# Switching loss calculation formula

I read an article talking about the power switching loss formula, and I am curious about one thing.

How to use math to prove the 1/6 and 1/2 in these two waveforms? Could someone give me some idea?

• Being proportional to FS is obvious... And simple integration of "straight line" definitions for Vds and Id. Nov 28 '21 at 15:07
• 1/2 in the second one is due to the two slopes coming after each other if you count the area under the graph. 1/6 is due to one rising while the other one is falling. Less intuitive. Derive the formula or count the area under the graph I * V. Nov 28 '21 at 15:44
• @winny Yes, but I'd like to know how to prove it.
– EEC
Nov 28 '21 at 23:07
• @EEC Are you familiar with integral calculus? The most straightforward way to derive the formula is to just integrate the product of V and I. Nov 29 '21 at 0:47

These expressions let you calculate the theoretical losses when a power switch turns on or off with overlapping current and voltage.

I have carried the calculations in my book for the two scenarios and the turn-off sequence in particular. The calculation is quite simple and involves an integral to average the instantaneous power $$\p(t)\$$ over a switching period. The below figure shows you a real shot taken on a flyback converter when the switch opens:

Please note that these are idealized waveforms and switching losses are extremely difficult to theoretically evaluate. This is because many parasitics (components, layout and so on) are involved which can significantly affect final waveforms. Same with simulation which usually leads to wrong results for switching losses.

• HI Verbal, Could you tell me which topic in your book talks about math equations in 1/2? I do not find this equation in your book.
– EEC
Nov 28 '21 at 23:06
• Hello, in the second edition, this is in the 20-W design example, equations (7.261) and (7.262). Nov 29 '21 at 6:28
• Hi, I do not find the 7.261 in the second edition. could you tell me which page?
– EEC
Nov 29 '21 at 14:09
• How come you can't find an equation number?? It is page 736, second edition, green book, in the 20-W design example. Nov 29 '21 at 14:35
• Verbal, I find it! thanks.
– EEC
Nov 29 '21 at 14:46

Here is the "starting" idea ...

Formulas being proportional to FS is obvious.

Integration done for ONE cycle. So, multiply by frequency.

Next case :

• Hi Antonio, Thanks. could you also tell me the other?
– EEC
Nov 28 '21 at 23:11
• @EEC Added in answer. Nov 29 '21 at 9:35
• it seems like you use the triangle formula right? the equation is very clear, thanks.
– EEC
Nov 29 '21 at 14:08
• Yes. In this linear case, Maple does so with operator "integrate" within two limits: int(f(t),t), but it is very general for other functions and so easily ... Nov 29 '21 at 16:08