Laboratory of Integrated
Activators (LAI) in Lausanne

1. behaviour

A permanent magnet (PM) is a ferromagnetic material, composed of iron, nickel or cobalt alloys. They are characterized by a very important hysteresis effect on the B-H curve.
Such a material, when submitted to a very strong external magnetic field
(created by a coil), presents a remanent field, opposite to the magnetization direction. The corresponding B-H curve is represented in Figure 1. The positive sign for H corresponds to the magnetization direction.

On most modern PM, the B-H curve is practically a straight line, except in the area of low B values.

In first approximation, the B-H curve can be described by the following equations :
B = B0 + µd H    (1)
B0 is the remanent induction Hr is the coercitive field

H0 is the field for the zero value of B on the straight line, with :

H0 = B0 / µd
µd = the internal permeability of the magnet, generally from 1.05 to 1.25 µ0

For the most common PMs, the following values are usual :

Ferrite
B0 = 0.25 to 0.42 T
µdr = 1.1 to 1.2
NdFeB
B0 = 1 to 1.25 T
µdr = 1.1 to 1.25
SmCo
B0 = 0.9 to 1.1 T
µdr = 1.05 to 1.15

For these different materials, reduced values of B0 can be obtained with plastic PM, made of a mixture of PM powder and epoxy resin.

The internal permeability of such materials is very low nearby the vacuum permeability.

Figure 1	Figure 2	Figure 3

2. modelling

Two laws can be applied to a PM placed in a magnetic circuit such as described in Figure 2. The flux created by the magnet is the same as the flux in the air gap :

The circulation equation for the magnetic field [1] can be written, considering a perfect iron material (Hiron = 0) :

Using equations (2), (3) and (4), it comes :

Λair is the air permeability or, in a more general way, the external permeability of the magnet circuit (see also Equation 8 in [2]). Equation (5) can be drawn on the B-H plane of Figure 1. Figure 3 defines the two characteristics for a PM :

* the internal PM magnetic characteristic (1) :
BPM = B0 + µd H(a)

* the external PM magnetic characteristic (5) :
BPM / HPM = - Λext lPM / SPM (b)

The intersection point between these two lines corresponds to the PM B-H values.

Figure 3 can be transformed using the following steps :

* to multiply the B scale by SPM
* to multiply the H scale by - lPM
* to rotate the figure of 90°, clockwise.

Thus, Equations (1) and (5) become :

Figure 4 describes the new representation of the PM. By analogy, the following equations correspond to a real voltage source with an external resistance :

U = U0 - Rint I source equation }(8)
U = Rext I load equation
These equations are described by Figure 5.

Figure 6 is an equivalent circuit of a real source with a load. For the PM, using
Equations (6) and (7), it is possible to define:

There is a total analogy between equations (9) and (8) with the following correspondence:

So it is possible to represent a PM as an equivalent magnetic circuit, similar to an
electrical one (Fig. 7).

Based on such an equivalent circuit, it is possible to determine forces or torques
created by a PM or by the interaction between a PM and a current.

In such a case, equations (6) and (7) of [2] become :

3. scaling laws

The scaling laws have the aim to compare two motors of different sizes, but totally similar in their structure. Out of this approach, it is possible to show the evolution of torque, power, efficiency and heating as a function of sizes. It is also possible to compare different
structures such as variable reluctance or PM motors and thus to define their application domains.

Considering two similar motors of respective length l and l' (respective diameters d and d'). All materials (laminations, copper, PM) are the same for both motors. The scaling
factor r* is defined as :

r*=1*/1=d'/d

The goal of the present analysis is to express the main motor relative characteristics as functions of r*. So, any characteristic F can be written as :

F*=F'/F=f(r*)

In the following paragraphs, the common characteristics for different actuator principles will be defined. In the other sections, these results will be applied to different structures.

3.1.2 Copper losses

The copper losses can be written as:

3.1.3 Permeances

A permeance can be written as [1]:

3.1.4 Heating

The heating in steady state can be described by an equivalent convection coefficient:

Δ θ = JL / αSc

PL are the total losses and ac the equivalent convection coefficient. (15)

For small actuators, the copper losses are the most important :

3.2 Variable reluctance actuator
3.2.1 Torque characteristic

For a variable reluctance actuator, the torque can be written using the energy derivative methodology [2]:

If a constant heating is imposed, the relation (16) can be introduced :

Tr* = r*4     (19)

3.2.2 Relative copper losses and efficiency

The copper losses can be related to the mechanical power :

Pj/Pmech = Pj/(ΩTr)

The above ratio can be written in a relative form as :

The efficiency does not depend on the current density J.

It is impossible to realize a small variable reluctance motor with a good efficiency. The above results can be extended to the asynchronous motors.

3.3 PM motors
3.3.1 Torque characteristic

The following developments can be applied to any kind of PM motor : brushless DC, DC, stepping. The main torque is created by the mutual inductance between the PM and the
stator phases. Thus, the torque can be written as [2] :

If a constant heating is imposed, the relation (16) can be introduced in (23):

Tm* = r*3.5

3.3.2 Relative copper losses and efficiency

The relative copper losses can be written, using (12) and (24) as :

(25)

For a constant speed, if J* = r*, the relative losses and the efficiency can be held constant. Thus, it is possible to build small PM actuators with a high efficiency. The efficiency can be written as :

(26)

3.4 Application
3.4.1 Reference

Starting from a reference motor of 1 kW, the goal is to apply these scaling laws in order to characterize a watch motor. As a reference, the following motor is chosen :

*mechanical power
Pmech = 1 kW
*rating speed
ΩN = 3'000 rpm
*characteristic active length
I = 100 mm or diameter
*relative copper losses
PJ/Pmech = 0.05

This value corresponds to an efficiency (copper only) of 35.2% The aim is to extrapolate to a watch motor, with the following characteristics :

*mechanical power
1µW
*average speed
30 rpm

3.4.2 Variable reluctance motor With a constant heating (19), the ratio r* becomes :

h'J = 6.32 . 10-5

3.4.3 PM motor

For a constant efficiency (hJ* = 1), using (23) and (25), the scaling factor is :

The corresponding sizes are too large to be built in a watch. With an efficiency of 33.3 %, the
corresponding values become :

P' j / P' mech =2
P*j / P* mech = 40

Using (26) :

Other factors, like the modification of the relative volume of copper and PM (it is not a scaling reduction any more), makes it possible to reduce the motor sizes. In this case, the PM solution will use 5'275 time less energy than the VR solution !

3.5 Comparison

Using the equations (19) and (24), it is possible to write :

Tr*= r*4 Tr = Krd31
Tn * =r*35 Tn = knd251

Moreover, if the length is proportional to the diameter, it is possible to write:

Tr = k'r d4
Tm = k'm d3.5

In Figure 9, both curves are drawn. They have a crossing point for the diameter d0 ant the torque T0. This point depends on the PM choice : d0 = 100 ÷ 150 mm for the bore
diameter T0 = 50 ÷ 150 Nm

3.6 Conclusion

The scaling factor methodology describes the characteristic evolution of motors and actuators, according to their size. The PM importance is enhanced for small sizes.

It is possible to keep a high efficiency, even for small sizes. T e m p e r a t u r e , l o s s e s
a n d performances may be expressed as a size function, for proportional size reduction.

This methodology also gives information on how to change the parameters as a size function.

LIST OF SYMBOLS INDEXES
B flux density 0 reference, vacuum C coefficient ai air d diameter c coil e thickness co copper f frequency d differential f, F function el electrical H magnetic field ext external i, I current h hysteresis J current density int internal k coefficient J Joule's or copper losses l length l length L inductance L losses	m mass m magnet, magnetic N turn number mech mechanical P power PM permanent magnet R resistance r reluctance, relative Rm reluctance tot total r scaling factor * f'/f, relative S section ' relative to the new actuator T torque u voltage V volume a angle F flux L permeance m permeability h efficiency r resistivity Q MMF