Technology Magnets

Imagine the magnetic field surrounding magnetized permanent magnets as a room filled with magnetic lines of force. The lines of force extend from the magnet’s north pole and expand through the room in a large arc to the south pole. In doing so, every line of force creates its own circle.

The magnetic flux is the sum total of all lines of force.

The symbol indicating the magnetic flux is Φ (phi), and represents the entire number of a magnet’s lines of force.

The stronger the magnetic effect of a magnetic field at any given location, the more dense one can imagine the progression of the lines of force there.

The remanence or induction is the density of the lines of force.

The symbol for remanence is B and stands for the number of lines of force that run vertically through a surface of 1 cm².

One unit of remanence or induction is:

1 Gauss = 1 line of force per cm² = 10^-8 voltseconds/cm²
Remanence = (Total magnetic flux in Maxwell)÷(Surface in cm², through which the magnetic flux exits)

To understand coercive force, imagine the resistance to demagnetization by an outer magnetic field. It is the durability to permanent magnets opposing a demagnetizing influence.

The symbol for the coercive force is the character H_c, which stands for the size of an outer field opposite to a permanent magnet, which is capable of pressing down the permanent magnet’s remanence to zero.

The unit for coercive force is:

1 Oersted ≅ 0,8 AW/cm (ampere windings per cm)

Inside each permanent magnet, after it has been magnetized, is a magnetic voltage, which can be seen as the cause for why a magnetic field is generated in the space surrounding the magnet.

The magnetomotive force (MMF) is the sum total of the magnetic voltage or the total magnetic driving force of a permanent magnet, the effect of which is that the space surrounding the magnet is filled with lines of force.

The symbol for magnetomotive force is Θ (theta) and stands for the sum total of the existing magnetic voltage.

1 Gilbert = 1 Oersted × cm ≅ 0,8 ampere windings
Magnetomotive force = coercive field strength ∗ length of the magnet in cm

The further out of the space the lines of force of a magnetic field are driven, the greater we can imagine the magnetic field strength to be.

The coercive field strength is the magnetic voltage per unit of length of a magnetic field.

The symbol for the coercive field strength is H and stands for the magnetic voltage or magnetomotive force, which is contained in 1 cm length of a permanent magnet.

1 Oersted ~ 0,8 AW/cm (ampere windings per cm)

The link to the size of the ampere winding typically used in electricity theory is created as follows:

The magnetic voltage of a coil through which electric current flows is equal to the magnetic voltage of a permanent magnet. The magnetic current of a coil or its magnetomotive force is calculated based on the formula l * n / L = AW/cm, whereas I equals the flowing current in ampere, n the number of the wire’s windings and L the length of the coil in cm, resulting in the product ampere windings/cm.

The hysteresis loop is the graphic representation of the behavior of a material in an outer magnetic field. For this purpose, the remanence B is entered in the ordinate and the coercive field strength H on the abscissa in a right-angle coordinate system. Now, the material from which the hysteresis loop is to be calculated is subjected to a gradually growing magnetic field. The remanence increases in the specimen as a result. If we enter this increase in the first quadrant of the coordinate system, the result is what is referred to as the initial curve, which at + H_S (saturation field strength) reaches a saturation value typical for any ferromagnetic material. This saturation is also referred to as maximum induction.
If we now allow the field strength of the outer magnetic field to decrease, the remanence in the permanent magnetic material does not return to the initial curve. The inductive values are higher and while the outer magnetic field is fully shut off (H = 0) a finite value Br is retained. This is the remanent induction or the residual magnetism in the remanence point.

If the outer field is now reversed and if we allow it to increase this way, then remanence B will continue to decrease until it fully disappears in what is referred to as the coercivity point B^Hc (B = 0).
Modern materials from the group of rare earths create a deeper anchoring of the magnetic dipoles and are not reversed until a magnetic field is reached, which in some cases is several times higher J^Hc . This value is decisive for a resilience to demagnetizing magnetic materials in strong demagnetizing fields and temperature resistance, for example in electric motors. In order to calculate magnetic circuits, the value B^Hc is used.
While the magnetic field continues to amplify, B, now with an inverted sign, approaches a negative branch towards the first opposing saturation point. If we now repeat the entire process again with changed polarity, then B returns to the first saturation point on a path that is symmetric to the first curve. The loop closed this way is referred to as the hysteresis loop. The area that it encloses represents the energy that was converted into heat during demagnetization.

This characteristic curve that is exceptionally important for permanent magnets is an excerpt from the hysteresis loop. It shows the progression of B in the second quadrant for an increasing field strength opposite to the magnetizing. As a criterion for the quality of a magnetic material and for calculating permanent magnets, it is extremely important.

The mechanics of how the operating point of a magnet adjusts itself is quite complicated. For practical purposes, the following qualitative explanation will suffice:

The location of a magnet’s operating point that was magnetized in a yoke or in a solenoid coil is found in remanence point B_r after the outer magnetic field generated with the current is switched off. After opening the yoke or after the magnetizing pulse, an air gap is now created in the magnetic circuit, which until this point has been completely closed. Magnetic poles are now formed on the surfaces of the air gap. These poles “magnetize” the air gap, i.e. they emit a magnetic flux and build up a magnetic field in the air gap. At the same time, the magnet demagnetizes itself as far as necessary in order to build up the magnetic field, and does so depending on the length of the air magnet.

The magnetomotive force necessary to build up the field is generated by the mechanical energy that is needed to open the magnetic circuit, or in order to pull out the magnet from the yoke. This magnetomotive force has the effect that the magnet demagnetizes itself, because it is facing the opposite direction of the magnetization. This is why remanence B wanders from induction point B_r on the demagnetization curve in direction H to a new point B_A, the operating point, and its location depends on how far the magnetic circuit was opened and how large the coercive force of the magnetic material is. Its location can be influenced by selecting the magnet longer or shorter depending on the coercive force.

The operating point of a magnet on a demagnetization curve shows the current magnetic state. The remanent operating point B_A and to coercive operating point H_A correspond to the operating point.

The (BH)_max point is on the position of the demagnetization curve where the corresponding values for remanence and for coercive force are multiplied by each other, resulting in the largest product.

This product is calculated based on the corresponding values for remanence and for coercive field strength, which can be read from the demagnetization curve. It shows the remanent energy of a magnetic material.

In order to properly utilize a magnetic material, it is expedient to design the magnetic circuit so that the magnet operates in the (BH)_max point, because in doing so, the maximum energy is available in the air gap for a given volume of magnetic material.

The (BH)_max-product indicates the maximum operating capacity of a magnetic material in relation to the spatial unit.

The symbol for the energy product is (BH)_max and stands for the maximum product (B_A * H_A) of a magnetic material.

The unit for the energy product is:

Gauss × Oersted ≅ 0,8 ∗ 10^-8 wattseconds/cm³
10⁶ Gauss ∗ Oersted = 7,958 ∗ 10-3 wattseconds/cm³
1 wattseconds= 1 Joule = 10⁷ erg = 0,102 mkg

Most permanent magnets do not operate in the same operating point along their entire length. The entire magnetic flow only goes through the magnet in the neutral zone (halfway between the two poles). This entire magnetic flow is composed of the useful flow in an air gap and the inevitable flux leakage. If we want to allow a magnet to work along its entire length in the same operating point, this magnet must become thicker towards the middle in order to counteract the flux leakage. This is one reason why the sides are tapered on many horseshoe magnets.

However, in practical applications, we assume that the magnets work along their entire length in the same operating point, i.e. that they show the same magnetic flux everywhere. In most cases, this is exactly enough, however it requires that the scattering loss is considered in all calculations, designs and measurements.

The reluctance is the magnetic resistant of a magnetic circuit. It is the resulting value of all magnetic partial resistances of different units from which the magnetic circuit is composed. The reluctance value is needed in order to determine the magnetic flux, which is required in a specific part of a magnetic circuit, for example in an air gap. The symbol for reluctance is R.

When calculating permanent magnets, the assumption is preferably based on the specific reluctance or its reciprocal value, the specific permeance. The most difficult part of determining the specific reluctance or permeance is calculating the scattering and potential losses. There are in fact formulas that can be used to calculate the most common magnetic circuits, yet these formulas are either much too complicated, or the calculations based on simplified formulas only result in approximated values. So, it does not make any sense to attempt to exactly calculate a magnetic circuit to five digits after the decimal point if in that case only approximately calculated factors are available for scattering and potential loss. Even for a supposedly exact calculation of these factors, the calculations will inevitably have to be verified with measurements on a model.

What can generally be stated is that the scattering loss factor ρ₁ (rho₁) is between 2 and 10 in most cases. The given cross-section of the useful air gap must be multiplied by this value to compensate for the inevitable scattering losses of every magnetic circuit. The potential loss factor ρ₂ (rho₂) is much smaller and is typically between 1.1 and 1.5. The length of the given useful air gap must be multiplied by this factor in order to compensate for the losses at the magnetomotive force (MMF) that arise as a result of the bending lines of force and at the joints of the magnetic circuit.

The scattering loss factor is referred to as σ (sigma) in modern literature, while the potential lost factor is referred to as τ (tau).

We differentiate between σ_r of σ. The practical scattering factor is σ_r = Φ_i ÷ Φ_en.The total scattering factor is σ = Φ_i ÷ Φ_e. For this the Φ_en  includes the net flux in the air gap, Φ_i the flux in the magnets and Φ_e the total flux that crosses the air gap.

Because it is so difficult to calculate the exact loss factors, permanent magnetic circuits should not be designed based on calculated values alone. Experience values and measurement results based on test models must also be considered. Because of these difficulties, the design of permanent magnetic circuits should not be a purely mathematical task, but should just as much be an art form, which, by the way like all art forms, requires talent and experience.

Modern computer-aided calculation programs may very reliably calculate defined permanent magnet systems in the set-up phase. However, it will become more difficult for systems that are set up with multi-pin magnets or magnets made up of several different poles.

The calculation programs are not yet able to suggest the basic set up and design of a magnet system. The planned magnet system must still be designed for a specific purpose or specific properties.

The many different nuances of isotropic, anisotropic and crystal-oriented magnetic materials would surely not have been produced if there had not been a need for them. For example, a magnet should only show an absolute minimum volume, or it must drive magnetic flux through a large air gap, or it must be able to withstand high temperatures, or it should be very affordable, or it must be possible to magnetize it on many poles along the circumference, or it must be able to withstand demagnetized fields. All of these requirements can only be met using a magnetic material coordinated for the intended purpose.

2. LAPLACE’s law

A current-carrying conductor with a length L in the field of other current-carrying conductors or permanent magnets s subjected to a force F.

I = current in amperes, L in cm, B = induction in Gauss apply

Valid for B perpendicular to L, whereas L is in turn perpendicular to L and to B. Here, the left-hand rule applies: B in index finger direction, current direction equals the middle finger, force equals the thumb.

3. KIRSCHOFF’s rules

In every point of a magnetic circuit, the sum total of the incoming flux ∑Φ_e equals the sum total of the outgoing flux ∑Φ_a

In a closed magnetic system, the sum total of the magnetic reduced voltage (Reduced voltage=Φ x R, i.e. flux times reluctance) is equal to the sum total of the effective magnetomotive forces in the system.

At times, these rules are also referred to as Houston’s rules for magnetic circuits.

4. Reluctance formula

A prismatic structure of length L and cross-section S has a magnetic resistance or reluctance R.
The following applies:

5. R.K. Tenzer formula

The practical scattering factor σ_r of a permanent magnetic circuit based on the “symmetrical magnetic system” pattern is approximated using the following formula:

1, 2 permanent magnets; 3 short-circuit yoke; A A’ C’ C’ = useful air gap:
A A’m + C C’n = scattering air gap;

U_a        = circumference of the magnet in cm
A_g     = cross-section of the magnet in cm²
x       = air gap in cm

6. Differential equation by C. Schick

For the total scattering factor σ of an ideal permanent magnetic circuit based on the “symmetrical magnetic system” pattern, the following differential equation applies:

x=Length from air gap, K=Constant
Φ_e=Flux in volume A m A^’ C^’ n C
Φ_i=Flux in the magnets

7. Formulas by C. Schick

The following relationships apply to the total scattering factor σ of a permanent magnetic circuit based on the “symmetrical magnetic system”:

8. Transformation formula by C. Schick

The following relationship exists between the total scattering factor σ and the practical scattering factor σ^r, which applies to the ideal magnetic circuit:

9. Force formula

Between a magnet with a cross section S and a pole shoe made of soft iron that touch each other, what we have is a magnetic induction B, which can be assumed to be constant. The force that the magnet applies to pull the pole shoe can be calculated using the following formula:

F = Force in kg
B = Induction in Gauss
S = Surface in cm²

(If a magnet and pole shoe do not touch each other, what needs to be done is to first calculate the scattering factors in order to obtain a more accurate value for the induction).

10. S. Evershed criterion for static magnetic systems

In order to take full advantage of the given volume of magnetic material, operating point H_a, B_a of the magnetic system must be selected so that the product H_a * B_a equals the maximum possible product.

11. E.A. Watson formula

For steel magnets approximately the following relationships apply:

B₀ and H₀ correspond to the optimum operating point and a is a constant.

12. S. Earnshaw theory

In the field of other permanent magnets, a permanent magnet cannot be in stable equilibrium.

Future of magnetic materials

In the near foreseeable future, no new magnetic material generation is expected. Current research is primarily focused on replacing hard to source raw materials and ways to optimally recycle magnetic material. The highly oxidizing rare earths cannot be easily extracted for recycling. This is why the focus must be on the ability to reuse the magnets. In the past, this approach was only successful in large generators used for wind turbines. Here, the large magnet segments are installed by way of mechanical screw connections so as to allow the magnet to be removed undamaged at a later time for reuse elsewhere.

Right now, a cold spray method is being developed in additive manufacturing. The NRC Research Institute in Canada expects that, as compared to sintered rare earths, the method will achieve a higher resistance and allow more complex configuration options as well as provide good magnetic values and higher corrosion resistance. In addition, system configurations combined with pole returns made of iron no longer need to be assembled or bonded.

Availability and development of magnetic materials

The main groups of magnetic materials are broken down into a groups of sub-varieties.
Reflected in these segments are mainly the development stages of each respective manufacturing option. The properties primarily aim to induce higher energy products (BHmax) as well as to increase remanence (Br) and the coercive force (Hcj). The energy content and the increased remanence are designed to increase the performance capability. For the coercive force (resistance), the objective is to improve or rather increase the application temperature and to sectorally demagnetize the magnets in electric motors, which is also referred to as footprint.