Precision differential capacitor

Precision Engineering 30 (2006) 347–352

Technical note

Precision differential capacitor J.V. Niekerk, P.P. Ambekar ∗ , J.G. Nichol, D. Lauben Stanford University, MC 4085, HEPL-STEP, 245 J, Stanford, CA 94305, United States Received 24 October 2004; received in revised form 19 September 2005; accepted 2 November 2005 Available online 15 May 2006

Abstract A 4-electrode multi-port precision differential capacitor for testing the gravitational reference sensor (GRS) electronics of the ST7 spacecraft has been built and tested. The differential capacitor has a tunable and intentional mismatch of 2 aF out of ∼2 pF total nominal capacitance, and holds stable for time-periods on the order of hours. The capacitor has two electrodes representing a pair of nominally identical drive plates, one electrode representing the metallic proof-mass, and one electrode acting as a non-contact pickup. The required adjustability has been achieved by introducing two pairs of grounded micrometers located between the drive and the ‘proof-mass’ electrodes for making coarse and fine adjustments. The capacitance is changed by advancing or retracting the micrometer spindles in the electric flux path to alter the capacitive coupling of each arm. The change in capacitance by coarse adjustment is 30 fF for each arm of the capacitor. The fine adjustment has a range of 1 fF with a setting resolution of 2 aF per arm. This mechanism of changing the capacitance achieves the fine control, repeatability and stability required to test the ST7 electronics. © 2006 Elsevier Inc. All rights reserved. Keywords: Differential capacitor; Macor; Adjustability; Attofarad

1. Introduction Future formation flying space missions such as laser interferometer space antenna (LISA) incorporate gravitational reference sensors (GRS) capable of nanometer displacement resolution over time scales greater than 1000 s [1]. As an example, the JPL Space Technology 7 flight (ST7) disturbance reduction system (DRS) incorporates a 4 cm3 Au/Pt proof-mass within an Au-coated BeO housing. Differential capacitive sensing is one proposed method to measure displacement to nanometer levels [2]. The 2 mm nominal proof-mass to housing gap and available electrode area and segmentation provide typically ∼2 pF nominal working capacitance, and the required displacement uncertainty limit of ∼3 nm dictates order 10−6 or 2 aF (2 × 10−18 F) sensitivity or 1 aF/pF. The capacitive displacement readout is accomplished by applying mirror-symmetric signals to two such paired capacitor plates, and measuring the residual signal appearing on the proof-mass using a third non-contact pickup electrode. In order to test and verify the electronics, a tunable differential

∗

Corresponding author. Tel.: +1 650 736 1311; fax: +1 650 725 5470. E-mail address: [email protected] (P.P. Ambekar).

0141-6359/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.precisioneng.2005.11.011

test capacitor having unprecedented resolution and stability is required to match the flight GRS sensor characteristics. Theoretical analysis [3–5] in the past has given some ideas to develop a precision capacitor and experimental attempts [6–9] have been made to build and stabilize a capacitor. This paper presents a differential capacitor which meets project specific requirements by stabilizing the capacitor using precision engineering techniques.

2. Design analysis 2.1. Design requirements Fig. 1 shows a morphological mapping from the ST7 proofmass/housing to an equivalent hypothetical multi-terminal precision differential capacitor, as an aide to discussion. The stated goal is to design and build a 4-electrode multi-port capacitance bridge having an intentional, tunable mismatch of 2 aF out of ∼2 pF total nominal capacitance for two otherwise identical primary bridge arms A–C and B–C. The capacitance of electrodes A or B to ground may be variable as permitted by the measurement technique, which uses an auto-nulling differential voltage drive having less than 1
output drive impedance. Electrode

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Fig. 1. ST7 cube housing and equivalent electrical circuit.

C, which represents the proof-mass, is kept at virtual ground by this bridge balancing circuit. A third bridge arm C–D with capacitance of ∼10 pF is required for non-contact sensing of the signal on C. The precise mismatch between A–C and B–C must be known to ±2 aF and this mismatch must hold stable to 10% over several hours. D must be completely shielded from A and B in order to provide an uncontaminated measure of the residual error. 2.2. Design solutions to given requirements In order to achieve the strict performance requirements, precision geometric engineering principles were incorporated into the capacitor design and construction in order to reduce the accuracy constraints on the manufactured tolerances of the capacitor. The design details are discussed in the following subsections. 2.2.1. Adjustability and required tolerances The required tuning resolution of 2 aF represents 1 × 10−6 of the nominal bridge capacitance of 2 pF. To illustrate the difficulty, a capacitance bridge with an exact mismatch of 2 aF would require manufacturing the capacitor to a 1 ␮m/m dimensional tolerance. For a capacitor on the scale of 100 mm this would require nanometer scale manufacturing tolerance. To overcome this difficulty and relax manufacturing tolerances, the capacitor is designed to be adjustable, with a resolution of 2 aF over a minimum tuning range of 1 fF consistent with likely machining errors. Adjustability of electrodes at this level would see to introduce nanometer scale tolerances. The solution adopted is a capacitance bridge with fixed electrodes and turning micrometers which introduce variable shunting of electric field flux to ground, rather than control position of the electrodes. Fig. 2(a) shows cross-section of the final design. The capacitor is constructed from a block of macor ceramic. Electrodes A, B and C are created by boring holes in the macor and coating the inner surface with conductive coating. Electrode D is provided by a sphere inside hole C, which satisfies the strict shielding requirement of D from A and B. The assembly is covered with an aluminum ground shield G. The adjustability of the capaci-

tance bridge is achieved by placing pairs of micrometer spindles at ground potential in the path of the electric field between the bridge electrodes A, B and the common bridge arm C. The inner micrometers are in a direct path of the electric field between A, B and C to block electric flux, for coarse adjustments, while the outer micrometers form an indirect shunt path to ground for fine adjustments. The range of the coarse adjustment allows the manufacturing tolerances to be relaxed to 10 ␮m. The required range of the coarse adjustment is determined by the amount of asymmetry that is likely. The total range of the coarse adjustment is 50 fF with 100 aF resolution, based on 10 turns with 50 graticules per turn. The fine adjustment was designed to have a range of 1 fF and a resolution of 2 aF. The placement of each micrometer for its desired range was determined by experimenting with the micrometer position in a 3-D electrostatic model and solving for the capacitances using a computer program, FastCap, as explained in Section 2.3. 2.2.2. Symmetric geometry and temperature stability The stability of the differential imbalance ratio between the capacitances of electrodes A to C and B to C is more important than the precise value of those capacitances. Operationally, the capacitor will be tuned for slight imbalance with electronics operating, and then not adjusted for several hours, during which time the drift in the electronics will be assessed. The requirement states that the capacitor mismatch must hold stable to 10% throughout the measurement period. This requirement implies a stability of 0.2 aF for the typical tuned mismatch of 2 aF. On the macroscopic scale, a mechanically symmetric bridge greatly reduces the requirements on the absolute stability of the bridge capacitance. If both of the bridge capacitances change by the same fraction (because of their symmetric response to a disturbance), the mismatch changes only by this same fraction, e.g. if both arms increase by 10% from 2 to 2.2 pF, the result is the mismatch increasing by 10% from 2 to 2.2 aF. The basic symmetry of the capacitance geometry reduces the stability requirement on the common mode capacitance of the bridge arms to 10% provided several related symmetry requirements are satisfied. First, the bridge must be symmetric in its temperature distribution. The maximum allowable imbalance in temperature between the arms is calculated as follows. From literature, CTEmacor = 9.4 × 10−6 K−1 . So for given stability requirement of 100 nm/m, l/l = 0.1 × 10−6 , the allowable temperature difference between arms will be T = 10 mK. Thus, the bridge has a strict requirement on the homogeneity of the temperature. A material with a high diffusivity is favorable. Silicon is a better choice than macor for its higher diffusivity, but macor was chosen for its machinability and availability. Note that this is not an absolute temperature stability requirement, but defines the maximum temperature difference between the two primary electrodes, A and B. The adjustment micrometers may introduce asymmetry. The worst case is one of the coarse adjustment micrometers at full extension, the other at minimum extension. The range of the micrometer travel is 8 mm, this corresponds to a change in capacitance of 50 fF (per side) for the coarse

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Fig. 2. Mechanical design: (a) cross-section of assembly, (b) 3-D cross-sectional view and (c) exploded view.

adjustment. The required length stability of the micrometer is: l = Cstability req × sensitivity = 0.1 aF ×

8 mm 50 fF

= 1.6 × 10−8 m CTEmicrometer = 1.2 × 10 T =

−5

K

−1

l/ l 1.6 × 10−8 /8 × 10−3 = = 170 mK CTEmicrometer 1.2 × 10−5

The temperature difference between the arms of the capacitor must be maintained to 170 mK to ensure that the maximum

asymmetry of the adjustment does not change the mismatch by more than 0.1 aF. Note that this requirement is less stringent than the 10 mK arm temperature balance requirement calculated above. 2.2.3. Electrode stability controlled by insulating block Three primary electrodes are defined by symmetrically bored holes in the macor block. There are many advantages to this design. The first is, the electrodes are rigidly constrained which provides good vibration immunity, which may have been a problem for mounted conductors. The second is, the non-porous ceramic provides an insulator permittivity which is less sensitive to changes in humidity. The third major advantage is, the temper-

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ature sensitivity is reduced. When the macor changes its size due to temperature changes, the capacitance scales linearly with the change in dimension. This was verified in the FastCap computations. The effect is described by analogy to a simple parallel plate capacitor having C = εA/d. For the macor block electrodes, two counter-balancing effects scale together, the projected area of the electrodes and the spacing between them. The macor block controls all these dimensions since the conductors are thinly coated holes in the block. The macor, which is much more rigid than the conductive layer, controls the overall geometries of the electrodes. 2.2.4. Kinematic mount It is important to keep the distance between the electrodes constant, which means, no stress should transfer from the aluminum base (e.g. stress due to temperature change) to the macor block, which could change its dimensions. So the macor is mounted kinematically on the base by three spheres resting in 3 V-grooves. Also, this ensures minimal transfer of heat from the base to the macor block, and repeatable placement of the macor block relative to the base. In this precision adjustable capacitor, stability and repeatability are most important thus a kinematic mount is also used for placing the electrode D (sphere) inside the electrode C (cylinder) in order to achieve stability in the positioning of D with respect to C. 2.2.5. Outer shield In order to achieve the required thermal and electrical stability, an aluminum outer shield is added. The shield is grounded and serves two purposes. Firstly, it shields the capacitor from any stray external electric fields. Secondly, it helps maintain thermal homogeneity of the capacitor by providing a good heat conduction path to heat input from various directions, thus making it less localized. 2.2.6. Shielding of D from A and B The sensing electrode D is shielded from A and B by the design geometry. This electrode is placed inside the electrode C. There is no path for the electric field from either electrode A or B to D. The connectors to these electrodes are located on opposite sides of the capacitor. This requirement is basic to the capacitive readout system employed in the spacecraft electronics. Fig. 2(b) shows a 3-D cross-section of the final design, and Fig. 2(c) shows the exploded view. 2.3. FastCap field solver FastCap [10] is a multi-pole accelerated capacitance extraction program for arbitrary 3-D structures, developed at MIT. It was used extensively in the design of the differential capacitor geometry. It was used to define dimensions which set the nominal capacitance of 2 pF and also used to calculate the placement and size of the micrometers for the desired adjustment resolution and range. FastCap requires input files that contain the coordinates of a quadrilateral or triangular mesh. MATLAB was used to generate the meshes of the capacitor geometry.

Fig. 3. Generation of mesh in MATLAB (used for FastCap program).

Fig. 3 shows an example of the MATLAB generated mesh that was used in the FastCap calculations. 2.4. Manufacturing and assembly The capacitor was manufactured as per the FastCap design. Fig. 4 shows the photographs of the final capacitor which was then tested. Gold-coated spring pins were used for connecting different electrodes in the macor block to the impedance analyzer. This helped in reducing the contact impedance in the circuit while preserving thermal and kinematic isolation. 2.5. Capacitance testing For the final testing, a container (Coleman Cooler, model 6251-707W) was used for temperature stability. The experiment was done at room temperature so no active heating or cooling system was needed. Plastic ‘Substitute Ice’ bags (Rubbermaid Co.) were added in order to increase the thermal inertia. Four RTDs were placed at four corners of the container in order to measure the temperature distribution inside the container. An agilent impedance analyzer (4294 A), with accuracy ∼1 fF, and an Andeen–Hagerling AH2007A with resolution 1 aF, were used to measure the capacitance. 3. Results 3.1. Primary driven capacitance The results are shown in the Fig. 5. Fig. 5(a) shows the variation of capacitance due to coarse adjustments. The change

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achieved was approximately 30 fF for each capacitor. The slight difference between the two capacitances is due to the uncertainties in manufacturing as explained above. The essential part to achieve in this was the overlap of 15 fF which becomes the usable tuning range. Fig. 5(b) shows the capacitance change due to fine adjustments. It can be seen that a range of 1 fF was achieved for the whole micrometer travel. On the micrometer dial, there are 500 units over the range of travel. So by changing the micrometer reading by one unit we can change the capacitance by 2 aF, repeatably, which achieves the requirement. There is a slight mismatch between the fine adjustments of the two arms due to manufacturing. For the needed change of 2 aF capacitance, this asymmetry is tolerable. The impedance analyzer showed a primary driven capacitance of mean value 1.9564 pF at 100 kHz frequency with a standard deviation of 249.5 aF. The reason for this large standard deviation is the impedance analyzer itself, which exhibits cyclic error likely due to cycling of its internal oven. The analyzer has a published stability of 1 fF. These results suggest that depending on the stability of the capacitor and successful implementation of precision engineering principles at fF level, performance can be extrapolated to the aF level and stability can be achieved. 3.2. Ratio stability In order to check the response of the capacitor to temperature changes, a known deliberate temperature change of ∼2.5 ◦ C is applied while tracking the arm ratio A–C to B–C. The response of each arm was measured. The capacitance ratio (A/B) versus time is plotted as shown in Fig. 6(a). The capacitances are changing by nearly the same amounts but there are some higher order effects. Fig. 6(b) shows capacitance ratio (A/B) versus temperature. This plot identifies the level of temperature control needed to achieve the stability requirement. There is hysteresis in the capacitance ratio. This hysteresis and other higher order effects may be caused by humidity changes, temperature coefficient of impedance analyzer, or mechanical hysteresis. Mitigation for these effects might be to perform the experiments in more inert and tightly controlled environment, and to complete thermal cycling above/below ambient. Testing for temperature shows that the employed precision engineering principles are helping in reducing common mode errors. It also confirms that the manufacturing errors are small. These results indicate that temperature stability of 0.2 mK will be sufficient to hold the required 0.2 aF stability. 4. Summary Fig. 4. Final parts: (a) base with grooves for kinematic mount, (b) coarse adjustment micrometers, middle groove and chrome steel ball in macor block and (c) aluminum shield and connectors.

A 4-electrode multi-port precision differential capacitor having tunable imbalance of 2 aF out of ∼2 pF total nominal capacitance has been successfully built and tested. Many precision design principles were used in order to achieve the desired accuracy and stability. The mismatch is adjustable with a resolution of 2 aF, and can hold stable over time-periods on the order of hours with reasonable thermal control.

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Fig. 5. Primary driven capacitor values: (a) coarse adjustment and (b) fine adjustment.

Fig. 6. (a) Capacitance ratio (A, C)/(B, C) vs. time and (b) capacitance ratio (A, C)/(B, C) vs. temperature.

Acknowledgements This project was done as a part of ‘Precision Engineering’ course (ME 324) at Stanford University. Authors are thankful to Prof. Daniel DeBra and Prof. David Beach for their valuable guidance and support throughout this project. Dale Gill, Lo Van Ho and the ST7 team was helpful in technical and non-technical matters. References [1] Folkner WM, et al. Disturbance reduction system: testing technology for precision formation control. Proc SPIE 2003;4860:221–8. [2] Heerens WC. Application of capacitance techniques in sensor design. J Phys E Sci Instrum 1986;19(November):897–906. [3] Castelli F. A proposal for toroidal cross-capacitor. IEEE Trans Instrum Meas 2000;49(August):721–6.

[4] Olyslager F, Lindell IV. Capacitance relations for a class of twodimensional conductor configurations. IEEE Proc Sci Meas Technol 1996;143(September(5)):302–8. [5] Jeffery A, Lee LH, Shields JQ. Model tests to investigate the effects of geometrical imperfections on the NIST calculable capacitor. IEEE Trans Instrum Meas 1999;48(April(2)):356–9 [CPEM98 special issue]. [6] McGregor MC. A simple three-terminal micrometer capacitor. J Sci Instrum 1954;31(May(5)):190–1. [7] Moodley SS, van den Berg W, Veldman CS. Improving the mechanical stability of a standard capacitor. IEEE Trans Instrum Meas 2003;52(April(2)):392–5. [8] Small GW, McGregor MC, Lee RD. Stable gas-dielectric capacitors of 5and 10-pF values. IEEE Trans Instrum Meas 1989;38(April(2)):372–7. [9] Ku Yi-sha, Tsai Yuan-lun, Chen G. An experimental study in the design of high-stability gas dielectric capacitors. Instrum Meas Technol Conf, Conf Proc IEEE 1994;May(2):931–2. [10] Nabors K, White J. FastCap: a multipole accelerated 3-D capacitance extraction program. IEEE Trans Comput Aided Des 1991;10(11):1447–59.