**1. Magnetic field in matter**

Prof. Roman Kurdziel „Podstawy elektrotechniki” wydanie II całkowicie zmienione. wrote:The magnetic field in a vacuum depends only on the electrical circuits that produce it. In material environments there is also the influence of molecular currents in particles of matter. An electron moving around the nucleus with an angular velocity of omega_zero on an orbit with a radius r represents the elementary magnetic dipole. Magnetic dipoles are generally chaotic in matter, so that the body does not exhibit a magnetic state unless it is exposed to an external magnetic field, i.e. created by external causes, e.g. current in any electrical circuit.

The elementary magnetic dipole placed in the external magnetic field is affected by a mechanical moment, which puts the electron, regardless of orbital motion, in precession motion, similar to whirligig. The precession axis is the H field strength vector. This creates an additional magnetic field that weakens the external field slightly. This phenomenon is called diamagnetism.

For the reason given above, all bodies should have diamagnetism. However, in many bodies, placed in the external magnetic field, the reverse phenomenon is observed, i.e. some amplification of the external magnetic field. The phenomenon can easily be explained if it is assumed that the electron, in addition to orbital motion, performs a rotational movement around its axis, called an electron spin. The spin of the electron is accompanied regardless of the magnetic moment resulting from orbital motion, the spin magnetic moment p_s.

In individual atoms, some of the electrons spin in one direction, while the others in the opposite direction, which corresponds to the opposite return of the spin moment. If the numbers of electrons rotating back and forth are equal, the sum of spin magnetic moments is zero and the body exhibits diamagnetic properties resulting from the orbital motion of the electrons. However, if the number of electrons with a certain direction of rotation prevails, the sum of spin magnetic moments is different from zero and the atom exhibits a certain resultant spin moment, which in the external magnetic field tends to take a position consistent with the direction of field strength. The spin dipole field and the external magnetic field add up, i.e. the presence of this type of matter increases the magnetic field relative to the field that a given electrical circuit would produce in a vacuum. This phenomenon is called paramagnetism.

The impact of the environment on the magnetic field is therefore marked: in diamagnetic environments, a decrease, and in paramagnetic environments, an increase in magnetic inductance B in relation to the inductance B_o, which a given external field of H intensity would cause in a vacuum. The ratio of Resultant B induction to external field strength H is called the magnetic permeability of the environment

$$u = \frac{B}{H}$$

(...) At certain values of the ratio of the distance D between atoms to the diameter of the atom d, namely when 1.5 < D/d < 3.5, there are conditions favorable for the spontaneous parallel arrangement of the resultant spin moments of adjacent atoms. Clusters of atoms are formed, i.e. domains with the same orientation of spin magnetic moments, 10 ^ 14 to 10 ^ 16 atoms, and behave like correspondingly large magnetic dipoles. Such bodies were called ferromagnetic bodies because the above properties were first observed in iron.

(...) In ferromagnetic bodies, the relationship B = f (H) is non-linear.

The above introduction illustrates phenomena occurring in matter under the influence of a magnetic field. Is the above knowledge about the properties of the magnetic field in matter required to design a transformer? No, but it makes it easier to understand the phenomena occurring in the transformer. The topic of winding selection and estimation of the power that the core of the given material and cross-section can transfer, is repeated on the forum many times, which is why I decided to bring this topic closer. Especially, that from year to year, less and less used transformers working with 50/60Hz are used, and switching power supplies are increasingly used.

In the following text I will focus only on transformers that do not have a crack in the core.

**2. Core material parameters**

Transformer core materials are not linear elements. Core parameters strongly depend, among others, on:

- temperature

- magnetic field strength

- frequencies and changes in the magnetic flux amplitude

For this reason, material documentation on one chart often presents material behavior for e.g. 25 ° C and 100 ° C to illustrate a certain range of material work.

Let's analyze the parameters given in the 3C90 core documentation:

https://elnamagnetics.com/wp-content/uploads/...ube-Materials/3C90_Material_Specification.pdf

$$\matrix{u_i \text{ – It is the amplitude permeability of the magnetic core when the intensity}\\

\text{of the magnetic field is unboundedly close to “zero”. }\\

\\

u_a\text{ – It is the relative permeability acquired from the maximum value of the magnetic flux}\\

\text{density and the maximum value of the intensity of the magnetic field when there are}\\

\text{periodic changes in a magnetic core that is in a demagnetized state and a magnetic field}\\

\text{that makes the average value of the intensity “zero” is applied. }\\

\\

P_v\text{ – Losses in core under certain conditions } (\frac{kW}{m^3} = \frac{mW}{cm^3})\\

\text{ρ – Material resistivity }\\

T_c\text{ – Curie temperature}\\

\text{Density – Material density }}$$

After the above parameters we can learn a bit about the material, but they are of not very usefull when designing the transformer. They are used mainly to quickly estimate the material's capabilities. Much more important values are those contained in the charts:

Fig. 1 shows magnetic permeability as a complex number (u'_s and already '' _ s). The ratio of these two values (u '' _ s / u'_s) determines the material loss tangent (the ratio of the power stored in the magnetic field to the dissipated power in the form of heat).

Fig. 2 Initial permeability graph versus core temperature.

Fig. 3 Graph of magnetic hysteresis loop

Fig. 4 Graph of permeability values relative to the peak of the magnetic flux (for a sinusoidal wave).

Fig. 5 Magnetic permeability of a material by constant magnetic field (H_dc) magnetization at zero value of alternating magnetic field strength.

Fig. 6 Very important graph. It describes to us a very important B * f relationship to material losses. The B * f relationship directly determines in what conditions we can "squeeze" the most power from the material at specific losses in iron. Keep in mind that this is a graph given for a 100 degree Celsius core.

Fig. 7 The graph shows the effect of temperature on iron losses for several points of the previous graph.

**3. Assumptions for transformers**

We already know a bit about the parameters describing the materials from which the transformers are made, but what do we need to start physical winding of the transformer? Where to begin? From the converter topology and its assumptions. Some topologies (e.g. flyback or LLC) use transformers with gap. However, I will focus here on calculations for transformers without a gap, which can be found in such topologies as LCC, half / full bridge (including phase shift, PWM and other variations), push-pull and others.

**3.1 Primary winding voltage, frequency and filling.**

The voltage applied to the primary winding forces the magnetizing current to flow on the primary winding. The parameter that we should specify is the definite integral of the voltage on the primary winding in the angle 0 to PI (for a classic bridge with 50% filling). We obtain the voltage surface area in the time domain, i.e. voltoseconds [V * s] or Weber (wb), i.e. the unit of magnetic flux. For rectangular voltage it is child's play - amplitude * time, for sine it is not difficult to determine. Let the integrals finally be useful for you in something practical

**3.2 Rms current (RMS current) of the primary and secondary windings**

Quite an obvious parameter, the power supply supplies something, so we need to determine the square root mean current that will flow through the primary and secondary windings. This value consists of the current consumed by the load and the magnetizing current.

**3.3 Secondary voltage**

As above, there is nothing to dwell on this.

**3.4 Current in the cable**

If we have already assumed what current will flow through our windings, now it is worth considering what cross-section of the wires. Giving too small a cross-section of wires, we will have a voltage drop across the winding resistance and additional losses. When we give too large a section, we will have to use a core with a larger window and / or section.

**3.5 Fill factor.**

This is a relatively difficult to estimate factor. In a real transformer we are not able to use 100% of the core window and we need to determine to what extent we will use it. This factor will be influenced, among others, by: bobbin dimensions, winding varnish thickness, amount of insulation used, technique and winding winding quality.

**3.6 Material, iron losses and Delta B**

It is sometimes difficult to talk about the choice of material here, because the fewest ones (3C90, 3F3, F-827, F-867) are the easiest available and we need to adjust our parameters to the material. You have to consider here the losses in the core, the frequency of the converter and the delta B at which the core will work. Of course, all 3 parameters are interrelated - the higher the frequency and/or delta B, the greater the loss in the core.

Fortunately, the following chart can help us find optimal values:

What is Delta B anyway? This is the difference between the minimum and maximum magnetic induction with which the core works. It is worth remembering that for symmetrical systems such as the classic half bridge it is the value between the negative and positive value of the magnetic induction amplitude.

Bmax for which we can determine iron losses is not the same as delta B. This is the maximum amplitude from the average value (in the case of a symmetrical waveform from zero), so if we have delta B at 100mT, then we consider losses for Bmax 50mT, but there are exceptions - when the magnetic induction constant component appears. When does it appear? For example, during the first dozen or so cycles of starting the power supply.

**4. Calculation**

Very accurate calculation of the transformer parameters requires the use of very complex formulas and not easy to use. In order to simplify the equations describing the transformer, a number of assumptions in the model have been introduced, which, unfortunately, have a large impact on the discrepancy between reality and calculations. Such simplifications include:

- uniform field strength in the core. At the same time, the winding is treated as a coil with a length of at least 20 times its radius.

- all magnetic flux passes through the core (no leakage flux).

- the core has no gap

**4.1 Ap**

Having the above assumptions, we can determine the power factor for our transformer:

$$A_p = \frac{\{V_1 * D_{on}\}*2*I_{1rms}}{f_s*ΔB*J*k}$$

As we would expect, increasing the transformer power (voltosecond * 2 * rms average current) increases the required transformer power factor. The denominator of this equation is much more interesting. We can conclude that by increasing the frequency, delta B, reducing the cross-section of the cable and improving the fill factor of the winding, we reduce the required value of the core power factor.

However, by increasing the value of the elements from the denominator, we increase the losses in iron and copper (in addition to the fill factor). The constructor and the requirements set for him are responsible for ensuring a proper balance between losses and core size.

Having the calculated power factor we can compare it to the real power factor of the core. If this parameter is not given, we can calculate it by the product of the cross-sectional area of the core and the window of this core.

**4.2. Number of turns and magnetizing current.**

$$n_1 = \frac{\{V_1*t_{on}\}}{ΔB*A_e}$$

The increase in the number of turns on the primary winding will have a higher value of magnetic flux induction (weber, voltosecond) and the decrease in the number of turns will be due to the higher value of delta B and the core cross-section.

To be sure, the magnitude of the magnetizing current can be calculated:

$$ΔI = \frac{\{V_1*t_{on}\}*l_e}{{N_1}^2 * A_e * µ_i * µ}$$

where:

$$\matrix{

l_e \text{ = average magnetic path }\\

A_e \text{ = effective core cross section }\\

µ_i \text{ = initial magnetic permeability of the material }\\

µ \text{ = magnetic permeability in a vacuum }

}$$

It must be remembered that this is the difference between the maximum and minimum current amplitude. As the system works symmetrically, the maximum amplitude in one direction is half of this value. The exception is starting the transformer. Before the work is settled, the calculated difference will be asymmetrical and may lead to saturation of the core. Some controllers (eg L6699) have a "safe-start" algorithm protecting against such saturation of the core in the initial cycles of the power supply.

If we are satisfied with the magnetizing current, we can go about the physical design of the transformer.

**5 Theory and reality**

The calculated number of windings is often not the end of design. In fact, there are still many elements that affect the final appearance of the transformer. Among others:

- magnetic coupling factor

- leakage inductance

- inter-turn capacities

- EMI emission

- safety standards

- material selection for the insulator and spacing (creepage, clearance)

- heat dissipation from the core

and probably many more.

It must be remembered that the above formulas need to be modified for certain specific conditions. For example, if we use a core made of silicon metal sheets, it should be remembered that in this case the effective cross-section is slightly smaller than the measured cross-section due to ... insulation between individual sheets.

The gap in the core causes a decrease in the magnetic permeability value of the core, which increases the magnetizing current and energy stored in the magnetic field.

A very important parameter, which unfortunately is not calculated here, is the leakage inductance. Estimating this value is very difficult, because you have to count practically the magnetic flux for the entire space around the transformer ... Especially since the location of the primary and secondary windings relative to each other is very important.

Unfortunately, the most effective and fastest method of checking the leakage inductance is by measuring on an existing transformer ...

In the network you can find a lot of good studies on the determination of mutual inductance by the method of shifting and counter-winding winding, so I will not rewrite them. Unfortunately, calculating this value is theoretically not easy and you have to rely on Biot-Savart law.

Designing transformers is not a simple thing, but I hope that the above article will help many beginners to enter this world more smoothly. When designing a transformer, you need to have an overview of the entire power supply (and often what it will supply), and not just focus on it as one element.

source:

https://www.youtube.com/watch?v=L0flpv8FAu8

https://www.youtube.com/watch?v=3nfqBzPMknY

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