THE INFLUENCE OF PARASITIC CAPACITANCES ON THE ERROR IN MEASURING THE ACTIVE RESISTANCE OF QUARTZ RESONATORS

V. K. Zhukov, P. A. Ovsyannikov

(Presented by the scientific seminar of the Department of Information-Measuring Technology and the Sector of Electromagnetic Methods of Testing)

Quartz resonators are currently widely used in many fields of science and technology where it is necessary to maintain high stability of physical quantities and to carry out measurements with high accuracy.

In this connection, there is an increasingly felt need to improve radio measurements associated with the production of quartz resonators.

It is known from the literature [1] that the greatest accuracy in measuring the active resistance of a quartz resonator is provided by the resonance method, in whose circuit implementation the quartz resonator is included in the circuit of a frequency-dependent divider composed of the quartz resonator and a resistor \(R\) (Fig. 1).

Fig. 1. Circuit of the frequency-dependent divider.

At the frequency of series resonance, when the resistance of the quartz is purely active, the voltage \(U_2\) is directly proportional to the magnitude of this resistance and is equal to

\[ U_2 = U_1 \frac{R_{\mathrm{kv}}}{R + R_{\mathrm{kv}}}. \tag{1} \]

Experimental studies have shown that, with this measurement method, the parasitic capacitance \(C_n\) introduces an error (Fig. 1).

Let us consider the influence of the capacitance \(C_n\) on the error in measuring the active resistance of a quartz resonator.

It follows from Fig. 1 that

\[ U_2 = U_1 \left( 1 - \frac{ \dfrac{R}{1 + j \omega C_n R} }{ \dfrac{R}{1 + j \omega C_n R} + j \omega L_{\mathrm{kv}} - j \dfrac{1}{\omega C_{\mathrm{kv}}} + R_{\mathrm{kv}} } \right). \tag{2} \]

\[ U_2 = U_1(1 - K) \]

where

\[ K = \frac{ \dfrac{R}{1 + j \omega C_n R} }{ \dfrac{R}{1 + j \omega C_n R} + j \omega L_{\mathrm{kv}} - j \dfrac{1}{\omega C_{\mathrm{kv}}} + R_{\mathrm{kv}} }. \tag{3} \]

After transformation we obtain

\[ K = \frac{R \omega C_{\mathrm{kv}}} {\omega C_{\mathrm{kv}} R + (1 + j\omega C_{\mathrm{p}}R)\left(j\omega^{2} L_{\mathrm{kv}} C_{\mathrm{kv}} - j + \omega C_{\mathrm{kv}} R_{\mathrm{kv}}\right)} , \tag{4} \]

At the series-resonance frequency, \(\omega_0\),

\[ \omega_0 L_{\mathrm{kv}} = \frac{1}{\omega_0 C_{\mathrm{kv}}}, \tag{5} \]

whence

\[ L_{\mathrm{kv}} = \frac{1}{\omega_0^{2} C_{\mathrm{kv}}}. \tag{6} \]

Substituting (6) into (4), we obtain

\[ K = \frac{R \cdot \omega \cdot C_{\mathrm{kv}}} {\omega C_{\mathrm{kv}} R + (1 + j\omega C_{\mathrm{p}}R) \left[ R_{\mathrm{kv}}\omega C_{\mathrm{kv}} - j\left(1 - \frac{\omega^{2}}{\omega_0^{2}}\right) \right]} . \tag{7} \]

Taking into account that, for small detunings,

\[ \left(\frac{\omega^{2}}{\omega_0^{2}} - 1\right) = 2\frac{\Delta\omega}{\omega_0}, \]

expression (7) is transformed to the form

\[ K = \frac{R \cdot \omega \cdot C_{\mathrm{kv}}} {\omega C_{\mathrm{kv}}(R + R_{\mathrm{kv}}) - 2\frac{\Delta\omega}{\omega_0}\,\omega C_{\mathrm{p}}R + j\left(\omega^{2}C_{\mathrm{p}}C_{\mathrm{kv}}RR_{\mathrm{kv}} + 2\frac{\Delta\omega}{\omega_0}\right)} . \tag{8} \]

Denoting \(\omega C_{\mathrm{p}}R = p_1\) and substituting it into expression (8), we find the modulus \(K\):

\[ |K| = \frac{R \cdot \omega \cdot C_{\mathrm{kv}}} {\sqrt{ \omega^{2}C_{\mathrm{kv}}^{2}\left[(R + R_{\mathrm{kv}})^{2} + p_1^{2}R_{\mathrm{kv}}^{2}\right] - 4\frac{\Delta\omega}{\omega_0}\omega C_{\mathrm{kv}}Rp_1 + 4\frac{\Delta\omega^{2}}{\omega_0^{2}}(1 + p_1^{2}) }} . \tag{9} \]

Dividing the numerator and denominator in expression (9) by \(\omega C_{\mathrm{kv}}\), after which, taking (5) into account, we obtain

\[ |K| = \frac{R} {\sqrt{ (R + R_{\mathrm{kv}})^{2} + p_1^{2}R_{\mathrm{kv}}^{2} - 4\frac{\Delta\omega}{\omega_0}R \cdot p_1\omega_0 L_{\mathrm{kv}} + 4\frac{\Delta\omega^{2}}{\omega^{2}}\omega_0^{2}L_{\mathrm{kv}}^{2}(1 + p_1^{2}) }} . \tag{10} \]

Since \(\Delta\omega \ll \omega_0\), then

\[ \frac{\omega_0}{\omega} = \left(1 - \frac{\Delta\omega}{\omega_0}\right)^{2} = 1 . \tag{11} \]

After this simplification we have

\[ |K| = \frac{R} {\sqrt{ (R + R_{\mathrm{kv}})^{2} + p_1^{2}R_{\mathrm{kv}}^{2} - 4\Delta\omega Rp_1L_{\mathrm{kv}} + 4\Delta\omega^{2}L_{\mathrm{kv}}^{2}(1 + p_1^{2}) }} . \tag{12} \]

For \(C_{\mathrm{p}} = 0\) we have

\[ |K_0| = \frac{R}{R + R_{\mathrm{kv}}}. \tag{13} \]

Since \(C_{\mathrm{p}} \ne 0\), a measurement error arises:

\[ \gamma' = \frac{(1 - |K|) - (1 - |K_0|)} {1 - |K_0|}. \tag{14} \]

Substituting their expressions for \(|K|\) and \(|K_0|\) and denoting

\[ \frac{R_{\mathrm{q}}}{R}=p_2, \]

we find

\[ \gamma = \frac{1+p_2-\sqrt{p_1^2 p_2^2+(1+p_2)^2}} {p_2\sqrt{p_1^2 p_2^2+(1+p_2)^2}} . \tag{15} \]

Expression (15) is presented graphically in Fig. 2, from which it follows:

Fig. 2. Dependence of the measurement error on the parameters \(P_1\) and \(P_2\).

1. At \(p_2=1\), the measurement error is maximal.
2. As the resistance of the capacitance \(C_p\) approaches the resistance \(R\), the measurement error becomes maximal.

In instruments being designed, in order to increase the accuracy of measuring the active resistance of quartz resonators, it is necessary to seek solutions that help reduce the influence of the capacitance \(C_p\) on the measurement error.

References

1. G. B. Altshuller. Frequency Control of Quartz Resonators. Moscow, “Svyaz,” 1969.