An inductor is designed for specific conditions. However, you can judiciously vary these conditions. In fact, you can use a given inductor for many applications. In order to do this, the designer must mathematically predict, or ‘extrapolate’ the inductor's performance for a set of conditions other than its original intended conditions.

**Fig. 1** shows the current through an inductor in continuous mode operation (upper waveform). One of its main elements, I_{DC}, which is the geometrical center of the ac/ramp component, is also the average (dc) value of the total inductor current, and the current into the load.

At maximum rated load I_{O}

The ac component of the current is

where:

I_{TROUGH} = I_{DC}-ΔI/2

The energy associated with I_{TROUGH} depends on the energy going into the load, but is itself not transferred into the load. It's therefore residual. We now define *r* as the ratio of the ac component to the dc component of the inductor current — calculated at the maximum load condition. It's the “current ripple ratio” with the converter delivering its max rated load:

By definition, *r* is a constant for a given converter/application (even though the actual output load varies). Also, a high inductance reduces ΔI and results in lower *r* (and lower rms current in the input/output capacitors), but may result in a very large and impractical inductor. For most buck regulators you would choose *r* to be in the range of 0.25-0.5 (at the maximum rated load). After selecting the inductance, as the load on the converter decreases (keeping input voltage constant), ΔI remains fixed but the dc level decreases and *r* increases. Ultimately, at the point of transition to discontinuous mode of operation (lower waveform in **Fig. 1**), the dc level is ΔI/2. The current ripple ratio, *r*, at the point of transition to discontinuous mode is 2. For the continuous conduction mode converters covered here, *r* can range from 0 to 2. The load where transition occurs is *r*/2 times the max rated load. So, choosing *r* to be 0.3 (at a maximum load of 2A), the transition to discontinuous mode of operation will occur at 0.15×2A=300 mA. This may be an additional consideration when choosing the inductance.

In a “swinging” inductor, inductance normally increases as load current decreases and the point of transition to discontinuous mode may be significantly lower. Here, consider only inductors with a “flat” characteristic over the entire current range.

**Table 2,** our main design table, is in terms of volt-μsec or “Et,” which is defined as the voltage across the winding of the inductor times the duration in msec for which it's applied. Together with I_{O}, Et almost completely defines the application itself, but solely from a magnetics viewpoint. Et determines the ac component, and I_{O} the dc component. Almost none of the parameters in **Table 2** even involve the frequency. The exception is the core loss term that depends on both Et and the frequency (f). As a corollary, all applications with the same Et, I_{O}, and f are the same application from the viewpoint of inductor design/selection. You can cross-utilize an inductor across all such applications without recourse to the extrapolation procedure presented later in this article. We require this procedure only when attempting to use the inductor for a different Et, I_{O}, or f than its original design.

Inductor size relates primarily to its energy-handling capability, which is ½LI_{PEAK}^{2}, the peak energy in the inductor. For a given application, if we reduce inductance, it seems that this would cause an increase in ΔI, and therefore I_{PEAK} would increase too. This would cause the energy requirement to increase as it depends on the square of current. However, a detailed calculation shows the inverse is true. In terms of *r* and Et, we can derive the energy handling capability:

For a given application, Et is fixed and so is I_{O}, so the term in square brackets gives “e” the shape shown in **Fig. 2**. We can see that the energy-handling requirement (size of inductor) decreases as *r* increases (L decreasing). The best value is the knee, so it's a good idea to target an *r*=0.3 to 0.4. Increasing *r* much more than this will provide no great improvement in size of inductor, but the rms current in the input and output capacitors (especially for large duty cycles) will increase substantially.

In fact, the choice of *r* (or L) is the only degree of freedom available to the designer. It impacts the inductor and also affects all the converter's power components in varying degrees. **Table 3** lists a complete set of optimization equations expressed as a function of *r* and **Fig. 2** is a visual guide (based on **Table 3**) of alterations of *r* from the chosen reference value. This leads to the best value for *r*.

One of the main points in the decision-making process is **Table 3**, which provides the general equations required for optimization. It also provides their respective values for *r*=0.3 in the right-hand column, which serves as a benchmark (equivalent to the “flat top/low current ripple/high inductance” approximation, often used for quick estimates). **Fig. 2** normalizes each displayed parameter to unity value at the benchmark value of *r*=0.3, and then plots their respective variations, as *r* changes on either side.

Cost is a key concern, depending on the absolute value of a given parameter, not necessarily its relative variation. For example, it may seem that the percentage change in the input capacitor's ripple current requirement (rms rating) is not of much concern, but in reality, the cost related to any change in this rating may be far more significant than that related to the output capacitor. The designer must note from the equations and curves provided that the variation of the input ripple current is maximum for largest duty cycles, but its absolute value is the largest for D=0.5. Switch dissipation calculation depends on whether the switch is a bipolar transistor or MOSFET. If it's a MOSFET, we need to apply I^{2}R where I is the rms switch current and R is the R_{DS(on)} of the MOSFET. If it's bipolar, we need to use V×I, where V is the saturation voltage across the switch, and I is the average switch current. **Table 3** provides both V, the saturation voltage, and I, the average switch current.

**Fig. 2** shows little change in the rms inductor current over a very wide range of *r*. For the purpose of evaluating copper loss, designers often use I_{O}^{2} instead of I_{RMS}^{2}. The core loss, though not plotted out (because of too many core material possibilities), increases substantially with increasing *r*. Even if we keep core size unchanged, the flux density swing, B_{AC}, will go up proportionally to the square root of *r*. Core loss for most inductors used in buck converters, especially ferrite, is very small compared to the copper loss. However, it's increase with *r* may not be a major concern.

RMS capacitor currents in the input and output are the main components to consider because they can increase rapidly with *r*. For example, from **Fig. 2**, increasing *r* from 0.3 to 0.6 cuts the energy-handling requirement of the inductor by about 35%, pointing to a smaller inductor, but output capacitor dissipation (assuming unchanged ESR) increases 400%. We now need to select a much larger output capacitor with twice the ripple current rating.

Keep in mind that there's an output voltage ripple ΔV=ESR×ΔI associated with the current ripple. For loop stability reasons, you cannot typically decrease output capacitor ESR below 100 mΩ (with voltage mode control). For high loads (high *r*), output capacitor dissipation will necessarily be high since we cannot reduce ESR further. This may call for physically large size output capacitors to handle the dissipation. In addition, the output voltage ripple will also be high. Because the output voltage ripple can't be decreased by reducing the ESR, we will need to add a post LC filter, which we would rather avoid. So, for high load currents, it may become necessary to decrease *r* substantially. This in turn will call for a large inductor with slower transient response, but it may be a trade-off worth considering.

As mentioned earlier, *r* also determines the current where the converter goes discontinuous. This may be of concern either for loop stability reasons and/or ensuring a fast step-load response at light loads, or if the minimum expected load is extremely low and requires a low output voltage ripple at this condition.

The implication of the physical size of the inductor relates only to its energy handling capability. This suggests inductor designs that are basically core-saturation limited. Although this is usually true for ferrite-based inductors, it may not be true for some powdered iron inductors. Core saturation and even more core loss limits the size of these. This depends on flux swing, or ΔI, not I (or energy “e”). So, while **Fig. 2** is still valid, the best choice criterion may change. It may be necessary to choose or restrict *r* to much smaller values than the knee (typically *r*=0.15 for powdered iron). Most powdered iron cores don't saturate abruptly, and in such cases, generally, core loss is set at about 10% of the overall inductor loss, the rest being copper loss. In most ferrite-based designs, the core loss is usually less than 5% of the total inductor loss.

For current-mode control, we may need to limit the maximum allowed value of *r* according to the slope compensation used, which avoids subharmonic instabilities. So, the inductance may have to be larger than a certain minimum value.

**Example:** For an LM2593HV voltage-mode controlled buck converter (**Fig. 3**) we require an output voltage ripple of 30mV peak-to-peak (±15mV).

Where:

V_{IN}=24Vdc

V_{O}=12Vdc

f=150,000 Hz

I_{O}=1A

V_{SW}=Voltage drop across the switch when it is on=1.5V

V_{D}=Estimated forward voltage drop across the Schottky catch diode=0.5V

For loop stability reasons, we shouldn't use an output capacitor with an ESR less than 100 mW, and we prefer not to use an LC post filter. Therefore, our ΔI must be:

This is a typical and acceptable value for *r*. For a buck regulator, we can show that the duty cycle, D, is:

Switch on-time is:

=3.62 μsec

Et for the application is:

=38.0Vμsec

From the general equation V=L dI/dt, Vdt=LdI. Here Vdt is the applied volt-seconds, so:

Solving for L:

μH

Next, we calculate the required energy handling capability. Every cycle, the peak current is:

The required energy handling capability “e” is

Most vendors don't provide the energy handling capability directly, but it's useful to calculate and check it, as it literally sizes the core. Therefore, it helps decide if we need to move to a physically larger or smaller inductor, regardless of inductance. We can compare the data for a single family of core-saturation limited inductors using the same ferrite core material and the same geometry from a given vendor. This will show that although the inductance and current ratings are widely different, the energy handling capability, as calculated above, turns out to be exactly the same for all. In powdered iron cores, the ac current component is the main size factor, so this is only partially true. In such cases, inductors designed for a smaller *r* than the rest of the family will have a larger “energy handling capability.”

### Choosing an Inductor

The inductance calculated above and the dc current rating (max load) is the basis of our first pass selection. We tentatively select a part from Pulse Engineering because its L and I_{DC} are close to our requirements, even though the rest does not seem to fit our application at first sight (**Table 1**, on page 18). In particular, the inductor's design frequency is 250 kHz, but our LM2593HV application is 150 kHz. We intuitively believe that since the frequency decreases that core losses will increase, and so will the peak flux density. In fact, the reverse happens in our case, and that is why it's important to follow the full extrapolation procedure. According to the vendor's information:

- 380mW dissipation corresponds to 50°C temperature rise.
- The core loss equation for the core=6.11×10
^{-18}× B^{2.7}×f^{2.04}mW

f in Hz and B in Gauss - Et
_{100}is the Vμsecs at which B is 100 Gauss.

It's conventional to use half the peak-to-peak flux swing for core loss equations. So, like most vendors, the B above actually refers to ΔB/2. Keep this in mind in the calculations that follow. The astute designer can recognize the exponents of B and f in the core loss equation to correspond to a ferrite (which this is). Powdered iron material 52 (from Micrometals Inc.) would have given us an equation of the form a×B^{2.11}×f^{1.26}. Most off-the-shelf commercial inductors specify a 40°C to 55°C temperature rise.

**Table 3** lists all the equations required to extrapolate the inductor's performance from its original intended design conditions (unprimed parameters) to a general application (primed parameters). Our chosen application conditions are: Et′=38Vμsec, f′ = 150,000 Hz, I′_{DC} = 1A. You can ignore T_{AMBIENT} assuming it unchanged.

Using **Table 2, Equation 2**, on page 20, the design value for *r*=0.438 extrapolated to our application, *r*′=0.277. We expected *r*′ to be slightly lower than 0.3 since the chosen inductor has a higher inductance than we required (137 μH instead of our calculated requirement of 127μH). This is acceptable, however, as the output voltage ripple will be less than demanded.

**From Table 2, Equation 7**, on page 20, Peak Flux Density, B_{PEAK}=3267 Gauss. Extrapolated to our application, B′_{PEAK}=3084 Gauss, which is less than B_{PEAK} and therefore acceptable.

To ensure the regulator will deliver the rated load, we must ensure the peak current is less than the internal current limit of the switcher IC. The design value, I_{PEAK}=1.21A, corresponds to a B-field of 3267 Gauss as calculated above. This gives an energy handling capability of e=100μJ. We required at least 84μJ, so the inductor seems sized right and we can proceed with the analysis.

You can find the value of I_{PEAK} from **Table 2, Equation 3**. Extrapolated to our application, I′_{PEAK}=1.14A, which corresponds to a B-field of 3084 Gauss, as calculated above. The minimum value of the LM2593HV current limit over temperature and tolerance is 2.3A. Since the peak value I′_{PEAK} is less than 2.3A, the controller can provide the desired output power without hitting the current limit.

Calculations for the copper losses, P_{CU}, and core losses, P_{CORE}, involve **Table 2, Equations 7** and **8**, respectively, on page 20. Plugging the P_{C}U value of 385mW and P_{CORE} value of 18.7mW into **Table 2, Equation 10** gives the temperature rise for the inductor, where Rth is the inductor's thermal resistance. The vendor stated 380mW dissipation corresponds to a 50°C rise. So the inductor's thermal resistance is DT=53°C. Extrapolated to our application μT′=51°C, which is acceptable.

The role of intuition should be clear here. For example, we may have thought that since the inductance and current rating of the available inductor was what we needed, if we lowered the inductor's 250 kHz design frequency to our 150 kHz application frequency, the peak current and fields would increase (since the switch on-time would increase). In fact, the calculations showed that the peaks actually decreased; therefore it was safe to use the inductor. The peaks diminished with decreasing frequency because the original design of the inductor was for a higher Et (59.4Vμsecs vs. our applied 38Vμsecs). So we did not just lower the frequency, we also decreased the applied voltage.

Looking at it in a different way, since Et (with I_{O} and f) defines the converter/application configuration itself, we did not just lower the frequency, we actually went to an entirely different input-output voltage combination from what the inductor had been originally designed for. By applying a lesser voltage than designed for, we seem to have more than compensated the expected increase in the peak fields due to the reduction in frequency. Intuition may have indicated otherwise, but the calculations correctly revealed the safe use of the inductor in the application.

### Acknowledgments:

*Thanks are due to Jon Cronk and Chester Simpson for valuable input and feedback, and to Dr. GT Murthy and Disha for the encouragement in making this possible*.

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