May 5, 2016

# Introduction

Capacitors are passive two-terminal components that resist the change of voltage.  There are so many different forms of capacitors (electrolytic, tantalum, film, ceramic, polymer, etc.)  They vary by construction, dielectric, size, and polarization.  Yet they all share one thing in common: they store energy.

These tiny devices have so many uses and there are many ways to take their characteristics into account when modelling them in SPICE software.  My focus has always been in the practice of using components in RF Engineering.  This post will be geared to those engineers, however, the model presented can be used in modelling power supply circuitry as well as others.

# Theory

Capacitance is (mathematically) how much charge across two plates is stored proportional to voltage.

This is a very simple equation, but it isn't the most useful in most electrical engineering fields (unless you are DEEP into theory).  We need a better equation to work with.  So we know that capacitors oppose the change in voltage.  When a change occurs, a current is produced within the capacitor across the dielectric causing the plates to be charged (or discharged).  Depending on the direction of the current, we could release current into the sinking circuit to maintain the voltage.  This can be given by the following equation:

This equation is much easier to work with.  And because we have the Laplace Transform (to transform to the frequency domain), we can find the impedance of an ideal capacitor.

Then to find impedance we move I(s) to the right side and sC to the left and we will get the following equation:

With the impedance in the frequency domain, we can do a lot.  If you have a desire for a transient calculation, a more formal approach is required.

# Real Capacitors

So now we can talk about real capacitors and how they act in live circuits.  Again, I'm focusing on their use in RF engineering and this list isn't exhaustive by any stretch. However, here are a few uses of a capacitor:

• Filters: RF/Microwave filters almost always have capacitors
• Resonant Circuits:When combined with inductors, capacitors can resonate at a very specific frequency to create a highly efficient matched circuit. Useful in power transfrom and filter applications
• Sensors: Capacitive touch screens work with principal of capacitors (although not necessarily RF)
• DC Blocks: Capacitors block DC current and can help to block DC from entering on an RF line
• Energy Storage: When DC is required to power an RF line (PIN Diode switches, etc), they can be used to keep AC current away from DC sources by places the capacitors close to the RF line.  At high frequencies, capacitors act like shorts and thus AC current will flow to ground.

Like with inductors, low frequency and high frequency uses of capacitors change parameters that we need to take into account when simulating a capacitor. As the frequency of a circuit goes up, the capacitor will start to behave differently from it's ideal implementation. This is caused by the dielectric, size, and composition of the capacitor.

Every real-world component has parasitics. And because of this, basic elements (like inductors, capacitors) will have resonant frequencies due to the parasitic parts.  To model a capacitor properly, we need to add these in.

There are two main parasitics to take into account: Equivalent Series Resistance (ESR) and Equivalent Series Inductance (ESL). With these two parameters we can model a capacitor extremely well up to its first resonant frequency.  The standard model is below:

The C represents the ideal capacitance (within a tolerance).  The ESR and ESL is what we defined above.  As you may guess, we have a series resonant circuit here. The capacitor and inductor will resonate at some frequency and that is what we need to figure out.  To find this value, let's get the impedance of this circuit.

This equation tells us two things:

1. At frequency 0 (or DC), the impedance "blows up" (or gets extremely high)
2. At some frequency greater than 0, we will see an impedance equal to R

After simplication and substitution we get the zeroes of the above equation:

It's a little hard to see how this gets us close to the resistance.  So let's reform our original impedance equation to simplify our equation a little bit.

Here we can see the same to be true as before.  When w = 0, the second half of the equation "blows up" and we have a very large impedance.  However, if we want to find out where the impedance is minimum, we solve for w in the numerator of the second half.

Now we know that when

we will zero out the second half of the equation and only have R left.  That was much simpler...

# Example

So let's go through an example now.

One of my favorite capacitor companies is ATC Ceramics.  They make a great line of ceramic capacitors perfect for RF/Microwave circuits. Of all of their capacitors the 100B series is the one I used the most (Reference: http://atceramics.com/Product/17/100_B_Series_Porcelain_Multilayer_Capacitors_(MLCs)).  If you look at their data, they provide the ESR directly.  However, they don't have ESL shown in any of their graphs. We must derive this from their Series Resonance map.  Let's pull this information.

I want a capacitor with a value of 100pF (100E-12F).

On the ESR chart, they show different values at different frequencies.  Unfortunately, we don't have the SRF yet.  So let's look at the SRF chart next.

We see that our 100pF capacitor resonates around 700 MHz.  If we go back to the ESR chart, we can see that around 700 MHz (roughly interpolated), we get a value of about 0.07 Ohms or 70 mOhms.  So now we need to solve for the ESL.

If we plug in our values we get:

Now we have all of our values we need to simulate our real world capacitor.  If we plot this on the smith chart (Data found at http://www.atceramics.com/design-support.aspx) we can see how it is very close.

This is the ATC100B101 S2P (S22) plotted up to 700 MHz.  The top of the line is where 700 MHz is located.  It is very close to zero on the smith chart which shows that at 700 MHz (with measured data) we resonate.

Conclusion

Capacitors and their real world application can be a little confusing if you go to simulate them and the measurements don't line up.  A lot of times it has to do with these parasitics not being included in the measurement.  This is why it is critical for high frequency engineers to make sure that when you move to simulate real world circuits that you include every factor available.  Many times you can just estimate these parameters.  In practice, I usually used R = 0.1 and L = 0.5 nH for ATC capacitors (100B, 600S, etc).  This gives a fairly good simulation and typically lines up with most measurements in larger circuits.  However, if you want to be more accurate, use the technique above to find the appropriate values to enter into the simulation.

Hope this helped!

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