ELECTRICAL POWER TRANSFORMER AND INDUCTOR DESIGN
BASIC PRINCIPLES, THEORY, CALCULATION
Before discussing the calculation of magnetic components for switching power supplies, let's just quickly go over the basic concepts and definitions.
Transformer is a passive device which transfers alternating (AC) electric energy from one circuit into another through electromagnetic induction. It consists of a ferromagnetic core and two or more coils (windings).
A changing current in the primary winding creates an alternating magnetic field in the core. The core multiplies this field and couples most of the flux through the secondary windings. This in turn induces alternating voltage (electromotive force, or emf) in each of the secondary coil according to Faraday's law.
Power transformer in SMPS is designed to change amplitude of highfrequency pulses by the turns ratio and to provide isolation between circuits. Note that it can't transfer a DC component of a pulse: in a steady state mode net
voltseconds across each winding should be zero, otherwise the core will saturate.
DC output voltage can be obtained only by using rectifiers. Nevertheless, an average voltage across a real coil's terminals can be nonzero due to nonzero wire's resistance. This DC offset can be used for
lossless sensing of an average current across an inductor or a transformer winding with unidirectional current: if you add an RC network parallel to the coil, the voltage across the capacitor is proportional to the coil's average current. For better thermal stability the wire can be made of low TCR material, such as a copper alloy.
MAGNETICS DESIGNING
normally involves tradeoffs between size, cost and power losses. The main constraint in all cases (except for saturable inductors) is that peak magnetic flux density
Bmax should not approach the core material's saturation flux value
Bsat. Note that in higher frequencies, core loss rather than saturation can become the main limiting factor for Bmax. The flux change is a function of the applied
voltseconds and the core geometry. Excessive voltseconds applied to a coil cause core saturation. When it happens, the windings are effectively shorted out.
The table below provides the formulas for Bmax in a steady state operation as function of
N×Ac product, applied voltage and frequency for common voltage waveforms. Contrary to popular misconception, Bmax does not depend on the magnetic material properties or air gaps. It does not depend on the transferred power neither. However, for thermal reasons, we have to limit ohmic losses in the wires. Most textbooks provide formulas for estimation of the core size based on the product of magnetic crosssection area by the window area available for the winding. Unfortunately, this method is not very helpful because these formulas are based on pretty much arbitrary selection of current density and on an assumption of a certain window utilization (fill) factor. Depending on the insulation grade and application requirements, current density in the copper can be selected anywhere between 180 and 500 A/sq.cm. The fill factor can be anywhere between 0.1 and 0.5 depending on the transformer construction and insulation. As the result, practical design may involve several iterations. Anyway, once you selected the magnetic material, you can find from datasheet its saturation flux Bsat and pick some derating, such as 70% or so. Then you determine minimum required
N×Ac product to assure that
Bmax<0.7×Bsat. Knowing "Ac" of a chosen core size, we can find primary turns N. Secondary turns are calculated based on the required output voltage for a given
SMPS topology.
Function 
Waveform 
Bmax, gauss 
Sine wave 

Vrms×10^{8}/4.44N×Ac×F 
Square wave 

Vpk×10^{8}/4N×Ac×F 
Bipolar pulses with D=Ton/T=Ton×F
(0<D<0.5) 

Vpk×D×10^{8}/2N×Ac×F 
Unipolar pulses
with passive reset 

Br+Vpk×Ton×10^{8}/N×Ac 
In these and other equations: V  voltage
(volts), N  coil's turns, Ac  core's crosssectional area (sq.cm), F frequency (hertz), Br  remanence (gauss) 
When you get an actual prototype you can measure the hot spot temperature by placing a thermocouple under the coil and adjust the design if neccesary. S.A.Mulder proposed [in 1990 Phillips App Note] an empirical formula for thermal resistance of a wirewound transformer measured at the hot spot:
Rth≈(53...61)×Ve^{0.54} ^{o}C/watt, where Ve core volume in cubic centimeters. From this, a "rule of thumb"
hotspot temperature rise in Celsius vs. total losses P(watt) is:
T_{rise}≈50/sqrt(Ve)×P. Note that the above thermal relationship as well as most textbook procedures related to the magnetics thermal management are applicable to natural convection cooling. For applications with forced airflow or conduction cooling these procedures results in an overdesigned part because of an overstated temperature rise.
In general, ideal SMPS transformers need to transfer all energy instantaneously from one winding to another while storing no or little energy in the process. Some topologies do need certain amount of energy stored in magnetizing inductance for a proper operation. Conversely, a
power inductor is used in SMPS as an energy storage device. It accumulates energy in the magnetic field as current flows through it, and then transfers it into another circuit during the alternate part of the switching cycle. In power supplies, the inductors are also used for filtering out high frequency currents (in which case they are often called
chokes). In power inductors the current is controlled. For such "
currentdriven" coils:
B=L×Ipk×10^{8}/N×Ac,
where L  inductance (in henry), Ipk  peak current in amps, B  flux in gauss. All units and formulas in this page are given in CGS (see CGS to SI
unit converter). Once you set Bmax and choose a core size, you can find from the above equation the number of turns for the desired inductance L:
N=L×Ipk×10^{8}/Bmax×Ac.
Note that "L" is not constant. If the current keeps increasing, at some point Bmax will be approaching Bsat and "L" will start dropping. To prevent the magnetic material saturation at a required current, an air gap can be introduced. The length of a net discrete gap:
lg≈0.4×π×N×Ipk/Bmax.
In practice, the gap may need to be selected slightly larger than the calculated value due to flux fringing. Combining the above equations for N and lg yields:
lg≈0.4×π×L×Ipk^{2}/(Bmax×Ac).
The gap is used not only in inductors. It is also often introduced in transformers to increase working flux swing in singleended topologies, to store more energy in magnetizing inductance, and to stabilize the inductance value. For powder metal materials with a distributed gap and soft saturation curve, the calculation process may take several iterations. In short, you can first pick a powder core based on desired L×Ipk
^{2} by using manufacturer's charts. Then determine the turns
, where AL  specific inductance in mH/1000 turns (which is nH/turn) from the data sheet. Then find peak bias
H=0.4×π×N×Ipk/le (Oersted), determine the rolloff in percentage of initial permeability, and correct the turns for the desired L.
Below you will find more magnetics theory, transformer and inductor design information, tutorials, tools and free downloads.