Crystal oscillator <1 MHz

Created: May 2019

Introduction

Crystal oscillators below 1 MHz tend to be sluggish when used with oscillator circuits designed for higher frequencies. The oscillation potential is inhibited by the crystal's high motional resistance (Rm) which causes excessive loss. For example, a  100 kHz crystal has an Rm ≈18 kΩ (vs. just 6 Ω for a 10 MHz crystal).

Material & methods

To enable the oscillator to be analyzed as a two-port, a virtual earth in inserted at the emitter and the loop is broken at the base (Alechno transform). The conditions for oscillation are: 1. transmission phase = 0 degree and 2. gain at 0 phase, |G0| > 1 [1].

The 100 kHz crystal was modeled using the parameters on Kraus' website [2]. Due to the unavailability of a BC549 model in my circuit simulator, it was replaced with BC547B. This substitution is not expected to significantly change the results because most of the pertinent parameters are similar except that the former has been test-binned for lower noise.

At zero phase crossing, the overall gain is less than 1. So, the cascade doesn't enough gain to sustain oscillation.

The second experiment changes the feedback capacitors C1 and C2 to the reactance values recommended by Hayward [3]; i.e. C1 = -j500 and C2 = -j200. DeMaw also recommended reactance values in the same ballpark; i.e. C1 = -j1000 and C2 = -j250. [4]

With the new capacitor values, |G0| dropped drastically. Therefore, scaling the feedback capacitors with frequency does not boost the potential for oscillation.

The third experiment inserts a 4 mH inductor (Le) in series with R2. 

At the zero phase, the simulated |G0| has increased to 1.4. Therefore, inserting a 4 mH inductor (Le) in series with R2 enables the oscillation conditions to be fulfilled. How does Le works its magic? R2 loads the resonator. Adding Le help to isolate R2, hence reducing the loading.