Comparison of the Brushless Doubly Fed Reluctance Machine and the Synchronous Reluctance Machine 1

Transcription

1 Comparison of the Brushless Doubly Fed Reluctance Machine and the Synchronous Reluctance Machine 1 R.E. Betz and M.G. Jovanović Department of Electrical and Computer Engineering University of Newcastle, Australia Technical Report: EE Created: May 1, 1998 Last Revised: January 14, 1999 Printed: January 14, Work supported by the Danfoss Visiting Professor Program, Aalborg University, Denmark, and the Australia Research Council. Liverpool John Moores University School of Engineering Byrom St., Liverpool, L3 3AF 3 This is a Department of Electrical and Computer Engineering, University of Newcastle, Australia technical report number Preliminary

2 Contents Preface Acknowledgments v vi I Theoretical Analysis 1 1 Control Properties 1.1 Introduction Maximum Torque per Secondary Ampere Comparison with the SYNCRE L Maximum Torque Per Total Ampere Comparative Analysis with the SYNCREL Power Constant Torque Operation Power Factor Input-Output Performance Limitations Main assumptions Torque E xpressions Voltage Expressions Power and Voltage Relationships Input/Output Relationships II Experimental Studies 67 III Appendices 69 A Relationship Between BDFRM and SYNCREL Inductances 70 A.1 Twin Air-gap Technique for Inductance Evaluation A.1.1 Primary to Secondary BDFRM Winding Inductances.. 77 A.1. Primary (Secondary) BDFRM Winding Inductances A.1.3 SYNCREL Inductances A. Comparison of the Twin Air-gap Approach and its Co-sinusoidal Counterpart

3 CONTENTS ii B Copper Losses Expressions 86 B.1 Introduction B. Basic E xpressions B.3 Copper Losses Comparison C Primary Flux Oriented Torque Expression 90 D Dynamic and Steady State Equations in the ω p Reference Frame 91 E Power Expressions for the BDFRM 93 F Space Vector Model for the SYNCREL 98 F.1 SYNCRE L Torque E xpression G Base Definitions and Normalised Expressions 101 H Matlab Files 108 H.1 BDFRM Torque vs Inductance Ratio H.1.1 File: torque per amp vs xi.m H. Real and Reactive Power Plots for Constant Torque H..1 File: constant torque.m H.. Constant Torque and Varying Reactive Power H.3 Maximum Torque per Total Amperes H.3.1 Main Program H.3. Optimization Function H.4 Input-Output Performance Limitations H.4.1 BDFRM-SYNCREL Inverter Currents Plots H.4. BDFRM-SYNCREL Torque Relationships H.4.3 BDFRM-SYNCREL Voltage Related Relationships

4 List of Figures 1.1 Various reference frames used for the BDFRM Normalised torque per ampere for the BDFRM Torque ratio of 6/ pole BDFRM and the 4 pole SYNCREL with the same copper losses, active copper and current angles of π/ α s and α p for maximum torque per (i pn + i sn ) i pn + i sn for maximum-torque-per-total-ampere current angles i pn and i sn for maximum-torque-per-total-amperes angles Maximum-torque-per-total-amperes for various torque levels Real and reactive powers under the maximum-torque-per-totalamperes condition with the secondary winding being supplied at the mains frequency (ω sn = 1) Primary power factor under maximum torque per total ampere conditions Secondary power factor under maximum-torque-per-total-ampere conditions MTTA torque ratio between the 6/ pole BDFRM and 4-pole SYNCRE L having the same copper losses MTTA total input current ratio between the 6/ pole BDFRM and 4-pole SYNCREL with equal copper losses MTTA ratio between the 6/ pole BDFRM and 4-pole SYN- CREL having the same copper losses The α p and α s angles for constant T n =1andω sn = The primary winding currents with T n =1andω sn = The secondary winding currents with T n =1andω sn = Range of current vector movement for the primary winding. Condition: T n =1andω sn = Range of current vector movement for the secondary winding. Condition: T n =1andω sn = Total current magnitude (i pn +i sn ) under the T n =1andω sn =1 condition Primary winding real (P pn )andreactive(q pn ) powers under the condition T n =1andω sn = Secondary winding real (P sn )andreactivepower(q sn ) under the T n =1andω sn = 1 condition Primary winding power factor when T n =1andω sn = Secondary winding power factor when T n =1andω sn = Primary winding volt-amps (VA) with T n =1andω sn = Secondary winding volt-amps (VA) with T n =1andω sn = Secondary current angles for the unity power factor of the windings 43

5 LIST OF FIGURES iv 1.7 BDFRM currents under the unity power factor and MTPSA conditions Power factors of the BDFRM windings under various control strategies Reactive power requirements for the BDFRM windings under the conditions of different control strategies and ω sn = Maximum power factor torque per secondary ampere (TPSA) normalised to its maximum value Maximum torque of BDFRM and SYNCREL with equal copper losses MTPSA secondary voltage of BDFRM and MTPA voltage of SYNCREL under the condition of equal copper losses and the same speed (1-pu) of both machines Ratio of the SYNCREL MTPA output power and the BDFRM MTPSA power output for the same copper losses and shaft angularvelocity BDFRM secondary frequency when the BDFRM and the SYN- CREL having the same copper losses run out of volts under maximum torque per inverter current conditions Rotor angular velocity for the BDFRM and the SYNCREL having the same copper losses, unity pu voltages and minimum inverter current rating control Output power when the machines having the same copper losses run out of volts under the conditions of maximum torque per inverterampere Optimised inverter-fed winding currents of the BDFRM and the SYNCRE L with the same copper losses BDFRM kva vs shaft power ratio with the unity pu voltage and MTPSA current angle Machines kva vs torque ratio under the conditions of maximum torque per inverter ampere and with unity applied voltages BDFRM secondary voltage when the equivalent SYNCREL runs out of volts Inverter kva ratings for unity SYNCREL voltage and the same speed of both machines kv A/T out for the machines rotating at the same speed with the unity applied voltage of the SYNCREL kv A/P out for the machines with the same speed and unity SYN- CRELvoltage A.1 Inverse air-gap function for a 4-pole salient rotor (p r = 4) A. L p /L ps for the BDFRM with an infinite q-axis air-gap A.3 L d /L q for the SYNCREL having an infinite q-axis air-gap

6 Preface This report examines the fundamental control properties of the brushless doubly fed reluctance machine (BDFRM). It compares the optimal control performance of the ideal BDFRM under various control strategies with the classical synchronous reluctance machine (SYNCREL). Whilst these machines both have saliency in their rotors the basic operational principles of the machines are quite different. Robert Betz Aalborg, Denmark, 1998.

7 Acknowledgments R.E. Betz would like to acknowledge the support of the Danfoss Visiting Professor Program at Aalborg University, Denmark, and in particular Professor Frede Blaabjerg. Much of the work in this report was carried out whilst R.E. Betz was on a 1 month sabbatical leave at the Department of Electrical Energy Conversion at Aalborg University in 1998.

8 Part I Theoretical Analysis

9 Chapter 1 Control Properties 1.1 Introduction The brushless doubly fed reluctance machine (BDFRM) is by no means a new machine design. The basic operational principles were established by Broadway in the early 1970s [1]. Most of this work concentrated on various issues related to the machines fundamental operation principles, some aspects of its construction, and its steady state performance. Since this time several papers have appeared on the dynamic performance of the machine [ 4], but as yet a truly comprehensive analysis of the control properties of the machine has not appeared. This report is intended to fill this void. The BDFRM is a machine worthy of study because it allows a variable speed drive system to be constructed with the inverter supplying a maximum of half the output power. One can get an even more lowly rated inverter by Trading off increased machine size for decreased inverter size. This could be a significant advantage in applications requiring large drive systems, since the inverter is a significant proportion of the system cost. Furthermore, the lower rating of the power electronics also helps minimise troublesome harmonics injected into the power system. Similar advantages can be obtained using slip energy recovery drives, but these involve the use of a motors with wound slip ring rotors. Such machines are expensive, but more importantly the rotor and its slip rings are not as reliable as say the squirrel cage induction machine. The BDFRM offers similar performance to these systems with a brushless machine. Furthermore the rotor does not involve any windings and therefore has reliability even better that the squirrel cage induction machine. Although the emphasis in this paper is on the development of the control properties of the BDFRM, it will also attempt to compare the performance of the BDFRM with the synchronous reluctance machine (SYNCREL) and the squirrel cage induction machine (IM) where appropriate. The reason for doing these comparisons are: 1. The rotor of the BDFRM is virtually identical to that of the SYNCREL, and it is interesting to see the advantages and disadvantages afforded to the BDFRM by virtue of its more complex stator winding.. The IM is the mainstay of most current industrial drive systems, therefore

10 1. Maximum Torque per Secondary Ampere 3 it is of practical interest to see what merits the BDFRM may have in relation to these systems. We shall investigate and the following properties of the BDFRM: 1. Conditions for maximum torque per secondary amperes.. Conditions for maximum torque per total amperes. 3. Power factor control and the trade-offs with other properties. 4. Maximum rate of change of torque. 5. Maximum output power for a given inverter rating. 6. Efficiency and losses. Remark 1.1 This report should be read in conjunction with [5] as this fills in many details on machine modelling not presented in this report. 1. Maximum Torque per Secondary Ampere This is a very important property for any machine, especially when the machine is used in motoring mode. In order to make comparisons between the BDFRM and the SYNCREL we need to use the inductance relationships established in Appendix A and the torque expressions for the two machines. Using (C.4) we can write the following expression assuming that the reference frame is aligned with the primary flux vector: T e = 3 p L ps r λ p i sq (1.1) L p This expression occurs because λ pd = λ p and λ pq = 0 due to frame alignment. Remark 1. Note that this frame alignment makes it obvious that one can control the torque in the machine in an independent manner if the flux is fixed by the primary winding connection to the grid supply (making λ p approximately constant). As we shall see, even though we cannot control the flux in the machine, we will be able to control the machines power factor in a decoupled fashion as well. Remark 1.3 The fact that a primary flux reference frame is being used means that the reference frame is rotating at ω p radians/sec. Remark 1.4 Note that this expression for torque does not give one any hint of the power factor of the primary and secondary windings. The flux λ p is constant because of the grid connection of the primary winding, but it is made up of components contributed from the primary and secondary d-axis currents. The mix of these in the λ p expression is not indicated by (1.1). Remark 1.5 It should be noted that we are working out the maximum torque per secondary winding ampere. These expressions are not taking into account the primary amperes and are in no way restricting the copper losses in the machine. These issues shall be addressed later in this report.

11 1. Maximum Torque per Secondary Ampere 4 q q s q p p i p s i s d p p s d s r p s r d Figure 1.1: Various reference frames used for the BDFRM From Figure 1.1 one can see that the secondary q axis current can be written as i sq = i s sin α s and consequently the torque per ampere of the torque producing current for the machine can be written as: T e i s = 3 p L ps r λ p sin α s (1.) L p Clearly the maximum torque per ampere occurs when α s = π/, therefore we have: T e i s = 3 max p L ps r λ p (1.3) L p In order to make a comparison of this value with the SYNCREL we need to get it into a form where the inductance values can be compared. This can be achieved using the relationships of (A.4). Substituting for L ps and L p we have: T e i s = 3 max p r (Ldp L qp )(L ds L qs ) L dp + L qp λ p (1.4) If we assume that n s = n p for the sake of a comparison, then we can say: L d = L dp = L ds (1.5) L q = L qp = L qs (1.6)

12 1. Maximum Torque per Secondary Ampere 5 Figure 1.: Normalised torque per ampere for the BDFRM. and therefore (1.4) becomes: T e i s = T pamax = 3 max p L d L q r λ p = 3 L d + L q p ξ 1 r ξ +1 λ p (1.7) where ξ = L d /L q (the inductance or saliency ratio as defined for the SYN- CREL). We can normalise the torque per ampere against the ideal torque per ampere achieved with an infinite saliency ratio. The ideal machine torque per ampere can easily be seen to be: T e i s = T paideal = 3 ideal p rλ p (1.8) If we plot the T pamax /T paideal versus the saliency ratio it generates a similar curve to the power factor versus saliency ratio for the SYNCREL [6]. The function to be plotted is: T pan = T pa max T paideal = ξ 1 ξ +1 (1.9) and it appears in Figure 1.. Remark 1.6 The curve shown in Figure 1. indicates that the rotor saliency ratio is very important in respect to allowing a machine to achieve its maximum possible torque per ampere for a given level of primary flux.

13 1. Maximum Torque per Secondary Ampere Comparison with the SYNCREL Let us consider the inductance expressions for the SYNCREL. The general form of the self inductance expressions (which form the basis of the d and q axis inductance expressions) is: L xx = L 1 + L cos θ d (1.10) where L xx the self inductance of phase xx, L 1 the inductance DC component, L the inductance oscillating component and θ d the electrical angle of the rotor d axis with respect to the xx phase. Since θ d is in electrical radians then we can say that: θ d = p p θ rm = pθ rm (1.11) where p p the pole pairs of the p-pole SYNCREL, and θ rm the mechanical angle of the rotor d axis with respect to the xx phase axis in radians for the SYNCREL. Substituting this into (1.10) we can write: L xx = L 1 + L cos(pθ rm ) (1.1) which is the same form as the BDFRM phase self inductance expressions shown in (A.8). As noted in Appendix A the SYNCREL and BDFRM inductances can be directly related to pseudo d and q axis inductances of a SYNCREL, and the above observations mean that the angles of the rotor for the two machines can also be considered as equivalent, since the self inductances vary in the same manner with rotor movement. The standard torque expression [6] for the SYNCREL is briefly derived in Appendix F.1 as: where: T esy = 3 p p(l d L q )i d i q (1.13) i the magnitude of the current vector = = 3 4 p p(l d L q )i sin θ i (1.14) i d + i q θ i the angle of the current vector relative to the d axis in elec rads. Now let us consider the torque expression for the BDFRM. In order to carry out the comparison between the BDFRM and the SYNCREL we need to convert the torque expression for the BDFRM (1.1) into a form that involves the inductances of the machine and the d and q currents. The expression for the BDFRM primary flux space vector in a stationary reference frame can be shown to be [5]: λ ps = L p i ps + L ps i s s e jθr (1.15) where the variables are defined in Appendix A. We need to convert this expression into the primary reference frame rotating at ω p rad/sec. Using the normal

14 1. Maximum Torque per Secondary Ampere 7 frame conversion expressions for space vectors [7] we can write the following expressions for the stationary frame vectors: Substituting into the primary flux expression we get: λ ps = λ pr e jθp (1.16) i s s = ĩ s r e jθp (1.17) λ pr = L p i pr + L ps ĩ s r e jθr (1.18) Since ĩ s r = i s s e jθp then ĩ s r e jθr = i s s e j(θr θp). In addition a condition that was in force when the expressions were derived was ω r = ω p + ω s, which implies that θ r = θ p + θ s. Therefore θ s = θ r θ p. Hence the current expression may be written as: ĩ s r e jθr = i s s e jθs = ( i ss e jθs) = i sr (1.19) Hence the primary flux vector expression may be written as: λ pr = L p i pr + L ps i s r (1.0) Remark 1.7 The primary current vector is in a frame rotating at ω p. Note that i s r vector is also a vector rotating at ω p with respect to the stationary frame, despite the fact that it appears to be the complex conjugate of the secondary frame vector. It just so happens that with the frame transformation used the dq components of this vector are the same as those for the i sr vector of the secondary equation (which rotates at ω s ). As a consequence of this the is r vector has a DC value in steady state. Remark 1.8 The choice of the reference frame rotating at ω p in the above expressions leads to expressions for both the primary and secondary equations that have DC values in steady state. However, it should be noted that the primary and secondary equations are in two different frames to achieve this. The primary equation is in the ω p reference frame, and the secondary equation is in the ω s reference frame. If the ω p frame is aligned with λ pr then there can be no imaginary components in (1.0). Writing (1.0) in Cartesian form we have: λ pr = λ pd + jλ pq = L p i pd + L ps i sd + j(l p i pq L ps i sq ) (1.1) and neglecting the imaginary component (because of the frame alignment) we can write 1 : λ pr = λ pd = L p i pd + L ps i sd = λ p (1.) Substituting this expression into the torque expression of (1.1) we can write: T ebm = 3 p L ps r (L p i pd + L ps i sd )i sq L p [ ] = 3 p r L ps i pd i sq + L ps i sd i sq L p (1.3) 1 Note that i sd in the following equation is in the ω s reference frame, and not the ω p reference frame.

15 1. Maximum Torque per Secondary Ampere 8 Using the expression of (A.4) and assuming the relationships in (1.5) and (1.6), we can write this expression as: T ebm = 3 [ p Ld L q r i pd i sq + (L d L q ) ] (L d + L q ) i sdi sq (1.4) Now since: p r = p pr + p s = pole pairs of the rotor = p p we can write the expression as: T ebm = 3 p p [(L d L q )i pd i sq + (L d L q ) ] i sd i sq (1.5) L d + L q which uses the same variables as the expression for the SYNCREL torque in (1.13). Remark 1.9 The first part of (1.5) has exactly the same form as the SYN- CREL expression of (1.13). Therefore the second term results in additional torque. The second term in (1.5) can be written as: (L d L q ) ( ) ξ 1 i sd i sq = (L d L q )i sd i sq (1.6) L d + L q ξ +1 and hence the BDFRM torque expression becomes: T ebm = 3 ( ) ] ξ 1 p p(l d L q )i sq [i pd + i sd (1.7) ξ +1 where ĩ d = i pd + = 3 p p(l d L q )ĩ d i sq (1.8) ( ) ξ 1 ξ+1 i sd. Remark 1.10 The term ξ 1 ξ+1 that appears in the torque equation is the normalised maximum torque per ampere for the BDFRM. It is also the power factor for the ideal SYNCREL. Forming a ratio of the BDFRM and SYNCREL torques assuming that i pd = i sd = i d and i sq = i q we get: [ ( ) ] 3 T ebm p p (L d L q )i d i q + ξ 1 ξ+1 (L d L q )i d i q = T 3 esy p =1+ ξ 1 p(l d L q )i d i q ξ +1 T e bm = ξ (1.9) T esy ξ +1 Remark 1.11 The presence of the second winding produces a larger pseudo ĩ d current for the BDFRM, allowing about 75% more torque to be produced from the BDFRM (for a typical value of ξ =7) for the same current in the secondary winding as the SYNCREL (i s = i) and with the same equivalent effective turns per pole for all the windings.

16 1. Maximum Torque per Secondary Ampere 9 Remark 1.1 Note that while i pd = i d (remember that these are in different reference frames) the current magnitude of the BDFRM primary winding is somewhat less compared to that in the SYNCREL winding (i p <i). This results from the fact that i pq = Lps L p i sq = L d L q L d +L q i sq = ξ 1 ξ+1 i q which follows from (1.1) due to λ pq =0(because of frame alignment). Therefore, ( ) ξ 1 i p = i pd + i pq = i d + ξ i ξ +1 +ξcos θ i +1 q = i ξ +1 For the maximum torque per ampere current angle of the SYNCREL (θ i = π/4), the above expression can be reduced to: ξ +1 i p = ξ +1 i If ξ =7, for instance, then i p 0.9i. Remark 1.13 The above comparison is not very fair to the SYNCREL as the BDFRM has twice the total number of turns and almost twice copper losses. A better comparison would involve ensuring that the same amount of active material is used in both the machines and seeing what the comparative torque levels are. In order to carry out a fairer comparison with the SYNCREL we shall assume that the copper losses in both the machines are equal. In order to make this comparison sensible we shall make the following assumptions: 1. The SYNCREL and the BDFRM have the same frame size and are wound with the same gauge wire.. The primary and secondary windings for the BDFRM are wound in the same stator slots. 3. Iron losses in the machines are not accounted for in the development below. We shall now assume that the number of turns per pole for both the primary and secondary windings of the BDFRM is half that for the SYNCREL winding: n p = n s = 1 n sy (1.30) where: n p amplitude of the primary winding function = N p primary turns/phase/pole p pr n s amplitude of the secondary winding function = N s secondary turns/phase/pole p s n sy amplitude of the SYNCREL winding function = N sy SYNCREL turns/phase/pole p p See [5] for details on this.

17 1. Maximum Torque per Secondary Ampere 10 Remark 1.14 The above turns condition ensures that the BDFRM and the SYNCREL windings have the same amount of copper in them. This is shown in Appendix B. Remark 1.15 The above condition also means that the slots of the BDFRM and the Syncrel would be roughly the same size. It also implies that the core maximum flux densities would be roughly the same for the same amount of iron in the stator and the same rotor. Under these conditions it was shown in Appendix B that for the same copper losses in both machines the following relationship must exist between the corresponding currents: (βp pr + π )i p +(βp s + π )i s =(βp p + π)i sy (1.31) where β the machines aspect ratio = l/d, wherel the stack length of the machines, and d the diameter of the machine from the centre of the stator slots (for further discussion of this see Appendix B). In order to proceed further with this analysis we need the torque expression in an appropriate form. Consider (1.1) and substitute in for the flux using (1.) (repeated in here for convenience) we can write: T ebm = 3 p L ps r (L p i pd + L ps i sd )i sq (1.3) L p In addition, from (1.1) we can also establish the following relationship since the imaginary part of the flux must be zero (λ pq = 0) because of frame alignment: L p i pq = L ps i sq (1.33) We can state the dq currents in terms of the amplitudes of the currents: i pd = i p cos α p (1.34) i pq = i p sin α p (1.35) i sd = i s cos α s (1.36) i sq = i s sin α s (1.37) Substituting these into (1.33) gives: L ps = i p sin α p (1.38) L p i s sin α s if i s sin α s 0. These expressions allow us to write the torque expression as: T ebm = 3 p L ps r (L p i p cos α p + L ps i s cos α s )i s sin α s L p [ = 3 (ip ) ] p sin α p r L p i p i s cos α p sin α s + L ps i s i s sin α s L cos α s sin α s p = 3 p r(l p i p cos α p sin α p + L ps L p i s cos α s sin α s ) T ebm = 3 4 p rl p (i p sin α p + L ps L i s sin α s ) (1.39) p

18 1. Maximum Torque per Secondary Ampere 11 Remark 1.16 Equation (1.39) has a very similar form to the SYNCREL torque expression in (1.14). Remark 1.17 If one first looks at (1.39) one is tempted to say that α p = α s = π 4 will maximise the torque produced by the machine. However, the primary and secondary current magnitudes and angles are inter-related and we are not free to arbitrarily choose the angles independently of the currents. This is due to the frame alignment and constant flux condition. As we shall see the α p = α s = π 4 condition DOES NOT lead to the maximum torque per ampere for the machine in any sense. To make a preliminary comparison of the torque production of the BDFRM and the SYNCREL we shall assume that the current angles are both equal to π/4. Using the α s = α p = π/4 condition (1.39) can be written as: T ebm = 3 4 p rl p (i p + L ps L i s) (1.40) p Substituting using (1.33) we can write: T ebm = 3 p rl p i p (1.41) Now consider (1.31) and again using (1.33) to substitute for i s we can write after a small amount of manipulation: i p = (βp p + π)l ps (βp pr + π )L ps +(βp s + π i )L sy (1.4) p Substituting this expression into (1.41) we can write: [ ] T ebm = 3 p (βp p + π)l ps rl p (βp pr + π )L ps +(βp s + π i )L sy p [ ] (βp p + π)l p L ps =3p p (βp pr + π )L ps +(βp s + π i )L sy (1.43) p using p r =p p. We now need to get the SYNCREL torque expression into an appropriate form to enable comparison with the BDFRM expression. Using (1.14) and setting θ i = π/4 for maximum torque we can write: Applying (A.4) we can further write: and hence: T esy max = 3 4 p p(l d L q )i sy (1.44) L d L q =L pssy (1.45) T esy max = 3 p pl pssy i sy (1.46) where L pssy denotes the BDFRM equivalent mutual inductance for the SYN- CREL winding (remember that it is different than the BDFRMs).

19 1. Maximum Torque per Secondary Ampere 1 Remark 1.18 The L ps = L d L q equivalence holds for both cosinusoidal and twin air-gap approximations for the air-gap between the rotor and the stator. Remark 1.19 Note that (1.45) follows from (A.4) by assuming that the number of turns/pole for the BDFRM windings is equal to that in the SYNCREL (n p = n s = n sy ). If one wants to have the same amount of copper in the machines, then this is not the case according to (1.30). Due to the fact that the number of turns per pole on the BDFRM windings is actually half that on the SYNCREL winding then the inductances have the following relationship (refer to (A.4)): L pssy =4L ps (1.47) This relationship allows us to write the SYNCREL maximum torque in terms of the primary to secondary mutual inductances for the BDFRM: T esy max =6p p L ps i sy (1.48) We are now in a position to make the comparison between the torque produced by the two machines by forming a ratio of the two torque expressions: [ ] T r = T (βp p+π)l pl 3p ps p e bm (βp pr+ = π )L ps +(βps+ i π )L sy p T esy max 6p p L ps i (1.49) sy = (βp p + π )L pl ps (βp pr + π )L ps +(βp s + π )L p (1.50) Using (A.4) it can be established that: L p = 1 8 (L d sy + L qsy ) (1.51) L ps = 1 8 (L d sy L qsy ) (1.5) where L dsy and L qsy are the actual SYNCREL inductances in the d and q rotor axis respectively. Substituting into the torque ratio expression and simplifying gives: T r = 1 ξ 1 ξ + β(ps ppr) β(p ξ +1 (1.53) pr+p s)+π where ξ = L d /L q = L dsy /L qsy. For a typical 6/ pole BDFRM design (p s = 3,p pr = 1) and a 4-pole SYNCREL (p p = ) assuming β = 0.5 theabove expression becomes: T r = 1 ξ 1 ξ (1.54) +0.39ξ +1 The plot of this function for various ξ s is shown in Figure 1.3. If we reverse the BDFRM poles so that p s =1andp pr = 3 then we get: T r = 1 ξ 1 ξ 0.39ξ +1 We have also plotted this in Figure 1.3. (1.55)

20 1. Maximum Torque per Secondary Ampere Prim poles > Sec poles Sec poles > Prim poles Saliency Ratio Figure 1.3: Torque ratio of 6/ pole BDFRM and the 4 pole SYNCREL with the same copper losses, active copper and current angles of π/4. Remark 1.0 It should be emphasised once again here that L d,q are the SYN- CREL inductances for n p = n s = n sy, whereas L dsy,q sy are for the case when the two machines have equal amount of copper (n p = n s = n sy /). Despite the fact that L dsy,q sy =4L d,q, the corresponding saliency ratios are however the same, as stated above. Remark 1.1 If ξ then T r 1 regardless of the pole numbers and machine dimensions. For obtainable values of ξ (around6to7typically),t r 0.5 for the machine designs considered previously (see Figure 1.3). Remark 1. The above T r value has important implications. If one assumes a maximum input frequency of ω s = ω p = ω e (the SYNCREL input frequency) the output mechanical angular velocity of the BDFRM is the same as that for the SYNCREL, then the output power of the machine is approximately half that of the SYNCREL with the same stator copper losses and active copper. This fact shall be confirmed in Section 1.5, and has been previously eluded to in the experimental results obtained in [8]. Therefore the price one pays for having a half sized inverter (compared to that required for the SYNCREL or the induction machine) is that the machine has to be larger to produce the same power. However, the situation is not quite as simple as I have outlined here. We will consider the situation in more detail in Section 1.5. Remark 1.3 The primary winding input voltage to the BDFRM, if wound as assumed in the above analysis, would have to be approximately one quarter that

21 1.3 Maximum Torque Per Total Ampere 14 of the SYNCREL. This is due to the fact that the halving the number of turns in the windings lowers the inductances by a factor of four. Remark 1.4 Note that the above comparison has been made for α s = α p = π/4, and as stated previously this is not the maximum torque per ampere angle for the machine. However the analysis does simplify the comparison and gives one some feel for the comparative torque production for the BDFRM compared to the SYNCREL. Remark 1.5 Note that the p pr > p s curve indicates larger torques over a range of saliency ratios compared to the p s >p pr curve because the losses in the secondary circuit are lower for a given current. Therefore the secondary current can be increased (and hence the torque) to achieve the same copper losses. Note that the increase in the secondary current is not directly reflected in exactly the same change in i p (because of the below unity secondary to primary coupling), hence the lower overall losses. Remark 1.6 It should be noted that the analysis that led to the T r ratio implicitly assumes that the primary flux linkage is variable. This is due to the fact that α s = α p = π/4 and i s =(L p /L ps )i p. Under the alignment conditions for the equations we have λ p = L p i pd + L ps i sd, which is clearly variable as i pd and i sd vary together. This fact leads to these results being at variance with other similar results later in the report, since these other results have the flux constant. 1.3 Maximum Torque Per Total Ampere Another measure of the maximum torque per ampere is to calculate the machine torque per total amperes flowing into the machine i.e. to maximise the torque per (i s + i p ). In order to make the following analysis as independent as possible of specific machine parameters we shall use the normalisations defined in Appendix G.9. The problem we are trying to solve can be stated as follows: min i tn = i pn + i sn = ζ sin α p +sinα s α s,α p sin(α s + α p ) (1.56) Remark 1.7 One is tempted to think that one simply has to minimise i tn over α s and α p in (1.56). However, α s and α p are not independent due to the flux and frame alignment conditions. Therefore the minimisation can be simplified to a one variable minimisation. In order to minimise the above expression we need to find the relationship that exists between the current angles. Let us therefore find the normalised expressions for the d and q axis currents for the primary and secondary windings. Using (G.45) we can write: i sqn = ζt n (1.57)

22 1.3 Maximum Torque Per Total Ampere 15 We also know from the frame alignment condition (G.4) that: therefore: ζ = L pn L psn = 1 L psn = i sqn i pqn (1.58) i pqn = i sqn ζ = T n (1.59) The normalised flux for the BDFRM is: λ pn = L pn i pdn + L psn i sdn (1.60) Realising that λ pn = L pn =1andL psn =1/ζ we can rearrange this as: i sdn = ζ(1 i pdn ) (1.61) Now taking the ratio of (1.57) and (1.61) we can write: i sqn i sdn = i sn sin α s i sn cos α s = ζt n ζ(1 i pdn ) (1.6) which can be simplified to give: i pdn = tanα s T n tanα s (1.63) We can then use this expression together with (1.59) to form a ratio as follows: which simplifies to: i pqn i pdn = i pn sin α p i pn cos α p = T n tanα s T n tanα s (1.64) or: tan α p = α p =tan 1 [ T n tan α s tanα s T n (1.65) Tn tan α s tanα s T n ] (1.66) Equation (1.65) is the fundamental relationship that must exist between the current angles, and it is this that allows us to convert the expression for i tn to be in one variable. We can manipulate (1.56) as follows: ζ sin α p +sinα s i tn = sin α s cos α p +sinα p cos α s = ζ tan α p cos α s + tan αs cos α p tan α s +tanα p (1.67)

23 1.3 Maximum Torque Per Total Ampere 16 Now substituting (1.65) we get after a little manipulation: i tn = ζt n + cos αs cos α p ( tan α s T n ) sinα s = ζt n sinα s + tanα s T n tanα s cos α p (1.68) where α p is as in (1.66). Equation (1.68) can further be manipulated using: sin α s = cos α p = tan α s 1+tan α s (1.69) 1 1+tan α p (1.70) and (1.65) to give an expression for i tn totally in terms of α s : i tn = ζt n 1+tan α s + ( tan α s T n ) + Tn tan α s (1.71) tanα s One can take the derivative of (1.71) with respect to tan α s, andthenequate the result to zero and solve for the tan α s that will minimise i tn. However, in practise the resultant derivative expression is very complex and can not be solved analytically. Therefore the optimal angle was calculated using a numerical minimisation procedure implemented in Matlab R. A copy of the program that generated the following plots can be found in Appendix H.3. In addition to plotting the optimal current angles for the primary and secondary, plots for the total current, torque per total ampere, and the real and reactive powers for the primary and the secondary appear. All the values have been calculated for a set of normalised torques ranging from 0.1pu to 1.5 pu. Note that the secondary frequency, ω sn, is equal to one for all the plots shown. The ω sn value makes a difference to the secondary winding power plots. Notice also that the plots are for the case where the primary and secondary windings have the same inductance (L p = L s ), and the equivalent SYNCREL inductance ratio ξ is eight (see Figures A. and A.3 for the relationship between the L d /L q ratio and ζ using the two air-gap modelling approach). From (A.76) it then follows that ζ =9/7. Another variable used in the program in Appendix H.3 is the coupling coefficient k ps. The relationship between the inductances of the mutually coupled system of the BDFRM inductances can be written as (using standard transformer theory): L ps = k ps Lp L s (1.7) The normalised form of this expression can be written as: L psn = k ps Lpn L sn (1.73) k ps = L psn Lpn L sn (1.74)

24 1.3 Maximum Torque Per Total Ampere Alpha [Rad] s p Normalised Torque [PU] Figure 1.4: α s and α p for maximum torque per (i pn + i sn ). We also have the relationship of (1.58) which allows us to write: ζ = L pn L psn = L pn k ps Lpn L sn = using L pn = 1 (because of the normalisation definition). In the special case of L pn = L sn =1thenwehave: 1 k ps Lsn (1.75) ζ = 1 k ps (1.76) For any particular machine the inductances L p, L s and L ps are known from the machine design. Therefore k ps and ζ can be determined using the above expressions. Figure 1.4 shows the angles for the maximum-torque-per-total-amperes (MTTA). Remark 1.8 One interesting feature of Figure 1.4 is that α s does not change in value as much as α p. In addition the values of α s are larger angles as compared to α p for all the torque levels. Therefore changes in the secondary current magnitude, i sn, will produce significant changes in i sqn. This makes sense since the torque is produced by the i sqn component. Remark 1.9 Note also from Figure 1.4 that α s does not approach π/, indicating that the secondary current is making a significant contribution to the primary flux via mutual coupling. However, if the coupling between the primary and secondary winding becomes less, then the secondary current does not make

25 1.3 Maximum Torque Per Total Ampere 18 Total Current Magnitude [PU] Normalised Torque [PU] Figure 1.5: i pn + i sn for maximum-torque-per-total-ampere current angles. as large a contribution to the primary flux. Therefore, it has less effect in lessening the primary flux producing current. Consequently α s tends more towards π/ under these conditions for all torque levels, since this angle will mean that the secondary current will be more directly contributing to torque output, without affecting the current into the primary too much. Figure 1.5 shows the total current flowing into the machine versus the torque output of the machine. Remark 1.30 Note from Figure 1.5 that if the torque doubles the total current flowing into the machine does not double. Therefore the torque/total ampere is not constant for different required output torques. This is different from the SYNCREL where the maximum-torque-per-ampere is a constant for a particular machine. Figure 1.6 presents the individual current magnitudes for the currents flowing in the machine under the same conditions as Figure 1.5. This plot shows that the change in the total current is largely due to changes in the secondary current magnitude. Since the secondary current angle is much larger than the primary current angle, then variations in the secondary current magnitude will reflect more in the q-axis current. As can be seen from (1.1), the q-axis current is responsible for the torque production, therefore it is the secondary current that mostly changes with changing torque demands. Figure 1.7 shows the maximum torque-per-total-amperes for the BDFRM for various torque levels. Notice that the maximum-torque-per-total-ampere

26 1.3 Maximum Torque Per Total Ampere Primary and Secondary Currents [PU] i pn i sn Normalised Torque [PU] Figure 1.6: i pn and i sn for maximum-torque-per-total-amperes angles Torque/Total Amps Normalised Torque [PU] Figure 1.7: Maximum-torque-per-total-amperes for various torque levels.

27 1.3 Maximum Torque Per Total Ampere 0 increases considerably as the required torque increases. This is due to the fact that the flux producing current is a fixed overhead in the primary current loading and does not by itself produce any torque. The machine has to run with constant flux due to the primary winding connection to the supply grid. When the torque demand increases most of the increased current is related to increasing the torque, and therefore the flux producing current becomes a smaller percentage of the total current. Figure 1.8 shows the real and reactive powers for the machine under the MTTA condition. Note that the real powers for both the primary and secondary windings are the same, and therefore both windings are equally sharing the real power being supplied. Remark 1.31 At low levels of torque the reactive power being supplied by the secondary is small. This is because only a small value of i sqn is required here to produce the torque, and therefore it is better to supply the necessary magnetising current directly from the primary, rather than from the secondary where one produces the flux inefficiently (in terms of the current required) via mutual coupling. As the torque level increases the i sqn value required increases. However, this value is with respect to the secondary reference frame which does not lie on the secondary flux axis. Therefore the i sqn current has a significant component along the λ s vector, and contributes significantly to the secondary reactive power. The final plots for the MTTA situation are the power factor plots for the primary and secondary windings. Figure 1.9 shows the variation of the primary power factor over the torque range. The power factor for the primary improves as the torque level increases since the secondary starts to take over more of a share of the magnetising current. From Figure 1.10 it can be seen that the secondary current power factor varies only a comparatively small amount Comparative Analysis with the SYNCREL The purpose of this section is to compare the maximum torque per total ampere (MTTA) performance of the BDFRM and the SYNCREL counterpart assuming equal copper losses and the same amount of active copper in both the machines. The following analysis shall be therefore carried out under the same conditions as that in Section Remark 1.3 Note that the MTTA for the SYNCREL is actually the maximum torque per ampere (MTPA) as the machine has a single winding. Using the base values definitions introduced in the Appendix G, the SYN- CREL maximum torque expression (1.48) in the normalised form can be written as : T nsy max = 4 ζ i n sy (1.77) where i nsy is the SYNCREL current magnitude in per unit (pu). The relationship (1.31) between the BDFRM and the SYNCREL winding currents that allows the machines to have the same losses has identical form in

28 1.3 Maximum Torque Per Total Ampere Real and Reactive Power [PU] Q pn P pn P sn Q sn Normalised Torque [PU] Figure 1.8: Real and reactive powers under the maximum-torque-per-totalamperes condition with the secondary winding being supplied at the mains frequency (ω sn =1).

29 1.3 Maximum Torque Per Total Ampere Primary Power Factor Normalised Torque [PU] Figure 1.9: Primary power factor under maximum torque per total ampere conditions Secondary Power Factor Normalised Torque [PU] Figure 1.10: Secondary power factor under maximum-torque-per-total-ampere conditions.

30 1.3 Maximum Torque Per Total Ampere Torque ratio Prim. poles < Sec. poles Prim. poles > Sec. poles BDFRM torque [pu] Figure 1.11: MTTA torque ratio between the 6/ pole BDFRM and 4-pole SYNCREL having the same copper losses pu and is reproduced here for convenience : (βp pr + π )i pn +(βp s + π )i sn =(βp p + π)i n sy (1.78) Considering a 6/ pole BDFRM and 4-pole SYNCREL designs with the same aspect ratio of β 0.5, then from the previous expression it follows that : i nsy = 0.4i pn +0.6i sn (p pr =1,p s =3) (1.79) i nsy = 0.6i pn +0.4i sn (p pr =3,p s =1) where i pn and i sn are the normalised currents of the primary and secondary BDFRM winding defined by (G.5) and (G.6) respectively. Using (1.77) and (1.79) and assuming ζ = 9/7 (corresponds to ξ = 8), one is now able to generate relevant BDFRM vs SYNCREL optimal torque performance plots from the MTTA results for the BDFRM presented in Figures 1.6 and 1.7. These plots are shown in Figures 1.11, 1.1 and It can be seen from Figure 1.11 that the BDFRM torque producing capabilities are significantly inferior to the SYNCREL s over the entire torque range despite the much higher total input current levels (Figure 1.1). The lower torque production of the BDFRM as compared to the SYNCREL can be explained by the smaller inductances (since the number of turns per pole of the BDFRM windings is half that in the SYNCREL) and relatively poor magnetic coupling between the windings which is inherent with this machine. This fact

31 1.3 Maximum Torque Per Total Ampere Current ratio Prim. poles < Sec. poles Prim. poles > Sec. poles BDFRM torque [pu] Figure 1.1: MTTA total input current ratio between the 6/ pole BDFRM and 4-pole SYNCREL with equal copper losses MTTA ratio Prim. poles < Sec. poles Prim. poles > Sec. poles BDFRM torque [pu] Figure 1.13: MTTA ratio between the 6/ pole BDFRM and 4-pole SYNCREL having the same copper losses.

32 1.4 Power 5 along with the greater total input currents results in the further MTTA reduction of the BDFRM relative to the SYNCREL as shown in Figure Another interesting observation from Figure 1.11 is that the BDFRM can develop more torque at lower torque levels with the two-pole winding being used as a primary winding. The situation is however totally opposite at torques above about 1-pu. This is consistent with the corresponding input current behaviour shown in Figure 1.1. The current magnitudes are obviously much higher at lighter loads in the case of a two-pole primary winding. Remark 1.33 At unity pu torque for the BDFRM, the SYNCREL is capable of producing almost twice as much torque with half the current into the machine compared to the BDFRM. The MTTA value of the BDFRM is consequently about one quarter that of the SYNCREL. Remark 1.34 A comparison of Figure 1.11 with Figure 1.3 seems to indicate an inconsistency. The T r from Figure 1.3 is greater than that from Figure This arises due to the fact that Figure 1.3 was generated by allowing λ p in the machine to vary. Therefore the losses in the BDFRM and the SYNCREL vary together with changing current magnitude. 1.4 Power In order to get some preliminary understanding of the power relationships for the BDFRM one is to look at the performance of various variables of the BDFRM under constant torque and frequency operation (i.e. constant power). These plots are also generally instructive when one is attempting to understand the operation of the machine Constant Torque Operation This section shall develop a number of plots for the situation where both the output torque and the primary and secondary input frequencies are constant. The equations developed will use the normalisations in Appendix G. Rewriting equation G.44 here for convenience: T nbm = sin α s sin α p +sin α p sin α s sin (α p + α s ) = sinα s sin α p sin(α s + α p ) (1.80) (1.81) If we choose a value for α s then we can solve for α p so that some desired torque (T ndesired ) is achieved by solving for the zero of: F (α p )=sinα s sin α p T ndesired sin(α p + α s ) = 0 (1.8) As a result one will have the α s and α p that will give the desired torque. Remark 1.35 Note that the constant flux condition for the primary winding is automatically taken care of in (1.8) since the normalised torque was derived by assuming that λ p was a constant. Therefore the solution to (1.8) is implicitly a constrained solution, since we are solving F (α p )=0subject to λ p = constant.

33 1.4 Power 6 Once we have α s and α p then we can solve for the other important characteristics of the machine the real and reactive powers and power factors for the primary and secondary windings, the individual winding currents and their dq components, and finally the total current magnitude flowing into the machine (i.e. the addition of the primary and secondary winding magnitudes). Remark 1.36 The secondary power parameters are affected by the secondary frequency being applied. The primary is not affected because the primary frequency is fixed by the grid supply frequency. As we shall see if the frequency of the secondary winding becomes greater than the frequency of the primary, the real power being supplied by the secondary becomes greater than the primary. The solution of (1.8) and the subsequent plotting was carried out in a Matlab R script which appears in Appendix H.. Before examining the plots there is an interesting point in relation to the solution range (1.8). The Matlab R fzero function has difficulty solving (1.8) at the point where the secondary winding is carrying all the magnetising current. This point can easily be found from equation (1.65), repeated here for convenience: tan α p = T n tan α s (1.83) tanα s T n Note that the denominator becomes zero when: tan α s = T n (1.84) or α s =tan 1 T n (1.85) and α p = π/. Notice from (1.63) that this angle means that there is no d- axis primary current, and therefore the secondary winding would be carrying all of the magnetising current for the machine. By careful programming in the Matlab R script one is able to get correct solutions around this critical point using (1.65). Remark 1.37 As we shall see later a more succinct, accurate, and simpler Matlab R formulation can be devised. This formulation does not involve using fzero and therefore avoids the above problem. Remark 1.38 We are able to generate solutions for (1.65) when α s < tan 1 Tn (the critical angle ). Note that this means that the secondary winding is supplying reactive power to the machine which is in turn being used to supply reactive power to the network (i.e. negative reactive power is being input into the primary). Remark 1.39 Most of the plots below are for the case when T n =1and ω sn = 1. Therefore the shaft speed of the BDFRM is the same speed as that for a SYNCREL being fed with the same frequency. However, in terms of the general usage of this machine this is probably not the normal situation, since the machine is often operated with low values of ω sn so that the amount of power that the inverter has to handle is small. This situation will be considered when we start to look at the effects of varying ω sn.