VXO design question (do you need to know entire equivelant circuit?)

When designing a VXO, do you have to know the values of all of the components of the equivelent circuit of the crystal, or is there some other approach that is taken? I noticed that the only values having to do with that, that seem to be given on most spec sheets for crystals are the circuit conditions (which apparently is the capacitance of the surrounding circuit), the series resistance and the parrallel capacitance.

In a VXO, the crystal acts as an inductor with a very small series capacitor (.02 pF?) and its ESR. So you could calculate the notional inductance if required. Should be enough to feed your simulator?
Frank

How would you calculate the inductance if you don't know what the series capaciatnce is? It seems as though there is not enough information.

I ran through the equations and I thought it makes sense.. But the more I look at it, if
C1 << C0 than C0 will basically have the effect dropping out of the equation for the impedance of the crystal. And the oscillation frequency would not be dependant on C0, but on L1 and C1, niether of which values are provided by the manufacturer. So how do you determine their values for designing the overall circuit and making a tuner circuit that will track it? Below are some calculations to show how I mean. Please tell me where I am going wrong. I didn't include the ESR in my calculations because I thought that it is only causing the damping and would not effect the value of resonant frequency. Is this incorrect? Please see below:

At some frequency W C1 = W L, then the circuit is ESR with Co across it. At a higher frequency W L - W C1 is still an inductive impedance ( Lx), then Lx will resonate with Co to provide a high impedance. In a conventional crystal oscillator a trimmer is used to change the value of Co to adjust the crystal frequency by say +- 200 parts/million.
Frank