# How does this modulation cause sidebands?

The following figure is an excerpt from this paper: Input Differential-Mode EMI of CRM BoostPFC Converter by Fei Yang.

Q#1 (first highlighted sentence): How does the modulation line frequency cause sideband harmonics? As far as I know, modulating a signal with a sinusoidal signal results in just frequency-shifting and mirroring of spectra like the following example:

Q#2 (second highlighted sentence): I can't get it correctly. Maybe a visual explanation by spectra would help. the following figure is my impression which is Contrary to the text.

• Is it a constant "on-time" frequency modulated PWM waveform that they are defining. If not, can you be clear about the type of PWM regime they are using for PFC? Also (for Q2), what is it that confuses you specifically? Jun 23 at 16:16
• @Andy aka, yes, it is a CRM fixed-on-time(FOT) PFC. maybe the answer for Q1 clarifies the Q2. Jun 23 at 16:32
• Those are sidebands. Jun 23 at 17:10

A PFC in CRM mode uses a variable frequency, depending on the input voltage: check eq. (1) and (5) of your paper. When input voltage changes, duty cycle changes (eq. 1) and switching frequency depends on duty cycle (eq. 5).

Moreover there is a second phenomenon not considered in that paper: the triangular waveform at variable fs (frequency modulation) has also an amplitude variation according to the input voltage (amplitude modulation).

So, when the input voltage changes, it modifies the triangular current waveform both in frequency and amplitude, 100 times per seconds, and this is a modulation, both FM and AM.

You can see the frequency modulation as the carrier frequency swinging up and down in the spectrum domain (naive interpretation) or as a fixed frequency carrier with many pairs of sidebands around it, each spaced 100Hz from the previous one (frequency modulation is mathematically complicated, a lot of Bessel's functions!)

If the frequency modulation index isn't too high you will have a limited number of sideband pairs, or you can imagine that the carrier frequency isn't walking up and down the spectrum too much and it will stay within the bandwidth of your EMI receiver, tipically 9 kHz wide for conducted emissions.

I don't think that the FM spectrum will stay within 9kHz, but I didn't put numbers in the equation and I didn't read all the paper.

In conclusion: you can see the current spectrum as a series of lines at an average switching frequency fs (so you have fs, 2fs, 3fs....) and around each of these carries there are sidebands at 100Hz above and belove, 200Hz, 300Hz... a very complicated spectrum!

My impression is that the spectrum around each carrier is wider than 9kHz, so the problem is more complicated, but you get some noise mitigation because your receiver doesn't measure the total power around each carrier.

Complicated stuff, not easy to explain!

• BTW, about the second question, The resultant sidebands are only 100 Hz away from the switching frequency and the RBW of the receiver is 9khz. So my impression is that these sidebands are well placed and measured in the receiver's field of view (Contrary to what is said in the text). (See the last figure I've just added). Can you clarify it for me? Jun 24 at 12:55
• In FM there are many sidelines, each one is at N*fs+/- M*100Hz, where coefficient M can have a large value when the modulation index is high. Sidebands can be quite wider than 9kHz. Consider for example in time domain, where intuition helps, a sinusoidal carrier at 100kHz that moves up and down from 70kHz to 130kHz with a frequency of 1Hz. Total energy is not all inside the 9kHz bandwidth centered at 100kHz, carrier is received only when it in the range 95.5kHz to 104.5kHz. If you look to this signal in frequency domain its sidelines are 1Hz apart, but many of them are not received. Jun 24 at 16:25

@Andy aka, yes, it is a CRM fixed-on-time(FOT) PFC

Because it's a fixed charge-time and a constant slope transfer-time, the PWM modulation frequency MUST change because it has no other option. This is because this type of PFC operates in the PWM boundary condition i.e. as soon as the charge phase ends, the transfer phase begins and, as soon as the transfer ends, the next charge begins immediately.

The constant slope of the transfer-time is because the PFC is trying to produce a constant DC voltage at the output (such as 400 volts DC) and, due to the inductor law: -

$$V_{OUT} = L\dfrac{di}{dt}$$

The ratio $$\di/dt\$$ must remain constant for a constant DC output voltage.

And, because the input voltage varies from virtually 0 volts to the peak of the AC voltage, it's slope must be proportional to $$\V_{IN}\$$. Consider this diagram I made: -

The picture above is an exaggeration of the waveforms and slopes in order to demonstrate the variable slope of the charging current (fixed duration) compared with the fixed slope (variable duration) of the transfer current. Maybe a more realistic picture would be this (same story but a tad harder to see): -

So, just to reiterate: -

• The charge current slope is proportional to $$\V_{IN}\$$ but fixed in duration
• The transfer current slope is proportional to $$\V_{OUT}\$$ (i.e. it is a constant and a variable duration)

Hopefully you can see that the PWM waveform is a high frequency when $$\V_{IN}\$$ is low and a much lower frequency when $$\V_{IN}\$$ is high.

This causes frequency modulation artefacts in the noise/EMI.

Consider also my answer here about the amplitude modulations produced when the duty cycle changes: -

This also causes A.M. noise/EMI artefacts.

• thanks andy, for your deep explanations, but I already knew these AM and FM modulations happening due to the CRM operation of PFC. my questions were 1. how does this FM modulation result in sidebands (mathematically in spectra ) and 2. The resultant sidebands are only 100 Hz away from the switching frequency and the RBW of the receiver is 9khz. So my impression is that these sidebands are well placed and measured in the receiver's field of view (Contrary to what is said in the text). (See the last figure I've just added) Jun 24 at 6:26
• Are you in fact asking a question about sine wave modulation and, in fact it has nothing really to do with PFC. If you are then standard texts from RF books will give you this answer. Jun 24 at 8:01
• how about the second question (Q2)? Jun 24 at 8:12
• @WeTech try this link from RFcafe - it might help you better if you learn this from a standard RF approach (rather then trying to piggy-back it on top of PFC): rfcafe.com/references/electrical/frequency-modulation.htm Jun 24 at 8:18