ELECTRICAL POWER TRANSFORMER AND INDUCTOR DESIGN

In power supplies, the inductors are also used for filtering out high frequency currents (in which case they are often called chokes).

Function	Waveform	Bmax, gauss
Sine wave		Vrms×108/4.44N×Ac×F
Square wave		Vpk×108/4N×Ac×F
Bipolar pulses with D=Ton/T=Ton×F (0<D<0.5)		Vpk×D×108/2N×Ac×F
Unipolar pulses with passive reset		Br+Vpk×Ton×108/N×Ac
In these and other equations: V - voltage (volts), N - coil's turns, Ac - core's cross-sectional area (sq.cm), F- frequency (hertz), Br - remanence (gauss)

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 "volt-second driven" coil, the flux change is a function of the applied volt-seconds and the core geometry. Contrary to popular misconception, it does not depend on the core's magnetic properties or air gaps. The table to the left provides the formulas for Bmax for common voltage waveforms. The N×Ac product should be selected so that Bmax<0.7×Bsat at maximum operating temperature. Excessive applied volt-seconds cause core saturation. 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 "current-driven" coils:
B=L×Ipk×10⁸/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×10⁸/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 lenght of an equivalent discrete gap: lg≈0.4×π×N×Ipk/Bmax . Note that the real gap should be selected slighly larger than the calculated value due to flux fringing. Combining the above equations yields: lg≈0.4×π×L×Ipk²/(Bmax×Ac). The gap is also often introduced to increase working flux swing in single-ended 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×Ipk² 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 Philips App Note (1990) an empyrical formula for thermal resistance of a wire-wound 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" hot-spot 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 over-designed component because of an overstated temperature rise.

Below you will find more magnetics theory, transformer and inductor design information, tools and free downloads.