ELECTRICAL POWER TRANSFORMER AND INDUCTOR

DESIGN PRINCIPLES, CALCULATION, THEORY, TUTORIALS,
AND OTHER INFORMATION

Magnetic components are necessary parts of switching power supply circuits.

A

transformer

is a passive device which transfers alternating (AC) electric energy from one circuit into another through electromagnetic induction. It normally 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 the 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.

A

power transformer

in SMPS is designed to change amplitude of high-frequency pulses by the turns ratio and to provide isolation between circuits. Note that it can't transfer a DC component of the pulse voltage: in a steady state mode net volt-seconds across each winding should be zero, otherwise the core will saturate. DC output voltage is obtained only by using rectifiers. Nevertheless, an average voltage across a real coil's terminals can be non-zero due to non-zero coil's resistance. This DC offset can be used for lossless sensing of an average current across an inductor or a transformer winding: 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 (some topologies do need some energy stored in magnetizing inductance for 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 all or a portion of this energy 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).

FIND POWER TRANSFORMER BY SPEC

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×2F (0<D<1)		Vpk×D×10⁸/4N×Ac×F
Unipolar pulses with passive reset		Br+Vpk×Ton×10⁸/N×Ac
In this and other equations: V - voltage (volts), N - winding's turns, Ac - core's cross-sectional area (sq.cm), F- frequency (hertz), Br - remanence (gauss)

The 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 should not reach the core material's saturation flux value Bsat. In a "volt-second driven" coil the flux change is related to the applied volt-seconds and the core geometry, but does not depend on the core's magnetic properties or air gaps. This table provides the formulas for maximum flux density Bmax for common voltage waveforms. The N×Ac product should be selected so that Bmax<0.7Bsat. Note that in higher frequencies, core loss rather than saturation normally becomes the main limiting factor.

In inductors the coil is normally driven in such a way that it carries a required current. 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. Note that B can't exceed Bsat. If the Ipk keeps increasing, at some point B 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 a discrete gap: lg≈0.4×π×N×Ipk/Bmax .
The number of turns can be calculated to provide the desired inductance L:
N=L×Ipk×10⁸/Bmax×Ac
Combining the above equations: lg≈0.4×π×L×Ipk²/(Bmax×Ac).
For powder metal cores with a distributed gap and soft saturation curve, the process may take several iterations. In short, you can first pick a core based on desired L×Ipk² by using manufacturer's recommendations, calculate turns Formula for inductor turns

, where AL - specific inductance in mH/1000 turns 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 desired L.

Another important constrain is a maximum "hot spot" temperature rise. Note that most textbook's procedures related to the temperature rise are written for natural convection cooling. For applications with forced airflow or conduction cooling these procedures may result in an over-designed component because of an overestimated temperature rise. Below you will find more magnetics theory, transformer and inductor design information, tools and other free downloads.

FREE INDUCTOR
AND POWER TRANSFORMER
DESIGN SOFTWARE

UNITRODE SEMINAR MAGNETICS HANDBOOK
(MAG 100A)

MAGNETISM PRINCIPLES,
EQUATIONS, TUTORIALS

Transformer turns and wire calculator
(includes skin effect)

SMPS PFC inductor calculation tool

Transformer calculation for various switching regulator topologies

Software to design electrical inductors with powder cores

Output inductor calculation tool

Current transformer design software

Ferrite magnetic calculation tool (includes skin and proximity effects)

Core loss calculator
for non-sinusoidal waveforms

Introduction and Basic Magnetics (Design for Switching Power Supplies)

Magnetic Core Characteristics

Windings data and skin effect

Power supply transformer design

Inductor and Flyback Transformer design

Magnetic Core Properties

Eddy Current Losses in transformer windings

Equivalent electrical circuit

The Effect of Leakage inductance

Coupled filter inductors

How to design a power supply transformer with fractional turns

Magnetic field unit conversion and equations - online calculator and table

The following three guides are instructor's slides: see the author's terms of use

Basic transformer theory

SMPS transformer design procedure and equations

Inductor design procedure

Planar power transformers basics and design guide

Electrical transformer: how it works and physical principles

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ELECTRICAL POWER TRANSFORMER AND INDUCTOR

DESIGN PRINCIPLES, CALCULATION, THEORY, TUTORIALS, AND OTHER INFORMATION