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Understand slope compensation in PCMC DC-DC converters

16 Dec 2015  | Sergio Franco

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Slope compensation
Looking back at figure 5, we observe that if we want figure 5b to retain the same IL value as Figure 5a, we need to reduce the iEA value of Figure 5b so as to "push down" the iL waveform till the respective ILs align. By how much do we need to reduce iEA? To answer, let us draw the desired iL waveforms for three different values of D. As depicted in figure 7, top, we start out by drawing the down-ramps for iL, all vertically

Figure 7: Constructing the compensated iL waveforms for D = 0.25, 0.5, and 0.75.

centred about identical ILs, and all with the same slope of Sf = -VO/L. Next, we complete the iL waveforms by drawing the up-ramps, as shown in figure 7, bottom. Finally, we superimpose the three figures as in figure 8, and observe that the locus of the peaks defines a ramp with a slope of Sf/2 = –VO/2L.

Figure 8: The locus of the peaks of Figure 7 is a ramp with a slope of Sf/2.

Figure 9: The locus of the peaks of Figure 7 is a ramp with a slope of Sf/2.

This shows precisely by how we must reduce iEA, hence the designation slope compensation.

Figure 9 shows one way of modifying the circuit of figure 3 so as to achieve slope compensation. The circuit now includes a saw-tooth generator operating at a frequency of fS, whose output vRAMP is then subtracted from vEA to produce the desired locus of peak values for iL. With slope compensation, the waveforms of figure 5 change as depicted in Fig. 10, where iEA(comp) = (vEAvRAMP)/Ri.

Figure 10: The inductor current of the circuit of Figure 9 for two different duty cycles.

As an added bonus, slope compensation also eliminates sub-harmonic oscillation, as depicted in figure 11. Using graphical inspection, we observe that a beginning-of-cycle disturbance il(0) will result in

Figure 11: Slope compensation prevents sub-harmonic oscillation regardless of D.

an end-of-cycle disturbance il(TS) of lesser magnitude, even though D > 0.5 (in fact, you can convince yourself that this holds for any value of D, 0 < D < 1). It is the case to say that with slope compensation we are in effect killing two birds with one slope – ops stone.

The error amplifier EA, shown in figure 9 as a mere triangle, serves two important functions: (a) to drive its inverting input voltage as close as necessary to the non-inverting one so as to approximate Eq. (4), and (b) to provide a frequency profile suitable to ensure a prescribed phase margin for the whole system. Not at all an ordinary amplifier, which can easily form the body of a future blog on stability analysis and error-amplifier design.

[1] Design with Operational Amplifiers and Analog Integrated Circuits by Sergio Franco

About the author
Sergio Franco is an author and (now emeritus) university professor.

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