Efficiency Optimization of PMSM Based DriveSystem

Waleed Hassan and Bingsen WangDepartment of Electrical and Computer Engineering

Michigan State University2120 Engineering Building

East Lansing, MI 48824Emails: hassanwa@msu.edu; bingsen@egr.msu.edu

Abstract—The proposed work in this paper is a combinationof analytical and numerical methods that are able to calculatethe harmonic losses of permanent magnet synchronous machine(PMSM) at any operating point with less execution time andadequate accuracy. Both the fundamental and harmonic lossesof the machine in conjunction with the inverter losses are modeledsuch that the inverter-motor efficiency can be tested for differentconditions. Numerical simulation of the field oriented controlhave been conducted and results include current, modulationindex, power factor, switching frequency and all the variablesin the drive system. Furthermore, Maximum efficiency controlstrategy has been chosen such that the drive system can work atthe most efficient condition at all operation points. The proposedwork is presented in the context of drive system that is basedpermanent magnet synchronous machine (PMSM). However, theproposed method can be extended to induction motor drivesystem or any motor use PWM in control of the inverter outputvoltage.

I. INTRODUCTION

Loss modeling and control of high performance motor drivesystem has attracted significant research attention [1]–[4]. Thereduced system losses not only can be translated to higherefficiency and lowered operating cost, but also can reducethe thermal stress on various components or sub-systems andin return will improve the reliability of overall system. Thereduction of life-cycle cost due to the improved reliability isof great practical significance in many of the cost-sensitiveapplications of the electric drive systems.

The efficiency and power density of permanent magnetsynchronous machines (PMSMs) have been studied by severalauthors [2], [4]–[6]. It has been shown that PMSM canachieve significantly higher efficiencies than induction motorswhen used in adjustable speed drives. Analytical methodfor estimating the fundamental losses of surface mountedPMSM has been proposed in [5]. Several researchers haveutilized electrical model of PMSM that includes a parallelresistance that accounts for core losses in high performanceapplications [2], [4]. It can be concluded that both copper andcore losses need to be accounted for in the analysis and controlof high performance motor drives. The core losses are due toboth the fundamental and harmonics excitations.

The existing optimization methods of the drive systemis focused on the fundamental losses of electric machines.The inverter losses are often not counted in the optimization

process although the modeling of inverter loesses is alreadyknown [7]. A study on the harmonics losses of transformershas been reported [8]. This paper proposes an integrativeapproach to optimize the efficiency of the drive that includingthe fundamental losses and the losses due to the switching ofthe inverter.

The rest of the paper is organized as the following. Sec-tion II provides analytical model of PMSM harmonic lossesbased on the armature reaction field produced by three-phase current of stator winding. Double Fourier series fornaturally sampled sinusoidal PWM (SPWM) has been utilizedfor evaluation and analysis. Section III presents analytical-numerical method of loss calculation for three-phase invertercontrolled by SPWM. The optimal-efficiency control strategyis implemented in section IV. The results and conclusion arepresented in section V.

II. HARMONICS LOSS MODELING OF PMSM

The core losses in the machine consist of two components,i.e., hysteresis and eddy current losses. Both types of core lossare due to time variation of the flux density in the core. Thecomplete expression for core losses in [W m−3] is:

Pcd = Ke

∞∑

n=1

(nωs)2B2

p,n +Kh

∞∑

n=1

nωsBβp,n (1)

where Ke is the eddy loss coefficient; Kh is the hysteresis losscoefficient; Bp,n is the peak flux density of the nth-order; andωs is fundamental angular frequency of applied voltage [9].

A. Harmonic Flux Density

In this section analytical model for surface mounted PMSMmotor harmonic loss is developed based on the work publishedin where the armature reaction field produced by the statorwinding is predicted. The armature reaction field produced bya three-phase stator winding is:

B3φ(α, r, t) = Bp sin(nωrt+ vα+ θn) (2)

where ωr = Ppωm is the electrical rotor angular speed in[rad s−1]; the amplitude of flux density Bp due to the nth-

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2012 IEEE 7th International Power Electronics and Motion Control Conference - ECCE Asia June 2-5, 2012, Harbin, China

978-1-4577-2087-1/11/$26.00@2012 IEEE

order armature reaction current is determined by:

Bp,n = µ02W

πδ

∑

n

In∑

v

1

vKsovKdpvFv(r) (3)

B. Eddy Current Loss

Once the maximum amplitude of the flux density has beendetermined, which results from the armature space vectorcurrent, the procedure of finding accurate model for core lossis shown in the following approximate expression.

Ped = Ke

∑

n

(nωr)2B2p,n (4)

Substituting (3) into (4) and defining the volume in which theloss is evaluated, the expression for eddy core loss as will bereached as

Pe = Kem

∞∑

n=1

(nωr)2I2n (5)

where ρi is the mass density of the steel core material in [kgm−3]; V is the steel core volume in [m3]; and their productgives the weight of the core material in [kg]. Kem is definedas:

Kem = (ρiV )Ke

(µ0

2W

πδ

)2(∑

v

1

vKsovKdpvFv(r)

)2

C. Hysteresis Loss

Following the same procedure as the last section, an expres-sion for hysteresis loss can be reached as presented in (6).

Ph = Khm

∞∑

n=1

nωrI2n (6)

where Khm is defined as:

Khm = (ρiV )Kh

(µ0

2W

πδ

)2(∑

v

1

vKsovKdpvFv(r)

)2

The final expression for the core loss in [W] is given by:

Pc = Pe + Ph = Kem

∞∑

n=1

(nωr)2I2n +Khm

∞∑

n=1

nωrI2n (7)

Close examination of Kem and Khm shows that they areconstant for a given machine design since they solely dependon the machine dimensions.

D. SPWM Harmonics Analysis

If the losses associated with fundamental is excludedfrom (7), the core harmonic loss can be expressed as:

Pc,h = Pe,h + Ph,h

= Kem

∞∑

n=2

(nωr)2I2n +Khm

∞∑

n=2

nωrI2n

(8)

It is clear that the motor harmonics loss can can be fullyevaluated if the amplitudes of current harmonics determined.

For three-phase inverter with two-level phase legs, a bal-anced set of three-phase line-line output voltages is obtained

if the references are displaced by 120. Under these conditions,the line to line voltages are given for naturally sampled PWMby [10]

vll =

√3Vdc2

M cos(ωot+ π/6)

+4Vdcπ

+∞∑

m=1

+∞∑

n=−∞n6=0

Jn(mπ2 M

)

msin

(m+ n)π

2sin

nπ

3× cos (mωct+ n(ωot− π/3))

(9)

The major significant side-band harmonics are at the frequencyof fh = fs ± 2f, fs ± 4f, 2fs ± 5f.

The surface mounted PMSM can be analyzed as a singlephase circuit that consists of series impedance and back emfsince the reactances on the d-axis and the q-axis are identical.At harmonic frequencies 3,6,9 stator resistance and back emfare neglected so that the motor modeled as synchronousinductance only. The motor phase current can be determinedby the phase voltage and the synchronous inductance of thestator winding. With the assumption of star connection ofPMSM, the phase current is given by:

I =vll/√3

ωLs(10)

Plugging equation (9) in equation(10) yields the harmonicsof the stator current. Once the stator harmonic current isdetermined, the harmonic losses are readily to be calculated.

III. LOSS OF VOLTAGE SOURCE INVERTER

The majority of harmonic losses of the motor, which iscaused by PWM carrier frequency, can be minimized byincreasing the inverter switching frequency. Nonetheless, theinverter switching loss will concurrently increase, as detailedin the subsequent section. Therefore, for achieving globaloptimization of the system, one needs to define the valueof the optimal frequency that minimizes the motor harmonicloss and inverter losses together. The aim of this section is toprovide analytical model for the calculation of power lossesin IGBT-based voltage source inverter used in PMSM drivesystem. The inverter parameters of the semiconductor switchesin the inverter have been extracted from the data sheets ofFAIRCHILD, RURG5060 and HTGT30N60A4 modules.

A number of different methods have been proposed to esti-mate the losses of voltage source inverters. The first is basedon the complete numerical simulation of the circuit by specificsimulation programs with integrated or parallel-running lossescalculations. The second is to calculate the electrical behaviourof the circuit based on analytical behaviour model [7]. Foraccurate estimation of the losses during the transient time andall operating conditions, a hybrid model has been developed. Inthis model, the operating conditions that are resulted from thenumerical simulation of the motor drive system are utilized inthe analytical model. The losses in a power-switching deviceconsist of conduction losses, switching losses and off-stateblocking loss.

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A. Switching Losses of VSI

The equation for the switching loss Pls of a VSI withsinusoidal ac line current and with IGBT switching devicesis given by:

Pls =6

π· fs · (Eon,I + Eoff,I + Eoff,D) ·

VdcVr· ILIr

(11)

where fs is the VSI switching frequency; Vdc is the dc linkvoltage; IL is the peak value of the ac line current that isassumed sinusoidal; Eon,I and Eoff,I are the turn-on and turn-off energies of the IGBT, respectively; Eoff,D is the turn-offenergy of the power diode due to reverse recovery current.

B. Conduction Losses of VSI

In contrast to the switching losses, the conduction losses aredirectly depending on the modulation function. For the carrierbased PWM method, the conduction losses of the IGBT andthe diode can be expressed as:

Plc,I =VCE,02π

· IL(1 +

M · π4· cos (φ)

)

+rCE,02π

·(π

4+M

(2

3· cos (φ)

)) (12)

Plc,D =VF,02π· IL

(1− M · π

4· cos (φ)

)

+rCE,02π

·(π

4−M

(2

3· cos (φ)

)) (13)

where M is the modulation index and φ is the displacementangle between the fundamental of modulation function and theload current. The total inverter losses at different operationpoints will be analyzed in the next subsection.

C. Total Losses of VSI

The total losses of the VSI have been evaluated based onthe parameters that are listed in Table I.

TABLE IPARAMETERS OF THE SEMICONDUCTOR DEVICES OF THE VSI.

Power Devices Parameters

IGBT Iref (A) 50 Vref (V ) 600Eon,I (mJ) 0.6 Eoff,I (mJ) 0.966VCE,0 (V ) 1.6 rCE,0 (mΩ) 15

Diode rF,0 (mΩ) 8 VF,0 (V ) 1.6Eoff,D (mJ) 0.7

A plot of the switching loss for a range of switching fre-quencies is shown in Fig. 1. It is clear that the switching loss isproportional to switching frequency and maintains constant fora given load current and switching frequency. The conductionloss depends on load displacement angle, modulation indexand load current. Fig. 2 illustrates the variations of the con-duction losses with the load displacement angle under differentloading conditions. The plot of conduction loss equation showsthat the loss is maximized when the displacement angle is zeroand decreases as the displacement angle increases or decreasesfrom zero. In field oriented control of variable speed drive

words, increasing the motor speed will increase the inverter losses since the current and

modulation index will be higher.

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 310

20

30

40

50

60

70

80

iL = 10 AiL = 9 A

iL = 8 A

104

Switching Frequency (Hz) ×

SwitchingPow

erLosses(W

)

Figure 3.1: Switching losses of VSI at different load currents.

-100 -80 -60 -20 -10 0 20 40 60 80 10024

26

28

30

32

34

36

MI=0.2, iL=8 A

MI=0.6, iL=9.3 A

MI=1, iL=10 A

Load Displacement Angle (degree)VSICon

ductionPow

erLosses(W

)

Figure 3.2: Conduction losses of VSI at different modulation indices.

21

Fig. 1. Switching losses of VSI at different load currents.

words, increasing the motor speed will increase the inverter losses since the current and

modulation index will be higher.

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 310

20

30

40

50

60

70

80

iL = 10 AiL = 9 A

iL = 8 A

104

Switching Frequency (Hz) ×

SwitchingPow

erLosses(W

)

Figure 3.1: Switching losses of VSI at different load currents.

-100 -80 -60 -20 -10 0 20 40 60 80 10024

26

28

30

32

34

36

MI=0.2, iL=8 A

MI=0.6, iL=9.3 A

MI=1, iL=10 A

Load Displacement Angle (degree)VSICon

ductionPow

erLosses(W

)

Figure 3.2: Conduction losses of VSI at different modulation indices.

21

Fig. 2. Switching losses of VSI at different modulation indices.

system, for a given speed command the controller will assignvoltage proportional to that speed by means of modulationaction. However, increasing the motor speed requires higherline to line voltage and causes the motor to draw higher currentfor constant load torque. In other words, increasing the motorspeed will increase the inverter losses since the current andmodulation index will be higher.

IV. LOSS MINIMIZATION CONTROL STRATEGY

The combined copper losses and iron fundamental lossesare minimized through control strategy developed in [4] Thecontrollable losses can be minimized by proper choice of thearmature current vector. It is a common practice to set thecurrent id = 0. As a result, the armature current vector that isin phase with the back EMF is applied. In addition, irreversibledemagnetization of permanent magnets can be avoided . Therecent development of the permanent magnets has broughtmaterials with high coercivity and high residual magnetism.Therefore, several control methods have been proposed toimprove the performance of the PM motor drives. In suchcontrol methods, the d-axis component of armature current isactively controlled according to the operating speed and loadconditions. In the next subsection, the basic equations of motormodel needed to develop the control algorithm are presented.

A. Modeling Fundamental Losses of PMSM

The steady state d, q model of PMSM is shown in Fig. 3.The fundamental iron loss consists of hysteresis loss and eddycurrent loss. Herein, these two loss components are lumped

1029

vrds

vrqs

Rs

Rs

voq

vodirds

irqs

iod

ioq

Rc

Rc

Ld

Lq

icd

icq

Lqωrioq

Ldωriod

ωrλaf

+

+

+

−

−

−

Figure 4.3: Dynamic equivalent circuit of PMSM that includes core resistance.

(4.3) : vrds

vrqs

= Rs

iod

ioq

+

(Rc + Rs

Rc

)

vod

voq

(4.1)

vod =

(Rc

Rc + Rs

)(vrds − Rs iod

)(4.2)

voq =

(Rc

Rc + Rs

)(vrqs − Rs ioq

)(4.3)

In equations (4.2) and (4.3), the output voltages are functions of the stator output currents

and voltages. The dynamic part of the circuit indicates that the output voltages are functions

of the output currents and the rotor speed, which is mathematically represented by:

vod

voq

=

Ld p −Lq ωr

Ldωr Lq p

iod

ioq

+

0

ωr λaf

(4.4)

where p denotes ddt ( ) . Rearranging of equation (4.4) leads to equations (4.5) and (4.6):

25

Fig. 3. Steady state equivalent circuit of PMSM including core lossresistance.

into a single quantity, which is represented by the core lossresistance Rc. As shown in Fig. 3, the voltage equations ofPMSM in steady-state are expressed as:

[vrdsvrqs

]= Rs

[IodIoq

]+Rc +RsRc

[vodvoq

]

[vodvoq

]=

[0 −Lqωr

Ldωr 0

] [IodIoq

]+

[0oλaf

] (14)

where

Iod = Irds − Icd; Ioq = Irqs − Icq

Icd = −(LqR c

)ωr Ioq; Icq =

1

Rc(λaf + Ld Iod)ωr

(15)

The armature current ia, the terminal voltage va and the torqueTe are expressed as:

ia =√i2d + i2q; va =

√v2d + v2q (16)

va =√

(Rsid − ωrLqioq)2 + (Rs iq + ωr(λaf + Ld iod))2

TE =3

2

P

2[λaf Ioq + (Ld − Lq) Iod Ioq]

(17)

The copper loss PCu and the iron loss PFe are determinedby:

PCu =3

2Rs

[(iod − oLqioq

Rc

)2+(ioq +

o(λaf+Ldiod)Rc

)2]

PFe =3

2

ω2r

Rc

[(Lqioq)

2 + (λaf + Ldiod)2]

(18)

B. Condition for Minimized Losses

The total electric power loss PE = PCu + PFe can beexpressed as a function of iod, Te and ωr. Under the steadystate, the electrical loss PE is function of iod. Nevertheless, forsurface mounted PMSM Ld = Lq . Hence, simple expressionfor PE can yield. By differentiation of PE with respect to iod,

Σ Σ

Σ

- -

-

θr

+ +

+

α, β

α, β

α, β

d, q

d, q

v∗sα

v∗sβ

isα

isβ

vr∗qs

vr∗ds

irqs

irds

dθrdt

PI

PIPI

PARK

TRANSFORMATION

TRANSFORMATION

CLARKE

TRANSFORMATION

ir∗qs

SPWM3-PHASE

INVERTER

INV.PARK

PMSM

isa

isb

Vdc

ω∗r

ωr

a, b

ir∗ds

LossMinimizationAlgorithm

Figure 6.3: Field orientation control diagram incldues loss minimization algorithm.

55

Fig. 4. Field oriented control diagram including loss minimization algorithm.

the condition for iod that minimizes the controllable lossesresults has been found.

Iod = −λaf (Rs +Rc)ω

2rLd

RsR2c + ω2

rL2d (Rs +Rc)

(19)

The role of the loss minimization algorithm in the fieldoriented control system is shown in Fig. 4. The simulationresults will be presented in the next section.

V. SIMULATION RESULTS AND DISCUSSION

The machine parameters of surface mounted PMSM thathave been used in the simulation are listed in Table II. Thefield oriented control system includes the loss minimizationalgorithm implemented in MATLAB/SIMULINK. The Motorharmonic losses of the machine are calculated by MATLABblock using the variables that is obtained from SIMULINK forvarious operation conditions. The converter losses are directlycalculated by use of SIMULINK system with parameters thatare generated by FOC subsystem. The results and discussionare presented in three subsections; FOC under loss minimiza-tion control strategy (LMCS), VSI-motor fundamental losses,and VSI-motor harmonic losses.

TABLE IIMACHINE PARAMETERS.

Parameter Value UnitsNumber of pole pairs 4Stator resistance 0.52 ΩCore loss resistance 450 ΩRated speed 4500 rpmRated torque 6 NmRotor flux linkages 0.08627 WbRated peak voltage per phase 150 VRotor inertia Jm 0.00036179 kgm2

Viscous friction coefficient Bm 9.444 × 10−5 NmsLd 0.0013 HLq 0.0013 H

A. FOC with Loss Minimization Control

The dynamic responses of currents, speed and torque atrated conditions are shown in Fig. 5 and Fig. 6, respectively.The system reaches the steady state with smooth dynamic

1030

linkage. Consequently, the voltage V , and the iron loss PFe, are smaller as compared with

−2

0

2

4

6

8

0 0.01 0.02 0.03 0.04 0.05 0.06−30

−20

−10

0

10

20

30

0−4

0.01 0.02 0.03 0.04 0.05 0.06

30

0.0600

5

10

15

20

25

0.01 0.02 0.03 0.04 0.05

ir qs(A

)ir ds(A

)is a

,is b

,is c

(A)

Time (sec)

irds

ir∗ds

ir∗qs

irqs

Figure 7.1: Dynamic response of phase currents, id and iq of surface mounted PMSM.

58

Fig. 5. Dynamic response of phase currents, irds and irqs of surface mountedPMSM.

response in approximate 0.14 second for the system parameterslisted in Table II. The current and speed controllers are de-signed by a symmetric optimum approach which is addressedin [4]. For calculation of the core loss resistance, it is requiredthat the motor runs at rated speed and torque of sinusoidalpower supply such that the harmonic losses are not generated.The fundamental iron loss PFe is calculated by subtracting themechanical and copper losses from the total output power. Theiron loss resistance, Rc, can be obtained from fundamentalcore losses as in (18), where the measured Rc representsthe iron loss resistance at the rated output power. PracticallyRc depends on operating conditions. However, it is assumedconstant in this work.

Fig. 7 shows the efficiency of surface mounted PMSM ver-sus speed for the two control strategies: zero id control strategyand (LMC) strategy. It is clear that the motor efficiencyunder maximum efficiency control strategy is higher than theefficiency under zero id control strategy. However, the differ-ence in efficiency is not as remarkable for surface mountedPMSM as for interior PMSM due to the following reasons.In interior PMSM The negative d-axis current produces apositive reluctance torque due to the saliency (Ld < Lq),and accordingly the stator current and the copper loss PCu issmaller compared with the id = 0 control. Moreover, in bothtypes of PMSMs the negative d-axis current reduces the fluxlinkage. Consequently, the voltage V , and the iron loss PFe,are smaller as compared with the id = 0 control strategy [4].Since the surface mounted PMSM has no saliency (Ld = Lq),there is no reluctance torque. However, the influence of thedecreased flux linkage and applied voltage is still effective.

the id = 0 control strategy [41]. Since the surface mounted PMSM has no saliency (Ld = Lq),

there is no reluctance torque. However, the influence of the decreased flux linkage and applied

voltage is still effective.

0 0.01 0.02 0.03 0.04 0.05 0.060

1000

2000

3000

4000

5000

0 0.01 0.02 0.03 0.04 0.05 0.060

2

4

6

8

10

12

14

16

Speed(rpm)

Torque(N

.m)

Time (sec)

Speed

Speed*

Te

Tl

Figure 7.2: Torque and speed dynamic response of surface mounted PMSM.

7.2 VSI and PMSM Fundamental Losses

As presented in Chapter 3, the losses of VSI are divided into two types; conduction losses

and switching losses. Equation (3.5) shows that the switching loss is proportional to switch-

59

Fig. 6. Torque and speed dynamic response of surface mounted PMSM.

ing frequency and motor phase current. Hence, if the motor is controlled under constant

switching frequency, the switching losses are proportional to motor phase current. However,

motor current increases/decreases for any increase/decrease in load torque or/and reference

speed. As a result, the switching loss increases/decreases for any increases/decreases in load

torque or/and reference speed. Alternatively, the conduction loss given in Equations (3.9)

and (3.10) is proportional to motor phase current, modulation index and the displacement

500 1000 1500 2000 2500 3000 3500 400083

84

85

86

87

88

89

90

91

92

93

4500

Speed (rpm)

Efficiency

(%)

at ir∗d = 0

at ir∗d given by

loss minimization algorithm

Figure 7.3: PMSM efficiency versus speed.

angle. The modulation index is controlled by load and speed since any increase in load or

speed requires higher current which is supplied via higher voltage. Therefore the modulation

index increases as well to supply the required voltage. Regarding the displacement angle, its

effect on conduction loss is discussed in Chapter 3. Figure 7.4 (a) illustrates the relationship

60

Fig. 7. PMSM efficiency versus speed.

B. VSI and PMSM Fundamental Losses

As presented in Section III, the losses of VSI are dividedinto conduction losses and switching losses. Since the switch-ing loss is proportional to switching frequency and motorphase current, the switching loss increases/decreases for anyincreases/decreases in load torque or/and reference speed. Inaddition, the conduction loss is proportional to motor phasecurrent, modulation index and the displacement factor. Themodulation index is controlled by load and speed since anyincrease in load or speed requires higher current which issupplied via higher voltage. Therefore the modulation indexincreases as well to supply the required voltage. Regarding thedisplacement angle, its effect on conduction loss is discussedin Section III.

Fig. 8(a) illustrates the relationship between motor funda-mental loss and ird under loss minimization control strategy.The curve is convex with global minimum loss point atird = −2.6A for the given parameter in Table II. Fig. 8(b)shows the relationship between the inverter losses and ird. Theresult shows that the minimum loss point occurs at ird = 0

1031

−20 −15 −10 −5 0 5 10 15 20

3

4

5

6

7

−20 −15 −10 −5 0 5 10 15 205

6

7

8

9

10

11

−20 −15 −10 −5 0 5 10 15 202

3

4

5

6

7

8x100

x10

x100

x:-3

y:223.9

x:-2

y:275.9

2

id reference (A)

id reference (A)

id reference (A)

PMSM

Fundam

entalLosses(W

)VSILosses(W

)Com

bined

Losses(W

)

(a)

(b)

(c)

Figure 7.4: Motor-inverter fundamental losses versus ir∗d at rated speed; (a) PMSM funda-

mental losses; (b) VSI losses; (c) Combined fundamental losses of PMSM and VSI.

61

Fig. 8. Motor-inverter fundamental losses versus ir∗d at rated speed: (a)PMSM fundamental losses; (b) VSI losses; (c) Combined fundamental lossesof PMSM and VSI.

. This result can be explained as following: the rms valueof the phase current is

√i2d + i2q . When id = 0, it is clear

that the phase current has the minimum value. However, itwas explained that most inverter loss is controlled by thecurrent. When the phase current is minimum the inverter losswill be minimum as well. Fig. 8(c) shows the relationshipbetween the combined losses of motor and inverter versusird. The relationship shows that the combined minimum lossvalue of inverter and motor occurs at a point of ird that liesbetween the zero ird and the optimal ird that is generated byloss minimization algorithm.

For all conditions of loads and speeds, it can be observedthat the inverter minimum loss value occurs at zero ird, and theird generated by loss minimization algorithm is always negativefor the same conditions. Moreover, the combined minimumloss value will strictly lie in the region between zero ird andird generated by minimum loss algorithm.

C. VSI and PMSM harmonic losses

The motor harmonic losses at any operating point canbe calculated as the following. First The machine operatesat rated speed and torque off sinusoidal power supply. Theinput power is then measured and the copper losses andoutput power are calculated, which gives motor efficiencyand fundamental core losses. The same procedure is repeatedexcept that the motor is supplied with SPWM-variable speeddrive controller under the same torque and speed conditions.The core losses that are calculated with SPWM represent thesummation of fundamental and harmonic losses at rated outputpower. Subtracting the sinusoidal-core losses from SPWM-core losses yields the motor harmonics loss at rated torqueand speed. At Te = 6 N ·m and N = 4500 rpm the calculatedharmonics loss is 90.8445 W. However, the motor efficiencyis reduced from 92.68 to 90 % when switching the sinusoidalsupply with SPWM-variable speed drive controller. In PMSMthe hysterics losses are very small and can be neglected. Inthis work the hysterics losses value is assumed to be 5% ofthe total harmonic losses.

Pe,h = 0.95Pc,h; Ph,h = 0.05Pc,h (20)

The next step is to calculate the coefficients Kem and Khm

:

Kem =Pe,h

∞∑m=1

∞∑n=−∞

(ωm,n|Im,n|ω 6=ω

)2

Khm =Ph,h

∞∑m=1

∞∑n=−∞

(ωm,n|Im,n|ω 6=ω

)2(21)

The calculated values are Kem = 3.8729 × 10−8 andKhm = 0.0013. The harmonic losses in [W] for any operatingcondition is:

Pc,h = 3.8729× 10−8∞∑

m=1

∞∑

n=−∞

(ωm,n|Im,n|ω 6=ω

)2

+0.0013∞∑

m=1

∞∑

n=−∞ωm,n

(|Im,n|ω 6=ω

)2 (22)

The motor harmonics loss, VSI loss and the total loss versusswitching frequency are shown in Fig. 9. The results showthat the PMSM harmonics loss decreases as the switchingfrequency increases. Nevertheless, the VSI losses increase asthe switching frequency increase. The summation of the abovetwo losses show that the curve is convex and has globalminimum loss value at the switching frequency of 19 kHz.However the specific optimal switching frequency will dependon particular set of design parameters of the VSI. In otherwords, when the parameters of the VSI such that the loss ofVSI higher than the motor harmonics loss at the same rangeof switching frequency, the curve will be no more convex andalmost will be linear.

Fig. 10 shows the relationship between PMSM harmonicloss and modulation index. It can be seen that up to M = 0.73

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50

52

54

56

58

90

92

94

96

98

100

102

80

143

144

145

146

147

x:1.9e+004

y:143.2

148

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

x 10

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

x 10

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

x 10

46

48

4

4

4

Switching Frequency (Hz)

Switching Frequency (Hz)

Switching Frequency (Hz)

PMSM

Harmon

icLosses(W

)VSILosses(W

)Com

bined

Losses(W

)

Figure 7.5: Harmonic losses of PMSM, VSI losses and the combined harmonic losses of

PMSM and VSI versus switching frequency at rated speed.

64

Fig. 9. Harmonic losses of PMSM, VSI losses and the combined harmoniclosses of PMSM and VSI versus switching frequency at rated speed.

the harmonics loss increases as the modulation index increases.As the modulation index passes M = 0.73, the harmonicsloss begins to decrease to locally minimum loss value at unitymodulation index. Because the loss is minimum near unitymodulation index, it is suitable to use variable dc link suchthat the modulation index is kept near the unity at all operatingconditions.

VI. CONCLUSIONS

The main results of the work can be summarized in thefollowing aspects:• The motor loss that is caused by the fundamental com-

ponent of phase currents can be minimized by using themaximum efficiency control strategy with the assumptionthat the core loss resistance and copper loss resistance areconstant under all operating conditions.

• The combined motor-inverter fundamental losses are min-imized under maximum efficiency control strategy in theregion in which ir∗d < id < 0.

convex and has global minimum loss value at the switching frequency of 19 kHz. However

the specific optimal switching frequency will depend on particular set of design parameters

of the VSI. In other words, when the parameters of the VSI such that the loss of VSI higher

than the motor harmonics loss at the same range of switching frequency, the curve will be

no more convex and almost will be linear.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

X=0.73

Y=112.2

Modulation Index

PMSM

Harmon

icsLoss(W

)

Figure 7.7: PMSM harmonics loss versus modulation index.

Figure 7.6 shows the combined loss of motor harmonics loss and VSI loss versus switching

frequency at different speeds and rated torque. The result shows that the total losses increase

as the motor speed increases. Also, at low speeds up to 50 % of the rated speed the VSI

loss is dominant. When the speed increases, the motor harmonic losses become dominant

such that the loss curve becomes convex. Figure 7.7 shows the relationship between PMSM

harmonic loss and modulation index. It can be seen that up to M = 0.73 the harmonics loss

increases as the modulation index increases. As the modulation index passes M = 0.73, the

66

Fig. 10. PMSM harmonics loss versus modulation index.

• PMSM harmonics loss can be minimized by increasingthe converter switching frequency with the assumptionthat SPWM inverter is used without any harmonics in-jection technique.

• For a given range of switching frequencies in whichthe motor can work with acceptable performance, thechoice of minimal switching frequency would minimizethe converter loss.

• For a given range of switching frequencies in whichthe motor can deliver acceptable performance, the choiceof the optimum switching frequency that will minimizemotor-inverter losses follows the parametric optimization.The optimal loss point is affected by the system param-eters. Hence, changing of the values of VSI parameterswould vary the optimal point of motor-inverter losses.

REFERENCES

[1] D. S. Kirschen, D. W. Novotny, and T. A. Lipo, “On-line efficiencyoptimization of a variable frequency induction motor drive,” IndustryApplications, IEEE Transactions on, vol. IA-21, no. 3, pp. 610–616,1985.

[2] R. Colby and D. Novotny, “Efficient operation of pm synchronousmotors,” IEEE Transactions on Industry Applications, vol. IA-23, no. 6,pp. 1048–1054, Nov./Dec. 1987.

[3] L. Xu, X. Xu, T. Lipo, and D. Novotny, “Vector control of a synchronousreluctance motor including saturation and iron loss,” IEEE Transactionson Industry Applications, vol. 27, no. 5, pp. 978–985, Sep./Oct. 1991.

[4] S. Morimoto, Y. Tong, Y. Takeda, and T. Hirasa, “Loss minimizationcontrol of permanent magnet synchronous motor drives,” IEEE TRAN.on Industrial Electronics, vol. 41, no. 5, pp. 511–517, Oct 1994.

[5] G. Slemon and X. Liu, “Core losses in permanent magnet motors,” IEEETransactions on Magnetics, vol. 26, no. 5, pp. 1653 –1655, sep 1990.

[6] R. Schifer and T. Lipo, “Core loss in buried magnet permanent magnetsynchronous motors,” Energy Conversion, IEEE Transactions on, vol. 4,no. 2, pp. 279 –284, jun 1989.

[7] M. Bierhoff and F. Fuchs, “Semiconductor losses in voltage source andcurrent source igbt converters based on analytical derivation,” ser. inProc. Power Electronics Specialists conference, vol. 4, 2004, pp. 2836–2842.

[8] C. Mi, G. R. Slemon, and R. Bonert, “Modeling of iron losses ofpermanent-magnet synchronous motors,” IEEE Trans. Ind. Appl., vol. 39,no. 3, pp. 734–742, May 2003.

[9] Z. Zhu and D. Howe, “Instantaneous magnetic field distribution inbrushless permanent magnet dc motors. ii. armature-reaction field,” IEEETransactions on Magnetics, vol. 29, no. 1, pp. 136–142, Jan. 1993.

[10] D. G. Holmes and T. A. Lipo, Pulse Width Modulation for PowerConverters. John Wiley & Sons, 2003.

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