The Permanent Magnet And It's Role

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 + md H
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 / md
md = the internal permeability of the magnet, generally from 1.05 to 1.25 m0

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

Ferrite
B0 = 0.25 to 0.42 T
mdr = 1.1 to 1.2

NdFeB
B0 = 1 to 1.25 T
mdr = 1.1 to 1.25

SmCo B0 = 0.9 to 1.1 T
mdr = 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.

2. Modeling

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:

F_PM = F_air
B_PM S_PM = B_air S_air
B_air = m₀ H_air

The circulation equation for the magnetic field [1] can be written, considering a perfect iron material (H_iron = 0):
= Ni = 0
H_air l_air + H_PM l_PM = 0

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

            

L_air 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

B_PM = B₀ + m_d H

• the external PM magnetic characteristic

B_PM / H_PM = - L_ext l_PM / S_PM

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:
Q = Q_PM - F
(6) Q = F

With :
Q = H l_PM Q₀ = H₀ l_PM F = B_PM S_PM F₀ = B₀ S_PM

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 - R_int I    source equation
U = R_ext 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:

R_mPM==internal PM reluctance
                           = 1 / L_PM
R_mext==external PM reluctance
                           = 1 / L_air
(6) --> Q = Q_PM - R_PM F
(7) --> Q = Rmext F

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:

• For a system with a PM:

Fx = 12 d PMdx 0 2
_PM = PM equivalent permeance
= 1/R_mtot
R_mtot = R_mPM + R_mext


• For a system with a PM and a coil:

Fx = 12 dLcdx ic 2 + dLcPMdx ic 12 d PMdx 0 2

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.

3.1 Scaling Methodology
3.1.1 Scaling factor

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* = l'/l = 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:

PJ = V J2 dV
* = 1 (same material)
V* = r*3

3.1.3 Permeances

A permeance can be written as [1]:

= permeance = dSl
V = r*

3.1.4 Heating

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

= PL Sc
PL are the total losses and c the equivalent convection coefficient.
= P*L / r*2 (15)

For small actuators, the copper losses are the most important:
P*J / r*2

Using (13):
= J*2 r*

For a constant heating ( = 1), the current density becomes :
J* = 1 / r*

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]:

Tr = 12 d d c2
c = Sco J = coil MMF
c = J* r*2

Using (14) and (17):

Tr* = J*2 r*5

If a constant heating is imposed, the relation (16) can be introduced:
Tr* = r*4

3.2.2 Relative copper losses and efficiency

The copper losses can be related to the mechanical power:
PJ / Pmech = PJ / ( Tr)

The above ratio can be written in a relative form as:
(PJ / Pmech )* = 1 / (r*2 *) (20)

The efficiency, related to the copper losses 'J is:
'J = 11 + (PJ/Pmech )* (1/ J - 1)
= 11 + (1/ J -1) / (r*2 *)

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]:

Tm = d mcd m c
m is the PM-MMF
m = Ho lm
H0 = coercitive field H0* = 1
m = r*

Using (14), (17) and (22):
T*m = J* r*4 (23)

If a constant heating is imposed, the relation (16) can be introduced in (23):
Tm* = r*3.5 (24)

3.3.2 Relative copper losses and efficiency

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

(PJ / Pmech )* = (PJ / Tm )*
= J* / (r* *) (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:
'J = 11 + (1/ J -1) J*/ (r* *) (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 N = 3'000 rpm
• characteristic active length l = 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 1 W
• average speed 30 rpm

3.4.2 Variable reluctance motor

With a constant heating (19), the ratio r* becomes:

r* = (Pmech*/ *)1/4
Pmech* = 10-9
* = 10-2
r* = 1.78 10-2
l' = r* l 1.78 mm

Using (21), the efficiency is:

'J = 6.32 . 10-5

3.4.3 PM motor

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

r* = (Pmech*/ 2)1/5 = .1

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):

r* = P *mech (PJ / Pmech )* *2 2 = 4.8 . 10-2

l' ~ 4.8 mm

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 = kr d3 l
Tm* = r*3.5 Tm = km d2.5 l

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

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. Temperature, losses and 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.

REFERENCES
[1] M. Jufer, "Rotating field and Torque". Mavilor, 2001
[2] M. Jufer, "Torque Generation and Permanent Magnet". Mavilor, 2002