DC-DC Converter Design and Magnetics--The Current Ripple Ratio "r", etc.

The Current Ripple Ratio 'r'

In Fgr. 2 we first introduced the most basic, yet far-reaching design parameter of the power supply itself - its current ripple ratio 'r.' This is a geometrical ratio that compares and connects the ac value of the inductor current to its associated dc value. So

r = delta I IL

= 2 × IAC IDC

Fgr. 5: BCM and Forced CCM Operating Modes: Boundary conduction mode, IAC is equal to IDC; Forced continuous conduction mode; negative current, negative slope negative current, positive slope.

Here we have used delta I = 2 × IAC, as defined earlier in Fgr. 2. Once r is set by the designer (at maximum load current and worst-case input), almost everything else is pre-ordained - like the currents in the input and output capacitors, the 'RMS' (root mean square) current in the switch, and so on. Therefore, the choice of r affects component selection and cost, and it must be understood clearly, and picked carefully.

Note that the ratio r is defined for CCM (continuous conduction mode) operation only. Its valid range is from 0 to 2. When r is 0, delta I must be 0, and the inductor equation then implies a very large (infinite) inductance. Clearly, r = 0 is not a practical value! If r equals 2, the converter is operating at the boundary of continuous and discontinuous conduction modes (boundary conduction mode or 'BCM'). See Fgr. 5. In this so-called boundary (or "critical") conduction mode, IAC = IDC by definition. Note that readers can refer back to Section 1, in which CCM, DCM, and BCM were all initially introduced and explained.

Note that an exception to the "valid" range of r from 0 to 2 occurs in 'forced CCM' mode, discussed in more detail later.

Relating “r” to the Inductance

We know that current swing is voltseconds per unit inductance. So we can also write:

delta I = Et/L µH (any topology)

Here 'Et' is defined as the (magnitude of the) voltµseconds across the inductor (either during the on-time or off-time - both being necessarily equal in steady state), and LµH is the inductance in µH. The reason for defining Et is that this number is simply easier to manipulate than voltseconds because of the very small time intervals involved in modern power conversion.

Therefore, the current ripple ratio is...(any topology)

Note also that from now on, whenever L is paired up with Et in any given equation, we will drop the subscript of L, that is, "µH." It will then be "understood" that L is in µH.

Finally, we have the following key relationships between r and L ...

Incidentally, the preceding equation, that is, the one involving VOFF, assumes CCM, because it assumes that tOFF (the time for which VOFF is applied) is equal to the full available off-time (1 - D)/f .

Conversely, L as a function of r is (any topology).

In subsequent sections we will often use the following easy-to-remember form of the previous equations. We’re going to nickname this the "L × I" equation (or rule): (any topology)

But perhaps we are still wondering - why do we even need to talk in terms of r - why not talk directly in terms of L? We do realize from the above equations that L and r are related.

However, the "desirable" value of inductance depends on the specific application conditions, the switching frequency, and even the topology. So it’s just not possible to give a general design rule for picking L. But there is in fact such a general design rule-of-thumb for selecting r - one that applies almost universally. We mentioned that it should be around 0.3 to 0.5 in all cases. And that is why it makes sense to calculate L by first setting the value of r. Of course, once we pick r, L gets automatically determined - but only for a given set of application conditions and switching frequency.

The Optimum Value of r

It can be shown that, in terms of overall stresses in a converter and size, r ˜ 0.4 represents an "optimum" of sorts. We will now try to understand why this is so, and later we will try to point out exceptions to this reasoning.

The size of an inductor can be thought of as being virtually proportional to its energy-handling capability (the effect of air-gap on size will be studied later). So for example, we probably already know intuitively that we need bigger cores to handle higher powers. The energy-handling capability of the selected core must, at a bare minimum, match the energy we need to store in it in our application - that is,

1/2 × L × IPK2.

Otherwise the inductor will saturate.

In Fgr. 6, we have plotted the energy, E = 1/2 × L × IPK2, as a function of r. We see that it has a "knee" at around 0.4. This tells us that if we try to reduce r much lower than 0.4, we will certainly need a very large inductor. On the other hand, if we increase r, there isn't much greater reduction in the size of the inductor. In fact, we will see that beyond r ~ 0.4, we enter a region of diminishing returns.

Fgr. 6: How Varying the Current Ripple Ratio r: Affects All the Components

In Fgr. 6, we have also plotted the capacitor RMS currents for a buck converter. We see that if r is increased beyond 0.4, the currents will increase significantly. This will lead to increased heat generation inside the capacitors (and other related components too).

Eventually, we may be forced to pick a capacitor with a lower ESR and/or lower case-to-air thermal resistance (more expensive/bigger).

Note: The RMS value of the current through any component is the current component responsible for the heat developed in it - via the equation P = IRMS 2 × R, where P is the dissipation, and R is the series resistance term associated with the particular component (e.g. the DCR of an inductor, or the ESR of a capacitor). However, it can be shown that the switch, diode, and inductor RMS current values are not very "shape-dependent." Therefore, the heat developed in them does not depend much on r, but mainly on the average value of the current. On the other hand, the RMS of the capacitor current waveforms can increase significantly, if r is increased. So capacitor currents are very "shape-dependent," and therefore depend strongly on r. The reason for that is fairly obvious - any capacitor in a steady state has zero average (dc) current through it. So since a capacitor effectively subtracts out the dc level of the accompanying current waveform, we are left with a capacitor current waveform that has a large "ramp portion" built-in into it.

Therefore, changing r changes this ramp portion, thereby impacting the capacitor current greatly.

Note that in Fgr. 6, though we have used the buck topology as an example, the energy curve in particular is exactly the same for any topology. The capacitor current curves though, may not be identical to those of the buck, but are similar, and so the conclusions above still apply.

Therefore, in general, a current ripple ratio of around 0.4 is a good design target for any topology, any application, and any switching frequency.

Later, we will discuss some reasons/considerations for not adhering to this r ~ 0.4 rule-of-thumb (under certain conditions).