# Magnetics - transformers, boost, buck - ferrite/iron



## PStechPaul (May 1, 2012)

Some basic concepts:

*Magnetic field strength* is defined as

H = I * N / length = Ampere-turns / meter

Thus you can produce any desired field strength by increasing current or number of turns, and by decreasing the length over which the current is applied. 

*Magnetic flux density* is defined as

B = Phi / A = Flux / Area = Vs / m^2 = Teslas

Page 122 of the Epcos catalog provides these formulas along with a description of hysteresis and the mechanism that describes the behavior of magnetic materials. The flux density (B) is related to field strength (H) by

B = u0 * ur(H) * H

The B-H curve shows that, at low levels of H, the flux density follows a curve close to that of a coil in air or vacuum, but soon the curve rises more sharply, corresponding to a higher permeability, which is caused by magnetic domains in the material aligning with the field. Eventually, all of these domains align, and the flux density flattens out, so that it does not increase with field strength, except perhaps a slight amount according to the permeability of vacuum u0. This is called saturation.

The B-H curve also shows the *hysteresis* of the magnetic material, which is caused by remanent magnetism or alignment of the magnetic domains, which do not reverse until a sufficiently strong opposite field is applied. This hysteresis varies considerably with the material, being less for "soft" materials such as iron, and greater for "hard" materials such as ferrite. The area of the hysteresis curve is proportional to the losses of the material.

An inductor is a component which stores energy in the form of current and the electromagnetic field it produces, which is analogous to a capacitor which stores energy in the form of voltage and an electrostatic field. The energy stored in the inductor is:

E = 0.5 * I^2 * L

*Inductance* (L) is related to the slope of the B-H curve, so a steep curve showing a fast rise of flux density with field strength is analogous to a fast rise of voltage with current, and this shows high permeability as well as high inductance. The flux density B may be thought of as voltage and the field strength H as current, so their product is power. It is not quite as simple as that, and the Epcos document goes into real and imaginary components of permeability (complex permeability), which may be thought of as resistive and reactive, or dissipated energy versus stored energy, and thus efficiency.

An air gap may be introduced in the magnetic circuit to lower the permeability but also allow a greater amount of stored energy. This is because the lower permeability allows a much higher current and magnetic field strength. The magnetic material will still saturate at the same flux density, which corresponds to voltage, but the current contributes energy according to a square function. Thus the lower inductance from the reduced permeability with an air gap allows higher current and greater stored energy.

A fixed air gap inductor exhibits a rather sharp saturation curve at the point where most of the magnetic material saturates at the same flux density, depending somewhat on its physical shape, whereas materials such as powdered iron have a distributed air gap, and variations in the size cause saturation to occur less sharply.

The amount of power that can be handled by an inductor or transformer depends on the amount of energy that can be stored in its inductance and how fast the field can be reversed to transfer it. This is determined a little differently for a transformer as compared to an inductor.

A transformer is based on volt-seconds and is limited by the saturation of the magnetic material. According to http://www.smps.us/magnetics.html this limit does not depend on the core's magnetic properties or air gaps, but on the frequency and wave shape (dV/dt). This will determine the maximum power based on heating of the core. Additionally, there will be resistive losses in the windings, and the combination will determine the maximum power for a given design.

An inductor is based more on current and saturation, at which point the effective inductance is reduced and the maximum energy storage is attained. There will typically be a period of time where the inductor will be storing energy applied to it, and another period of time where it will be released into a load. Unlike a transformer, there will be a net DC current flow, and the effective inductance depends on this value.

To determine the power that is to be handled by an inductive component, one may use the amount of energy that can be stored and the maximum frequency that can be applied. For instance, a 100 uH inductor rated at 10 amps can store 0.5*100*10^2 = 5000 uJ or 0.005 Watt-seconds. If you can drive this inductor at 100 kHz, you can transfer 0.005*100 = 0.5 kW or 500 watts. It may actually be 1/2 that amount because part of the time will be storing energy and the remainder will be releasing it.

I'll go into details on selecting a core material, shape, and size, and windings, in a subsequent post.

Note that this is based on my own understanding of magnetics, and I may not have it quite right. Please correct me or clarify as needed.


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## PStechPaul (May 1, 2012)

The core geometry makes a great difference in the magnetic characteristics, and can be complex to calculate, but here are some basics. The inductance of a toroidal core is:

L = u0 * ur * N^2 * A / length

An important concept here is that inductance is proportional to the enclosed area, and this is useful to consider when routing the wiring of a high current circuit, especially when it carries high frequency AC. 

For most coil geometries, a value (AL) is provided that makes the calculation of inductance fairly easy:

L = AL * N^2

AL is often given in terms of nanoHenries / sqrt(N), although sometimes it is given as nH/N or nH/N^2, and sometimes it is in terms of uH. I have some E55/28/21 cores made from type N27 ferrite, and AL is given as 5800 nH, and ue is 1610. A set of two is 55x56x21 mm, the effective length is 124 mm, cross-sectional area is 354 mm^2, and volume is 43900 mm^3. Maximum core losses are 8W/set at 200mT and 25kHz at 100C.

The specifications also show a performance factor, or PF, which is based on the maximum power that can be transferred:

PF = f * Bmax

For the E55/28/21 core of N27 material, PF = 538 at 25 kHz with a copper fill factor of 0.4 (40%). It is possible to get greater fill factor and thus higher power by using square cross-section magnet wire or copper tape, and also by using a higher frequency. The N27 material is optimized for about 25 KHz, while there is also N87 material which will handle 2396 watts at 100 kHz in the same core size. Note that this is 11% higher than 4 times that of N27, which may be due to the u=1780 vs 1610 which is about the same ratio. The N27 cores are in stock at Newark for $2.21 each, while the N87 cores are in UK stock at $4.57 each. Price is one of many trade-offs and design considerations.

To see how the N27 material will perform at twice the rated frequency, or 50 kHz, there is a chart showing the PF vs F. At 25 kHz it is about 6500, and at 50 kHz it is about 8000. So you can get about 23% higher power by doubling the frequency. For N87 material, at 25 kHz, the PF is about 8500, at 50 kHz it is 12500, and at 100 kHz it is 18000. Other materials, such as N49, have a PF of 22000 at 200 kHz and 42000 at 1 MHz. 

To design a transformer, as I am doing for my DC-DC converter using the E55/28/21 N27 cores, I might start by assuming a 50 kHz operational frequency, and a 24 VDC input for 300 VDC output at 3 amps, or 900W. I think this is achievable with this core. Using a simple push-pull center tap topology, I can enter these values in the on-line calculator: http://www.poweresim.com/Xformer/Select_Application_Front_Page.jsp

The application recommends a boost choke topology, but I want to use a transformer, which will provide isolation. The core recommended is EE55B, which are about the same size as what I have (316 mm^2 vs 354 mm^2), and it shows a total of 10 primary turns and 86 secondary turns. The ratio should be 300/24 = 12.5 while this is 86/5 = 17.2. This probably allows for some "headroom" for regulation.

Another way to figure out the primary turns is by using the formula given by http://www.smps.us/magnetics.html for a square wave.

Bmax = (Vpk * 10^8) / (4 * N * Ac * F) = 400 mT (milliTeslas)

Thus, for Ac = 3.54 cm^2 and a maximum flux density of 400:

N = 24V * 10^8 / (400 * 4 * 354 * 50 * 10^3) = 8.5

Another way to determine the number of turns is to use the inductance and a magnetization current of 1% of the continuous current at the intended power level. In this case the input can be considered 24 VRMS and 37.5 amps for 900 watts. A reasonable current is 0.375 amps at 24 volts which is 64 ohms and the inductance would be:

L = 64 / (2 * PI * F) = 200 uH

With AL = 5800 nH = 5.8 uH

N = sqrt(200/5.8) = 5.87

So, 6 turns at 24 volts is 4 volts per turn, and the 300V secondary would be 75 turns.

For a square wave, you can figure the peak current by using the voltage and the inductance and the time it is applied (volt-seconds). A 50 kHz square wave applies the voltage for 10 uSec, and peak current will be:

I = 24 * 10 uSec / 200 uH = 1.2 amps

The field strength is 1.2 amps * 6 = 7.2 A-t / 124 mm (length) = 58 A/m which corresponds to about 200 mT according to the graph. It appears that saturation will occur at about 100 A/m so this is a safe value.

Note: I am not really certain about these calculations but I think they are at least a good first order approximation. The best way (IMHO) to determine the correct values is to build a prototype and subject it to testing. Saturation can be detected by increasing voltage or decreasing frequency until the I/V starts to increase sharply. The size of the wire used for windings was not discussed, but there are standard transformer winding tables that can be used.


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## PStechPaul (May 1, 2012)

There is another good reference on magnetics, contributed by Tim Williams, a regular on the newsgroup sci.electronics.design:
http://webpages.charter.net/dawill/tmoranwms/Elec_Magnetics.html

Some other useful links:
http://www.giangrandi.ch/electronics/trafo/trafo.shtml (line voltage power transformers)

http://schmidt-walter.eit.h-da.de/smps_e/smps_e.html (switchmode supplies)

http://www.vias.org/eltransformers/lee_electronic_transformers_03_03.html (core materials)

http://sci.tech-archive.net/Archive/sci.electronics.design/2005-07/msg01591.html (my own post on SED on induction motor winding)

http://laminationspecialties.com/products.asp?productId=6 (laminations)

http://www.laserlaminations.com/index.html (laser machined laminations)

http://www.phasemotorparts.com/ (electric motor components - China)

http://multimechatronics.com/images/uploads/mech_n/Motors_for_Mechatronics_An_Introduction.pdf (Motors for Mechatronics)

http://ludens.cl/Electron/trafos/trafos.html (transformer winding)

http://tahmidmc.blogspot.com/2013/02/ferrite-transformer-turns-calculation_22.html (ferrite transformer turns calculation)

http://www.thomas-skinner.com/index.html (transformer laminations)

http://sound.westhost.com/xfmr.htm (transformers - section 1 of 3)


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## PStechPaul (May 1, 2012)

While searching for info on magnet wire and windings, I found this very comprehensive text on magnetics and design of inductors and transformers:

http://coefs.uncc.edu/mnoras/files/2013/03/Transformer-and-Inductor-Design-Handbook_Chapter_1.pdf

There are 21 chapters, but I could not find an index, so just substitute numbers to get the various chapters. There is even a chapter on rotary transformers!


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## eldis (Sep 3, 2013)

Thanks for putting all this together, Paul! Very nice summary.


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## Tony Bogs (Apr 12, 2014)

Yes, never knew that there was a formula for calculating the increase in inductance due to fringing flux in a transformer. 
I've seen that many of Paul's posts have SPICE simulation results in them. 
www.femm.info has a free finite element tool for magnetics.


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## PStechPaul (May 1, 2012)

I got an email from ECN for a download of a "DC-DC Book of Knowledge" by RECOM. It looks pretty good.

http://whitepapers.ecnmag.com/20161...rs.ecnmag.com/20161114_recom_bookofknowledge/

https://abm-website-assets.s3.amazonaws.com/ecnmag.com/s3fs-public/Book of Knowledge_WEB_EN_1611.pdf


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