Control of a Permanent Magnet Synchronous Machine for use ...

THE 5th STUDENT SYMPOSIUM ON MECHANICAL AND MANUFACTURING ENGINEERING

Control of a Permanent Magnet Synchronous Machine for use in

an Active Suspension System, with a Magnetic Lead Screw

E. Jensen, J. Seegert, P. Sørensen, R. Kæseler, T. Nielsen

Department of Materials and Production, Aalborg UniversityFibigerstraede 16, DK-9220 Aalborg East, Denmark

Email: emje13, jseege16, pjsa13, rkasel13, tornie13 @student.aau.dkWeb page: http://www.mechman.m-tech.aau.dk/

AbstractThis paper investigates how a Permanent Magnet Synchronous Machine (PMSM) can be used in combinationwith a Magnetic Lead Screw (MLS) as an active automotive shock absorber. The PMSM utilizes a hollowrotor, encircled by permanent magnets on the outside with which the stator field interacts. The interior of thehollow rotor is equipped with helical shaped magnets, facilitating rotation relative to the MLS translator. Afterassembly of the PMSM, test are performed to evaluate parameters for a mathematical model. The model isvalidated through various tests, and a field oriented control scheme is developed. Furthermore, vehicle dynamicsare investigated to estimate the conditions the shock absorbing unit will be subjected to.

Keywords: Active Suspension, Permanent Magnet Synchronous Machine, Magnetic Lead Screw

1. IntroductionThe automotive industry have had greater focuson suspension systems in recent years, and threegeneral terms have been made to distinguish betweenthem: Passive-, active- and semi-active suspensionsystems. Traditionally, the passive system consistsof a steel spring and a viscous damper, providingsuspension with constant damping. This is still themost commonly used system implemented in carstoday, but newer and more advanced systems areclosing this gap. Active systems inputs power tocontrol roll, pitch, and heave, whereas semi-activeonly changes the suspensions energy dissipation.General Motors have developed a semi-active system[1], while Mercedes [2] and Bose [3] have madeactive systems. The success of these are highlydependent on weight and cost, which leads tocontinuous interest in the development, why AalborgUniversity (AAU) have researched these suspensionsystems.

Previous work from AAU investigated a MagneticLead Screw’s (MLS) efficiency and ability to func-tion as a wave energy converter, documented in[4] and [5]. It was outlined in [4] that the MLShas a high efficiency of more than 90 % due tothe reduction of contact elements. In [6] a singleMLS was implemented in a Vehicle Test Platform(VTP) in conjunction with a Permanent Magnet

Synchronous Machine (PMSM) and an air spring,which combined constituted the new shock absorberdesign. A PMSM has the advantage of being com-pact, making it ideal in this regard due to spacelimitations. The PMSM can work as both generatorand motor at high efficiencies, which is essential foran energy efficient system. In order to fully explorethe potential of shock absorbers in vehicles, a VTPwith four shock absorbers are of interest. One nativefront suspension unit in a VW Passat B5 has alreadybeen replaced with this new system, why this paperassembles and examines a second suspension unit forimplementation in the VTP. A 3D representation ofthe completed assembly is displayed on Fig. 1.

Fig. 1 3D illustration of an assembled shock absorber [6].

1

2. ManufacturingThe construction of the shock absorber included theassembly of three major parts; the MLS (1), thePMSM (2) and an air spring (3). The PMSM statoris shown in Fig. 2. To prevent the stator iron fromdamaging the windings (4), mylar paper (5) is placedin each slot, and a fiber cloth (6) at the ends ofthe stator cross section. Finally, glass fiber sticks (7)is placed in the slots as an enclosure, keeping thewindings in place.

Fig. 2 Cross sectional illustration of the stator.

After assembly, PMSM stator is tested with a 1000 Vmegger, to ensure the integrity of the field windings.The windings are connected together to form threephases as indicated on Fig. 3. The three phasesare connected together in a star point, long wiresfor connections are soldered and exposed areas areisolated with heat shrink tubing. Finally, waxedcotton string is applied to tie down loose wires.

The assembly of the PMSM rotor, the MLS rotorand the MLS translator is initialized by gluing per-manent neodymium magnets onto each component.The PMSM utilizes a hollow rotor design with 20magnetic poles mounted on the outside perimeterin a shifting pattern. Rectangular shaped magnetsare used, whereas the MLS rotor and translatorutilizes helically shaped magnets instead. To obtainan optimal flux field without compromising the man-ufacturing complexity, the neodymium magnets havebeen magnetized in three different directions. Theyare then arranged in a spiralling pattern throughoutthe translator shaft, following the schematic of aHalbach array illustrated on Fig. 4. This pattern is

Fig. 3 Illustration of field windings.

chosen to minimize iron loses by concentrating fluxtowards away from the back iron, into the airgap.

Fig. 4 Halbach schematic for optimal flux density.

The final preparations, before the complete leadscrew is assembled, includes sealing the air gapsbetween each magnet with an epoxy based filler. Thisserves to reinforce the structure, locking the magnetsin position during the final milling process. Here theouter diameter is reduced to in order to smoothenthe surface and prepare it for installation within therotor. A SolidWorks drawing of the assembled MLSis shown on Fig. 5.

Fig. 5 SolidWorks drawing of MLS translator and rotor[6].

2

3. Mathematical ModelThe entire suspension system can advantageously bedivided into smaller segments, which can then bemathematically described individually. The PMSMrepresent one of these subsystems, why an electricalequivalent circuit of the is established. It consistsof three phases, each with an inductor, a resistor anda counter-electromotive force (back-EMF) connectedin series. The phases are joined in a star formationas illustrated on Fig. 6. The back-EMF is the voltageinduced in the windings, by the permanent magnets,as the rotor spins.

Fig. 6 Equivalent circuit of the PMSM

A set of voltage equations can be derived forthe stationary abc-reference frame, based upon theequivalent diagram. The matrix form of these areshown in Eq. 1.

uaubuc

=

La Lba LcaLba Lb LcbLca Lcb Lc

d

dt

iaibic

+

Ra 0 00 Rb 00 0 Rc

iaibic

+

eaebec

(1)

The phase resistors may have small asymmetries dueto the manual manufacturing, but they are assumednegligible. The inductors may also deviate from eachother, but this is neglected as well. The assumptionof equal mutual inductance are also made, which isshown in Eq. 2.

R = Ra = Rb = Rc

L = La = Lb = Lc

M = Lab = Lbc = Lca

(2)

Since the star connection must follow Kirchhoff’scurrent law, Eq. 3 must also apply for the system.

ia + ib + ic = 0 Mib +Mic = −Mia (3)

This simplifies Eq. 1 to Eq. 4.

uaubuc

=

L−M 0 00 L−M 00 0 L−M

d

dt

iaibic

+

R 0 00 R 00 0 R

iaibic

+

εaεbεc

(4)

This phase plane representation can be simplifiedfurther. By transforming the equations into a statorfixed α, β coordinate system, and then into a rotorfixed d, q coordinate system, some mathematicalsimplifications emerge. AC quantities become DCproviding a static reference frame for the controllerto operate on. The stator fixed coordinate system isobtained by the Clarke transformation and the rotorfixed coordinate system is obtained with the Parktransformation. The Clarke transformation matrix isshown in Eq. 5.

[xαxβ

]=

2

3

[1 −1

2 −12

0√32 −

√32

]xaxbxc

(5)

The Park transformation matrix is shown in Eq. 6.

[xdxq

]=

[cos(θe) sin(θe)−sin(θe) cos(θe)

] [xαxβ

](6)

3

Fig. 7 Illustration of different coordinate systems.

The rotor fixed voltage equations can now berewritten as Eq. 7. Here ωe represents the angularvelocity of the electric field, while λpm denotes theflux linkage in the air gap, generated the permanentmagnets.

ud = Rid + Ldiddt− ωeLiq

uq = Riq + Ldiqdt

+ ωe(Lid + λpm)

(7)

Torque can be calculated using Eq. 8, if q- and d-axis inductances are assumed to be identical. In thecase of surface mounted permanent magnets, thisassumption is a fair approximation. The number ofmagnetic poles on the rotor is represented with p.

τe =3p

4λpmiq (8)

From Eq. 8 it can be seen that torque is proportionalto the q-axis current. The d-axis current is alignedwith the rotors magnetic pole and provides no torque.Therefore, the d-axis current should be controlled to0, as it only contributes additional iron and copperlosses.The angular velocity ωm of the PMSM can bedetermined using newtons 2nd law for rotatingbodies. This is an ordinary differential equation asshown in Eq. 9. Here Jm is mass moment of inertia,Bm is viscous friction, τdf is dry friction and loadtorque is τL.

τe = Jm2

p

dωedt

+2

pBmωm + τdf + τL (9)

The Variable Frequency Drive (VFD) primarilyconsists of two subsystems; the inverter and themicro controller. The purpose of the inverter isto distribute voltage from the DC-bus across eachphase using the six internal switches illustrated onFig. 8. The micro controller generates the Pulse-Width Modulation (PWM) signals used to controlthe switches during operation. The duty cycles arecalculated by comparing the modulation signal witha triangular carrier wave with a predefined frequency.Simplifications regarding the modelling includes theassumption of infinitely fast transistor switches, nodead time when switching and a constant DC-busvalue.

Fig. 8 Inverter circuit diagram.

The modulation signal is generated by space vectormodulation. This modulation technique relies onswitching between different vectors as seen on Fig. 9The reference vector is shown by Us. Fixed vectorsare eg. V100 where the three subscripts indicatesphase A,B and C respectively. 1 indicates the switchis connected to the high leg of the DC bus and 0 thelow leg. In addition to the vectors shown, two zerovectors (000) and (111) exist. On the zero vectors, allthree phases have the same potential making the lineto line potential 0. To realise the reference vector, themodulator would switch between two active vectorsand a zero vector. The time spent on each of theactive vectors is determined the by the angle ofthe reference vector, while the time spent on thezero vector is determined by the amplitude of thereference vector.

4

Fig. 9 Space vector diagram from [7].

4. ParametersFor accurate modelling and control of the PMSM,the electromechanical characteristics must be deter-mined. This was done through various tests, and theresults is displayed in Tab. I.

Parameter Symbol Value UnitPhase resistance R 0.41 [Ω]Phase inductance L 1.13 · 10−3 [H]Back-EMF constant λpm 2.97 · 10−2 [V·s/rad]Torque constant KT 0.4455 [Nm/A]Iron and friction losses B 2.9 · 10−3 [Nm·s/rad]

Tab. I Measured PMSM parameters.

The phase resistance was found using Ohm’s lawR = U/I . Constant voltage was applied across onephase, while the current was measured. This processwas repeated on the other two phases after which anaverage was calculated.

The inductance is determined by method of instan-taneous flux linkage [8]. The rotor shaft is locked inplace with a steel plate throughout the test, in orderto exclude any influence from back-EMF. A voltstep is given while measuring voltage and current.One phase is connected to the positive terminal, andthe other two to the negative. The instantaneous fluxlinkage is given by Eq. 10 and the inductance by Eq.11. The equivalent inductance for a single phase isdesired, hence the 2

3 in Eq. 11.

λ(t) =

∫ t

t0

U(t)−Ri(t)dt (10)

L =2λ(t)

3i(t)(11)

In order to determine the back-EMF constant, ahand drill is connected to the rotor shaft to bringit up to speed. This induces a voltage in the PMSMcoils which is measured with an oscilloscope. Bymeasuring the line-to-line voltages together withthe electrical frequency, the back-EMF constant iscalculated using Eq. 12. Conversion factors

√2 and√

3 are used to express the back EMF constant inunits [Vpeak/(rad/s)].

λpm =

√2ul−l,rms√

3ωe(12)

Finally, the power loss due to viscous friction andiron losses (hysteresis and eddy current losses) aredetermined. Since they are difficult to separate, theyare expressed together as a single coefficient Bm,calculated with Eq. 13. In this test, the mechanicalvelocity ωm of the load motor is increased inintervals of 100 RPM. Meanwhile, the resistingtorque τvf from the PMSM is measured with thetransducer. The setup for tests and validation isdisplayed on Fig. 21.

Bm =τvfωm

(13)

Fig. 10 Block diagram of field oriented control.

5. Test and validationBefore a controller can be designed, a validationof the model is performed to ensure is replicatesthe real system. Closed loop tests with a previouslydesigned PI controller [6] is conducted to analyze thedynamic response. Fig. 11 displays the response froma 2 Ampere q-axis current step, which are nearlyidentical both in amplitude and transient response.

5

Fig. 11 Step response comparison between the nonlinearmodel and measuremed data.

Some open loop experiments are conducted aswell, but these revealed inconsistencies between themodel and the test data. Specifically, the cross-couplings showed significantly different behaviour insimulation and test data. To examine the couplingscloser, a Relative Gain Array (RGA) analysis isconducted on the linearized model.

6. LinearisationTransfer functions are derived through a lineariza-tion by Taylor series approximation. The followinglinearization point is used:

ωe,0 = 200rads

, iq,0 = 2A , id,0 = 0.6A(14)

Yielding linearisation coefficients:

Kd = Kq = ωe,0L = 0.1242

Kwd = iq,0L = 0

Kwq = id,0L = 0.0023

(15)

Applying the coefficients to the voltage and mechan-ical equations yields the following state space model:

d

dt

idiqωe

=

A︷ ︸︸ ︷−RLd

Kq

Ld

Kwq

Ld−Kd

Lq

−RLq

−Kwd−λpm

Lq

03P 2λpm

2J −Bm

J

B︷ ︸︸ ︷ idiqωe

+

1Ld

0 0

0 1Lq

0

0 0 −PJ

UdUqτL

idiqwm

=

C︷ ︸︸ ︷1 0 00 1 00 0 1

P

idiqωe

(16)

This can be transformed to transfer functions by:

G(s) = C(sI −A)−1B (17)

yielding a 3x3 system. As it is desired to controlthe currents iq and id with the voltages, Uq and Udrespectively, the transfer functions concerning theseinputs and outputs are all considered, as follows:

[idiq

]=

[G(1, 1) G(1, 2)G(2, 1) G(2, 2)

] [UdUq

](18)

The RGA analysis investigates, as the name suggests,the relative gain between inputs and output, andspecifically the cross couplings, i.e. what influencevoltage Uq will have on id and Ud on iq. As this isthe subject of the analysis only the transfer functionsdependent on the voltages should be included. TheRGA elements can therefore be calculated as:

RGA(H) =

[λ11 λ12λ21 λ22

]=

[λ11 1− λ11

1− λ11 λ11

](19)

λ11 =1

1− h12h21h11h22

6

Fig. 12 RGA numbers and elements.

Sweeping through frequencies from 10−1 to 104

yields the RGA elements and numbers on Fig. 12.A fully decoupled system is RGA(H) = I, meaningλ = 1 and 1−λ = 0. The element values can be seento be approximately 0.8 and 0.2 respectively, whichindicates cross couplings between inputs and outputs,which was also proven by a open loop voltage step,shown on Fig. 13.

Fig. 13 Voltage step with current responses.

The linear model was compared to measured datawith a voltage step of 28 V, resulting in the currentresponse shown on Fig. 14 and the velocity responseon Fig. 15. This was an insufficient response, but again of 1/3 of the DC Bus voltage proved to givean adequate response, why this was applied whendesigning controllers.

Fig. 14 Voltage step with current responses.

Fig. 15 Voltage step with current responses.

7. ControlAs the couplings was considered disturbances andthe linear model adequate for control design pur-poses, a root locus revealed a dominant pole and zerowhich was utilized to simplify the transfer functionto:

Gq,a =K(s+ β)

τs+ 1(20)

Where,

τ = 0.0372β = 2.048K = 17.886

A previously designed PI controller [6] gave anovershoot of 50 % but due to the maximum poweroutput of the inverter, of 1.5kW, and the desire tominimise energy consumption, it was decided todesign a controller with no overshoot. This was done

7

with Internal Model Control (IMC) which builds onthe idea, of only feeding back model inaccuraciesand disturbances. A schematic of the IMC approachis shown on Fig. 16.

Fig. 16 Internal Model Control schematic from [9].

Describsions of the used parameters can be found inTab. II.

Parameter SymbolReal plant G0(s)Modelled plant G(s)Compensator Q(s)Closed Compensator C(s)Disturbances W (s)Input R(s)Output Y (s)

Tab. II Optimized parameters for the quarter car model..

The model G is separated as:

G = G+ ·G− (21)

Where G+ is chosen as;

G+(s) =Kβ

τ + 1(22)

A filter implemented to apprehend singularity of C.It is chosen as;

Gf =1

λs+ 1(23)

The compensator is calculated as:

Q = (G+)−1 ·Gf =

τs+ 1

Kβ(λs+ 1)(24)

C(s) =Q

1−G+Q=

τs+ 1

K(βλ− 1)s(25)

The compensator is thus a PI controller, with the PIcoefficients:

Kp =τ

K(βλ− 1), Ki =

1

K(βλ− 1)(26)

This controller results in the closed loop system;

Y =G0(τs+ 1)

K(βλ− 1)s+G0(τs+ 1)R

+K(βλ− 1)s

K(βλ− 1)s+G0(τs+ 1)W (27)

As Ga gets closer to the real system G, the closedloop system becomes:

limGa→G0

Y =

1β s+ 1

λs+ 1R+

(λβ − 1)s

β(λs+ 1)W (28)

As s goes towards 0, the influence of the disturbancebecomes zero, while the output Y becomes thereference value R.

Requirements for the controller is stated as:

1

β< λ BW < 3, 000

rad

sBW > 1, 500

rad

s

Where BW is the closed loop bandwidth. To meetthese requirements, λ is chosen to be 0.55, whichis found through bode plots. The controller is thentuned by increasing the the integrator gain, todecrease the rise time without creating an overshoot.The closed loop step response of the system with theold, the new and the tuned new controller is plottedand compared in Fig. 17.

Fig. 17 Step response.

Even though it looks like the new controller withouttuning have a steady state error, this is not thecase, as it slowly moves to the desired value. Thetuning has significantly improved the new controller,having decreased the rise time immensely without

8

any noticeable drawbacks. This tuned controller isalso considered superior to the old controller, sinceit has no overshoot and a shorter settling time. Thecontrol design is thus considered a success. It isimplemented in the real system, and a step of 2 ampis performed.

Fig. 18 Step response in real system.

The result of the test can be seen on Fig. 18. The oldcontroller did not overshoot in this example, unlikeprevious real system test. It is assumed, that thereason for this, is due to a lower load torque in thenewest test. The new controller still proved superiorhowever, especially considering reduction of noise.This is because it is less aggressive near steady state,thus being less affected by noise and disturbances.

8. VTP modelA VTP model is made to simulate the suspensionunit. It is modelled as a quarter car, with either apassive or a skyhook suspension control scheme.

Fig. 19 Car model with passive (left) and skyhook (right)

Skyhook uses the idea of implementing a fictivedamper between the suspended mass, m1 andthe sky. Due to limited access to the real VTP,an optimization algorithm based on data from afrequency sweep of the real system, was used tocalculate model parameters, which is available inTab. III.

Parameter Symbol Value UnitBody mass m1 698.5 [kg]Wheel mass m2 69.9 [kg]Suspension damper c1 6.84 · 103 [Ns/m]Suspension spring k1 4.9 · 106 [N/m]Tire spring k2 5.89 · 108 [N/m]

Tab. III Optimized parameters for the quarter car model.

These parameters seems pretty high, but are stilldeemed realistic. The model is then used to simulatethe passive suspension versus the skyhook. It simu-lated the vehicle driving over a roadbump at 10km/h.The acceleration response of chassis can be seen onFig. 20.

Fig. 20 Acceleration response of chassis

This showes the advantage of skyhook, however italso required more power from the PMSM. Dueto this huge power demand, a passive suspensionsystem is recommended for this VTP.

Fig. 21 Load motor (left), torque transducer (center) andPMSM (right).

9

A passive damper was tested on the PMSM on atest bench, see Fig.21 where opposing motor wasrunning with a sinusoidal velocity trajectory, and thePMSM was dampening it with different dampingcoefficients. The torque output was measured versusthe reference torque, see Fig. 22.

Fig. 22 Torque response

The motor followed the torque reference at allfrequencies, with highest measured being 15Hz. Thistest concluded that the FOC control for the PMSMwas adequite for high frequency torque trajectories,i.e. it is suitable for passive suspension usage.

9. ConclusionThis paper has investigated the design and conceptof an electromagnetic shock absorber developedby Aalborg University. This included manufacturingthe PMSM and MLS, which provided a deeperunderstanding of the system. Starting out withthe governing equations, a nonlinear model wasestablished. Hereafter, a series of test has beencarried out to determine the parameters requiredfor this model. This resulted in values for phaseresistance, phase inductance, back-EMF constant,torque constant, friction and iron losses.

Open loop simulations was carried out with theobtained parameters and subsequently comparedwith test data acquired from the experimental setup.Unfortunately, the results proved unsatisfactory. Thesame study was executed with closed loop controlinstead. This provided better results as the transientresponses matched.

In spite of deviations, a linear model has beenderived using Taylor series approximations on non-linear expressions in the governing equations. Acontroller has been designed from the internal model

control principle. Subsequent tests of this has proventhe design holds in a test environment. The controllerhas been compared with a previously designed con-troller, and the overshoot has been reduced fulfillingthe design goal. Investigations into the vehicle dy-namics suggest the PMSM is suitable for use in asuspension system.

Acknowledgement The authors of this work grate-fully acknowledge Sintex for sponsoring the 5th

MechMan symposium.

References[1] GMAuthority, “General Motors Magnetic Ride

Control Technology,” 2014. Available:http://gmauthority.com.

[2] Micro-tronik, “Mercedes Benz 216 Suspensionsystem Control Unit,” 2004. Available:http://www.micro-tronik.com.

[3] T. C. Connection, “Bose Unveils "ProjectSound",” 2004. Available:http://www.thecarconnection.com.

[4] N. Berg, R. Holm, and P. Rasmussen,“Theoretical and Experimental Loss andEfficiency Studies of a Magnetic Lead Screw,”IEEE Trans. on Industry Application, 2015.

[5] N. Berg, R. Holm, M. Walkusch, P. Rasmussen,and R. Hansen, “Design of a Magnetic LeadScrew for Wave Energy Conversion,” IEEETrans. on Industry Application, 2013.

[6] N. Berg, “Active Suspension - WP1&2,” tech.rep., Aalborg University, 2016.

[7] L. Mathe and P. O. Rasmussen, “AC MotorDrives: Converters and Control,” 2017.[Course].

[8] W. Soong, “Inductance Measurements forSynchronous Machines,” Power EngineeringBriefing Note Series, 2008.

[9] L. Schmidt and P. Johansen, “Multi variableand non-linear control methods,” 2017.[Course].

10