This note is an appendix to the basic work, which derives formulas for calculating the core saturation time. Based on these results, it is possible to calculate the performance of a transformer with a closed magnetic circuit if its reference parameters, such as saturation induction and cross-sectional area, are known. This work not only performs such a calculation but also demonstrates a direct relationship between the Stoletov curve [1], its approximation parameters, and the core saturation time.

From this work, we know that for a high-Q coil, in which active resistance can be neglected, the core saturation time is found as follows:

Moreover, \(B_s\) and \(S\) in this formula are reference data for a specific core.

At this stage, we will calculate the transformer's primary winding. The secondary winding will be easily recalculated later. From formula (1), we immediately obtain the number of turns of our primary winding: \[ N = {\tau\, U \over B_s S} \tag{2}\] It remains to determine \(\tau\)—the core saturation time. We can take this from the frequency of the voltage applied to our winding. First, let's imagine that we have an ideal transformer and that we are applying ideal rectangular pulses to it, and then we will recalculate all this for a real transformer and a sinusoidal current waveform.

Figure 1 schematically shows the voltage waveform applied to the primary winding of a transformer. We are currently considering rectangular pulses. It's clear that in one period, we must saturate the core twice (in the positive and negative directions), and then return it to its initial state twice more. For now, we assume that our core is ideal, and its saturation time is equal to the demagnetization time.

Thus, over a period, we get a total of four \(\tau\), hence the oscillation period: \(T = 4\tau\). Therefore, our formula changes to: \[ N = {T\, U \over 4\, B_s S} \tag{3}\] where: \(T\) is the oscillation period.

As we know, frequency is the reciprocal of the period. Then the final formula for calculating the number of turns of the primary winding of a transformer with rectangular oscillations will be as follows: N = {U over 4, f, B_s S} ag{4}] where: (f) is the oscillation frequency.

To convert to sinusoidal oscillations, it is sufficient to introduce the form factor [2], which is found as the ratio of the effective value to the average (rectified) value. For a sine wave, it is approximately 1.11. This parameter can be simply added to our formula to obtain the final calculation for the number of turns in the primary winding of a transformer with sinusoidal oscillations:

The last formula is exactly equal to the expression for the official transformer calculation, of course, without taking into account some adjustments for the actual design. For example, it does not take into account the fill factor of the magnetic circuit with ferromagnetic material, but this parameter is only relevant for sheet (ribbon) cores [3]. Also, for a real core, \(B_s\) is taken approximately 1.5 times smaller than the reference value to avoid complete saturation.

Let's consider two typical cases at once: the first is a conventional network transformer with a transformer iron core, the second is a high-frequency transformer with a ferrite core. This will allow us to immediately see how the number of turns is affected by frequency, operating magnetic induction, and the effective cross-sectional area of ​​the magnetic circuit.

For a transformer iron core, we will use a formula taking into account the fill factor of the magnetic circuit with ferromagnetic material. Then the effective cross-sectional area is:

\[ S_{eff} = k_z S \tag{6} \]

where: \(k_z\) is the magnetic circuit fill factor, \(S\) is the geometric cross-sectional area. Then the formula for the number of turns of the primary winding for a sinusoidal voltage will be:

\[ N = {U \over 4.44\, f\, B\, k_z S} \tag{7} \]

For a ferrite core, the fill factor is usually not introduced, since the core is solid, and the working formula remains the same as before:

\[ N = {U \over 4.44\, f\, B\, S} \tag{8} \]

In all further calculations, \(B\) will be understood not as the reference saturation induction, but as the selected working magnetic induction. In practice, it is set significantly below saturation to prevent the core from entering a hard mode.

Suppose we need to estimate the number of turns in the primary winding of a 220 V, 50 Hz network transformer. Let's take the following initial data:

\(U = 220\,\text{V}\),
\(f = 50\,\text{Hz}\),
\(B = 1.2\,\text{T}\) is a reasonable operating value for transformer iron,
\(S = 4\,\text{cm}^2 = 4\cdot10^{-4}\,\text{m}^2\),
\(k_z = 0.92\).

Substitute all this into formula (7):

\[ N = {220 \over 4.44 \cdot 50 \cdot 1.2 \cdot 0.92 \cdot 4\cdot10^{-4}} \]

First, calculate the denominator:

\[ 4.44 \cdot 50 \cdot 1.2 \cdot 0.92 \cdot 4\cdot10^{-4} = 0.0980352 \]

Then:

\[ N = {220 \over 0.0980352} \approx 2244 \]

Therefore, the primary winding should contain approximately:

\[ N_1 \approx 2240\text{...}2250\, \text{turns} \tag{9} \]

If we ignore the fill factor and simply substitute the geometric area \(S\), we get a slightly smaller number of turns:

\[ N = {220 \over 4.44 \cdot 50 \cdot 1.2 \cdot 4\cdot10^{-4}} \approx 2065 \]

That is, the magnetic circuit fill factor makes a quite noticeable correction. That is why it is essential to take it into account for sheet and tape cores.

Now let's consider a high-frequency transformer with a ferrite core. To make the example more realistic, let's take not a 220 V mains supply, but, for example, the primary winding of a 24 V converter. We'll choose the following initial data:

\(U = 24\,\text{V}\),
\(f = 10\,\text{kHz} = 10^4\,\text{Hz}\),
\(B = 0.2\,\text{T}\) is a typical conservative operating value for ferrite,
\(S = 1\,\text{cm}^2 = 1\cdot10^{-4}\,\text{m}^2\).

Substitute these values ​​into formula (8):

\[ N = {24 \over 4.44 \cdot 10^4 \cdot 0.2 \cdot 1\cdot10^{-4}} \]

Calculate the denominator:

\[ 4.44 \cdot 10^4 \cdot 0.2 \cdot 1\cdot10^{-4} = 0.888 \]

Then:

\[ N = {24 \over 0.888} \approx 27 \]

Therefore, the primary winding should contain approximately:

\[ N_1 \approx 27\, \text{turns} \tag{10} \]

In the first case, the line transformer required approximately 2244 turns, while in the second case, only about 27 turns. The difference is enormous, and it is determined primarily by the frequency: the higher the frequency, the fewer turns required for the same magnetic induction and core area. This is why high-frequency transformers are significantly more compact than line transformers.

Furthermore, for a transformer iron core, the fill factor \(k_z\) had to be taken into account, since the magnetic circuit is assembled from individual sheets or tape, and not its entire geometric cross-section is occupied by the ferromagnetic material. For ferrite, this correction is no longer necessary, since its magnetic circuit is solid.

Once the number of turns of the primary winding has been determined, calculating the secondary winding is much simpler. For an ideal transformer, the voltage ratio is equal to the number of turns:

\[ {U_2 \over U_1} = {N_2 \over N_1} \tag{11} \]

From this we immediately obtain:

\[ N_2 = N_1 {U_2 \over U_1} \tag{12} \]

where: \(U_1\) is the primary winding voltage, \(U_2\) is the secondary winding voltage, \(N_1\) is the number of turns of the primary winding, \(N_2\) is the number of turns of the secondary winding.

Of course, in a real transformer, one must take into account the voltage drop across the winding resistance, losses in the magnetic circuit, and sag under load. However, as a first approximation, formula (12) is quite sufficient.

Let's continue the previous example with a 220 V, 50 Hz network transformer. We obtained for the primary winding:

\[ N_1 \approx 2244\, \text{turns} \]

Now let's say we need to obtain the voltage on the secondary winding:

\[ U_2 = 12\,\text{V} \]

Then, using formula (12):

\[ N_2 = 2244 \cdot {12 \over 220} \]

Let's perform the calculation:

\[ N_2 \approx 122.4 \]

Rounding to the nearest whole number, we get:

\[ N_2 \approx 122\, \text{turns} \tag{13} \]

In practice, the number of turns of the secondary winding is often slightly increased if it is necessary to compensate for voltage drop under load. Therefore, the actual winding may be, for example, 124...128 turns, depending on the transformer power and the required accuracy.

Now let's continue with the ferrite transformer example. In the previous section, we obtained:

\[ N_1 \approx 27\, \text{turns} \]

Suppose we want to obtain the output here:

\[ U_2 = 12\,\text{B} \]

Then:

\[ N_2 = 27 \cdot {12 \over 24} \]

Therefore:

\[ N_2 = 13.5 \]

Nearest integer value:

\[ N_2 \approx 14\, \text{turns} \tag{14} \]

As in the previous case, in practice this number may be slightly adjusted depending on the voltage drop across the switches, the pulse shape, the rectification circuit, and the required stabilization. However, for a preliminary calculation, this accuracy is quite sufficient.

Once the number of turns has been determined, the wire thickness can be estimated. To do this, you first need to know the currents in the windings. If the transformer power is equal to \(P\), then for the ideal case:

\[ I_1 = {P \over U_1}, \qquad I_2 = {P \over U_2} \tag{15} \]

Next, the permissible current density \(J\) in the wire is selected, after which the required cross-sectional area of ​​the core is found as:

\[ A = {I \over J} \tag{16} \]

If the wire is round, its diameter can be estimated from the area of ​​the circle:

\[ A = {\pi d^2 \over 4} \]

from:

\[ d = \sqrt{ {4A \over \pi} } \tag{17} \]

Here \(A\) is expressed in mm², and \(d\) is expressed in mm.

For a rough estimate, we can use:

For a 50 Hz power transformer: (J approx. 2.5 A/mm^2);
For a 10 kHz high-frequency transformer: (J approx. 4 A/mm^2).

These are not strict values, but rather reasonable guidelines for preliminary calculations. For a more precise design, the current density is selected taking into account cooling, operating mode, permissible heating, and winding method.

To make the estimate more specific, let's assume the power of our network transformer is:

\[ P = 50\,\text{W} \]

Then the primary winding current:

\[ I_1 = {50 \over 220} \approx 0.227\,\text{A} \]

Secondary winding current:

\[ I_2 = {50 \over 12} \approx 4.17\,\text{A} \]

Now let's find the required wire area for the primary winding. At (J = 2.5, A/mm^2):

\[ A_1 = {0.227 \over 2.5} \approx 0.0908\, \text{mm}^2 \]

Then the diameter:

\[ d_1 = \sqrt{ {4\cdot 0.0908 \over \pi} } \approx 0.34\, \text{mm} \]

Therefore, a wire with a diameter of approximately:

\[ d_1 \approx 0.35\,\text{mm} \tag{18} \]

Similarly for the secondary winding:

\[ A_2 = {4.17 \over 2.5} \approx 1.67\,\text{mm}^2 \]

\[ d_2 = \sqrt{ {4\cdot1.67 \over \pi} } \approx 1.46\,\text{mm} \]

That is, for the secondary winding, a wire with a diameter of p is requiredApproximately:

\[ d_2 \approx 1.5\,\text{mm} \tag{19} \]

If a wire of this diameter is inconvenient to fit into the magnetic circuit window, instead of one thick core, several thinner wires can be used, connected in parallel, with the same total cross-sectional area.

Now let's perform the same assessment for a ferrite transformer. For clarity, let's take its power:

\[ P = 24\,\text{W} \]

Then the primary winding current at 24 V will be:

\[ I_1 = {24 \over 24} = 1\,\text{A} \]

Secondary winding current at 12 V:

\[ I_2 = {24 \over 12} = 2\,\text{A} \]

For a high-frequency transformer, we take \(J = 4\,\text{A/mm}^2\). Then for the primary winding:

\[ A_1 = {1 \over 4} = 0.25\,\text{mm}^2 \]

\[ d_1 = \sqrt{ {4\cdot0.25 \over \pi} } \approx 0.56\,\text{mm} \]

That is, the primary winding wire will be approximately:

\[ d_1 \approx 0.56\,\text{mm} \tag{20} \]

For the secondary winding:

\[ A_2 = {2 \over 4} = 0.5\,\text{mm}^2 \]

\[ d_2 = \sqrt{ {4\cdot0.5 \over \pi} } \approx 0.80\,\text{mm} \]

Therefore:

\[ d_2 \approx 0.8\,\text{mm} \tag{21} \]

At a frequency of 10 kHz, this calculation can still be used as an estimate. But as the frequency increases further, the skin effect must be taken into account, causing the current to be forced toward the surface of the conductor. In such cases, Litz wire or several parallel small-diameter insulated conductors are often used.

So, after calculating the number of turns of the primary winding, the secondary winding is determined by a simple conversion based on the voltage ratio. Then, using the known transformer power, the currents in the windings can be found, the required wire cross-section can be estimated, and its diameter can be selected.

For the examples considered, the following approximate values ​​were obtained:

Power transformer, 220/12 V, 50 Hz, 50 W:
Primary winding: (N_1) approximately 2244 turns, wire (d_1) approximately 0.35 mm;
Secondary winding: (N_2) approximately 122 turns, wire (d_2) approximately 1.5 mm.

Ferrite transformer, 24/12 V, 10 kHz, 24 W:
Primary winding: (N_1) approximately 27 turns, wire diameter (d_1) approximately 0.56 mm;
Secondary winding: (N_2) approximately 14 turns, wire diameter (d_2) approximately 0.8 mm.

Of course, this is only a preliminary calculation. During practical transformer design, additional checks are made to ensure that both windings fit within the magnetic core window, that the insulation between the layers is sufficient, that the permissible current density is not exceeded, and that the heating during long-term operation is not excessive. Nevertheless, even at this stage, a very plausible estimate of the future design can be obtained.

This paper examined a simple and intuitive method for calculating transformer windings, based on the core saturation time. Unlike the classical approach, where formulas are introduced as empirical, here they follow directly from the physics of the process—the accumulation of magnetic flux in the core under the action of an applied voltage.

The key result is the expression:

which relates the voltage on the winding to the time it takes for the magnetic induction to reach a given level. In fact, this is the integral form of the law of electromagnetic induction, written in terms of the finite change in magnetic induction.

It is from this relationship that all practical formulas for calculating the number of turns are naturally derived: for both rectangular pulses and sinusoidal voltage. Moreover, the coefficients 4 and 4.44 are not "magic numbers", but a direct consequence of the shape of the applied voltage.

It is especially important to note that the transition from a rectangular to a sinusoidal signal is accomplished through the form factor. This factor reflects the difference between the effective and average voltage values, and therefore directly affects the rate of increase of the magnetic flux in the core.

Here, a direct connection with the Stoletov curve is evident. As is well known, the Stoletov curve describes the dependence of magnetic induction \(B\) on the magnetic field strength \(H\), and in parametric termsThis form can be approximated using a set of coefficients. These coefficients determine how quickly the magnetic flux density increases with increasing field strength, and, therefore, how quickly the core approaches saturation.

On the other hand, the saturation time \(\tau\) is determined by the integral of the applied voltage, that is, the rate of change of magnetic flux. Thus, the Stoletov curve approximation coefficients effectively determine the dynamics of magnetic flux density change over time.

In other words:

— the Stoletov curve defines the static dependence of \(B(H)\);
— its coefficients determine the "steepness" of this dependence;
— and through the induction equation, this steepness directly translates into the saturation time of the core.

This is the fundamental relationship underlying all the formulas given: saturation time is not a separate parameter, but a consequence of both the properties of the material (via the Stoletov curve) and the operating mode (via the shape and magnitude of the applied voltage).

Practical examples have shown that:

— the number of turns is inversely proportional to frequency;
— increasing the working induction reduces the number of turns, but brings the core closer to saturation;
— the magnetic circuit fill factor has a significant effect on laminated cores;
— high-frequency transformers require orders of magnitude fewer turns than grid transformers.

It has also been shown that after determining the number of turns of the primary winding, the secondary winding can be found by simply recalculating the voltage ratio, and then, based on the known power, the current can be estimated and the wire cross-section selected.

Thus, the entire transformer calculation comes down to three basic steps:

1) selecting the operating magnetic induction;
2) calculating the number of turns to avoid saturation;
3) selecting wires based on the permissible current density.

Despite its simplicity, this approach yields the correct order of magnitude and is consistent with practice. And most importantly, it allows us to see the physical meaning of what is happening: how voltage, signal shape, and material properties together determine the transformer's operation.