Recent Developments of Electrical Drives - Part 15 pot

These two components can be represented by the global transformation ratio,ρ, between the motor shaft angle and the linear load displacement: This association between a speed reducer of

I-12 OPTIMIZATION OF A LINEAR BRUSHLESS DC MOTOR DRIVE

Ph Dessante1, J.C Vannier1and Ch Ripoll2

1Service EEI Ecole Sup´erieure d’´electricit´e (Supelec), Plateau de Moulon,

91192 Gif sur Yvette, France,

philippe.dessante@supelec.fr jean-claude, vannier@supelec.fr

2Renault Research Center—Guyancourt,

christophe.ripoll@renault.com

Abstract The paper describes the design of a drive consisting of a voltage supplied brushless motor

and a lead-screw transformation system In order to reduce the cost and the weight of this drive an optimization of the main dimensions of each component considered as an interacting part of the whole system is conducted An analysis is developed to define the interactions between the elements

in order to justify the methodology A specific application in then presented and comparisons are made between different solutions depending on different cost functions (max power, weight, cost, ) With this procedure, the optimization is no longer limited to the fitting between separated elements but is extended to the system whose parameters are issued from the primitive design parameters of the components

Introduction

The system studied in this paper is a linear electrical drive system realized with a voltage supplied brushless motor whose shaft is mechanically connected to a lead-screw drive device The aim of this system is to drive a load along a linear displacement

The specifications concerning the load consist mainly in two parts Firstly, it has to apply

a rather high static force at standstill as for instance to overcome some static friction force Secondly, it has to be driven from one point to another point in a given time This second part implies a dynamic force and a maximum speed depending on the kind of displacement function is chosen

A discussion to chose the displacement function is important because as the motor will have a limited torque capacity, it may be necessary not to accelerate neither too early nor too late when it is entering the constant power region Consequently this definition can have consequences on the system size At the very beginning it may be considered sinusoidal or corresponding to a bang-bang acceleration

System modeling

A general presentation of the system is given in Fig 1 where the power source is supposed

to be a battery bank

S Wiak, M Dems, K Kom˛eza (eds.), Recent Developments of Electrical Drives, 127–136.

2006 Springer.

Speed Reducer

Lead-Screw

Battery Bank

Power Supply

Figure 1 System main components.

Concerning the kinematic model, the lead-screw is represented by its transformation ratio deduced from the screw pitch while the speed reduction system introduces a speed transformation ratio These two components can be represented by the global transformation ratio,ρ, between the motor shaft angle and the linear load displacement:

This association between a speed reducer of a given ratio, N , and screw of a given pitch, τ,

gives the resulting value for the transformation ratio:

ρ =2π N τ (2) This ratio is used to convert the load specifications in motor specifications The load displacement is directly changed in angle variation and the forces are converted in torques taking into account the efficiency of each component During acceleration the motor inertia leads to a difference between the output torque and the electromagnetic torque

In this application two sorts of torques are to be generated by the drive system A static torque (at zero speed) can be necessary to reach the breakaway force on the load just before

it starts to move It can either represent the torque needed to maintain the load in a position

when an external force is applied With a given force, F

sta , the static torque is given by:

C sta= F sta

When the load speed is increased, generally the motor has to generate a torque with two components This second sort of torque is called the dynamic torque It contains a part corresponding to the force required to accelerate the load and a second part to accelerate

the rotor and the transmission system This part is represented by the inertia, J mot , of the

motoring part With a given dynamic resistive force, F

d yn , an equivalent mass of the load

m, a friction coefficient f, the dynamic torque for an accelerationγ at a speed v on the

load can be expressed as follows (4)

C dyn=



J mot ρ + m ρ



γ + fv + F



dyn

The two types of torque are dependent on the transformation ratio level

For the static torque it is obviously interesting to use a high value of the transformation ratio because the corresponding torque value will decrease and this will reduce the motor constraint (Fig 2)

Motion transformation ratio

Static

Dynamic

Figure 2 Torques vs transformation ratio (5).

For the dynamic torque, the increase of the transformation ratio will reduce the component

of the torque needed to drive the load but it will increase the torque required to accelerate the motoring parts mainly constituted of the rotor of the electrical motor

Consequently a first limitation appears when choosing the value of the transformation ratio It is not possible to retain a high value without having to generate a high dynamic torque

If we first consider the situation illustrated in Fig 2 for a fixed rotor inertia, it corresponds

to the case of the total force required by the load in dynamic mode, F

d yn tot , whose value

is referenced to the static force as:

F

dyn tot < F

In this case, a good value forρ could the one observed at the intersection between the two

curves [1–3]:

ρ i=



F

sta − F

dyn tot

J mot γ (6)

With that value the torque to be generated by the motor is minimal

Secondly we consider the case of a greater relative value for the total dynamic force needed by the load as:

F

sta /2 < F

dyn tot < F

As it can be observed in Fig 3, the dynamic torque will be minimal after the intersection between the two curves For this reason, a good value for the transformation ratio could be

in that case the one corresponding to the minimization of the dynamic torque:

ρ0 =



F

dyn tot

Motion transformation ratio

Static torque

Dynamic torque

Figure 3 Torques vs transformation ratio (7).

In that case, the torque to be generated by the motor is minimal with this choice

As the dynamic torque also depends on the value of the rotor inertia which will be defined during the motor design the situation is more complex and will be discussed

Other constraints [4] are also to be considered The load duty cycle is generally defined

and leads via the rms and the average values of the load dynamic to the definition of the corresponding rms torque:

C rms2 =



F rms

ρ

2

+



fv a ve

ρ

2

+2F dyn fv ave

ρ2

+ γ2

rms



ρ J mot+m

ρ

2

(9)

Among the limits concerning the motor, there could be a maximal rotor speed and the actual speed has to be considered:

This expression clearly indicates that the augmentation of the mechanical transformation ratio will need higher rotor speed for the motor

The motor supply and the battery tank characteristics introduce a limitation of the power consumption This finally depends on the efficiency reached by the motor and on the power consumed by the load The efficiency of a motor can be estimated from its main character-istics and the peak consumed power can then be defined:

P dyn = C dyn  = (Jˆ mot ρ2+ m)γ v + ( fv + F

For the motor design, different levels of complexity in modeling are available To simplify,

it is possible to define the main dimensions by using the peak torque, the rms torque, and

the rotor inertia as follows [5]:

J mot= 1

Consequently, these three relationships introduce three main dimensions parameters for the

design: the rotor radius R, the rotor length L, and the permanent magnet thickness E

The remaining parameters are more or less constant or weakly dependent on the motor size

They are defined as:

p = pole’s number.

H0= magnet’s peak magnetic field

B = airgap flux density.

A = stator excitation level.

γ p= pole’s overlapping factor

μ v= mass density

Concerning the converter, the volume of silicon can be linked to the maximum power value needed by the motor to drive the load

Optimization

The established relationships are used to define the constraints in the optimization procedure The motor peak torque has to be greater than the static and the dynamic torques The

nominal torque is also greater than the required rms torque.

C p > C sta (15)

C p > C dyn (16)

C n > C rms (17) The maximum power consumption is to be kept below the maximal value supplied by the battery bank The mechanical transformation device introduces inertia in the system equations Furthermore it needs a volume that will be a part of the total volume allowed to the system

P max > P dyn (18) Some technical constraints have to be added in order to be able to define a feasible motor It concerns the maximal rotor speed and the ratio between the rotor length and the diameter

 max > ˆ (19)

a R > L > bR (20)

A minimum relative value is needed for L to be kept in the domain of validity of the

previous expressions (12–14) A maximum value is settled to avoid the definition of a too thin rotor with a high length to diameter ratio as it could be required to reduce the rotor

and b.

Depending on the application, different cost functions can be minimized For instance,

if the weight is the principal criteria, the motor size will be reduced If the volume is

to be kept as low as possible, the mechanical transformation system size will be an issue

Results

We present here the results concerning the definition of the motor and the motion transfor-mation ratio whose dimensions are optimized for a given load

In this example, the load characteristics are the followings:

F

sta= 900 N γ = 1 m/s/sˆ

F

d yn= 450 N v = 35 mm/sˆ

F r ms = 90 N γ r ms = 0.1 m/s/s

f= 0 N/s v a ve = 28 mm/s

m= 1 kg

The optimization procedure uses the constraints (15)–(20) and searches a set of values for R,

L, E , and ρ which minimizes the motor peak torque It appears that the mass is minimized

as a consequence

As boundaries are used to limit the variation of these parameters to feasible values it appears that the result is always for the upper boundary value for the transformation ratio

In Fig 4, the evolution of the main rotor dimensions with the maximum authorized transformation ratio value are presented

x 104 0

5 10 15 20

25

R (O): L(*) & 10 × E (+) in mm

Figure 4 Rotor dimension R, L, E vs. ρ per meter

0 0.5 1 1.5 2 2.5 3

x 104 0

0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.4 0.45

Csta (o); Cdyn (*) & 2xCrms (.) in Nm

Figure 5 Motor torques vs.ρ maxper meter

The rotor mass as its inertia are decreasing as long as the maximum value for ρ is

increased

In Fig 5 the evolution of the torques is presented too

As it was observed before, the static torque diminishes when the ratio increases But in that procedure, it is observed for the dynamic torque as well and the good value forρ is the

maximum permitted value

This main difference is due to the fact that the rotor inertia changes its value when the ratio does so This could be a very important constraint for the motor design In the presented design procedure, some constraints (20) have been introduced to avoid such design difficulties

For every value of the maximum ratio, the rotor inertia can be evaluated and the previous good values forρ (6) and (8) can be calculated too They give the corresponding torques

presented in Fig 6

In that particular case the values are almost the same because the total dynamic force is near half the static force We can notice that the “good” ratio value is much more important than the permitted ratio value Consequently, the torques values are lower than the values obtained at the boundary of the domain

Finally, among the different values proposed by the design procedure, it is necessary to retain one of them to design the motor A criterion can be the maximum rotor speed

In Fig 7, the evolution of the maximum rotor speed with the maximum transformation ratio is presented

These speed values are rather common values for electrical motors For small motors the choice of a maximum speed of 6,000 or 9,000 rpm is reasonable

When the optimization procedure succeeds in defining a feasible motor, a more complex model is used to calculate all the dimensions

In Fig 8 is presented a view of one of these motors

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10 5

0 0.005

0.01 0.015

0.02 0.025

0.03 0.035

Figure 6 Former minimum torques vs.ρ optper meter

The airgap diameter is 8 mm and the outer diameter is close to 19 mm The rotor length

is 12 mm and the inertia is 0.022 kgmm2 NdFeB magnets are used to magnetize the airgap with gives a flux density equals to 0.8 T The resulting active mass is 20 g With the housing the resulting mass will be slightly higher The original commercial motor used to drive this application had a mass equal to 100 g

x 104 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Omegamax (rpm)

Figure 7 Maximum rotor speed vs.ρ per meter

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01 -8

-6

-4

-2

0

2

4

6

8

x 10-3

Figure 8 Resulting motor dimensions.

In Fig 9, a simulation of the flux lines distribution is obtained with FEM analysis This permits to verify the values expected from the design procedure

The nominal torque is 10 mNm and the peak torque is at least 50 mNm The maximum speed should be 6,600 rpm to drive the load at its maximum speed At maximum power, the motor efficiency is about 50% if it is assumed that the joule losses are predominant

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 9 Flux lines at load.

screw pitch of 3 mm and a speed reduction gearbox with a 9.5 ratio.

It can be observed that when the motor size decreases, the rotor speed increases which leads to the definition of a larger mechanical transformation system This is another con-straint which can be considered

Conclusion

In this paper, an electromechanical conversion system is analyzed resulting in a modeling of the components The model has to be inversed to link the dimensions to the performances for each component involved in the power conversion system Consequently the whole system dimensions are available for the aggregate optimization of the system This procedure permits a correct association between the components and can lead to a smaller volume or

a smaller weight than it could be defined with a separated element optimization The results presented have shown the interest to optimize simultaneously the rotor main dimensions and the transformation Actually, this procedure avoids the risk of having to design a nonfeasible motor with a too low inertia for a given torque

As it needs the complete specific design of a dedicated motor, it is reserved for rather expensive application (aircraft, space, ) with severe criteria or for very large scale appli-cation (automotive, )

As for this type of application, the total mass of the system is to be considered, a complete modeling of the transformation system is needed as for the electronic converter This could

be presented in a further work

References

[1] E Macua, C Ripoll, J.-C Vannier, “Optimization of a Brushless DC Motor Load Association”, EPE2003, Toulouse, France, September 2–4, 2003

[2] E Macua, C Ripoll, J.-C Vannier, “Design, Simulation and Testing of a PM Linear Actuator for a Variable Load”, PCIM2002, N¨urnberg, Germany, May 14–16, 2002, pp 55–60

[3] E Macua, C Ripoll, J.-C Vannier, “Design and Simulation of a Linear Actuator for Direct Drive”, PCIM2001, N¨urnberg, Germany, June 19–21, 2001, pp 317–322

[4] M Nurdin, M Poloujadoff, A Faure, Synthesis of squirrel cage motor: A key to optimization, IEEE Trans Energy Convers., Vol C6, pp 327–335, 1991

[5] C Rioux, Th´eorie g´en´erale comparative des machines ´electriques ´etablie `a partir des ´equations

du champ ´electromagn´etique, Revue g´en´erale de l’Electricit´e (RGE), Vol t79, No 5, pp 415–

421, mai 1970