Fourier Series in Power Supplies

In previous chapters we have seen that differential and common mode noise (DM and CM) emissions are basically either voltage or current sources with arbitrary waveshapes. What they have in common is that they are repetitive, and therefore involve the basic switching frequency of the converter. When we design a filter to suppress these emissions, we have to be conscious of the fact that the efficacy of a filter is best characterized in terms of the impedance it presents to a sine wave of a given frequency passing through it. In other words, to know the efficacy of the filter to CM and DM emissions, it is best to “decompose” the DM and CM waveforms into a sum of infinite sine wave components of varying amplitude and phase. As per the litmus test, when all the components are summed up, we should get back the original waveshape we started off with. The procedure of decomposition and reconstruction of an arbitrary waveform, either current or voltage, into sine wave components that are multiples of a basic (fundamental) frequency, is called Fourier series analysis. In general, we start with a “fundamental frequency” or “first harmonic” (invariably the switching frequency of the converter in our case), and add components of varying (and invariably diminishing) amplitudes. The frequencies are multiples of the fundamental frequency f, that is, f, 2f, 3f, 4f, and so on. The nth harmonic corresponds to a frequency nf.

In Figure 18.1, we have included a short course to refresh our collective memory on Fourier series. We remember in school we were used to dealing with Fourier series in terms of a certain angle “θ” expressed in radians (radians being a dimensionless quantity). All the functions we dealt with were based on θ, and repeated every 2π radians. We have a very similar situation in power supplies, where all voltages and currents are functions of “t” (time), repeating every time period (T) in steady state. We realize we can replace θ in all our familiar Fourier series expansions with the dimensionless quantity “2πt/T.” This way, we will achieve migration of all the well-known Fourier series textbook equations to the world of power supplies. This is the required substitution to remember as explained in Figure 18.1

Figure 18.1: Applying Fourier series in power supplies.

Note that angle θ itself may be rather confusingly called “x” or “2α” in textbooks, but nevertheless, we recognize it is an angle (in radians).

The Rectangular Wave

We use the above-mentioned mapping procedure to analyze a rectangular waveform of amplitude “A” in terms of its Fourier series. Assume we are, for example, talking about the rectangular voltage “Vds” (same as “V_DS”) across the switching FET. The results are shown in Figure 18.2. We have first analyzed the wave by assuming it is “ideal,” that is, with zero crossover (switch transition) time. We then get the following Fourier coefficients (with 2πt/T replacing θ, and pulse width designated as t_ON):

where and |c_n| represents the magnitude of the amplitudes of the nth harmonic. We are interested in magnitudes because a conducted EMI scan doesn’t care about the signs. However, we do need the signs to correctly reconstruct the original waveform. In general, we can sum only a few terms of the Fourier series to get fairly close to the original waveform. Summed up to “nmax” harmonics, we get Vds (as a function of time)

Figure 18.2: Fourier series in power supplies and plotting the Fourier coefficients.

Note that the first term is AD (amplitude times duty cycle), is just the average (DC) value of the waveform over the full time period T. We should remember that the average value is always the first term (the “n=0” term) in any Fourier expansions, and also that it is in fact irrelevant to EMI calculations since the CISPR-22 range starts at 150 kHz. However, besides the signs, the DC term is also required to correctly reconstruct the original waveform, as we have done in Figure 18.2 with a unity amplitude wave of 50% duty cycle as an example. We have summed over 1, 5, 10, 50, and finally 10,000 harmonics (not shown). We see how we move from an offset sine wave (when we sum only the n=0 and n=1 terms) to a perfectly rectangular waveform (with a very large n, we get the exact waveform we started with).

However, so far we have assumed zero crossover time. If we have a real-world situation with nonzero crossover times, then, instead of a rectangular waveform, we get a trapezoid. The Fourier coefficients get modified by an additional term below

We now have a product of two functions of the type sin(x)/x, instead of one. We need to understand what the properties of this function are.

The Sinc Function

The function sin(γ)/γ is called the sinc function. Its key properties are shown in Figure 18.3. On a log versus log plot (lowermost plot), it appears “flat-topped” at lower frequencies, with a unity value initially. In reality it is actually sloping rather gently downward, and at γ=1 its value is sin(1)=0.84. We call that a “breakpoint” because right after that, the slope changes dramatically. At higher frequencies it falls off with a slope of −20 dB/decade, a straight line on a log versus log plot. Note that the sinc function can be considered “clamped” to 1, for all frequencies less than the break frequency.

Figure 18.3: Understanding the sinc function.

When we have two breakpoints, as in the equation for the real-world c_n’s above (one for t_ON, the other for t_CROSS) we eventually see a downward slope of −40 dB/decade (obvious in the lowermost right-hand plot of Figure 18.2). Each breakpoint has clearly contributed −20 dB/decade to that slope. The actual breakpoints in terms of frequency are obtained by setting γ to 1. We thus get

In terms of the harmonic index “n” – imagining “n” varies smoothly, rather than just having integral values

Note that nbreak1 is less than 1 for duty cycles greater than 32%. Which means for all practical purposes it then becomes “invisible” since we start off with n=1.

In other words, the duty cycle needs to be below 0.32 for the first breakpoint to be “visible” (in terms of affecting any amplitudes of the Fourier terms). Otherwise, it occurs at frequencies below the fundamental frequency, and therefore can’t determine any of the actual harmonic amplitudes (i.e., n≥1). Of course, the effect of this first breakpoint is felt at almost all frequencies, since it imparts a downward slope of −20 dB/decade for all frequencies exceeding it (and likewise, clamps harmonic amplitudes below it, if any).

The Envelope of the Fourier Amplitudes

For designing the conducted EMI filters, we are not concerned with the granularity of the actual harmonic amplitudes. What matters to us is their envelope. That is what we really try to adjust and try to keep within the EMI limit lines, because even a single point in excess of the limits represents a failed EMI test. We see from Figure 18.2 (D=0.5) and from Figure 18.4 (D=0.2) that there is one breakpoint which is dependent on the switching frequency. After that breakpoint, the envelope of the amplitudes falls off as per 1/n, which is equivalent to a slope of −20 dB/decade. As per the equations in Figure 18.2, we have described the envelope as

Figure 18.4: Fourier coefficients for a unity amplitude trapezoid with D=0.2.

This equation is easy to understand since we know that the actual harmonic amplitudes (for a square waveform) are

Since for all angles (and all duty cycles), the maximum value of the sine term (in the rectangle border above) is unity, the envelope must be described by “2A/nπ.” And that is the envelope equation we have provided in Figure 18.2.

However, we see that there is a second breakpoint, after which the envelope falls off as per 1/n², which is basically equivalent to −40 dB/decade. The location of this breakpoint depends on the crossover time. We can visualize the second breakpoint as adding another −20 dB/decade to the already falling −20 dB/decade curve, leading to a cumulative slope of (−20)+(−20)=−40 dB/decade. The envelope after the second breakpoint is

Note that the original equation for c_n included duty cycle, whereas the envelope equations do not (because we have “maxed out” the terms). So yes, the harmonic amplitudes do go up and down with respect to D, but the envelope remains fixed and independent of D, as we can see in Figures 18.2 and 18.4. Note that D does enter the picture if it is below 32% as we see in the example below.

From Chapter 3 we know that V_OR=turns ratio×V_O=19×5=95 V. The duty cycle at high line (270VAC, rectified 382VDC) is D=V_OR/(V_IN+V_OR)=0.2. In Figure 18.4, we have plotted out the results of the Mathcad file originally used for Figure 18.2, but this time performed for D=0.2. Note that we have assumed unit amplitude in these plots, knowing that everything will scale proportionally to the amplitude.

Note that if the converter is operating in “continuous conduction mode” (CCM) at high line (unlikely) we can assume a simple rectangular V_DS waveform. That assumption will give accurate results for conducted mode EMI calculations even if the converter is in DCM.

The height (amplitude) of the rectangular V_DS (same as Vds in this chapter) pulse is V_IN+V_OR=382+95=477 V (ignoring the leakage inductance spike). From Figure 18.2 we know that the magnitude of the harmonic Fourier coefficients is

We thus get three harmonic amplitudes based on the exact Fourier coefficient equations above

That is what we essentially plotted in Figure 18.4 for unit amplitude. We can confirm the graph matches the results of the calculations above, by scaling the numbers “eye-balled” from the graph (i.e., 0.38, 0.3, and 0.2 for the first three harmonics, i.e., the solid black dots), for the case of A=477 V as follows:

We see the graphical-based results are close enough to the accurate calculation above.

However, now, coming to the equation for the envelope (before the second breakpoint), we get

Using this simplified equation, we get for the three harmonics

We see a close match with the terms n=2 and n=3, but not for n=1. The actual harmonic amplitude is only about 180 V based on our exact and graphical calculations above, not 304 V. In other words, the envelope equation is now giving misleading results for the first (fundamental) harmonic. To avoid overdesign of the filter we should understand the reason for this carefully. It is explained in Figure 18.4 too. The basic reason is that since the duty cycle was lower than 32%, the first breakpoint is “visible” at an effective n of 1.592, and it therefore clamps the amplitude of the n=1 term. In Figure 18.4, as per the envelope equation, we would have got the solid gray dot as the first harmonic amplitude (roughly between 0.6 and 0.7). In reality, we get the solid black dot (a little below 0.4).

However, we can actually rather cleverly, use the envelope equation itself to predict the amplitude of the first harmonic. This is shown as follows. The equivalent “breakpoint index” is

Note that in effect we are now implicitly assuming that “n” can be a continuum of values, not just a range of integers. It helps us in the math that follows, but keep in mind it has no physical significance in terms of the Fourier series. At this point, the coordinate of the envelope is

For unit amplitude, the amplitude needs to be 190.7/477=0.4. That is close enough to the actual calculation (we had gotten 0.374 for unit amplitude based on the graph in Figure 18.4 for the first harmonic). Our conclusion is just from the two equations below, we can estimate the amplitude of the first harmonic quite accurately. To re-emphasize: it is important to know the amplitude of the first harmonic most accurately, since our conducted EMI filter design is usually based on it. Here is a summary of what we need to do to quickly and generally (for any D) estimate, based on simple envelope calculations.

Practical DM Filter Design

The trapezoid we have looked at so far is a voltage waveform. But it can equally well be a current waveform if we use the “flat-top approximation” (for large inductance). In DM filter design as applied to a Buck or a Buck-Boost (or their AC–DC equivalents, the Forward converter and the flyback converter), the input current switch current is of the shape we had previously indicated in Figure 16.2. So, the DM noise source is

where I_SW here refers to the center of ramp. The Boost is not considered here as its DM filter design is almost trivial, since in effect, it has a natural LC filter on its input. In Table 18.1, we have provided the height of the switch current trapezoid for all the relevant topologies (with the flat-top approximation).

Table 18.1. Switch Currents (Center of Ramp) for the Relevant Topologies.

If there was no EMI filter present, the switching noise current received by the LISN would be

(since the LISN has an impedance of 100 Ω for DM noise).

However, the analyzer itself only measures the noise across one of the two effective series 50 Ω resistors in the LISN. So, the measured level of noise is

We have assumed that C_BULK is very large, and that it has no ESL, and also that its ESR is much less than 100 Ω. All reasonable assumptions of course.

Note that to be more accurate and avoid overdesign, we need to split I_SW into harmonic components as in the following example.

See Figure 18.5 for this entire example. Also read the section in Chapter 16, titled “The Road to Cost-Effective Filter Design.”

Figure 18.5: DM filter calculation for universal input flyback (see solved example).

DM Calculations at High Line

See Figure 18.5 for this entire calculation.

Practical CM Filter Design

Having understood that the worst-case for DM filter design is at low line, on account of the higher currents, we can sense that the worst-case common mode noise will occur at high line since the voltages are highest; and as described in Figure 16.5 that will lead to the highest injected currents into the enclosure. Let us therefore do this calculation at high line only. Let us also ignore the second breakpoint as we have already seen it is too far away to affect the filter design per se. Note that we are assuming that by using the X-cap C1 in Figure 16.3, this truly is CM noise, not mixed mode (MM) noise.