2k + 2 − y) pr pr where6 k = 0, 1, 2, . . . , pr − 1,

0

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