# The Math Behind the Electromagnetic Puzzle: Part 7 (the last) in Planet Analog's tutorial series on EMI

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Note that the trapezoid is also similar to the current in the Fet (assuming a very high inductance i.e. the 'flat-top approximation'). This current determines the DM noise. It is therefore helpful to see from the Figure 7 and Figure 8, what a trapezoidal spectrum looks like for different duty cycles, frequencies, rise and fall times etc. In all cases, we have also provided in the form of a gray line, the extreme case, i.e. with zero rise and fall times (the rectangular wave). This gray line always falls at 20dB/decade as we know. Each curve is displayed in the window of concern from the point of view of conducted EMI limits, i.e. from 150 kHz to 30 MHz. Note that each point in a spectrum is an actual mathematically generated harmonic amplitude. As mentioned, we are actually only concerned with the envelope of these clusters of points. One observation to make is: if tRISE is very small but tFALL is large, and vice versa, the envelope falls with a slope somewhere between 20dB/decade and 40dB/decade.

**Note:** The envelopes in Figure 7 and Figure 8 show a slight rise at the high frequency end. This is an artifact of the computational resolution used (to improve calculation speed), and no meaning should be drawn from it.

**The Road to Cost-effective Filter design**

If the converter has a switching frequency of fSW, the harmonics are fSW, 2fSW, 3fSW etc. The harmonics tend to have lower and lower amplitudes than the fundamental frequency (first harmonic). For simplicity at this point, let us assume that the amplitudes of these harmonics are flat with respect to frequency. We normally always start by trying to achieve compliance at the lowest frequency i.e. fSW. Suppose fSW is less than 500 kHz, then above this frequency there are factors working in our favor and some against. So should we try to use a complicated multi-stage filter to say achieve an attenuation of 60 or 80 dB/decade, or should we stick to the usual filters which provide only about 40dB/decade? If the switching frequency is less than 150 kHz, how low should we try to keep the EMI level at the fundamental frequency, since it is out of the range of the EMI limits anyway? These questions can only be answered if we are aware of the fact that there are factors working in our favor and some against, as we look at higher and higher frequencies. Unfortunately, if we design the filter with only the information presented so far, we will probably end up with an over-designed filter. We need to see the interplay of the forces arraigned against us, especially in the low-frequency region (150 kHz to 500 kHz). The facts are that

- The limit lines allow for progressively higher emissions below 500 kHz.
- In addition, the sensitivity of the LISN also decreases as the frequency falls off. This effectively allows us more noise. Roughly we can say that the LISN impedance falls from 50?- at about 500 kHz to about 5?- at very low frequencies, at an approximate rate of 10db/decade below 500 kHz. See Figure 9. The vertical axis is ohms.
- However, we note that the EMI filter becomes less effective at low frequencies, since it is naturally a low pass type. Typically, its attenuation rolls off at the rate of 40dB/decade.

Let us see what all this nets us. Suppose we have, by suitable design achieved compliance at the lowest frequency. If the switching frequency is less than 150 kHz, that would mean that we have about 2mV (66dBuV) of noise emissions at 150 kHz. Now let us go in the reverse direction i.e. from low-frequency to high-frequency. This is what happens (see Figure 10)

1. The LISN sensitivity increases (at the rate of ~10dB/decade). So we would start getting higher and higher noise readings.

2. But the EMI filter starts becoming more and more effective, attenuating the signal at a typical rate of 40dB/decade.

3. This swamps out the increasing LISN sensitivity, and so our measured noise actually falls at the rate of 40-10=30dB/decade.

4. But the limit lines are asking for us to decrease the noise level at the rate of only 20dB/decade.

5. Therefore the measured noise level continues to fall below the limit lines with an increasing headroom of 30-20=10dB/decade. This is why we need to achieve compliance at the lowest frequency first.

In reality, there will be some additional spikes in the EMI scan due to parasitics we didn't model, but we should deal with them individually, at a board level, rather than try to bring the entire EMI spectrum down by a brute-force over-designed filter. It is therefore important to be aware of the trends described above. However In the solved examples to follow we will actually ignore the change in the LISN sensitivity, mainly for simplicity. In effect we are just assuming that the fall in LISN sensitivity provides us some headroom for unexpected spikes.