CRYSTAL IMPEDANCE

WHAT YOU ARE LOOKING AT

The notebook this comes from opens with a confession: "I have a problem with calculating input impedances and transfer functions. I don't trust my own brain when it comes to the algebra." It then uses sympy to expand the input impedance of the standard crystal model and compare the answer against a published one, to settle whether the textbook or the reader is wrong. (Neither, as it turns out — the published version drops a term that barely moves near resonance.)

That expansion is one line, so this page does it directly rather than shipping a computer algebra system to your browser:

Z_in = 1 / ( 1/(R + sL + 1/(sC_F)) + sC_P )

A quartz crystal is a mechanical resonator that happens to be reachable through electrodes. The motional arm — R, L, C_F in series — is the mechanical resonance in electrical clothing, and C_P is the plain capacitance of the electrodes. Because the electromechanical coupling is weak, C_F comes out in femtofarads and L in millihenries: values you could never build from actual components, which is exactly why the Q is in the tens of thousands and why crystals are worth their cost.

Two resonances fall out. At f_s the motional arm's L and C_F cancel and the impedance collapses to R. Slightly above it, the now-inductive motional arm resonates with C_P and the impedance peaks at f_p. Between the two, and only between the two, the crystal looks inductive — and a Pierce oscillator needs it to look inductive. That window is a few kilohertz wide on a ten megahertz part, which is the entire reason crystal oscillators are accurate.

THINGS WORTH TRYING

• Look at the width of the shaded inductive region against the frequency axis. On the default values it is about 5 kHz out of 10 MHz. An oscillator has to land inside it, and everywhere else the crystal is capacitive and the loop will not start.
• Move C_P from 1 to 30 pF and watch f_p slide while f_s does not move at all. That is crystal pulling, and it is the one adjustment you get: f_s is fixed by the quartz, f_p by the quartz and your circuit. The pulling panels quantify it — note that sensitivity falls off as 1/C_P², so trimming is far more effective at small loads.
• Drop R to 1 Ω. Q goes to the millions and the series notch becomes narrower than the plot's frequency grid, so the drawn minimum stops reaching R. The dashed markers are computed analytically rather than by searching the curve, so they stay correct where the trace does not — a good reminder that a sampled plot of a very high Q resonance lies to you.
• Raise C_F towards 30 fF. The gap between f_s and f_p widens — it goes as √(1 + C_F/C_P) — and the crystal becomes easier to pull but less stable. This is the trade a MEMS resonator makes against quartz.

THE NOTEBOOK

from sympy import symbols,expand,simplify,solve

Gin,Zin, s,Cf,L,Cp,R,Z = symbols("G_{in} Z_{in} s C_F L C_P R Z")

Gin = 1/(R + s*L + 1/(s*Cf)) + s*Cp
Zin = expand(simplify(1/Gin))

Z = Zin.subs({R:50,Cf:5e-15,Cp:5e-12,L:50e-3})
func = lambdify(s, Z,'numpy')
Znum = func(1j*2*np.pi*f)

Source: jupyter/xosc.ipynb. The notebook finds the peak by np.argmax on a 1000-point grid; this page uses the closed forms f_s = 1/(2π√(L·C_F)) and f_p = f_s·√(1 + C_F/C_P) instead, which do not depend on how finely the curve happens to be sampled.