# How can I solve these two equations to determine resonant capacitors for a wireless charging coil?

My questions:

• What are S, D, and the apostrophe above Inductance?
• Is the equation looking for the inductance of the coil or the circuit?
• Once I know what S and D stand for how can I determine their frequencies?

The following is an excerpt from the datasheet for Texas Instruments bq51221 Single Chip Wireless Power Receiver.

10.2.1.2.9 Series and Parallel Resonant Capacitors

Resonant capacitors C1 and C2 are set according to WPC specification. Although this is a dual mode solution, the PMA does not specify an exact resonance frequency for the resonant capacitors and in fact does not specify that resonant capacitors are indeed needed.

The equations for calculating the values of the resonant capacitors are shown:

If you look at the main circuit on page 1 of the data sheet they show C1 and C2 in series: -

However, they forgot to indicate what $$\L_S\$$ is so, it's somewhat of a guess on my part. Firstly however, we have to think of pins AC1 and AC2 as current sources in opposition i.e. one is the inversion of the other.

This then permits us to be able to say that C1 and C2 are truly in series and, excitation from the AC pins to C2 is in the form of a current and, that current doesn't screw around with the impedances and formulas.

The resonant frequency of a parallel inductor and capacitor is this: -

$$F_R = \dfrac{1}{2\pi\sqrt{LC}}$$

And, it appears that C1 is the dominant reactance i.e. the highest reactance compared to C2 and, I suspect that this means you can rearrange the above formula to find C1: -

$$C_1 = \left(\dfrac{1}{2\pi F_R\sqrt{L}}\right)^2$$

Hopefully you can see that it is equivalent to your first formula (apart from the inductor being called $$\L'_S\$$). I think they are calling it $$\L'_S\$$ to fudge the math to make it work. So, I would call $$\L'_S\$$ the equivalent inductance that makes the circuit resonate at the correct frequency if we ignored C2.

Again, it seems an incompetent way of doing it but, TI no doubt have their reasons.

Then, for the 2nd formula TI are doing the right thing and solving my first formula where C is replaced with: -

$$C = \dfrac{1}{\frac{1}{C_1}+\frac{1}{C_2}}$$

• Thanks for the answer. My only comment is that the datasheet refers to them as series and parallel. Are you saying that they are in series to themselves but parallel to the coli? May 9, 2023 at 18:39
• It's a compound connection; C1 is definitely in series with the inductor but C2 is parallel to the current source produced by AC pins. Sometimes there isn't a clear distinction between the two capacitors because it involves think about the driving source and whether it is a voltage or a current source. I would tend to call it a series connection for calculation of tuning. May 9, 2023 at 19:03
• Turns out I have another comment. How do I calculate LC? May 9, 2023 at 19:49
• What frequency? May 9, 2023 at 20:04
• Sorry for stumbling thru this. In order to find resonant frequency you need to know capacitance (chose a capacitor) right. If that is true then doesn't that make your second equation kind of circular? I'm confused. May 9, 2023 at 20:13