Magnetics Design Primary Constraints: Peak Flux Density (B field) in the core : Bmax (T or Wb/m2) Core losses Saturation Peal Allowable Current density in any winding : Jmax (A/m2) Resistive losses Heat Wire with cross section Acond , carrying Irms amperes of current, has current density : Wire Cross Section: Acond (m2) If we have a limit on J < Jmax, then for each winding we must choose a wire gauge with :

Core Window Area: AW (window through which all windings must pass) Torroid E Cores N Turns of wire with cross section Acond AW AW Fill Factor kw, the fractional part of the window actually occupied by conductor cross sections. Applying our previous constraint to Acond, for a single winding (inductor):

Inductor Core Cross Section: Acore The core cross section must accommodate the peak induced flux without exceeding the maximum allowable flux density. For an inductor, the peak flux is proportional to peak current and inductance:

Inductor Air Gap: lgap The total reluctance of the magnetic path must be: The inductor air gap length is computed to provide the appropriate reluctance in the magnetic path for the desired inductance with the chosen number of turns: Since Ag ~ Acore, and m0 << mcore, the gap length can be approximated as:

Transformers (multiple windings) The different windings often must accommodate different currents, in which case they must have different gauges (different Acond): However, maximum current density must not be exceeded for any winding. Recall: Applies to all windings. Rated Current for winding “y”. Thus:

Transformer Core Cross Section: Acore In a transformer, the flux density may not exceed +/- Bmax over one AC operating cycle at the worst case operating condition. Let DT be the maximum excitation time for the primary. This will be at most, half the period, or kc /fs , where kc is parameter reflecting the operating duty cycle of a power converter, typically 0.5 for a bipolar square wave (worst case flux density). For a constant (bipolar square wave) primary voltage, Faraday’s Law tells us that the change in magnetizing flux will be : Therefore, the maximum change in flux density (-Bmax to +Bmax) for a voltage applied in one polarity is: This places a lower limit on the core cross section:

Additional Notes on Transformer Cores In conventional transformer applications, the flux in the core reverses completely during each half cycle of the primary voltage waveform, as reflected in the above discussion. The author chose to derive the relations and present an example using the forward converter topology. The forward converter is an unconventional transformer application, in that the core is magnetized in one direction and then completely demagnetized during each cycle, as illustrated in Figure 9.2 of the text. As a result, each cycle begins with zero flux in the core, increases to a maximum, and decreases back to zero. Note that power is delivered to the secondary circuit by transformer action only while the flux is increasing, and the maximum allowable change in flux is only half of that allowable in a conventional transformer application. For sinusoidal AC operation, and Vpri is RMS, a value of kconv = 0.45 provides the equivalent magnetizing volt-seconds. Since author omitted the “2” in the denominator of 9.9, he suggests a value of kconv = 0.225 be used in equations 9.6, 9.9, 9.12, and 9.14 In problem 9.8, if the author’s convention is followed, the values of kconv will be half of the values obtained when the above convention is used.

Area Product We now have expressions for the minimum core window and cross section, as functions of independent maximum ratings and operating parameters, which tells us how big the device must be. These expressions are used to express a useful magnetic core design parameter we call the Area Product: For conventional transformer applications For Inductors Note that Area product is independent of the number of turns!! ** For sinusoidal operation, since Vpri is RMS, the Vy are also RMS.

Core Selection From vendor data, we select a core with Ap, Aw, and Acore which meet our minimum requirements. For an inductor, the number of turns is computed using: For a conventional transformer, the number of turns for each winding is computed using : Note typo, eqn 9.14 The inductor air gap length is computed using the approximation derived previously: