226 Nuclear Instruments and Methods in Physics Research A269 (1988) 226-229 North-Holland, Amsterdam

T H E I N T E R - E L E C T R O D E C A P A C I T A N C E O F W E D G E

A N D S T R I P P O S I T I O N S E N S I N G E L E M E N T S

J o h n T H O R N T O N

Department of Electronic and Electrical Engineering, University of Surrey, Guildford, Surrey, GU2 5XH, England *

Received 4 August 1987 and in revised form 19 November 1987

A method of calculating the inter-electrode capacitances of wedge and strip position sensing elements is presented. Estimates made using the method are compared to measured values and found to be in reasonable agreement. An alternative formula is shown to be semiempirical, requiring calibration with similar, already manufactured, wedge and strips before it can be used.

1. Introduction

A wedge and strip (WS) is a position sensitive ele- ment which has been employed in many types of X-ray detector, for example: channel plate devices (CPD) and multiwire proportional counters (MWPC) [1]. Normally a WS is manufactured using lithographic techniques to form insulating channels in the metal coating of a planar insulator, by etching away the metal. Fig. 1 shows an example of the type of pattern produced. Three separate electrodes have been created: the strip, the wedge and the Z.

Fig. 2 shows an example of how a wedge and strip is used as a position sensitive cathode in a MWPC. The photoelectric absorption of an X-ray in the counter gas results in an electron avalanche onto an anode wire. As the ions created in the avalanche drift away from the anode wire, under the influence of the electric field, an induced surface charge distribution will appear on the WS and increase in amplitude. The centroid of the the induced charge distribution will lie directly beneath the centroid of the ion distribution. In the x-direction, which is defined as parallel to the anode wires, this corresponds to the position of X-ray absorption - as- suming negligible primary electron effects. In the y-di- rection this corresponds to the anode wire. The position x-ray absorption can be determined by measuring the signal amplitudes, due to the induced charge, from the three WS electrodes and employing suitable algorithms. For example, x coordinate information can be obtained from:

X = 2 s / ( s + w + z ) , (1)

* Some work performed at Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey, England.

0168-9002/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

where s, w and z are the amplitudes of the signals from the strip, wedge and Z electrodes respectively. The strip electrode's signal amplitude becomes larger with in- creasing x because the strip fingers become wider in this direction. Similarly the tapering of the wedge fingers enables y coordinate information to be obtained.

The accuracy with which a coordinate can be de- termined has been called the position resolution, it can be affected by many factors, for example: primary electron diffusion in M W P C and channel diameter in CPD [1]. Many of these factors depend on the type of detector employed, however, electronic noise is always present and is often the most significant. Electronic noise from the preamplifiers will affect the measured signal amplitudes in a random manner and thus worsen the position resolution. The position resolution limit in

l i

Channels

x

Fig. 1. A section of a typical wedge and strip pattern, where p is the pattern's period, D is the width of the WS and g is the width of the insulating gaps separating the three electrodes. Examples of the wedge, strip and Z electrodes' fingers are

denoted W, S and Z.

J. Thornton / The inter-electrode capacitance of wedge and strip position sensing elements 227

Window

Cathodes

Induced [harge

- - ~ WS

×

Fig. 2. A schematic diagram of how a multiwire proportional counter works. The incident X-ray was absorpted at position 1. The resulting electron avalanche, onto an anode wire, pro- duced a torus of ions (+). The ions migrated away from the anode wire producing induced surface charge distributions on both of the cathodes. A cross section of the surface charge

density (q) is also shown.

the x-direction, due to electronic noise, may be given as

[~1: 2 0.5

A x = B L [ C N ~ + F ( N Z w + N ~ ) ] / Q , (2)

where A x is the position resolution limit, B, C and F are constants for a given geometry, L is the length of the WS in the x-direction, Q is the total signal ampli- tude from the WS, and N s, N w and N z are the average (rms) electronic noise amplitudes from the preamplifiers connected to the strip, wedge and Z electrodes respec- tively.

The electronic noise from a charge sensitive pre- amplifier increases monotonically with an increasing input capacitance [2]. Noise versus capacitive load curves are normally supplied by the manufacturer. The major source of input capacitance to the three electrode pre- amplifiers in a WS based detector system are normally the inter-electrode capacitances. An example of this is presented in section 4. To decrease the inter-electrode capacitance and hence obtain better position by reduc- ing the electronic noise, much work has been done on optimising WS [3,4]. If the inter-electrode capacitance could be calculated, it would be possible to estimate the electronic noise limited performance of a WS based detector prior to manufacture. This paper presents a method of estimating these values for arbitrary WS on very thin or very thick substrates and compares some estimated and measured values. Furthermore, an alter- native approach is examined and shown to be semiem-

pirical, requiring calibration with similar, already manufactured WS, before it can be employed.

2. Theory

The inter-electrode capacitance of a three electrode WS is a formidable problem in 3-dimensional electro- statics, however, by using several approximations it was reduced to the 2-dimensional problem of the capaci- tance of two coplanar strips of unequal widths, for which the solution is readily available [5]. The ap- proximations were: (1) the periodic approximation, (2) the nearest neighbour approximation, (3) the infinitely long finger approximation, (4) the two electrode ap- proximation, and (5) the two half finger approximation.

The relative widths of the wedge, strip and Z elec- trodes' fingers vary over a WS to enable position infor- mation to be obtained, however, as this complicates the capacitance calculation considerably the mean width values were used. This was equivalent to assuming the WS to be fully periodic.

The capacitive coupling between say, a strip finger and a wedge finger in different periods was considered negligible, because of the large separation compared to those in the same period. This was the nearest neighbour approximation. The total capacitance was then N times the period capacitance, where N was the number of periods.

If the fingers were infinitely long the 3-dimensional problem would become a 2-dimensional problem [5]. This was considered to be a reasonable approximation because to ensure linearity the width, D, of most WS are much larger than their periods, p [6]. The width is in the direction of the strip fingers, as shown in fig. 1.

The two electrode approximation was to treat two electrodes as one while estimating the capacitance seen by the third electrode.

The result of the above approximations was to re- duce the problem to that of three coplanar strips, where the two outer strips are electrically connected. The geometry is symmetric and is shown in fig. 3a. The two half finger approximation was to consider the capaci- tance of this arrangement equal to twice that of the two electrode arrangement obtained by cutting down the line of symmetry and removing one half (fig. 3b). Be- cause there are 2 N of these sections on a WS, the total capacitance seen by the wedge or strip electrodes is 2 N times the capacitance from a section:

where k ~ p g / ( p - 2u) (u + g), p is the WS period, g is the width of the WS insulating gaps, u is half the width of the average strip finger or half the average width of a wedge finger (fig. 3b), N is the number of

228 J. Thornton / The inter-electrode capacitance of wedge and strip position sensing elements

i . r _ L i

i I

21 I 2 u I

C

Fig. 3. The various stages of pattern simplification required by the theory. The two outer electrodes in (a) are electrically

connected, hence the imaginary wire joining them.

periods on the WS, D is the width of the WS, and K is an elliptical integral of the second kind [8]; for very thin substrates (substrate thickness, h < g)

( = c 0 ; (4)

for very thick substrates (h > p /4 )

E =~0(1 + Cr) /2 , (5)

where c r is the relative permitivity of the substrate. The extra factor of 2 required to produce the coefficient of 4 is inherent in the coplanar strip expression because of the two WS faces (front and back).

The Z electrode has two fingers in each period therefore it was necessary to divide each period into quarters (fig. 3c). The number of these sections on a WS is 4N therefore the total capacitance seen from the Z electrode, Cz, is 4N times the capacitance from a sec- tion:

where k = p g / ( p - 4u)(u + g) and u is half the aver- age width of an average Z finger. Note that although there are more sections involved in the calculation of C z than C the value of u is correspondingly smaller.

For most WS g is much smaller than p / 8 and the areas of the wedge and strip electrodes are equal and are both half of the area of the Z electrode. For these

WS k is approximately 11g/p for the wedge and strip electrode calculation, and approximately 16g/p for the Z electrode calculation.

3. Experimental

A WS based MWPC, reported on elsewhere [1], was examined. Capacitance measurements were made, using a Boonton, model 72B, capacitance meter, of the inter- electrode capacitance of the WS and the other contrib- uting capacitances,

4. Results

The parameters of the WS are listed in table I and the capacitance results in table 2. The contributions, f rom the M W P C anode array and body, to the total capacitance seen by the strip electrode were found to be negligible in comparison to the inter-electrode contribu- tions. The values of the calculated inter-electrode capa-

Table 1 The WS parameters a)

Period, p 5 Length 120 Width, p 52 Substrate thickness, h 1.5 Substrate relative permitivity, ~r 5.25 Interelectrode gap, g 0.11 Width of average strip finger, 2 u 1.30 Average width of a wedge finger, 2 u 1.46 Average width of an average Z finger, 2 u 0.90

a) Dimensions in mm.

Table 2 Capacitance results a)

Measured capaci- tance [pFI

Predicted capaci- tance [pF]

Geometry

s to all 180 s to w+z 175 w to s + z 160 z to w + s 295

s to w+z 170 w to s+z 155 z to w + s 290

s to anode array and frame 18

s to MWPC body 26

200 200 330

WS inside MWPC

WS outside MWPC

a) Measurements accurate to +_2%.

J. Thornton / The inter-electrode capacitance of wedge and strip position sensing elements 229

citances are shown in table 2 for comparison with the measured values. These were calculated using eqs. (3), (5) and (6), and the parameters listed in table 1.

5. Discussion and conclusions

Another model that has been used [3,7] treats the WS as the sum of a parallel wire capacitor and a parallel plate capacitor [5]. The wires' centres are sep- arated by 2g and have diameters equal to the thickness of the conducting film. The parallel plates have a total area a factor ( 1 - 4 g / p ) less than that of the WS to take account of the insulating gaps. The distance be- tween the plates is the WS period. For the MWPC WS the parallel wire and plate capacitance contributions are 40 and 15 pF, respectively, when no extra multiplication constants (weighting factors) are employed. The small size of the parallel wire contribution is due to the thinness of the conducting film and the small size of the parallel plate contribution is due to the large separation employed, p. The total is less than a third of that measured experimentally. Therefore the success of this model relies on the choice of weighting factors, the values of which were not reported, also no rules were given for their calculation. Weighting factors can be obtained by calibration of the reported equation with measured WS values, however, the factors were said to be dependent on the relative magnitudes of the sub- strate thickness, pitch and inter-electrode gap. There- fore different weighting factors would be required for different WS [3]!

In comparison, the dependence of this paper 's method on the WS parameters is covered by the equa- tions given. The inter-electrode capacitance values for other WS can therefore be readily calculated.

An alternative to the parallel wire capacitor model for the evaluation of the edge contribution would be a parallel plate capacitor model, with plates of area 2 N D t (where t is the conductor 's thickness). However, this edge parallel plate contribution is negligible ( = 0.06 pF for the WS described). Even if the conductor was very thick (t = g) the contribution would still be only a small part of the measured value (5%).

Consider two adjacent fingers on a WS, they will be from different electrodes. Divide the fingers into rib-

bons of equal width, parallel to the gap. The capaci- tance contribution from the ribbons bordering the gap will be the largest because their separation is small - only just greater than g. The capacitance contribution from the ribbons in the centres of each finger will be the smallest because their separation is large - comparable to p. The variation of capacitance contribution across a finger is taken into account by the coplanar strip model but not by the parallel wire and plate model.

The predicted values were all over estimates, the mean error being about 17%. This is small considering the approximations made and supports the suitability of the simple coplanar strip model. The formulae, eqs. (3) and (6), are simple and easy to use. The necessary table or graphs of the elliptical integral of the second kind being widely available (for example see ref. [8]). Calibration constants may be required for WS with substrates of intermediate thicknesses (g < h < p / 4 , where h is the substrate thickness).

Acknowledgements

The author is grateful to the staff of Mullard Space Science Laboratory and in particular 1.M. Mason, H.E. Schwarz, J.S. Lapington and J.L. Culhane for useful discussions.

References

[1] O.H.W. Siegmund, S. Clothier, J. Thornton, J. Lemen, R. Harper and J.L. Culhane, IEEE Trans. Nucl. Sci. NS-30 (1983) 503.

[2] G.F.G. Delaney, Electronics for the Physicist, 1st ed. (Penguin, London, 1969).

[3] H.E. Schwarz and J.S. Lapington, IEEE Trans. Nucl. Sci. NS-32 (1985) 433.

[4] O.H.W. Siegmund, M. Lampton, J. Bixter, S. Bowyer and R.F. Malina, IEEE Trans. Nucl. Sci. NS-33 (1986) 724.

[5] W.R. Smythe, Static and Dynamic Electricity, 3rd ed. (McGraw-Hill, New York, 1968).

[6] J.G. Thornton, Ph.D. Thesis, University of London (1986). [7] J.S. Lapington and H.E. Schwarz, IEEE Trans. Nucl. Sci.

NS-33 (1986) 288, [8] Handbook of Mathematical Functions, 9th ed., eds. M.

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