One of the most difficult issues arising in the process of designing an SMPS is the issue of calculating transformers and inductors, including chokes. As you know, an inductor is an inductor designed in such a way that it is capable of withstanding high currents and has negligible operating losses. Most often, inductors are called inductors that operate with a large level of direct current flowing through the winding. The transformer is also a kind of inductor. For brevity, hereinafter, inductance coils will be denoted by KI.

The material outlined below makes it possible not only to create CIs on their own. The author also hopes that readers will be able to use this information to check and change the parameters of the CI during the repetition and repair of radio amateur or industrial structures. Indeed, often the main obstacle to this is the difficulty in acquiring ferrite cores of the specified type or a winding wire of a certain diameter.

It should be noted that the formulas and tables given below can be used when calculating any KI, and not only when calculating chokes and transformers for SMPS. The accuracy of calculating the parameters of the CI based on the method described below is 25-35 %, which in most cases is sufficient for practical purposes. Claims for higher calculation accuracy sometimes found in literary sources raise some doubts, since the reference data of the core manufacturers themselves have an accuracy of the order of 25 % and only some ferrites for signal circuits are determined more accurately.

The main electrical characteristics of the IC are inductance, ohmic resistance of the winding, maximum operating current and the amount of core loss. In addition, important characteristics are dimensions and weight, as well as price; and the complexity of manufacturing.

CI requirements vary depending on the specific application. For example, for many buck converters and most EMC filters, the choke inductance may be chosen higher than the design requires. At the same time, the quality of the converter or filter does not deteriorate, but, on the contrary, becomes better. At the same time, chokes for inverting and boost converters must have a certain, fairly strictly specified by calculation, inductance value. In such cases, a significant deviation of the inductance of the applied CI from the required one - both its decrease and increase - leads to undesirable modes of operation of the SMPS, unnecessary losses and overloads of semiconductor devices. A similar picture is observed for transformers. In some applications, such as push-pull converters and single-stroke converters with "forward key stroke" power transmission, the primary inductance of the transformer is not critical and can always be increased or even decreased if certain conditions are met. At the same time, single-ended converters "on the reverse stroke of the key", which in their essence are inverting converters, are very sensitive to the magnitude of the inductance of the transformer. In this case, the transformer is actually a modified choke. As for the maximum operating current and winding resistance, there is no limit to improvement: almost any choke or transformer can be successfully replaced with a choke or transformer with a higher maximum operating current and lower winding resistance.

Inductance

The inductance of the KI is calculated by the formula:

L = AL * N2 (μH), (1)

where AL is the reference parameter of the core, μH;

N is the number of turns in the winding.

For a ring core with a closed magnetic core without a gap, the AL parameter can be easily calculated independently using the formula:

where, u1 is the initial magnetic permeability of the core material;

u0 - absolute magnetic permeability of vacuum, physical constant having a value of 1.257 × 10-3 μH / mm;

Se is the effective cross-sectional area of the magnetic circuit, mm2;

Ie - effective length of the core, mm.

Reference data for a number of cores without clearance are shown in Tables 1-4. It also indicates the effective geometric parameters of the cores Ie and Se, as well as the relative magnetic permeability of ferrite. When using a material with a different value of the magnetic permeability, the value of the AL parameter should be recalculated:

AL = AL (tab.) * U1 / u1 (tab.) (3)

where AL (table) is the tabular value of the inductance factor of the core;

u1 (table) - magnetic permeability of ferrite, indicated in the table;

u1 is the magnetic permeability of the material used.

It is known that the designation of the brand of domestic ferrites includes an indication of their initial magnetic permeability, for example, ferrite 1000NM has a magnetic permeability mi = 1000, and so on. Typical ferrite permeability ranges are 100-10000. Almost all split cores for power electronics are made of high permeability ferrites. 1500 and more. It should be borne in mind that the higher the magnetic permeability of the ferrite, the higher the core loss at high frequencies. Split cores made of low permeability material are intended for signal circuits and are not recommended for use in SMPS power circuits.

Technical data of some foreign ferrites are given in table. 5. Due to the lack of space, a relatively detailed list is given only for ferrites from Philips, for other companies the author limited himself to popular power ferrites for split cores for SMPS.

Manganese-zinc ferrites of the following grades are most often used for split cores of SMPS:

• ЗС85, ЗС90, 3F3 from Philips;

• N27, N41, N47, N67 from Siemens;

• MLRS, RS40 from TDK;

• B50, B51, B52 by Thomson-LCC;

• F44, F5, F5A from Neosid, etc. Nickel-zinc ferrites are preferred for use at frequencies above 2 MHz, which is outside the operating frequency range of most modern SMPSs. As you can see from the above table, ferrites from different manufacturers have similar parameters and form interchangeable families. They can be replaced, among other things, with domestic ferrites of the 1500MM, 2000MM, 2500MM brands.

Rings from Philips and Siemens have a plastic shell, the color of which indicates the grade of ferrite or powdered iron. On split cores, the material grade is usually indicated in text form. Unfortunately, not all magnetic cores are properly labeled. A rough estimate of the magnetic properties of ferrite can be as follows: as a rule, ferrites with a higher permeability are dark, almost black, they show a noticeably granular structure on chips and fractures, while ferrites with a relatively low permeability are gray in color and a more uniform structure.

The AL value for gapped cores can also be obtained from the tabular data. As the gap increases, the effect is the same as if the magnetic permeability of the core material was decreased. Even relatively small gaps reduce the core permeability by tens and hundreds of times. The resulting effective magnetic permeability te depends mainly on the geometric dimensions and is almost independent of the magnetic permeability of the material:

where Ie is the effective length of the average magnetic line of the core, mm;

g is the total thickness of the gap, mm.

Formula (4) is valid when the following conditions are met: those are much less than the permeability of the core material mi, and the gap g is much less than the dimensions of the cross-section of the core.

Please note that for split cores in tab. 2-4, in addition to the value of the magnetic permeability of ferrite μ, the value of the effective magnetic permeability te for the core without a gap, which has a smaller value, is also given. The fact is that a really split core always has a certain gap, albeit a very small one. In addition, some of the magnetic lines pass by the core, especially if its dimensions are small and the shape differs significantly from the circular one. With very small gaps or low permeability of ferrite, relation (4) is inaccurate, because even with a zero gap, the effective magnetic permeability cannot exceed the magnetic permeability of the core material. At very large gaps, the shape of the magnetic field in them is distorted, which leads to additional errors when using formula (4). The expression "much less" implies a ratio of 10 or more times. Let the readers not be confused by the seeming limited scope of the formula (4), it covers the vast majority of practical cases.

For example, take a core consisting of two E20 / 10/5 W-shaped magnetic wires made of ZS85 material, that is, from ferrite with a permeability μ = 2000. The length of the middle magnetic line of the core is 42.8 mm, the cross-sectional dimensions are 3.5 * 5.0 mm in the thin part of the magnetic wire. Introduce a 0.25 mm thick non-magnetic gasket into the core, the gap width will turn out to be 2 × 0.25 = 0.5 mm. The effective magnetic permeability of the core with a gap (13 = 42.8 / 0.5 = 85.6. In this case, the conditions of applicability of formula (4) are met: m = 85.6 is much less than 2000; gap d = 0.5 mm much less than 3.5 mm.

The final formula for calculating the AL parameter of the gap core is:

where AL (table) and ue (table) are tabular values, and the conditions of applicability are the same as for formula (4).

Let's continue the above example with the E20 / 10/5 core made of ZS85 ferrite. Its tabular values: АL (tab.) = 1.3 MCGN, ue (tab.) 1430. After introducing a gap of 0.5 mm, formula (5) gives the result AL = 0.074 μH.

The limited volume of the journal article does not allow to place data on all types of cores available on the market. The way out is suggested by the following reasoning.

The AL value depends only on two factors: magnetic permeability and core geometry. Almost any closed core can be viewed as a “deformed ring”. For example, a core consisting of two W-shaped halves can be represented as follows: a large ring was cut lengthwise into two thin rings, then these thin rings were deformed into rectangles and made together in the form of a "figure eight". It is very important that with such a geometric (topological) transformation, the AL parameter changes insignificantly. Consequently, any closed core of a complex shape can be mentally subjected to the reverse transformation into a ring. Thus, it becomes clear how to deal with cores that are not described in the tables: it is necessary to measure their geometric dimensions, calculate the length of the average magnetic line and the average cross-section of the magnetic circuit, and then find the AL of the core by formula (2).

For example, for the same core E20 / 10/5, which has an average magnetic line length of approximately 45 mm and an average cross-section of the magnetic circuit of approximately 5 × 6 = 30 mm2, the calculation by formula (2) gives the result AL = 1.257 μH. This is not far from the “true” table value AL = 1.3 µH, which itself has an accuracy of 25 %.

There is also another way. It is not difficult to find the AL value from the results of measuring the inductance of the test winding. Wind a small winding around the core to be tested, for example 10 turns (N = 10). Then, with a measuring bridge or LC meter, measure the resulting inductance L and calculate AL using the formula:

You can find how many turns a winding must have to obtain a given inductance using the formula:

It is easy to see that both last formulas are simple transformations of formula (1).

Core saturation

When a large current flows through the core coil, the magnetic material of the core may saturate. When the core is saturated, its relative magnetic flux

The value decreases sharply, which entails a proportional decrease in inductance. A decrease in inductance causes a further accelerated increase in the current through the IC, etc. In most SMPSs, core saturation is extremely undesirable and can lead to the following negative phenomena:

- an increased level of losses in the core material and an increased level of ohmic losses in the winding wire lead to an unreasonably low efficiency of the SMPS;

- in additional losses cause overheating of the CI, as well as the radio components located nearby; it will be appropriate to mention that the reliability of electronic equipment is usually halved with an increase in temperature for every 6 degrees;

—Strong magnetic fields in the core, combined with a decrease in its magnetic permeability, are many times stronger than in normal operation, a source of interference and interference to small-signal circuits of the SMPS and other devices;

- the rapidly growing current through the IC causes shock current overloads of the SMPS switches, increased ohmic losses in the switches, their overheating and premature failure; e abnormally large pulse currents of the IC entail overheating of the electrolytic capacitors of the power filters, as well as an increased level of noise emitted by the wires and tracks of the SMPS printed circuit board.

The list can be continued, but it is already clear that the core should be avoided in saturation mode. Ferrites enter saturation if the value of the flux density of the magnetic induction exceeds 300 mT

(millitesla), and this value does not so much depend on the brand, ferrite. That is, 300 mT is, as it were, an innate property of ferrites, other magnetic materials have different values of the saturation threshold. For example, transformer iron and powdered iron are saturated at a magnetic flux density of about 1T, that is, they can work in much stronger fields. More accurate values of the saturation threshold for different ferrites are shown in table. 5. The value of the flux density of the magnetic induction in the core is calculated by the following formula:

where u0 is the absolute magnetic permeability of vacuum, 1.257 * 10 ″ 3, μH / mm;

ue is the relative magnetic permeability of the core (not to be confused with the permeability of the core material);

I is the current through the winding, A;

N is the number of turns in the winding;

Ie is the length of the middle magnetic line of the core, mm.

A simple transformation of formula (8) will help to find an answer to the practical question: what is the maximum current that can pass through the inductor before the core enters saturation?

where Bmax is a tabular value, instead of which you can use 300 mT for any power ferrite.

For cores with a gap, it is convenient to substitute expression (4) here. After the abbreviations, we get:

The result is, at first glance, rather paradoxical: the value of the maximum current through the IC with a gap is determined by the ratio of the gap size to the number of winding turns and does not depend on the size and type of the core. However, this seeming paradox can be easily explained. The ferrite core conducts the magnetic field so well that the entire drop in the magnetic field strength falls on the gap. In this case, the magnitude of the flux of magnetic induction, the same for the gap and for the core, depends only on the width of the gap, the current through the winding and the number of turns in the winding and should not exceed 300 mT for conventional power ferrites.

To answer the question of what value the total gap g must be introduced into the core so that it can withstand the specified current without saturation, we transform expression (10) to the following form:

To illustrate the effect of the gap more clearly, we will give the following example. Take a core E30 / 15/7 without a gap, ferrite ZS85, magnetic permeability te = 1700. Let's calculate the number of turns required to obtain an inductance of 500 μH. The core, according to the table, has AL = 1.9 μH. Using formula (7), we get a little more than 16 turns. Knowing the effective length of the core Ie = 67 mm, using the formula (9) we calculate the maximum operating current: Imax = 0.58 A.

Now we introduce a 1 mm thick spacer into the core, the gap will be d = 2 mm. The effective magnetic permeability will decrease. Beyond that, reduce the number of turns in the winding to reduce copper losses and at the same time reduce the core gap. It is important to emphasize that this recommendation does not apply to transformers in which the current flowing through the primary winding is determined by two components: the current transmitted to the secondary winding and a small current that magnetizes the core (magnetisation current).

As you can see, the gap in the choke core plays an extremely important role. However, not all cores allow the insertion of spacers. The ring cores are made one-piece, and instead of "adjusting" the equivalent magnetic permeability by means of a gap, it is necessary to choose a ring with a certain magnetic permeability of the ferrite. This explains the fact of a wide variety of types of magnetic materials used by the industry for the manufacture of rings, while split cores for SMPS, where it is easy to insert a gap, are almost always made of ferrites with high magnetic permeability. The most common when used in SMPS are two types of rings: with low permeability (in the range of 50 ... 200) - for chokes, and with high permeability (1000 and more) - for transformers.

Powdered iron turns out to be the most preferred material for annular solid core chokes operating at high bias currents. The permeability of powdered iron is usually in the range of 40 ... 125, most often there are rings made of materials with a permeability of 50 ... 80. Table 6 shows the reference data of the annular cores made of powdered iron of the company. Philips. It is not difficult to check whether the core enters into saturation during the operation of a conventional SMPS: it is enough to control the shape of the current flowing through the IC using an oscilloscope. The current sensor can be a low-resistance resistor or a current transformer. The KI operating in normal mode will have a geometrically regular triangular or sawtooth current shape. In the case of saturation of the core, the current shape will be bent.

Losses in the winding wire

The example considered in the previous issue of the journal shows that the introduction of a gap into the core makes it possible to significantly increase the maximum current through the IC. The larger the gap, the more current the coil can handle. In order to keep the inductance constant, the winding must contain more turns. However, by increasing the number of turns, we increase the winding resistance. This leads to additional power losses in the wires ("copper losses"):

Rob. = Robm. * 12 (W), (12)

where Rbm. - winding resistance, Ohm;

I is the current through the winding, A.

To calculate the losses in the winding, it is required to take into account the shape of the current through the KI. For example, virtually constant current flows through the chokes in the power supply filters and in many buck converters. For them, the AC component of the current through the KI is relatively small and amounts to 10-20% of the DC current through the winding. To calculate losses in copper, the alternating current component can be neglected and formula (12) can be used directly, substituting into it the average value of the current flowing through the inductor.

The shape of the current in the primary winding of the transformer of the push-pull converter has a shape close to rectangular. If the winding has two halves, then each of them will dissipate 1/2 of the power found by formula (12).

In an SMPS with an intermittent choke current, the current will have a triangular shape with pauses. In this case, the losses in the wire must be calculated using the formula:

where 1ampl is the peak value of the current, A;

t1 - time during which a triangular current flows through the winding, μs;

t0 - time during which there is no current through the winding, μs.

By using a thicker winding wire, the winding resistance can be reduced. Table 6 shows the parameters of the winding wires. In particular, for thick wires, their resistance is indicated at a frequency of 40 kHz, which is a fairly typical operating frequency of an SMPS. The increase in resistance with increasing frequency is due to the so-called skin effect: at high frequencies, the flowing current is displaced onto the outer surface of the wire. The skin effect is most noticeable for thick wires with a high ratio of the cross-sectional area to the length of the outer surface of the wire cross-section. For wires with a diameter of less than 0.5 mm, the effect of the skin effect at frequencies up to 100 kHz is negligible. As a practical measure to combat the skin effect, it is possible to recommend winding in several wires, and it is advisable to choose a diameter of each conductor no more than 1 mm. At the same time, it will also make the winding process easier, since it is not so easy to cope with thick wires. But one should not go to the other extreme, picking up a lot of thin conductors in a bundle, since in this case the winding process becomes excessively complicated, and the gain is insignificant. In SMPSs operating at frequencies below 100 kHz, the use of litz-rat, that is, factory-made wires consisting of a bundle of thin insulated conductors in a common silk braid, which is intended for radio-frequency circuits, does not provide practical advantages. Again, the shape of the current through the winding must be taken into account, and for most chokes, the effect of the skin effect can be ignored.

It is impossible to increase the cross-section of the winding wires infinitely, otherwise the winding cannot be placed on the core. In this case, a larger core must be used. A larger core will have a larger wire winding window and generally a larger AL value. This means that fewer turns will have to be wound to get the same inductance. Fewer turns - less flux of magnetic induction in the core, which means that you can reduce the size of the gap (in the case when the gap is necessary). This will increase the equivalent magnetic permeability of the core and give an even higher AL value, etc. The opposite is also true: if the core is too large, then few wires are required, but the dimensions and cost of the CI will turn out to be high.

In general, the degree of filling the core with a wire can serve as a good indirect indicator of the quality of the design of a transformer or choke. If the core is less than half filled with wire, then, most likely, this indicates that the design of the CI is far from optimal.

The equivalent circuit of the transformer is shown in Figure 1. Without taking into account the ohmic resistance of the windings and losses in the core, the transformer can be represented in the form of the primary inductance L, leakage inductance Ls, primary capacitance C1 and reduced secondary capacitance C2 ″.

When a transformer is used to transfer power directly from the primary to the secondary, it is sought to design it so that L is as large as possible. Generally speaking, the inductance L does not play any "positive" role in such cases. By increasing the inductance, thereby reducing the intrinsic current of the KI, which makes it "less noticeable" for the circuit. A larger inductance has a higher reactance and shunts the pulses transmitted through the transformer to a lesser extent. The magnetization of the transformer core occurs only by the current that is branched off into the inductance of the primary winding. The electrical energy in the transformer is transferred from the primary winding to the secondary winding directly, as if bypassing the core and not magnetizing it. Accordingly, even relatively small transformers are capable of transmitting significant power to the load if they have a high primary inductance and low wire losses.

To obtain the highest primary inductance, gapless cores and high permeability magnetic materials are used for transformers.

This ensures the maximum AL values of the core. In addition, transformers generally need to have a relatively large number of turns in the primary winding. However, some SMPS control circuits operate in a 'hard start' mode when the power is turned on, and the pulse duration can be much longer than in the operating mode. As a result, when the SMPS is started, the core saturates without a gap, the power transistors may fail, and the operation of the SMPS feedback circuits is disrupted. A simple solution to the hard start problem is to introduce a small gap into the transformer core. However, in no case should such a solution be considered as universal, since the gap, helping at start, in normal mode causes additional losses in the copper of the windings and in the power switches of the SMPS. A well-designed control circuit will provide a “soft start” and allow the SMPS to operate reliably without core gaps.

The initial stages of calculating transformers are detailed in the literature. The value of the minimum required inductance of the primary winding obtained as a result of such calculations should be used when creating a transformer based on the method described above for the CI, that is, select the core from the table, calculate the required number of turns using formula (7) and select the winding wires for the primary and secondary windings.

After that, you should check if the core is in saturation. Knowing the magnitude of the inductance, the maximum pulse duration and the maximum operating voltage of the primary winding, you can calculate the maximum current through the inductance of the primary winding of the SMPS (magnetization current):

where U is the voltage across the primary winding. V;

t is the pulse duration, μs;

L is the inductance of the primary winding, μH.

Substituting the obtained value into expression (8), we find the value of the flux density of the magnetic induction in the core. As noted above, for ferrites, it should not exceed ZOOMT.

Expression (14) can be transformed in such a way as to determine the required value of the inductance of the primary winding at a given magnetization current:

where U is the voltage across the KI, V;

t is the pulse duration, μs;

I is the current through the KI, A.

However, it is not enough just to avoid core saturation. This is a necessary condition for the normal operation of the IC, but in addition to this, an acceptable level of losses in the core material ("iron losses") should be ensured.

No magnetic material is perfect. Some ferrites have relatively low resistivity, which causes eddy current losses in the core. In addition, during magnetization reversal, the magnetic material does not return exactly to its original state; the magnetization curve always has a hysteresis loop. Therefore, in each cycle of operation, the core takes some of the energy of the SMPS and converts it into heat. The smaller the width of the hysteresis loop, the lower the losses in the magnetic material.

At the same time, the lower the frequency of the SMPS, the fewer re-magnetization cycles and less losses. In addition, the smaller the core volume, the less the amount of losses in it at the same amplitude of the magnetic field change.

The width of the hysteresis loop depends on the grade of the material, as well as on the amplitude of the change in the flux of magnetic induction in the core. For chokes operating at high, but predominantly direct, winding currents, core losses can often be neglected. The magnetic field of the core of such chokes is almost constant, and the re-magnetization occurs along the so-called private hysteresis loop, which has a small area and, accordingly, low losses.

However, this is not always true, and, for example, some of the simplest buck converter circuits re-magnetize the core of their choke over a large cycle, from zero to peak value. For transformers, the flux of magnetic induction varies either from zero to an amplitude value (single-cycle converters), or from negative to positive amplitude value (push-pull converters). In such cases, ferrite losses can be very high. I have come across unsuccessful designs of transformers, in which, during prolonged operation, the plastic frame of the winding melted due to heating of the ferrite.

Alexey Kuznetsov

Many thanks to Alexey for this material. It helped me in understanding the value of AL - the inductance factor. There were difficulties in calculating matching transformers for HF (1-30MHz). Now everything is quite simple - it is easy to recognize the AL of the existing core through the inductance, which I measure with the device. And knowing the inductance L I need, now, through AL, I can calculate the number of winding turns. Thanks again!