Micromechanical oscillator circuits: theory and analysis

Abstract

In this paper, detailed theoretical analysis of micromechanical Transresistance oscillator is presented. Analytical expressions are derived for the frequency pulling, critical transimpedance, maximum negative resistance, and start-up time constant of the Transresistance oscillator circuit which are useful for the design of micromechanical oscillators. These results are then used to study the frequency stability of Transresistance oscillator circuit and compare its operating conditions with that of the Pierce oscillator circuit which is widely used in micromechanical oscillators. The results conclusively show that the Transresistance oscillator has less start-up problems and better frequency stability than the Pierce oscillator. These results are then verified with a well-established circuit theory that compares the phase-frequency plots of the Pierce and Transresistance oscillator.

Appendices

Appendix 1: Derivation for circuit impedance

From Fig. 5, Z _m can be replaced by a test voltage source (V _T) to derive Z _c as follows.

From Fig. 15, the following two equations can be obtained.

$$ V_{\text{T}} = r_{\text{m}} I_{\text{i}} - I_{\text{i}} Z_{\text{o}} - I_{\text{i}} Z_{\text{i}} $$

(27)

$$ I_{\text{i}} = \frac{{V_{\text{T}} }}{{Z_{\text{a}} }} - I_{\text{T}} $$

(28)

Fig. 15

Transresistance circuit with test source

Z _c from (9) can be derived by combining and rearranging (27) and (28) to make the circuit impedance (V _T/I _T) the subject.

Appendix 2: Derivation for maximum negative resistance

From (10), the real part of Z _c can be simplified in terms of N and D as follows:

$$ \text{Re} (Z_{\text{c}} ) = \frac{N}{D} $$

(29)

where, $N = - (r_{\text{m}} - R_{\text{i}} - R_{\text{o}} )$ and $D = \omega^{2} C_{\text{a}}^{2} (r_{\text{m}} - R_{\text{i}} - R_{\text{o}} )^{2} + 1$

To find −Re(Z _c)_max, Re(Z _c) has to be differentiated with respect to r _m and equated to zero as follows:

$$ \frac{d}{{dr_{\text{m}} }}\left(\frac{N}{D}\right) = \frac{DN' - ND'}{{D^{2} }} = 0 $$

(30)

where, $N' = - 1$ and $D' = 2\omega^{2} C_{\text{a}}^{2} (r_{\text{m}} - R_{\text{i}} - R_{\text{o}} )$

The optimum r _m at Point X can be derived by rearranging (30) to make r _m the subject.

$$ r_{\text{mopt}} = \frac{1}{{\omega C_{\text{a}} }} + R_{\text{i}} + R_{\text{o}} $$

(31)

The −Re(Z _c)_max of (12) is derived by combining (31) and (10).