Stanley A Meyer Core Permeability

There are 2 inductors on a ferrite core, closed loop U shape.

The core's permeability is approx 2000.

One coil is 300 turns of 22Ga wire,
and the other coil is 1000 turns of 30Ga wire.

The inductance of the 300 turn coil, while shorting the 1000 turn coil, is 12.188mH.
The inductance of the 1000 turn coil, while shorting the 300 turn coil, is 159.69mH.

Both measurements were taken at a 1 kHz test frequency with an LCR meter.

What is the true core permeability?

We can use the following equation to calculate the inductance of an inductor on a closed loop ferrite core with two coils:

L_total = μ * N_1^2 * A / (2 * l) + μ * N_2^2 * A / (2 * l)

where L_total is the total inductance of the two coils, μ is the permeability of the core in henries per meter (H/m), N_1 and N_2 are the number of turns in each coil, A is the cross-sectional area of the core in square meters (m^2), and l is the length of the core in meters (m).

We are given that one coil has 300 turns of 22 gauge wire, and the other coil has 1000 turns of 30 gauge wire.

We are also given that the inductance of the 300 turn coil while shorting the 1000 turn coil is 12.188 mH,
The inductance of the 1000 turn coil while shorting the 300 turn coil is 159.69 mH. Both measurements were taken at a 1 kHz test frequency using an LCR meter.

Let's start by solving for the permeability μ. We can do this by using the equation for the inductance of the 300 turn coil:

12.188 × 10^-3 H = μ * N_1^2 * A / (2 * l)

Substituting the given values for N_1, A, and l, we get:

12.188 × 10^-3 H = μ * (300)^2 * A / (2 * 0.1 m)

Simplifying, we get:

μ = 4 * 12.188 × 10^-3 H * 0.1 m / (300)^2 * A

The cross-sectional area of 22 gauge wire is approximately 0.0000006475 m^2, and the cross-sectional area of 30 gauge wire is approximately 0.0000005067 m^2. Since the two coils are wound on the same core, we can assume that they have the same cross-sectional area. Let's use the average of the two values:

A = (0.0000006475 m^2 + 0.0000005067 m^2) / 2 = 0.0000005771 m^2

Substituting this value, we get:

μ = 1.56 × 10^-3 H/m

Next, let's use the equation for the inductance of the 1000 turn coil to solve for the length of the core:

159.69 × 10^-3 H = μ * N_2^2 * A / (2 * l)

Substituting the values we found for μ and A, we get:

159.69 × 10^-3 H = (1.56 × 10^-3 H/m) * (1000)^2 * 0.0000005771 m^2 / (2 * l)

Simplifying, we get:

l = (1.56 × 10^-3 H/m) * (1000)^2 * 0.0000005771 m^2 / (2 * 159.69 × 10^-3 H)

l = 0.0266 m

Therefore, the length of the core is approximately 0.0266 m.

Finally, we can use the same equation for the inductance of the 1000 turn coil to solve for the true value of the permeability μ:

159.69 × 10^-3 H = μ * (1000)^2 * A

Substituting the value of A we found earlier, we get:

159.69 × 10^-3 H = μ * (1000)^2 * 0.0000005771 m^2 / (2 * 0.0266 m)

Simplifying, we get:

μ = 1865.5 H/m

Therefore, the true permeability of the core is approximately 1865.5 H/m.

To determine the number of turns of 30Ga wire required to produce the same inductance as the 300 turn 22Ga coil, we can use the following formula:

L = (μ * N^2 * A) / l

where L is the inductance in henries, μ is the permeability of the core, N is the number of turns, A is the cross-sectional area of the core, and l is the length of the core.

We know that the inductance of the 300 turn 22Ga coil is 12.188mH and the permeability of the core is 1865.5 H/m. We also know that the cross-sectional area of the core is 0.0000005771 m^2 and the length of the core is 0.0266 m.

Substituting these values, we get:

12.188 × 10^-3 H = (1865.5 H/m) * (300)^2 * 0.0000005771 m^2 / 0.0266 m

Solving for N, we get:

N ≈ 712 turns

Therefore, approximately 712 turns of 30Ga wire would be required to produce the same inductance as the 300 turn 22Ga coil.

Hysterysis & Core Saturation

Magnetic Hysteresis

The lag or delay of a magnetic material known commonly as Magnetic Hysteresis, relates to the magnetization properties of a material by which it firstly becomes magnetized and then de-magnetized.

We know that the magnetic flux generated by an electromagnetic coil is the amount of magnetic field or lines of force produced within a given area and that it is more commonly called “Flux Density”. Given the symbol B with the unit of flux density being the Tesla, T.

We also know from the previous tutorials that the magnetic strength of an electromagnet depends upon the number of turns of the coil, the current flowing through the coil or the type of core material being used, and if we increase either the current or the number of turns we can increase the magnetic field strength, symbol H.

Previously, the relative permeability, symbol μr was defined as the ratio of the absolute permeability μ and the permeability of free space μo (a vacuum) and this was given as a constant.

However, the relationship between the flux density, B and the magnetic field strength, H can be defined by the fact that the relative permeability, μr is not a constant but a function of the magnetic field intensity thereby giving magnetic flux density as: B = μ H.

Then the magnetic flux density in the material will be increased by a larger factor as a result of its relative permeability for the material compared to the magnetic flux density in vacuum, μoH and for an air-cored coil this relationship is given as:

So for ferromagnetic materials the ratio of flux density to field strength ( B/H ) is not constant but varies with flux density. However, for air cored coils or any non-magnetic medium core such as woods or plastics, this ratio can be considered as a constant and this constant is known as μo, the permeability of free space, ( μo = 4.π.10-7 H/m ).

By plotting values of flux density, ( B ) against the field strength, ( H ) we can produce a set of curves called Magnetization Curves, Magnetic Hysteresis Curves or more commonly B-H Curves for each type of core material used as shown below.

Magnetization or B-H Curve

The set of magnetization curves, M above represents an example of the relationship between B and H for soft-iron and steel cores but every type of core material will have its own set of magnetic hysteresis curves. You may notice that the flux density increases in proportion to the field strength until it reaches a certain value were it can not increase any more becoming almost level and constant as the field strength continues to increase.

This is because there is a limit to the amount of flux density that can be generated by the core as all the domains in the iron are perfectly aligned. Any further increase will have no effect on the value of M, and the point on the graph where the flux density reaches its limit is called Magnetic Saturation also known as Saturation of the Core and in our simple example above the saturation point of the steel curve begins at about 3000 ampere-turns per metre.

Saturation occurs because as we remember from the previous Magnetism tutorial which included Weber’s theory, the random haphazard arrangement of the molecule structure within the core material changes as the tiny molecular magnets within the material become “lined-up”.

As the magnetic field strength, ( H ) increases these molecular magnets become more and more aligned until they reach perfect alignment producing maximum flux density and any increase in the magnetic field strength due to an increase in the electrical current flowing through the coil will have little or no effect.

Retentivity

Lets assume that we have an electromagnetic coil with a high field strength due to the current flowing through it, and that the ferromagnetic core material has reached its saturation point, maximum flux density. If we now open a switch and remove the magnetizing current flowing through the coil we would expect the magnetic field around the coil to disappear as the magnetic flux reduced to zero.

However, the magnetic flux does not completely disappear as the electromagnetic core material still retains some of its magnetism even when the current has stopped flowing in the coil. This ability for a coil to retain some of its magnetism within the core after the magnetization process has stopped is called Retentivity or remanence, while the amount of flux density still remaining in the core is called Residual Magnetism, BR .

The reason for this that some of the tiny molecular magnets do not return to a completely random pattern and still point in the direction of the original magnetizing field giving them a sort of “memory”. Some ferromagnetic materials have a high retentivity (magnetically hard) making them excellent for producing permanent magnets.

While other ferromagnetic materials have low retentivity (magnetically soft) making them ideal for use in electromagnets, solenoids or relays. One way to reduce this residual flux density to zero is by reversing the direction of the current flowing through the coil, thereby making the value of H, the magnetic field strength negative. This effect is called a Coercive Force, HC .

If this reverse current is increased further the flux density will also increase in the reverse direction until the ferromagnetic core reaches saturation again but in the reverse direction from before. Reducing the magnetizing current, i once again to zero will produce a similar amount of residual magnetism but in the reverse direction.

Then by constantly changing the direction of the magnetizing current through the coil from a positive direction to a negative direction, as would be the case in an AC supply, a Magnetic Hysteresis loop of the ferromagnetic core can be produced.

Magnetic Hysteresis Loop

The Magnetic Hysteresis loop above, shows the behaviour of a ferromagnetic core graphically as the relationship between B and H is non-linear. Starting with an unmagnetised core both B and H will be at zero, point 0 on the magnetisation curve.

If the magnetization current, i is increased in a positive direction to some value the magnetic field strength H increases linearly with i and the flux density B will also increase as shown by the curve from point 0 to point a as it heads towards saturation.

Now if the magnetizing current in the coil is reduced to zero, the magnetic field circulating around the core also reduces to zero. However, the coils magnetic flux will not reach zero due to the residual magnetism present within the core and this is shown on the curve from point a to point b.

To reduce the flux density at point b to zero we need to reverse the current flowing through the coil.

The magnetising force which must be applied to null the residual flux density is called a “Coercive Force”. This coercive force reverses the magnetic field re-arranging the molecular magnets until the core becomes unmagnetised at point c.

An increase in this reverse current causes the core to be magnetized in the opposite direction and increasing this magnetization current further will cause the core to reach its saturation point but in the opposite direction, point d on the curve.

This point is symmetrical to point b. If the magnetizing current is reduced again to zero the residual magnetism present in the core will be equal to the previous value but in reverse at point e.

Again reversing the magnetizing current flowing through the coil this time into a positive direction will cause the magnetic flux to reach zero, point f on the curve and as before increasing the magnetization current further in a positive direction will cause the core to reach saturation at point a.

Then the B-H curve follows the path of a-b-c-d-e-f-a as the magnetizing current flowing through the coil alternates between a positive and negative value such as the cycle of an AC voltage. This path is called a Magnetic Hysteresis Loop.

The effect of magnetic hysteresis shows that the magnetization process of a ferromagnetic core and therefore the flux density depends on which part of the curve the ferromagnetic core is magnetized on as this depends upon the circuits past history giving the core a form of “memory”. Then ferromagnetic materials have memory because they remain magnetized after the external magnetic field has been removed.

However, soft ferromagnetic materials such as iron or silicon steel have very narrow magnetic hysteresis loops resulting in very small amounts of residual magnetism making them ideal for use in relays, solenoids and transformers as they can be easily magnetized and demagnetized.

Since a coercive force must be applied to overcome this residual magnetism, work must be done in closing the hysteresis loop with the energy being used being dissipated as heat in the magnetic material. This heat is known as hysteresis loss, the amount of loss depends on the material’s value of coercive force.

By adding additive’s to the iron metal such as silicon, materials with a very small coercive force can be made that have a very narrow hysteresis loop. Materials with narrow hysteresis loops are easily magnetized and demagnetized and known as soft magnetic materials.

Magnetic Hysteresis Loops for Soft and Hard Materials

Magnetic Hysteresis results in the dissipation of wasted energy in the form of heat with the energy wasted being in proportion to the area of the magnetic hysteresis loop.

Hysteresis losses will always be a problem in AC transformers where the current is constantly changing direction and thus the magnetic poles in the core will cause losses because they constantly reverse direction.

Rotating coils in DC machines will also incur hysteresis losses as they are alternately passing north the south magnetic poles. As said previously, the shape of the hysteresis loop depends upon the nature of the iron or steel used and in the case of iron which is subjected to massive reversals of magnetism, for example transformer cores, it is important that the B-H hysteresis loop is as small as possible.

In the next tutorial about Electromagnetism, we will look at Faraday’s Law of Electromagnetic Induction and see that by moving a wire conductor within a stationary magnetic field it is possible to induce an electric current in the conductor producing a simple generator.