1. Magnetics Design Primary Constraints: Peak Flux Density (B field) in the core : B max (T or Wb/m 2 ) Core losses Saturation Peal Current density in the windings : J max (A/m 2 ) Resistive losses Heat Wire Cross Section: A cond (m 2 ) Wire with cross section A cond, carrying I rms amperes of current, has current density : If we have a limit on J < J max, then we must choose a wire gauge with :

2. Core Window Area: A W (window through which all windings must pass) AWAW AWAW Torroid E Cores Fill Factor k w, the fractional part of the window actually occupied by conductor cross sections. Applying our previous constraint to A cond, N Turns of wire with cross section A cond

3. For transformers with multiple windings, the different windings must accommodate different currents, thus they must have different gauges (different A cond ): However, maximum current density must not be exceeded for any winding, thus:

4. Inductor Core Cross Section: A core 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:

5. Transformer Core Cross Section: A core In a transformer, the flux density may not exceed +/- B max over one AC operating cycle at the worst case operating condition. Let  T be the maximum excitation time for the primary. This will be at most, half the period, or k c /f s, where k c is parameter reflecting the operating duty cycle of a power converter, typically 0.5 for worst case flux density. Faraday’s Law tells us that the change in magnetizing flux density will be : Therefore, the change in flux density is: This places a lower limit on the core cross section: **For sinusoidal AC operation, a value of k c = 0.225 is appropriate; V pri is RMS.

6. Area Product We now have expressions for the minimum core window and cross section, as functions of 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 Transformers For Inductors

7. Quiz Draw the waveform, and compute the peak and RMS values for a 0.75 A P-P triangular wave riding on a DC value of 5.0 A.

8. Core Selection From vendor data, we select a core with A p, A w, and A core which meet our minimum requirements.vendor data For a transformer, the number of turns for each winding is computed using : For an inductor, the number of turns is computed using: 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 A g ~ A core, and  0 <<  core, the gap length can be approximated as:

9. Design Example: Inductor Design Specs L=100  H I DC = 5 A  I = 0.75 A p-p B max = 0.25 T = 0.25 x 10 -6 W/mm 2 J max = 6.0A/mm 2 f s = 100kHz k w = 0.5

10. This would be 18 gauge wire, which is pretty stiff. The author suggests 5 strands of 25 gauge (0.162 mm 2 )

11. Design Example: Transformer Design Specs v 1 = v 2 = v 3 = 30 v I RMS = 2.5 A B max = 0.25 T = 0.25 x 10 -6 W/mm 2 J max = 5.0A/mm 2 f s = 100kHz k w = 0.5 k c = 0.5

12. This would be 20 gauge wire. The author suggests 3 strands of 25 gauge (0.162 mm 2 )