Non-Isolated Switch-Mode Power Supplies

1.1 Buck converters

The buck converter is a single-quadrant chopper, as studied in Chapter 1 of Volume 2 [PAT 15b]. The “load” is made up of an inductance L in series with the association of the actual load (presumed to be a current source Is) in parallel with a filtering capacitor C (see Figure 1.1).

In these conditions, for a correctly dimensioned power supply, the assembly (Is,C) may be considered to be analogous to the electromotive force (e.m.f.) Ea of a direct current (DC) machine, and the inductance L may be considered to play the same role as the armature inductance in the machine. Consequently, the results established in Chapter 1 of Volume 2 [PAT 15b] are applicable here. In the case of continuous conduction, an output voltage of s=α.Ve is obtained, where α is the duty ratio of the transistor control. Moreover, in cases of discontinuous conduction (i.e. for a current which cancels out in the inductance), the output voltage will be higher than in the continuous conduction case, in accordance with the characteristic shown in Figure 1.4 (see Chapter 1 of Volume 2 [PAT 15b]). A summary of the characteristics of this converter is shown in 1.1. The constraints applicable to the switches are similar to those for a one-quadrant chopper powering a DC machine, but we should also analyze the quality of the voltage supplied to the load. The waveforms produced are the same as those shown in Figure 1.2 (Volume 2, Chapter 1 [PAT 15b]), as the ripple of the output voltage Vs is considered to be a second-order phenomenon, negligible when calculating the ripple of current iL in inductance L (constant Vs, as for the e.m.f. Ea of a DC machine). Thus, this current may be considered (as in the case of a machine power supply) to be a time-continuous, piecewise-affine function, which may be written (presuming that the load current is is constant and equal to Is) as:

Lt=Is+i˜Lt

where ˜Lt is a signal with an average value of zero, with a “peak-to-peak” ripple ΔiL expressed as:

iL=α.1−α.VeL.Fd

where Fd = 1/Td is the switching frequency and α the duty ratio of the control of transistor T.

Second, given the current ripple ˜Lt, the (low) ripple of the output voltage υs(t) may be deduced, insofar as:

st=Vs+υ˜st

with:

s=α.Ve

and:

˜st=υ˜st0+1C∫t0t0+tiCτ.dτ=υ˜st0+1C∫t0t0+ti˜Lτ.dτ

Using this result, it is then easy to deduce the peak-to-peak ripple Δυs of voltage υs (t):

υs=1C.⋅12⋅Td2⋅ΔiL2=α.1−α.Ve8L.C.Fd2

These results are illustrated in Figure 1.2.

Note that the calculation of the output voltage ripple (used in dimensioning the filter capacitor C) corresponds to a continuous mode of operation. This is not strictly applicable for discontinuous mode, but this choice presents certain advantages:

– simpler calculations using this operating mode;

– the results obtained allow satisfactory dimensioning of capacitors (on the condition that a minimum safety margin is respected) for all possible cases;

– continuous conduction is often the most critical case with regard to the output ripple, although this is not true for buck converters. The output ripple is proportional to the charge current (for boost and buck-boost converters, described in the following sections), and is higher in continuous mode (discontinuous conduction = low charge).

1.2 Dimensioning a ferrite core inductance

For iron core windings (as discussed in Chapter 5 of Volume 1 [PAT 15a]), a simple magnetic circuit was considered, characterized on the sole basis of three geometric parameters (reduced to two parameters) which needed to be established. While the equation model of the inductance and the applicable usage constraints remain identical, the geometry of a ferrite core is required, and a core must simply be selected from the lists supplied in manufacturer catalogs. The first stage in this process is to choose a family of ferrite cores: this choice depends on the application and the available space. Note the existence of “E,I” structures (alongside double E structures); in power electronics, however, the RM and PM families are most interesting in terms of electromagnetic compatibility, as they are relatively “closed” and produce limited radiation into the immediate environment. Two examples of these families of cores are shown in Figure 1.3.

Once a family of cores has been selected, we need to choose a specific model in accordance with a given specification. To do this, the expressions of Ae (iron section) and Sb (windable section) are used. Note that two surfaces are linked to constraints relating to ferrite (magnetic flux density Bmax) and copper (current density...