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Analysis of Electric Machinery and Drive Systems Editor(s): Paul Krause, Oleg Wasynczuk, Scott Sudhoff, Steven Pekarek
15 INTRODUCTION TO THE DESIGN OF ELECTRIC MACHINERY 15.1. INTRODUCTION The majority of this text concerns the analysis of electric machinery and drive systems The focus of this chapter is the use of these concepts for design In particular, the design of a permanent-magnet ac machine is considered In doing this, the prevalent design approach based on design rules coupled with detailed numerical analysis and manual design iteration is not used Instead, the machine design problem is posed in a rigorous way as a formal mathematical optimization problem, as in References 1–3 The reader is forewarned that the approach has been simplified Structural issues, thermal issues, and several loss mechanisms are neglected, and infinitely permeable magnetic steel is assumed, though saturation is considered Even so, the design problem is nontrivial and provides an organized and systematic approach to machine design This approach may be readily extended to include a wide variety of design considerations The machine design problem is made easier if given context To this end, our problem is to design a three-phase, wye-connected, permanent-magnet ac machine to * produce a desired torque Te* at a desired mechanical speed ω rm It is assumed that the Analysis of Electric Machinery and Drive Systems, Third Edition Paul Krause, Oleg Wasynczuk, Scott Sudhoff, and Steven Pekarek © 2013 Institute of Electrical and Electronics Engineers, Inc Published 2013 by John Wiley & Sons, Inc 583 584 Introduction to the Design of Electric Machinery bs-axis Magnetically Inert Region Stator Backiron q-axis Rotor Backiron φr φs Tooth as-axis Slot Shaft Permanent Magnet Air Gap cs-axis d-axis Figure 15.1-1. Surface-mounted permanent-magnet synchronous machine inverter driving the machine is current controlled as discussed in Sections 12.8–12.11, and operated from a dc bus voltage of vdc It is desired to minimize mass, to minimize loss, to restrict current density (since it is closely related to winding temperature), to avoid heavy magnetic saturation, and to avoid demagnetization of the magnet Figure 15.1-1 illustrates a diagram of a two-pole version of the machine (we will consider a P-pole design) The phase magnetic axes are shown, as well as the q- and d-axis The stator is broken into two regions, the stator backiron and the slot/tooth region The rotor includes a shaft, a magnetically inert region (which could be steel but need not be), a rotor backiron region, and permanent magnets Arrows within the permanent magnet region indicate the direction of magnetization Also shown in Figure 15.1-1 is the electrical rotor position, θr, position measured from the stator, ϕs, and position measured relative to the rotor ϕr Since a two-pole machine is shown, these angles are identical to their mechanical counterparts, θrm, ϕsm, and ϕrm for the device shown Again, our approach will be to formulate the design problem as a formal optimization problem Hence, our goal will be to predict machine performance based on a geometrical machine description The work will proceed as follows First, Section 15.2 will set forth the details of the geometry Next, the winding configuration will be discussed in Section 15.3 Needed material properties will be outlined in Section 15.4 The current control philosophy will be delineated in Section 15.5 At this point, attention will turn to finding an expression for the radial flux density of the machine in Section 15.6, and a derivation of expressions for the electrical parameters of the Machine Geometry 585 machine in Section 15.7 The implications of the air-gap field on the field within the steel and permanent magnet is addressed in Section 15.8 As this point, the primary analytical results required for the machine design will have been put in place Thus, in Section 15.9, the formulation of the design problem is considered Section 15.10 provides a case study in multiobjective optimization-based machine design Finally, Section 15.11 discusses extensions to the approach set forth herein 15.2. MACHINE GEOMETRY Figure 15.2-1 illustrates a cross-section of the machine As can be seen, the machine is divided into regions Proceeding from the exterior of the machine to the interior, the outermost region of the machine is the stator backiron, which extends from a radius of rsb to rss from the center of the machine In this region, flux enters and leaves from the teeth and predominantly travels in the tangential direction The next region is the slot/ tooth region, which contains the stator slots and teeth and stator conductors as discussed in Chapter The slot/tooth region extends from rst to rsb The next region is the air gap, which includes radii from rrg to rst Proceeding inward, the permanent magnet includes points with radii between rrb and rrg and consists of one of two types of material, either a permanent magnet that will produce radial flux, or a magnetically inert spacer that may be air (as shown) The rotor backiron extends from rri to rrb Flux enters and leaves the rotor backiron predominantly in the radial direction; but the majority of the flux flow through the rotor backiron will be tangential It serves a purpose similar to the stator backiron The inert region (radii from rrs to rri) mechanically transfers torque from the rotor backiron to the shaft It is often just a continuation of the rotor Stator Backiron Tooth g Rotor Backiron rsb rst Slot rrg rss d st rrb di rrs dm rri 2pα pm P d rb Shaft Permanent Magnet Air Gap d sb Magnetically Inert Region Figure 15.2-1. Dimensions of surface-mounted permanent-magnet synchronous machine 586 Introduction to the Design of Electric Machinery backiron (possibly with areas removed to reduce mass) or could be a lightweight composite material Material in this region does not serve a magnetic purpose, even if it is a magnetic material Variables depicted in Figure 15.2-1 include: dsb—the stator backiron depth, dst—the stator tooth depth, g—the air-gap depth, dm—the permanent magnet depth, drb—the rotor backiron depth, di—the magnetically inert region depth, and rrs—the rotor shaft radius The active length of the machine (the depth of the magnetic steel into the page) is denoted as l The quantity αpm is the angular fraction of a magnetic pole occupied by the permanent magnet All of these variables, with the exception of the radius of the rotor shaft, rrs, which is assumed to be known, will be determined as part of the design process In terms of the parameters identified in the previous paragraph, the following may be readily calculated: rri—the rotor inert region radius, rrb—the rotor backiron radius, rrg—the rotor air-gap radius, rst—the stator tooth inner radius, rsb—the stator backiron inner radius, and rss—the stator shell radius A stator shell, if present, is used for protection, mechanical strength, and thermal transfer It will not be considered in our design Figure 15.2-2 depicts a portion of the stator consisting of one tooth and one slot (with ½ of a slot on either side of the tooth) Variables depicted therein which have not been previously defined include: Ss—the number of stator slots, θtt—the angle spanned by the tooth tip at radius rst, θst—the angle spanned by the slot at radius rst, rsi—the radius to the inside tooth tip, θti—the angle spanned by the tooth at radius rsi, θtb—the angle spanned by the tooth at radius rsb, wtb—the width of the tooth base, dtb—the depth of the tooth base, dtte—the depth of the tooth tip edge, and dttc—the depth of the tooth tip center at θt /2 For the purposes of design, it will be convenient to introduce the tooth fraction αt and tooth tip fraction αtt The tooth fraction is defined as the angular fraction of the slot/tooth region occupied by the tooth at radius rst Hence, rsb θ st rst rst π / Ss dttc wtt d st wtb θ tt / θt / θ tb / rsi dtb θ ti / dtte Figure 15.2-2. Slot and tooth dimensions rss Machine Geometry 587 αt = Ssθ t 2π (15.2-1) The tooth tip fraction is herein defined as the angular fraction of the slot/tooth region occupied by the tooth tip at radius rst It is defined as α tt = Ssθ tt 2π (15.2-2) As previously noted, not all the variables in Figure 15.2-1 and Figure 15.2-2 are independent, being related by geometry One choice of variables sufficient to define the geometry is given by T G x = [rrs di drb dm g dtb dttc dtte α t α tt dsb α pm l P Ss φss1 ] (15.2-3) where a “G” is used to denote geometry and the subscript “x” serves as a reminder that these variables are considered independent Note that we have not discussed the last element of Gx, namely ϕss1, in this chapter; it is the center location of the first slot as discussed in Chapter Given Gx, the locations of the slots and teeth may be calculated using (2.2-8) and (2.2-9); next, the remaining quantities in Figure 15.2-1 and Figure 15.2-2 can be readily calculated as rri = rrs + di (15.2-4) rrb = rri + drb (15.2-5) rrg = rrb + dm (15.2-6) rst = rrg + g (15.2-7) rsi = rst + dttc (15.2-8) θ t = 2πα t / Ss (15.2-9) θ tt = 2πα tt / Ss (15.2-10) θ st = 2π / Ss − θ tt (15.2-11) wtb = 2rst sin(θ t / 2) (15.2-12) wtt = 2rst sin(θ tt / 2) (15.2-13) rsb = (wtb / 2)2 + (rst cos(θ t / 2) + dtb + dttc )2 (15.2-14) w θ tb = 2a sin tb 2rsb (15.2-15) w θ ti = 2a sin tb 2rsi (15.2-16) dst = rsb − rst (15.2-17) 588 Introduction to the Design of Electric Machinery rss = rsb + dsb (15.2-18) Another geometrical variable of interest, although not shown in Figure 15.2-2, is the slot opening, that is, the distance between teeth This is readily expressed as θ wso = 2rst sin st 2 (15.2-19) In addition to the computing the dependent geometrical variables, there are several other quantities of interest that will prove useful in the design of the machine The first of these is the area of a tooth base, which is the portion of the tooth that falls within rsi ≤ r ≤ rsb and is given by atb = wtb dtb + ( )) rsb θ rsbθ tb − wtb cos tb 2 − ( )) rsi θ rsiθ ti − wtb cos ti 2 (15.2-20) The area of a tooth tip, which is the material at a radius rst ≤ r ≤ rsi from the center of the machine, may be expressed as wtt dtte + (wtb + wtt )(rsi cos(θ ti / 2) − rst cos(θ tt / 2) − dtte ) att = (15.2-21) − rst θ tt + rst wtt cos (θ tt / ) + rsi θ ti − rsi wtb cos (θ ti / ) The slot area is defined as the cross-sectional area of the slot between radii rsb and rsi This area is calculated as aslt = π 2 (rsb − rsi ) − atb Ss (15.2-22) The total volume of all stator teeth, vst, the back iron, vsb, and the stator laminations, vsl, may be formulated as vst = Ss (att + atb )l (15.2-23) sb vsb = π (r − r )l (15.2-24) vsl = vst + vsb (15.2-25) ss The total volume of the rotor backiron, denoted as vrb, rotor inert region, vri, and permanent magnet, vpm, are readily found from 2 vrb = π (rrb − rri )l (15.2-26) 2 vri = π (rri − rrs ) l (15.2-27) v pm = π (r − r )α pm l (15.2-28) rg rb Machine Geometry 589 wttR dttR wstR d wR wsiR d siR wtbR Figure 15.2-3. Rectangular slot approximation For purposes of leakage inductance calculations, it is convenient to approximate the slot geometry as being rectangular, as depicted in Figure 15.2-3 There are many ways such an approximation can be accomplished One approach is as follows First, the width of the tooth tip is approximated as the circumferential length of the actual tooth tip wttR = rstθ tt (15.2-29) The depth of the rectangular approximation to the tooth tip is set so that the tooth tip has the same cross sectional area In particular, dttR = att wttR (15.2-30) Next, the width of the slot between the stator tooth tips is approximated by circumferential distance between the tooth Thus wstR = rstθ st (15.2-31) The width of slot between the base of the tips is taken as the average of the distance of the chord length of the inner corners of the tooth tips at the top of the tooth and the chord distance between the bottom corners of the teeth This yields π θ π θ wsiR = rsi sin − ti + rsb sin − tb Ss Ss (15.2-32) Maintaining the area of the slot and the area of the tooth base, the depth of the slot (exclusive of the tooth tip) and width of the tooth base are set in accordance with dsiR = aslt wsiR (15.2-33) wtbR = atb dsiR (15.2-34) 590 Introduction to the Design of Electric Machinery Note that this approach is not consistent in that it does not require wsiR + wtbR = wttR + wstR However, this does not matter in the primary use of the model—the calculation of the slot leakage permeance as discussed in Appendix C The final parameter shown in Figure 15.2-3 is the depth of the winding within the slot, dwR This parameter will not be considered a part of the stator geometry, but rather as part of the winding Before concluding this section, it is appropriate to organize our calculations in order to support our design efforts In (15.2-3), we defined a list of “independent” variables that define the machine geometry, and organized them into a vector Gx Based on this, we found a host of related variables which will also be of use It is convenient to define these dependent variables as a vector G y = [rri rrb rrg rst rri θ t θ tt wtb wtt rsb θ tb θ ti dst rss wso aslt att atb vst vsb vsl vrb vri v pm wttR dttR wstR wsiR dsiR wtbR ]T (15.2-35) where again “G” denotes geometry and the “y” indicates dependent variables We may summarize our calculations (from 15.2-4 to 15.2-34) as a vector-valued function FG such that G y = FG (G x ) (15.2-36) This view of our geometrical calculations will be useful as we develop computer codes to support machine design, directly suggesting the inputs and outputs of a subroutine/ function calls to make geometrical calculations Finally, other calculations we will need to perform will require knowledge of both Gx and Gy; it will therefore be convenient to define G = [ GT x GT ] y T (15.2-37) 15.3. STATOR WINDINGS It is assumed herein that the conductor distribution is sinusoidal with the addition of a third harmonic term as discussed in Section 2.2, and given, in a slightly different but equivalent form, by (2.2-12) In particular, the assumed conductor density is given by P P * * nas (φsm ) = N s1 sin φsm − α sin φsm 2 (15.3-1) 2π P P * * nbs (φsm ) = N s1 sin φsm − − α sin φsm 2 3 (15.3-2) 2π P P * * ncs (φsm ) = N s1 sin φsm + − α sin φsm 2 3 (15.3-3) Stator Windings 591 * * where N s1 and α are the desired fundamental amplitude of the conductor density, and ratio between the third harmonic component and fundamental component, respectively The goal of this chapter is to design a machine that can be constructed, which means that we need to specify the specific number of conductors of each phase to be placed in each slot To this end, we can use the results from Section 2.2 Using (2.2-24) in conjunction with (15.3-1)–(15.3-3) yields * * N s1 P π P α3 3Pπ 3P N as,i = round sin φss,i sin − sin φss,i sin 2 Ss Ss P (15.3-4) * * 2π π P α N s1 P 3P 3Pπ N bs,i = round sin φss,i − sin S − sin φss,i sin S s s P (15.3-5) * * 2π π P α N s1 P 3P 3Pπ N cs,i = round sin φss,i + sin S − sin φss,i sin S s s P (15.3-6) where Nas,i, Nbs,i, and Ncs,i are the number of conductors of the respective phase in the i’th slot and where ϕss,i denotes the mechanical location of the center of the i’th stator slot, which is given by (2.2-8) in terms of ϕss,1, which is the location of the center of the first slot This angle takes on a value of if the a-phase magnetic axis is aligned with the first slot or π/Ss if it desired to align the a-phase magnetic axis with the first tooth The total number of conductors in the ith slot is given by N s,i = N as,i + N bs,i + N cs,i (15.3-7) For some of our magnetic analysis, we will use the continuous rather than discrete description of the winding Once the number of conductors in each slot are computed using (15.3-4)–(15.3-6), then from (2.2-12), (15.3-1), and (2.2-20), the effective value of Ns1 and α3 are given by N s1 = α3 = − π Ss ∑N as ,i cos (φss,i ) (15.3-8) i =1 π N s1 Ss ∑N as ,i cos (3φss,i ) (15.3-9) i =1 It is also necessary to establish an expression to describe the end conductor distribution The end conductor distribution for each winding may be calculated in terms of the slot conductor distribution using the methods of Section 2.2 In particular, repeating (2.2-25) for convenience, the net end conductor distribution for winding “x” is expressed 592 Introduction to the Design of Electric Machinery M x ,i = M x ,i −1 + N x ,i −1 (15.3-10) Using (15.3-10) requires knowledge of the net number of end conductors Mx,1 on the end of tooth This, and the number of cancelled conductors in each slot, Cx,i (see Section 2.2), determines the type of winding (lap, wave, or concentric) For the purposes of this chapter, let us take the number of canceled conductors to be zero and require the end winding conductor arrangement to be symmetric in the sense that for any end conductor count over tooth i, the end conductor count over the diametrically opposed tooth (in an electrical sense) has the opposite value Mathematically, M x ,i = − M x ,Ss / P +i (15.3-11) From (15.3-10) and (15.3-11), it can be shown that (Problem 4) M x ,1 = − Ss / P ∑N x ,i (15.3-12) i =1 Thus, once the slot conductor distribution is known, (15.3-12) and (15.3-10) can be used to find an end conductor distribution The distribution chosen herein corresponds to a concentric winding Note that (15.3-12) can yield a noninteger result In this case, minor alterations to the end conductor arrangement can be used to provide proper connectivity with an integer number of conductors In addition to the distribution of the wire, it is also necessary to compute the wire cross-sectional area To this end, the concept of packing factor is useful The packing factor is defined as the maximum (over all slots) of the ratio of the total conductor cross-sectional area within the slot to the total slot area, and will be denoted by αpf Typical packing factors for round wire range from 0.4 to 0.7 Assuming that it is advantageous not to waste the slot area, the conductor cross-sectional area and diameter may be expressed as asltα pf N s max ac = dc = (15.3-13) ac π (15.3-14) where ‖NS‖max denotes the maximum element of the vector NS If desired, ac and dc can be adjusted to match a standard wire gauge In this case, the gauge selected should be the one with the largest conductor area that is smaller than that calculated using (15.3-13) Finally, it will be necessary to compute the depth of the winding within the slot for the rectangular slot approximation This may be readily expressed as dwR = N s max ac α pf wsiR (15.3-15) 608 Introduction to the Design of Electric Machinery Br , S r Bi , Si Bt1 , St1 Bt , St Figure 15.8-3. Flux density in rotor backiron Now let us consider the field intensity in the permanent magnet Our interest is computing the minimum (most negative) field intensity in the positively magnetized region Combining (15.6-15) with (15.6-25) in a region of positive magnetization yields rr Fm, pk + Fs − Br r Rpm + Rg H= µ0 (1 + χ m ) (15.8-22) Assuming that Fm, pk + Fs > 0, it follows that the minimum field intensity occurs where Fs is a minimum and where the radius is a maximum Thus H mn rr Fm, pk + Fs,mn − Br rrg Rpm + Rg = µ0 (1 + χ m ) (15.8-23) where the minimum stator flux density over the positively magnetized region of the permanent magnet can be shown to be − Fs, pk − α pm ≤ mod (2 + 2φi / π , ) ≤ + α pm (15.8-24) Fs,mn = min(Fst1, Fst ) otherwise i At this point, it is convenient to define the functional dependence of our field calculations These calculations require no inputs beyond those previously defined The output of the analysis of this section is T F = [θ rm Bt1c Bb1s Brbt ,mx Brbr ,mx H mn ] (15.8-25) The calculations of this section may thus be organized as F = FF (M, G, W, I) (15.8-26) Formulation of Design Problem 609 15.9. FORMULATION OF DESIGN PROBLEM At this point, we are ready to consider the design of the machine There are many ways to formulate the design problem Herein, an approach will be given that is readily tailored to a specific application and can be readily expanded to include a large variety of considerations We will specifically consider the problem of designing a machine to * produce a positive torque Te* at a mechanical speed ω rm using an inverter with a given dc bus voltage, vdc Our approach will be to formulate the design problem as an optimization problem in which we attempt to minimize mass and loss, and then to use an optimization algorithm to arrive at the best choice of parameters Design Space An important early step in the design process is to identify the parameters we are free to choose We will use θ to denote the vector of free design parameters One possible choice of a parameter vector is T q = [ st rt ct mt G x Wx I x ] (15.9-1) Using this choice, every other quantity of the machine can be calculated However, it is often the case that some of these parameters are fixed prior to the initiation of the design process For our purposes, it is assumed that the shaft radius rsh is known Other variables that we will consider to be fixed include the number of poles P and the number of slots Ss Generally speaking, if one neglects frequency dependent losses, increasing the number of poles of the machine will result in decreased mass and loss Since we have not included these effects here, we will fix this parameter The number of slots should be an integer multiple of the number of phases times the number of poles so that the machine can be constructed in a symmetric fashion We will also assume a packing factor αpf, end winding offset leo, and the location of stator slot 1, ϕss1, to be fixed In order to simplify the design problem further, we will assume that we will not have tooth tips Thus, dttc = dtte = 0 and θtt = θt This further reduces the design space In our design process, we will also require the mass be less than mlim and have a power less than Plim We will base our magnetic analysis on J rotor positions Let us denote the vector of fixed design parameters as D, where * D = [ vdc Te* ω rm P Ss α pf le rsh v fs J φss1 mlim Plim ] T (15.9-2) With the variables associated with D set a priori, and in the absence of a tooth tip, the parameter vector that we must find becomes where T ˆ ˆ q = st rt ct mt G x Wx I x (15.9-3) 610 Introduction to the Design of Electric Machinery T ˆ G x = [ di drb dm g dtb α t dsb α pm l ] (15.9-4) * * T ˆ Wx = [ N s1 α ] (15.9-5) and Note that even with the fixed parameters, our design is left with four discrete variables (the material types) and 13 continuous variables for a total of 17 degrees of freedom Design Metrics While there are many possible design metrics of interest, herein we will focus on two—mass and loss The mass may be readily expressed as m = vsl ρs + vrb ρr + v pm ρm + 3vcd ρc (15.9-6) We will consider two components of the loss First, we will consider the resistive loss in the machine, which is readily expressed as Pr = Rs I s2 (15.9-7) Next, we will consider the semiconductor conduction loss As an estimate of the loss, we will treat both the diode and transistor as having a forward voltage drop of vfs With this assumption, it can be shown that the total semiconductor power loss in the three phase bridge is given by Ps = v fs I s π (15.9-8) The sum of these two loss components is given by Ploss = Ps + Pr (15.9-9) We will attempt to select the parameter of the machine to minimize the total mass m and the power loss Ploss, subject to operating constraints These constraints are our next topic Design Constraints In order to ensure proper operation of the machine, we will enforce a number of constraints on the design As an example, one constraint will be that the machine must produce the desired torque Let ci denote a variable that describes whether the i’th constraint is satisfied We will formulate ci in such a way that 0 ≤ ci ≤ 1, and that if Formulation of Design Problem 611 ci = 1, the constraint will be said to be satisfied In order to evaluate the constraint variable, we will define less-than-or-equal-to and greater-than-or-equal-to functions as x ≤ xmx lte( x, xmx ) = x > xmx 1 + x − xmx x ≥ xmn gte( x, xmn ) = x < xmn 1 (15.9-10) (15.9-11) + xmn − x Both of these functions have an appropriate range, and evaluate to if their respective constraints are satisfied As their respective constraints become further from being satisfied, they tend toward zero Our first two constraints will be related to the machine geometry and structural issues First, it is desired to keep the length of the teeth reasonable compared with their width Thus, we will require that c1 = lte ( dst , α tar wtb ) (15.9-12) where αtar is the allowed tooth aspect ratio (depth to width) In order to make the machine easier to wind, we will next require that the wire diameter multiplied by a slot opening factor αso (which is greater than one) be less than the width of the slot opening wso Thus c2 = lte ( dcα so, wso ) (15.9-13) The next five constraints are related to the field analysis, and in particular to ensuring that we not overly saturate the material nor demagnetize the permanent magnet c3 = lte ( Bt1c max , Bs,lim ) (15.9-14) c4 = lte ( Bb1s max , Bs,lim ) (15.9-15) c5 = lte( Bbrt ,mx, Br ,lim ) (15.9-16) c6 = lte( Bbrb,mx, Br ,lim ) (15.9-17) c7 = gte( H mn, Hlim ) (15.9-18) Another constraint is that the current density within the wire does not exceed an acceptable value Thus we choose * c8 = lte( I s / ac, J c,lim ) (15.9-19) * where J c,mx is the maximum allowable rms current density for the material, mc In essence, this is a thermal limit, which, if rigorously treated, should be a function of 612 Introduction to the Design of Electric Machinery how well the machine conducts heat away from the winding However, the allowed current density will be treated as a constant herein Our next constraint will be that the desired torque is obtained The torque is readily calculated from the lumped parameter model From Chapter 4, torque may be expressed as Te = 3P r r r (λmiqs + ( Ld − Lq ) iqsids ) 22 (15.9-20) Assuming motor operation at positive speed, the torque constraint may be expressed as c9 = gte(Te, Te* ) (15.9-21) The next constraint we will consider is designed to ensure that there is adequate dc link voltage The q- and d-axis voltage at our operating point may be expressed as r r r * vqs = rsiqs + ω r ( Ld ids + λ m ) (15.9-22) r r r * vds = rsids − ω r Lq iqs (15.9-23) whereupon the peak line-to-line voltage may be expressed as v pk ,ll = r r ( vqs )2 + ( vds )2 (15.9-24) From our work in Chapter 12, the peak line-to-line voltage must be less that the dc voltage if the desired current is to be obtained This leads to the constraint c10 = lte(vll , pk , vdc ) (15.9-25) Our two final constraints will be designed to focus our attention to reasonable mass and loss Thus, we will constrain the mass and loss as c11 = lte(m, mlim ) (15.9-26) c12 = lte( Ploss, Plim ) (15.9-27) where mlim and Plim are limits on the allowed loss and mass, respectively Before concluding this section, it is convenient to define an aggregate constraint as ca = Nc Nc ∑c i (15.9-28) i =1 where Nc is the number of constraints, in this case 10 The aggregate constraint has properties similar to the individual constraints It has a range of [0,1], and is if all constraints are satisfied, and less than if not Formulation of Design Problem 613 Design Fitness The fitness of the design refers to a mathematical function to be maximized It is desired to create the function so that when maximized, the resulting design finds the extrema of the desired metrics subject to all constraints being satisfied In this case, the fitness will be a vector-valued function since we desired to minimize both mass and loss A suitable choice in fitness function is given by ε (ca − 1) [1 1]T T f (q, D) = 1 m Ploss ca < ca = (15.9-29) where the notation f(θ,D) is used as a reminder that although not explicitly indicated, the mass and constraints values are functions of the design parameters and design specifications, and where ε denotes a small positive number used for scaling purposes for plotting; its value does not affect the outcome of the optimization If constraints are not met, then both elements of the fitness will be equal to a small negative number As the constraints become closer to being satisfied, both elements of the fitness function will tend to zero (and will be equal) If all constraints are met, the elements of the fitness are the inverse of the mass and the inverse of the power loss Note that maximizing f(θ,D) does not yield a single design, but rather a set of designs known as the Pareto-optimal front The designs on this front have the property that it is impossible to improve one metric without sacrificing the other In particular, maximizing f(θ,D) yields a set of designs, where each design in the set has the property that loss cannot be decreased without increasing mass Before concluding this section, let us consider the problem of evaluating the fitness function—something that is clearly required for optimization Given θ as defined by (15.9-3) and the fixed parameters D given by (15.9-2), then the material types st, rt, ct, and mt, as well as all parameters associated with Gx, Wx, and Ix are known Using st, rt, ct, and mt, the material parameter vectors S, R, C, and M are readily determined using (15.4-1)–(15.4-4) Based on the parameters in Gx, the remaining geometrical parameters are calculated using (15.2-35) and (15.2-36), which encapsulate the calculations of Section 15.2, whereupon of the geometry vector G is known From the input variables to the winding variable vector Wx, the winding variable vector W is readily calculated using (15.3-20) and (15.3-21) using the expressions in Section 15.3 Using the geometry and winding parameters G and W, as well as material parameters C, and M, the parameters of the lumped parameter machine model E are readily calculated using (15.7-12) and the analysis of Section 15.6 The currents stored in Ix can then be used to calculate the current vector I using (15.5-8) and (15.5-9) and the analysis of Section 15.4 Finally, information in the field vector F may be computed in accordance with (15.8-26) and the analysis of Section 15.8 At this point, all information needed to evaluate the design metrics and constraints is available The fitness can thus be calculated by sequentially evaluating (15.9-6)–(15.9-9) and (15.9-12)–(15.9-29) 614 Introduction to the Design of Electric Machinery 15.10. CASE STUDY In this section, we will undertake a case study in machine design The design requirements and fixed parameters of this design are enumerated in Table 15.10-1 Our goal is to determine a set of designs that define the performance possibilities in the trade-off between loss and mass In order to solve this optimization problem, any optimization method can be used However, in this work, we will use a genetic algorithm Genetic algorithms use the principles of biological evolution to evolve a population of designs The reader is referred to [4, 5] for introductions to genetic algorithms The effectiveness to this technique relative to other for this class or problem is discussed in Reference When using genetic algorithms, the parameters are encoded into “genes.” In this problem, the public-domain open-source MATLAB-based GOSET package is used [6] The encoding process consists of mapping the parameter values to normalized numbers between and 1, where corresponds to the minimum allowed parameter value and corresponds to the maximum allowed parameter value Thus, a minimum and maximum value for each parameter must be specified In addition, each parameter has a type indicating how it is mapped; denotes integer values that are linearly mapped to the range, denotes real numbers, but that are also linearly mapped, and denotes real numbers that are mapped in an logarithmic fashion Each parameter is assigned to a gene, and each gene is assigned to a chromosome (though only one chromosome is used herein) The ranges and types of all parameters used in this example are listed in Table 15.10-2 The ranges are determined by experience and engineering judgment Fortunately, only crude estimates of the ranges are needed The final step needed to perform the design using a genetic algorithm is to select a population size and the number of generations, both of which are set to 3000 in this study Figure 15.10-1 illustrates the evolution of the design Consider Figure 15.10-1a, which has an upper and lower trace The upper trace depicts the parameter distribution of the final population In this trace, each design is marked with a point in each of the 15 columns, each of which corresponds to a normalized parameter value (gene) The point placement along the width of the column is in order of the fitness of the first objective, in this case, mass; points on the left side of the column have relatively low fitness, while those at the right side have high fitness The distribution of points gives an indication of the sensitivity of fitness to a parameter within its range and the appropriateness of the parameter limits For example, if all point were clustered to the top of a column, it would indicate that the parameter range needs to be increased TABLE 15.10-1. Fixed Design Specifications Parameter vdc Te* * ω rm Ss P Value Parameter Value Parameter Value 400 V 20 Nm 2000 rpm 24 αpf leo rsh vfs J 0.5 1 cm 2 cm 2 V ϕss1 mlim Plim αtar αso π/24 25 kg 500 W 10 1.5 Case Study 615 TABLE 15.10-2. Parameter Ranges Parameter Description Max Gene 1 1 3 3 3 3 1 1 10−3 10−3 5·10−4 10−3 0.05 10−3 0.05 10−2 10 0.1 0.1 −50 4 10−1 5·10−2 5·10−2 10−2 5·10−2 0.95 5·10−2 0.95 0.5 103 0.7 50 10 11 12 13 14 15 16 17 Normalized Value 0.5 0.2 0.1 –0.1 0.5 10 11 12 13 14 15 16 17 Parameter Number best mean median 500 f:B(b) Md(g) Mn(r) Normalized Value f:B(b) Md(g) Mn(r) Min Stator steel type Rotor steel type Conductor type Magnet type Depth of inert region (m) Depth of rotor backiron (m) Magnet depth (m) Air gap (m) Depth of tooth base (m) Tooth fraction Depth of stator backiron (m) Magnet fraction Length (m) Pk fund cond density (cond/rad) Coefficient 3rd harmonic cond density q-axis current (A) d-axis current (A) st rt ct mt di drb dm g dtb αt dsb αpm l * N s1 * α3 r iqs r* ids Type 1000 1500 Generation (a) 2000 2500 3000 15 10 11 12 13 14 15 16 17 Parameter Number × 10–3 best 10 mean –5 median 500 1000 1500 Generation 2000 2500 3000 (b) Figure 15.10-1. Design evolution (a) Mass objective; (b) loss objective The lower trace depicts the fitness in terms of the first objective (reciprocal of mass) of the most fit individual versus generation, as well as the median and average inverse mass of the population versus generation Figure 15.10-1b is similar, although in this case, the fitness being considered is the second objective, in particular the reciprocal of the loss As previously described, the output of a multiobjective optimization is not a single design, but a set of designs referred to as a Pareto-optimal front For this case study, the Pareto-optimal front is shown in Figure 15.10-2 Therein, each “x” marks a specific design Each design is better than all the others in terms of either mass or loss; 616 Introduction to the Design of Electric Machinery Pareto-Optimal Front 500 450 400 Loss, W 350 300 250 200 150 100 50 10 12 14 16 18 20 22 24 26 Mass, kg Figure 15.10-2. Mass-loss tradeoff however, no design is better than any of the others in both mass and loss In optimization terms, the solutions are described as being “nondominated.” In terms of machine design, this set of designs defines the performance trade-offs between the competing objectives Again, each “x” in Figure 15.10-2 corresponds to an individual design The parameters and some performance data for machine design 1, which corresponds to the leftmost “x” in Figure 15.10-2 and is the design with the least mass, is given in Table 15.10-3 Note that the mass does not include the machine housing, and the losses not include core losses or windage However, the basic approach may be readily extended to include these effects There are many points of interest in Table 15.10-3 First, note that the conductor used is aluminum rather than copper This may surprise the reader in that copper in normally considered a better conductor However, this is only true on a volumetric basis On a per-mass-basis aluminum is a superior conductor This advantage is tempered, however, in that the increased volume requires an increase in steel to accommodate larger slots, so on some points on the front, copper is used Looking at the electrical model parameters, it is noted that the machine is slightly salient This is due to the fact that the permanent magnet has a susceptibility that is slightly larger than that of air In the operating point performance data, it is interesting that a small d-axis current is being used, suggesting a slight amount of flux weakening This is compatible with the fact that the peak line-to-line voltage is essentially equal to the maximum value allowed for the dc voltage of 400 V It is also interesting to observe that the stator tooth, stator backiron, and rotor tangential flux densities are all at the maximum value allowed Observe that the permanent magnet is not close to its demagnetization limit Case Study 617 TABLE 15.10-3. Machine Design from Pareto-Optimal Front Design data Outside diameter: 16.6 cm Total length: 11.5 cm Active length: 5.33 cm Number of poles: Number of slots: 24 Stator material type: M47 Rotor Material type: M36 Conductor type: Aluminum Permanent magnet type: SmCo17-R28 Permanent magnet fraction: 71.5% Permanent magnet depth: 0.896 cm Shaft radius: 2 cm Inert radius: 2 cm Rotor iron radius: 3.48 cm Air gap: 2.25 mm Slot depth: 2.38 cm Tooth fraction: 56.2% Stator backiron depth: 1.32 cm Rotor backiron depth: 1.48 cm Fundamental conductor density: 153 cond/rad 3rd harmonic conductor density: 36.5% Conductor diameter: 1.38 mm Stator Iron mass: 4.12 kg Rotor iron mass: 0.953 kg Conductor mass: 0.914 kg Magnet mass: 0.7 kg Mass: 6.69 kg a-phase winding pattern (1st pole): 0 18 49 49 18 Minimum conductors per slot: 67 Maximum conductors per slot: 67 Packing factor: 50% Electrical model Number of poles: Nominal stator resistance: 1.54 Ω q-axis inductance: 11.8 mH d-axis inductance: 12.1 mH Flux linkage due to PM: 489 mVs Operating point performance data Speed: 2000 rpm Frequency: 66.7 Hz q-axis voltage: 220 V d-axis voltage: −69.7 V Peak line-to-line voltage: 399 V q-axis current: 13.7 A d-axis current: −1.19 A Peak line current: 13.8 A Current density: 6.53 A rms/mm2 Torque: 20.1 Nm Semiconductor conduction loss: 52.6 W Machine resistive losses: 438 W Total loss: 490 W Machine efficiency: 90.6% Inverter efficiency: 98.9% Machine/inverter efficiency: 89.6% Stator tooth flux density/limit: 98.6% Stator backiron flux density/limit: 99.6% Rotor peak tangential flux density/limit: 98.8% Rotor peak radial flux density/limit: 85% Permanent magnet demagnetization/limit: 36.1% A cross section of the machine is depicted in Figure 15.10-3 Therein, the darkest region is the machine shaft The stator and rotor steels shapes are also indicated It is interesting that the machine used two different steels—M47 in the stator and M36 in the rotor The M47 has a higher saturation flux density that is particularly advantageous in the stator where the flux becomes concentrated in the teeth The M36 grade considered has a lower density that is advantageous in regions where the higher flux density is not needed 618 Introduction to the Design of Electric Machinery PMSM Radial Cross-Section 0.08 0.06 0.04 0.02 E –0.02 –0.04 –0.06 M47 M36 Aluminum –0.06 –0.04 –0.02 SmCo17-R28 0.02 0.04 0.06 0.08 Figure 15.10-3. Machine design cross section Figure 15.10-4 depicts the flux density in a tooth and in a backiron segment It is interesting that although the machine will have a sinusoidal back emf waveform, neither of the flux density waveforms is sinusoidal The tooth flux density waveform can be viewed as a trapezoidal contribution due to the permanent magnet added to a sinusoidal component due to the stator current Nevertheless, the approximately sinusoidal conductor distribution yields a machine with a sinusoidal back emf waveform (at least in terms of line-to-line measurement) It is interesting to compare this design with a second design from the front Table 15.10-4 lists the parameters and performance data from machine design 100 (the 100th machine design in order of increasing mass) This machine has significantly lower loss than the first machine (252 W vs 490 W), and significantly higher mass (9.71 kg vs 6.69 kg) Although many features of the design have changed, one of the most significant changes is an increase in wire diameter from 1.38 to 1.87 mm This lowers the machine resistance, but increases conductor mass from 0.914 to 1.64 kg and stator mass from 4.12 to 6.40 kg The increase in stator mass is in order to accommodate larger slots for the larger diameter wire The changes in magnet and rotor mass are less significant As a result of the increased conductor size, the current density drops from 6.53 to 3.32 A rms/mm2 15.11. EXTENSIONS The objective of this chapter was to provide an introduction to the design of permanentmagnet ac machines Although the approach considered was fairly detailed, several Acknowledgments 619 Backiron Flux Density, T Tooth Flux Density, T –1 –2 60 120 180 qr, Degrees 240 300 350 60 120 180 qr, Degrees 240 300 350 –1 –2 Figure 15.10-4. Flux density waveforms refinements are clearly necessary before using this approach for commercial designs Shortcomings include addressing stator core loss, incorporating thermal analysis, and addressing structural issues, to name a few Nevertheless, the general approach presented here may be readily extended to include these issues Indeed, some of these extensions are described in References 2, 3, and 7–10 Another straightforward extension is the consideration of multiple operating points This may be readily incorporated by requiring that all operational constraints be satisfied at every desired operating point and computing a weighted loss metric in terms of the individual operating points Finally, it should be observed that there are many configurations of permanent-magnet ac machines A discussion of many types, as well as other aspects that may be desirable to include in the design, such as core and proximity effect loss, are discussed in Reference 11 ACKNOWLEDGMENTS The authors would like to thank the U.S Office of Naval Research for supporting the development of this section as Grant N00014-08-1-0080 In addition, we would to thank the Grainger Center for Electromechanics for support of numerous machine design grants 620 Introduction to the Design of Electric Machinery TABLE 15.10-4. Machine Design 100 from Pareto-Optimal Front Design data Outside diameter: 18.3 cm Total length: 12.9 cm Active length: 6.58 cm Number of poles: Number of slots: 24 Stator material type: M47 Rotor material type: M47 Conductor type: Aluminum Permanent magnet type: SmCo27-R32 Permanent magnet fraction: 58.9% Permanent magnet depth: 0.866 cm Shaft radius: 2 cm Inert radius: 2.03 cm Rotor iron radius: 3.27 cm Air gap: 1 mm Slot depth: 3.68 cm Tooth fraction: 64% Stator backiron depth: 1.23 cm Rotor backiron depth: 1.23 cm Fundamental conductor density: 135 cond/rad 3rd harmonic conductor density: 36.1% Conductor diameter: 1.87 mm Stator iron mass: 6.4 kg Rotor Iron mass: 1.02 kg Conductor mass: 1.64 kg Magnet mass: 0.647 kg Mass: 9.71 kg a-phase winding pattern (1st pole) 0 16 43 43 16 Minimum conductors per slot: 59 Maximum conductors per slot: 59 Packing factor: 50% Electrical model Number of poles: Nominal stator resistance: 813 mΩ q-axis inductance: 12.3 mH d-axis inductance: 12.7 mH Flux linkage due to PM: 526 mVs Operating point performance data Speed: 2000 rpm Frequency: 66.7 Hz q-axis voltage: 219 V d-axis voltage: −67.5 V Peak line-to-line voltage: 397 V q-axis current: 12.7 A d-axis current: −2.19 A Peak line current: 12.9 A Current density: 3.32 A rms/mm2 Torque: 20 Nm Semiconductor conduction loss: 49.2 W Machine resistive losses: 203 W Total loss: 252 W Machine efficiency: 95.4% Inverter efficiency: 98.9% Machine/inverter efficiency: 94.3% Stator tooth flux density/limit: 99.4% Stator backiron flux density/limit: 99.9% Rotor peak tangential flux density/limit: 99.3% Rotor peak radial flux density/limit: 86% Permanent magnet demagnetization/limit: 42.4% REFERENCES [1] S.D Sudhoff, J Cale, B Cassimere, and M Swinney, “Genetic Algorithm Based Design of a Permanent Magnet Synchronous Machine,” 2005 International Electric Machines and Drives Conference, San Antonio, TX, May 15–18, 2005 [2] B.N Cassimere and S.D Sudhoff, “Analytical Design Model for Surface Mounted Permanent Magnet Synchronous Machines,” IEEE Trans Energy Conversion, Vol 24, No 2, June 2009, pp 338–346 Problems 621 [3] B.N Cassimere and S.D Sudhoff, “Population Based Design of Permanent Magnet Synchronous Machines,” IEEE Trans Energy Conversion, Vol 24, No 2, June 2009, pp 347–357 [4] J.H Holland, Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence, The MIT Press, Cambridge, MA, 1992 [5] K Deb, Multi-Objective Optimization Using Evolutionary Algorithms, John Wiley & Sons, Chichester, UK, 2001 [6] S.D Sudhoff, “Genetic Optimization Engineering Tool (GOSET).” Available at: https:// engineering.purdue.edu/ECE/Research/Areas/PEDS (accessed on January 21, 2013) [7] M Bash, S.D Pekarek, S.D Sudhoff, J Whitmore, and M Frantzen, “Comparing the Pareto-Optimal Fronts of Machine-Rectifier Systems,” Proceedings of the Electric Machinery Technology Symposium, Philadelphia, PA, May 19–20, 2010 [8] M Bash, S.D Pekarek, S.D Sudhoff, J Whitmore, and M Frantzen, “A Comparison of Permanent Magnet and Wound Rotor Synchronous Machines for Portable Power Generation,” 2010 Power and Energy Conference and Illinois, Urbana, IL, February 12–13, 2010 [9] J Krizan and S.D Sudhoff, “Modeling Semiconductor Losses for Population Based Electric Machinery Design,” 2012 Applied Power Electronics Conference, Orlando, FL, February 5–9, 2012 [10] J Krizan and S.D Sudhoff, “Theoretical Performance Boundaries for Permanent Magnet Machines as a Function of Magnet Type,” 2012 Power Engineering Society General Meeting, San Diego, CA, July 22–26, 2012 [11] D.C Hanselman, Brushless Permanent-Magnet Motor Design, McGraw-Hill, New York, 1994 PROBLEMS 1. Derive (15.2-4)–(15.2-18) 2. Derive (15.2-20) and (15.2-21) 3. Consider (15.2-32) Another approach that has been suggested [2] for this width is such that the reciprocal of wsiR is set to the average of the reciprocals of the chord lengths between inner corners of slot at the top of the slot (under the tooth tip) and the outer corners of the slot at the bottom of the tooth Derive an expression similar to (15.2-32) using this approach 4. From (15.3-10) and (15.3-11), derive (15.3-12) 5. Suppose round conductors are arranged in a grid such that center points are aligned with the vertices of the grid What is the highest packing factor that can be obtained? 6. Suppose that round conductors could be placed arbitrarily What is the theoretical limit on packing factor? 622 Introduction to the Design of Electric Machinery 7. Derive an expression analogous to (15.6-14) and (15.6-21) if the radial variation in the field is neglected 8. Derive (15.7-6) 9. Derive (15.7-7) and (15.7-8) 10. Assuming that the machine current is sinusoidal, derive (15.9-8) ... st (15. 2-31) The width of slot between the base of the tips is taken as the average of the distance of the chord length of the inner corners of the tooth tips at the top of the tooth and the. .. by the tooth at radius rsi, θtb? ?the angle spanned by the tooth at radius rsb, wtb? ?the width of the tooth base, dtb? ?the depth of the tooth base, dtte? ?the depth of the tooth tip edge, and dttc? ?the. .. that the reciprocal of wsiR is set to the average of the reciprocals of the chord lengths between inner corners of slot at the top of the slot (under the tooth tip) and the outer corners of the