Deriving resonance frequency of RLCC circuit

Question

I am trying to determine the resonance frequency of an RLC resonator with added capacitor, as shown below:

^{simulate this circuit – Schematic created using CircuitLab}

Two cases are easy to determine:

$$Rp=0 \rightarrow \frac{1}{\sqrt{LsCs}}$$ $$Rp=\infty \rightarrow \frac{1}{\sqrt{Ls\left(\frac{1}{1/Cs+1/Cp}\right)}}$$

Now I'd like to derive a formula that includes Rp. How should I approach this?

Answer 1

When Rp is infinite you have your equation shown incorrectly. The two capacitors are NOT in parallel; they are in series and as such add up like this: -

\$C_T = \dfrac{1}{\frac{1}{C_S}+ \frac{1}{C_P}}\$

When Rp is zero ohms there is no resonance so your first equation is meaningless.

So, here's the next problem - what do you mean by resonance? Do you mean: -

1. The frequency at which the impedance is maximum (parallel tuning)?
2. The frequency at which impedance is minimum (series tuning)?
3. The frequency that produces the right phase shift to make a pierce oscillator oscillate?

So, when you have decided what to do with the formula in your question and then decided what it is you are actually looking for you might see the wood from the trees.