We present the main mathematical analysis and tools needed to finally cap a successful filter design. First we should recall our hazy 'Fourier Series' high school class. Fourier analysis is often avoided by power supply engineers, but it can go a long way towards understanding and tackling several key issues like EMI/noise, transformer proximity losses, PFC etc.
Math Backgrounder: Fourier Series
Collecting some of the basic definitions first:
For a function f(x) with Time Period expressed as an angle (2-pi) we can write
For a function f(t) with a Time Period 'T' (units of time)
Note: The designer should not be confused that the first term is traditionally called 'ao/2', but also often called 'ao' , or something else. Either way, in any Fourier expansion of any arbitrary periodic function, it is always the area under the waveform over one time period (i.e. its arithmetic average).
Note: If the waveform is 'moved' up or down or side to side on the graph, while keeping its basic shape unchanged, the appearance of the Fourier series may seem to also change drastically, but the magnitude of the amplitude of each harmonic (i.e. |cn|) remains the same (except for the average term which we don't really care about anyway). It's the magnitudes of the harmonics that relate to measurable physical effects. And these change only if the peak to peak value of the waveform changes (or of course its basic shape).
Tip: We could calculate the cn for the simple case of a waveform of unity height (or peak to peak value). Then the cn for say a waveform with a height (or peak to peak value) of 'A' will simply be A times the value of cn for the unity waveform.
Tip: In most math school books the period is expressed as 2?, but in power supplies we know that the period we are interested in is in units of time not angle i.e. T=1/f. The way to convert angle ? to t, is to use the equivalence
The Rectangular Wave
Let us find the series for the unity height rectangular wave shown in Figure 1. Textbooks give the expansion as
This is in terms of angles. To avoid confusing x with distance, it helps to change x to a more familiar symbol for angle like θ . We can also use the summation sign to write the above expansion in a more compact form (ignoring the signs of the coefficients)
Applying our conversion rule to get from angles to time ('t-plane') we get
The cn in this equation can also be explicitly written out, using the same conversion rule. We must remember though, that 2α in the θ-plane corresponds to tON=DT in the t-plane (the switch ON time in power supplies). So
Finally we get what we are looking for
in the t-plane
Let us apply this to the Flyback in Figure 2.
We have provided the full expansion of the voltage on the Drain of the FET (ignoring leakage spike and ringing).
Note that we have now multiplied all the cn's by the actual wave amplitude, as we know how the coefficients scale. Also, the first term is the average value of the waveform. We could have also written this expansion with alternating signs as follows
In Figure 3 we have plotted both these expansions out (with same signs or alternating signs), so that the designer can see a) how including more harmonics helps to reconstruct the original waveform better, and b) the effect of the alternating signs (dotted curve) is just to shift the phase of the entire waveform.
Tip: When using a math crib-sheet to help in an expansion, we can avoid mistakes if we first scale the math function in the book down to unity peak-to-peak value (if needed). Then we apply the equivalence rule to map it to the t-plane, then multiply all the cn's by the actual peak-to-peak value. The DC value (first term) can be ignored, as it is usually irrelevant, but if required, it can always be added on later by just examining the final waveform and taking its average.
Analysis of Rectangular wave
We are interested only in the coefficients cn not in the actual Fourier series. We see that the coefficients have the form sinx/x. Plotting them out in Figure 4, first on linear graph scale, then on a log scale, we make the following key observation
- For EMI suppression it doesn't matter if say the odd harmonics are present or the even, or both. We are only concerned with the envelope of the emissions as that is what we need to design the filter and to keep below the EMI limit lines.
- For the rectangular wave, the cn's have the form sinx/x.
- On the log plot we can see that till x=1 this function is flat, then it rolls-off as x increases. The envelope falls at the rate of 20dB/decade (-20 dB being 1/10th, and a decade being 10 times).
Now we take the rectangular wave discussed above and make it a little more realistic by introducing non-zero rise and fall times. By a similar procedure as for a rectangular wave, we can get the following equation (for the case of equal rise and fall times)
where tRISE=tFALL=tR, and A is the amplitude (peak to peak). We are again ignoring any signs as being essentially irrelevant.
Clearly we will get two break points now. The first break point occurs at
Since n=frequency of harmonic/fundamental frequency, i.e. n=f?T, we get the corresponding break frequency to be
The second break point is at
We know what to expect too, that after the second break point, the net roll-off will be at 20+20=40dB/decade. See Figure 5.
Note that n must be an integer to have any physical meaning. The first breakpoint therefore may not be visibly apparent. What we will perceive is that the envelope ramps down almost from the lowest frequency, at the rate of 20dB/decade.
When does n1 get to be higher than n=2? We can solve to get
When does n1 get to be between n=1 and n=2? Solving
So only for very narrow duty cycles we might 'see' the first break point.
Below the first break frequency, the envelope of the harmonics becomes flat. The first break point should therefore be calculated and the envelope should be 'guillotined' (truncated) below this frequency. Other than that, we can use the following equations to describe the cn (note that in these equations, the cn are no longer the actual coefficients of the Fourier expansion, rather they represent the envelope)
The first of the two equations above is valid between the first and second break points, and the second equation is valid for all frequencies higher than the second break point. Note that the switching frequency is fSW=1/T. The trapezoid rise and fall times are described in Figure 6.