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I am trying to pick an inductor for use with switching regulator with switching frequency of 500KHz.

I looked at single inductor manufacturer which has two line up. One saying "Inductors for Decoupling Circuits", and the other saying "Inductors for Power Circuits."

The links to datasheets are below.

[Datasheet] Inductors for Decoupling Circuits

[Datasheet] Inductors for Power Circuits

What are the difference between the two other than the current capacity?

May I use any of them if both fits my current budget?

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Those two data sheets make it difficult to compare like with like, as the power inductors have two rated currents (inductance and thermal), whereas the decoupling inductors have a single rated current, and a minimum Q.

In my designs, I have two different applications, both of which require things called inductors. They both require a certain current to be supported while maintaining a minimum inductance.

The first application is in a power supply filter network, where some degree of dissipation in the ferrite at high frequency is positively beneficial. It damps down resonances between components, and absorbs RF signals that I don't want crawling between one module and another. Even those inductors look a bit 'lively' for my tastes, and I usually end up using 'impeders', which have a specified loss at RF.

The second application is energy storage, in dc-dc converters for instance. Here, any power dissipation in the ferrite is a bad thing, it all contributes to overall inefficiency in a converter design.

You can't really optimise designs for either of these applications from a brief datasheet alone. You need a detailed data sheet, and then investigation on a bench with the appropriate test gear. Though using components from the manufacturer's suggested range is a good start.

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As you have already observed, the current rating appears to be the main difference. I would argue that "Inductors for Power Circuits" is a description of a particular kind of inductor, while "Inductors for Decoupling Circuits" is a particular application for inductors.

But, as you also conclude, it would be perfectly appropriate to select an inductor that meets your requirements from either category.

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