Introduction

Inductors are passive two-terminal components that resist the flow of current.  Most inductors are wound in some way, shape, or form.  Energy is stored in a magnetic field for the duration of current flow and when the flow is cut, will induce a voltage across the inductor.  This leads us to the use of inductors to tune and protect circuits.  But before we go into detail on how it can be used and the real world applications, let's get the basics out of the way.

Theory

Inductance is (mathematically) how much magnetic flux is created based on a given current:

Although, this equation is beautiful in its simplicity, it's not real useful in the real world circuit theory.  For us, we desire a better equation.  So let's re-cap: an inductor opposes a change in current.  When a change occurs, a proportional voltage is induced across the inductor.  This brings up a better equation we can use (although it is saying the same thing as above, it is easier to understand in terms EEs already know).

Now this I can understand.  The change in current multiplied by the inductance produces a voltage.  Perfect!  Since we know that 

, can figure out the proper impedance of an inductor by turning this into it's laplace transform and solving for v(t) / i(t).

Reference: https://en.wikipedia.org/wiki/Laplace_transform

where,

and thus:

With this impedance across frequency, we can do a lot.  If you have a desire for a transient calculation, a more formal approach is necessary.

Real Inductors

So let's talk about real inductors.  How are inductors used?  Inductors are used in many many applications.  Here are a few:

• Filters: RF/Microwave filters almost always have inductors.
• Resonant Circuits: When combined with capacitors, inductors can resonate a very specific frequency to create a highly efficient matched circuit for the given frequency.  Useful in power transform and filter applications
• Sensors: Inductors can be used to detect magnetic fields or detect magnetically permeable fields.  Traffic lights use this to detect traffic.
• Transformers: Turning one voltage into another via paired inductors is very common in power applications
• Motors: Inductors create magnetic fields when current passes through the coils and thus motors and generators can be formed.  Think alternator in your car.
• Energy Storage: Like its counterpart (the capacitor) it can store energy in the magnetic field.  Mainly used in power supplies.

Low frequency uses of inductors and high frequency uses change the parameters that engineers look at when determining what type of inductor to use and with what stats.

I worked in primarily the high frequency RF range (30 MHz to 2000MHz).  In this range there a lot of factors to keep in mind.  The parasitics of the inductor is what caused me the most grief.  Parasitics are the innate imperfections in a component that results as a property of size, material, and frequency.  Inductors are no exception.  

Depending on the frequency range, inductors can be modelled in various different ways.  We usually modelled inductors based on their first resonant frequency.  One thing every engineer should realize is that components have many many resonant frequencies.  As you move up in frequency, inductors will keep circling the smith chart (so to speak).  After the first resonant frequency though, it can become unpredictable.  In this case, the first resonant is usually the most important.  This first resonant frequency is also called a Self Resonant Frequency (SRF).

To model this type of inductor (up to the SRF) is the model below:

The L represents the actual inductance.  This is usually very close to the advertised value within some tolerance.  Note that this is usually measured at a specific frequency as inductance L can change over frequency.

R represents the equivalent series resistance (ESR or DCR: DC Resistance) of the component and occurs due to the wire thickness and material.  It can also be affected by the ferromagnetic material if not an air core inductor.

The C is the equivalent parallel capacitance (EPC) of the component which occurs due to the space between coil turns.  There is a natural capacitance that forms here.

Turning this into an equivalent equation and solving to find the impedance produces the following equation:

This equation gives us some incite into how an inductor works.  Let's say we want to know what an inductor does at w = 0 (or DC).  By substituting w with 0 we get:

This definitely matches our DCR conclusion above where we have an R in the circuit.  Many times this is in the milliOhms but still it can cause issues in RF circuits.

Now what about the resonant point (SRF)?  This is a bit trickier but this would be the other place in the equation where the imaginary part of the equation becomes 0.  We need to solve this for w.  

With these we can call 

 the SRF and solve the equation for 

 to give us the impedance at resonance:

This tells us that at the SRF, the reactance disappears.  This can be seen because we have no reactance component (imaginary) in the result.  So the resistance at this point maximizes to inhibit current flow the most. Let's look at an example real quick.

Coilcraft is a leading maker of inductors.  Specifically they make great Air Core Inductors.  Let's take the 1111SQ-27N inductor (Datasheet: http://coilcraft.com/pdfs/1111sq.pdf).

It has an inductance of 27nH, DCR of 7mOhm, and an SRF of 2.6GHz.  The datasheet does NOT tell us what the parasitic capacitance is.  So let's figure this out.

By inverting the equation above (where we found 

), we can find C.

By substitution, we find C = 0.139 pF (1.387E-13 F).  Now we can find out 

 by substitution. (Remember

)

Now this is pretty big for an inductor!  Now with the values of R, L, and C we found, we can model this inductor fairly accurately in a SPICE program.  Below is a Smith chart plot of the inductor.  (S2P Data: http://coilcraft.com/zip/1111sq.zip)

(Plot Generated at http://quicksmith.engineerrf.com/)

Conclusion

So there ya go.  Inductors.  They are much more complicated than most engineers might think.  Here we have shown how a basic inductor can be modelled more realistically by using basic parasitic components (R and C).  These values, once determined from inductor datasheets (or measurement) can be thrown into spice programs and used for a realistic model of the inductor.  Unfortunately, if you model everything in a spice program with the "ideal" inductor, your simulations will always seem off (unless of course you work in kHz/low MHz range).  Once you get to higher frequencies (GHz), it becomes critical to take these things into consideration.

Note: I have always been a huge fan of Coilcraft Inductors.  During my time as an RF/Microwave Engineer, I have always preferred them over anyone else.  They offer free samples to try (within reason) to professionals and students alike.  So try them out, measure them, enjoy them.  If you have any questions, email me!

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