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. Normally 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.
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. However, 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).
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 reach the core material's saturation flux value
Bsat. In a "
voltsecond driven" coil, the flux change is a function of the applied voltseconds and the core geometry. Contrary to popular misconception, it does not depend on the core's magnetic properties or air gaps.
The table below provides the formulas for
Bmax for common voltage waveforms in a steady state operation. The
N×Ac product should be selected so that
Bmax<0.7×Bsat at maximum operating temperature. Excessive applied voltseconds cause core saturation.
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 it happens, the windings are effectively shorted out. Note that in higher frequencies, core loss rather than saturation normally becomes the main limiting factor.
In inductors normally the coil 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). From this we can find the number of turns for the desired inductance L:
N=L×Ipk×10^{8}/Bmax×Ac.
Note that L is not constant. If the Ipk keeps increasing, at some point Bmax will be approaching Bsat and "L" will start dropping. To prevent the core saturation at a required current, an air gap is usually introduced. The length of an equivalent discrete gap:
lg≈0.4×π×N×Ipk/Bmax .
Note that the real gap should be selected slightly larger than the calculated value due to flux fringing. Combining the above equations yields:
lg≈0.4×π×L×Ipk^{2}/(Bmax×Ac). The gap is also often introduced to increase working flux swing in singleended topologies, to store more energy in magnetizing inductance, and to stabilize the inductance value.
For powder metal cores with a distributed gap and soft saturation curve, the calculation process may take several iterations. In short, you can first pick a core based on desired L×Ipk
^{2} by using manufacturer's recommendations. Then determine the turns
, where AL  specific inductance in mH/1000 turns (which is µH/turn) from the data sheet. Then find peak bias
H=0.4×π×N×Ipk/le (oersteds), determine the rolloff in percentage of initial permeability, and correct the turns for the desired L.
Another important constrain to be aware of is a maximum "hot spot" temperature rise. S.A.Mulder proposed in a Phillips App Note (1990) 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 this relationship as well as most textbook's procedures related to the mags thermal management are applicable to natural convection cooling. For applications with forced airflow or conduction cooling these procedures may result in an overdesigned component because of an overstated temperature rise.
Below you will find more magnetics theory, transformer and inductor design information, tutorials, tools and free downloads.
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