IEEE TRANSACTIONS ON INSTRUMENTATION A N D MEASUREMENT. VOL. 39. NO. 5. OCTOBER 1990 73s

Noise Analysis for the Triaxial Capacitive Displacement Transducer

GREGORY A. BERTONE. Z. H. MEIKSIN, FELLOW, IEEE, A N D NORMAN L. CARROLL

Abstract-A computer simulation and experimental study of electric noise in the triaxial capacitance displacement transducer, when com- bined with the measurement instrument, leads to an optimal design range for the value of the reference capacitor of the transducer. The range is defined in terms of the decrease in noise, relative to the in- crease in capacitance.

I. INTRODUCTION N A previous paper [ 11, we discussed prior art capac- I itance-based displacement transducers and we intro-

duced the triaxial transducer which consists of a moveable shaft along the axis of two concentric capacitors. A cross section of the device is shown in Fig. 1. This drawing shows the intricate design which is needed to eliminate the effects of fringe capacitances and humidity [ 11. A sim- plified cross-sectional view which clearly depicts the elec- trodes of the capacitors, and the electrical equivalent cir- cuit of the transducer are shown in Fig. 2.

In this figure, C, is a fixed capacitor. The value of the variable capacitor, C,, , is determined by the position of the shaft which is electrically connected to ground, thus making inactive the portion of the variable electrode which it shields. The position of the shaft represents the dis- placement. In principle, the displacement measurement is done by arranging the two capacitors in a half-bridge con- figuration connected to two voltage sources, one of fixed value and one variable, with a common, grounded ter- minal. By controlling the value of the variable voltage source, the common electrode, C, is forced to virtual ground potential. Under this condition, the voltage of the variable source is equal to the voltage across the variable capacitor. A detailed description of the displacement measurement is given in the next section.

11. INSTRUMENTATION The block diagram of the instrumentation apparatus is

given in Fig. 3. Modulator 2 chops a precision dc refer- ence voltage VR, producing a square wave voltage V 2 ( r ) of fixed amplitude VR and frequency synchronized to the clock. Modulator 1 chops the dc output voltage V , of the error amplifier, producing a square wave VI ( r ) of variable

Manuscript received November 24, 1989; revised April 14, 1990. G. A. Bertone is with Electronic Instrument Research, Ltd., Irwin. PA

Z. H. Meiksin is with the Department of Electrical Engineering. Uni-

N . L. Carroll is with Measuretron, Inc.. Butler, PA 16001. IEEE Log Number 9037398.

15642.

versity of Pittsburgh. Pittsburgh. PA 15261.

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Fig. I . Triaxial capacitive displacement transducer.

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Fig. 2. Simplified cross-sectional view of (a) transducer and (b) electrical equivalent circuit.

amplitude V, and frequency synchronized to the clock but 180" out of phase with respect to V 2 ( t ) .

Voltages V , ( t ) and V, ( r ) are applied to the transducer capacitors C'and C,. as shown. The preamplifier, the syn- chronous demodulator, and the low-pass filter provide a dc error voltage proportional to the full-wave rectified amplitude of the "off-null" signal present at terminal C. This dc error voltage is then inverted and amplified by the error amplifier, generating a correction voltage VE which controls the amplitude of the modulator output waveform VI ( r ) and forcing point C back to null (i.e., virtual ground). Due to this servo action, the bridge is main- tained in a balanced condition as C,, is varied.

After examining Fig. 3 , we can write

where K represents the amplifier gain constant.

0018-9456/90/1OO0-0735$01 .OO O 1990 IEEE

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136 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT. VOL. 39. NO. 5. OCTOBER 1990

Let V , and V2 be, respectively, the amplitudes of the modulator output voltages VI ( t ) and V2 ( t ) . Then VE = VI and VR = V2 which results in

Since the bridge is maintained in a balanced condition by controlling V I , the following relationship is main- tained:

Combining (1) and (2) yields

( 3 )

A convenient zero displacement shaft position is defined for which C,, = C,. We note that the gain of the measure- ment circuit can be controlled by the value of V2.

111. NOISE ANALYSIS The transducer, being a capacitance device, is inher-

ently noiseless. Since the capacitance of C,, is a continu- ous function of the shaft position, the resolution of the device is, in principle, infinite. But the noise generated in the measurement circuitry, as it interacts with the trans- ducer, limits the resolution of the system. Since the preamplifier receives, as its input, the unamplified signal (i.e., the bridge output voltage), noise introduced in the preamplifier stage is the largest contributor to the noise- to-signal ratio at the output of the system. Also, since the preamplifier is connected directly to the transducer, its interaction with the transducer is of major importance, es- pecially as it is affected by the fixed capacitor C' which constitutes part of the measurement servo control loop.

The preamplifier, chosen to have the 1 /f noise comer frequency around 50 Hz, operates in the circuit at 2.5 kHz. Therefore it is necessary to include only the white noise of the amplifier. The error amplifier is used in the dc to 100 Hz range and therefore includes 1 /f noise. Noise for all resistors was included by means of noise current sources in parallel with the resistors.

Wl c+

Fig. 4. Preamplifier subcircuit model for SPICE analysis.

CPlO 14 I5 1 1

RCPII

PREAMP

The noise analysis was carried out using SPICE, a com- puter aided electronic circuit analysis program [2]. Each block in Fig. 3 is modeled by an equivalent circuit suit- able for SPICE noise analysis and all the circuits are in- terconnected to model the complete measuring circuit. The preamplifier and modulator models are shown in Figs. 4 and 5 , respectively. The noise voltage source and noise current in the preamplifier model utilize thermal noise from two separate resistors to provide uncorrelated white noise. The thermal noise voltages are transformed by the appropriate controlled source into the desired value of voltage or current noise. The value for each noise source was determined from the manufacturer's specification sheets for the particular amplifier used in the actual cir- cuit. The same applies to the independent controlled noise sources for the error amplifier mode. The noise sources are shown in Fig. 6.

The SPICE circuit model for the complete instrument is shown in Fig. 7. Components R50, C50, and R51 in this figure comprise the low-pass filter of Fig. 3. Note that when a SPICE noise analysis is run on this circuit the modulators do not function as they do normally because they are locked in one position. This was taken into ac- count as follows. Refer to Fig. 7 and consider the true operation. The output signal from the error amplifier, which extends from dc to 100 Hz, controls the amplitude of the 2.5-kHz square wave applied to node 52. This dc to 100 Hz baseband signal is translated up to 2.5 kHz, it passes through C, and is applied to the input of the pream- plifier circuit via node 5 1. Noise from the preamplifier is

BERTONE er U / . : NOISE ANALYSIS FOR TRANSDUCER 737

(J&-J&-rq CLK

ELK error amp

61 YBAT E INWZ

Fig. 7. Electronic measurement circuit model for SPICE analysis.

added to the signal which is then translated back down to the baseband frequency and applied to the error amplifier where it is band limited via the low-pass filter, from dc to 100 Hz. Therefore, the low-pass filter sets the loop band- width affecting the amplitude of the 2.5-kHz signal ap- plied to the transducer.

It can be seen that during a SPICE noise analysis when the modulators are not functional, the baseband signal from the error amplifier is not frequency translated and the amplitude of the signal applied to the transducer is dominated by the characteristics of the low-pass filter thereby changing the essence of circuit performance. This was overcome by disabling the low-pass filter (i.e., set- ting R51 to a very large value) and running the SPICE noise analysis over a 100-Hz wide bandwidth centered at 2.5 kHz. The total output noise was then obtained by mul- tiplying the noise spectral density (nv/( Hz) ' /*) by the square root of the bandwidth ( ( H Z ) ' ' ~ ) .

The analysis was carried out for two separate cases. First, C' was fixed. The value of C,. was varied and the corresponding output noise was determined for each value of C,,. Then, C,, was fixed and the analysis was repeated for different values of Cy The analysis results showed that noise output varied inversely with C, and was independent of C(,. Referring to (3), note that increasing the value of Cf not only lowers the noise but also lowers the sensitivity (volts per displacement) of the overall system. Therefore. to make a meaningful comparison, the system gain was

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0.00 I 4.0 8.0 12.0 16.0 20.0

C, (normalizedl Fig. 8. Output noise versus reference capacitance. C,

adjusted for each value of Cf to maintain system sensitiv- ity at a constant value for all conditions. The results of the noise computations are shown by the solid line in Fig. 8. The noise values in the figure are normalized with re- spect to the value for an arbitrarily chosen value of Cp

IV. EXPERIMENTS

A displacement transducer of the configuration shown in Fig. 1, together with a measurement system as shown in Fig. 3 , was tested for noise. The total output noise for the system was measured with an rms-responding meter using the 100-Hz bandwidth of the system under test to provide band limiting.

Measurements were performed for different values of C,. As was shown in the previous section, both noise out- put and signal output of the measurement system depend on the transducer parameter C,. Therefore, to obtain a meaningful basis for signal-to-noise comparison, for each selected value of C', the electronics gain was adjusted such that a displacement of 12 mm always produced an output voltage V,,, of 10 volts from the overall system. The dis- placement was then reset to zero and residual output noise from the system was measured. The nominal value of Cf was 0.4 pF. The curve displays Cfvalues normalized with respect to the nominal value.

The experimental setup was as follows. The transducer under test was installed in a calibration fixture where the displacement of the shaft was set by a micrometer. The accuracy of the displacement measurement setting was within f25.4 pm. This was only important for determin- ing the resetting of the overall system gain (i.e., a dis- placement of 12 mm corresponded to 10-V output) for the purpose of comparing noise at the same overall system sensitivity. Errors in setting the 12-mm displacement would translate to +0.2% error in the gain setting and since the system noise reading was dependent on the gain setting, the noise output measured at each value of C, would be in error by the same amount (+0 .2%) . Noise was always measured at zero displacement (although noise output was found to be independent of displacement).

738 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT. VOL. 39. NO. 5. OCTOBER 1990

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Fig. 9. Percentage noise reduction with respect to percentage increase in reference capacitance, C, versus C,

The experimental results are shown in Fig. 8 by the squares, connected with a dashed curve. The correspon- dence with the results of the computer simulation is evi- dent.

V. OPTIMIZATION

Fig. 8 shows that the noise output, for constant signal output, decreases with an increase in C,. However, even- tually, at several times the nominal value, Cf approaches a lower limit. Increasing Cf also increases the overall length of the transducer, which is undesirable considering practical manufacturing tolerances and application con- straints. Therefore, a value of Cf is reached beyond which further increase cannot be justified in terms of the addi- tional reduction in noise. The criterion for such a value is a subjective engineering decision.

To develop a guide for making this decision, we ex- amine the value of

dn l n (4)

as a function of C' In Fig. 8, the abscissa gives Cf/Cf,,, where C,, is an

arbitrarily chosen reference value. The ordinate gives n / n , where n, is the noise value corresponding to Cf,. The derivative of the curve gives

dn l n n

dCf/ c, . For the same curve, we compute

By multiplying equation (5) and (6), we obtain

dn / n

dCf / Cf which is identical to (4). Equation (7) is plotted as a func- tion of Cf/Cfn in Fig. 9. For values greater than 1 along the ordinate, as Cf is increased, the percentage reduction in noise exceeds the percentage increase in capacitance, but even for values less than 1, considerable noise reduc- tion is still apparent. The curve levels off quickly for val- ues smaller than 0.5. Therefore, a reasonable design guide is to choose a value of Cf in a range defined by

VI. CONCLUSIONS The SPICE program was used to predict the noise char-

acteristics of the triaxial displacement transducer. It was found that the signal-to-noise ratio improved as the ref- erence capacitance of the device was increased, but also that the rate of improvement decreased with increase in capacitance. Accordingly, an optimum range of capaci- tance values was defined for which there is considerable reduction in noise without excessive increase in capaci- tance; the ratio of the percentage noise reduction to per- centage capacitance increase being between 0.5 and 1. The computer simulation modeled predicted noise char- acteristics very close to those measured.

REFERENCES [ I ] G. A. Bertone et a l . , "Investigation of a capacitance based displace-

ment transducer," IEEE Trans. Instrum. Meas. , vol. 39, pp. 424-428, Apr. 1990.

Engle- wood Cliffs, NJ: Prentice-Hall. 1989.

121 W . Banshaf. Computer-Aided Circuit Analysis Using SPICE.