# A fast method for computing current distribution on printed circuit boards and microwave integrated circuits using the method of moments

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Ing., University of Belgrade, Serbia, 1990 A thesis submitted to the Faculty of the Graduate School of the University of Colorado at Boulder in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Electrical and Computer Engineering May 1999 I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9925398 Copyright 1999 by Jevremovic, Vladan All rights reserved. UMI Microform 9925398 Copyright 1999, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48183 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This thesis entitled: A Fast Method for Computing Current Distribution on Printed Circuit Boards and Microwave Integrated Circuits Using the Method of Moments written by Vladan Jevremovic has been approved for the Department of Electrical and Computer Engineering Edward F. Kuester K. C. Gupta The final copy of this thesis has been examined by the signators, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. iii Jevremovic, Vladan (Ph.D., Electrical and Computer Engineering) A Fast Method for Computing Current Distribution on Printed Circuit Boards and Microwave Integrated Circuits Using the Method of Moments Thesis directed by Professor Edward F. Kuester In order to determine current distribution on Printed Circuit Board (PCB) or Microwave Integrated Circuit (MIC) structure using the full-wave Method of Moments (MoM), the impedance of the structure must be computed first. A full-wave impedance matrix is frequency dependent in a non-trivial way; Green’s function and consequently impedance matrix [Z] must be recomputed at each frequency separately to solve for the unknown current. A large percentage of CPU time required to solve the matrix equation [Z](Tj=[V] is spent on calculating the impedance matrix, which becomes less and less efficient as the number of frequency points increases. Analysis of circuits with electrically thin substrates is even less efficient because these circuits require bigger impedance matrix in order to compute the unknown current [I] accurately. We propose a way to speed up the computation o f the impedance matrix by using a static approximation to Green’s function and further simplifying the image part of the Green’s function as a single term. Frequency independent inductance and capacitance elements L™ and Cnn' are computed using analytic expressions, which are exact, even for electrically thin substrates. We multiply Lnn' and Cnn' with phase shifts, which are linear functions of wave number ko, to get quasistatic inductance and capacitance matrices [Leq] and [Ceq]. Modified quasistatic impedance matrix [Z] is then assembled from j£D[Leq] and 1/jcoCCeq]. Impedance matrix [Z] at other frequencies Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. is computed by multiplying Lnn' and Cnn' with appropriately scaled phase shifts and with jo). Approximations made in computing impedance matrix cause error in current. We use an error bound not widely used in literature (but more accurate than better known bounds) to estimate current error. We show the improved accuracy of fullwave MoM solution when the substrate is electrically thin. CPU time needed to calculate current distribution on a MIC over frequency range using the approximate and full-wave MoM approach is compared, as well as accuracy of radiation pattern prediction. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V ACKNOWLEDGMENTS I would like to dedicate this work to all those who guided me through life and who gave me precious moral support throughout the graduate school, wholeheartedly to my wife Zana, mother Milica, and most affectionately to the memory of my father Milutin. I would like to thank my thesis advisor Dr. Edward Kuester, who provided inspiration for the approximation MoM work. His technical insight, guidance and wisdom had integral part in shaping this research effort. I also wish to express appreciation to Dr. K. C. Gupta, my second reader, and to the rest of the committee: Dr. Melinda Picket-May, Dr. Richard Booton and Dr. Karl Gustafson for their insightful suggestions regarding the thesis work. My special thanks goes to my former colleagues at Electromagnetics laboratory: Mike Spowart, Djordje Jankovic, Tom Schwengler and Darija Tomic, for their discussions and assistance. The years I spent in graduate program at CU Boulder will always remain in my memory. Last but not least, I am grateful for financial support I got from U S WEST Advanced Technologies, and for the moral support I received from my technical directors and coworkers. Working full time and conducting a Ph.D. research was an experience that was challenging at times, but, as a wise man once told me, “A little bit of discomfort builds character”. This doctoral thesis is all about character building, persistence and invaluable joy that quadruple integrals bring to our everyday lives. I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. VI CONTENTS CHAPTER I. Introduction.................................................................................................. 1 1.1 Research g o als...................................................................................... 1 1.2 Thesis form at........................................................................................ 3 II. Mixed Potential Integral Equation............................................................. 6 EH. The Method of Moments ........................................................................... 8 EV. Green’s function .......................................................................................... 12 4.1 Green’s function approximation ......................................................... 12 4.2 Green’s function error .......................................................................... 16 V. Impedance matrix approximation.............................................................. 19 VE. Error estimate and computation.................................................................. 26 6.1 Error Estimate ...................................................................................... 26 6.2 Impedance matrix error ........................................................................ 28 6.3 Current error ......................................................................................... 31 VEI. Applications of the proposed m ethod........................................................ 36 7.1 Current computation for electrically thin substrates......................... 36 7.2 Current computation over frequency range........................................ 38 7.3 Radiation pattern prediction and its accuracy...................................... 41 7.4 Unwanted radiation from P C B ............................................................. 44 v m . CONCLUSION .......................................................................................... 48 Bibliography .......................................................................................................... 51 Ii i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vii Appendix A. Analytic derivation of double line integrals............................................. 53 A- 1 Snow White (Integral 1 )....................................................................... 53 A- 2 Grumpy (Integral 2 ) .............................................................................. 55 A -3 Bashful (Integral 3) .............................................................................. 61 A- 4 Evil Stepmother (Integral 4 ) ................................................................ 63 A- 5 Happy (Integral 5) ............................................................................... 69 A- 6 Dopey (Integral 6) ............................................................................... 70 A -7 Doc (Integral 7) ................................................................................... 71 A- 8 Sneezy (Integral 8) .............................................................................. 80 A- 9 Sleepy (Integral 9) ............................................................................... 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. viii TABLES Table 6.1: Eigenvalues and condition number K(Z) for circuits in Fig. 6 .1 ...... 30 6.2: Comparison between the bounds in Eq. 6.1 and Eq. 6.3 andthe real current error from full-wave MoM solution...................................... 32 7.1: Current error as a function of substrate thickness and mesh grid s iz e ............................................................................................... 37 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ix FIGURES Figure 2 .1: Microstrip printed circuit geometry.................................................... 6 3.1: Subdomain rectangular and triangular c e lls....................................... 9 3.2: Roof top functions defined over a two pairs of cells: (ct|, ai) and (ocr.oz') ............................................................................................. 10 3.3: Pulse functions defined over two pairs o f cells (oil, (X2 ) and(cti', (X2 ') ...................................................................................... 10 4.1 Definition o f maximum and minimum distances between cells a and a ................................................................................................... 14 4.2 Error in image term of electric part of the Green’s function 5.1 Definition of a distance p between surfaces S and S ' and vectors associated with Eq. 5 .2 ........................................................................ 20 5.2 Collinear pair of cells over which test functions are defined 6.1 Microstrip circuits used in examples - top v iew ................................ 28 6.2 Impedance error distribution for various lengths of the three circuits shown in Figure 6.1 ........................................................................... 29 6.3 The maximum impedance error as a function of condition number of microstrip circuits in Figure 6 .1 .................................................... 16 21 31 6.4 Current error variation as a function of voltage source placement... 33 6.5 Current distribution along open-ended straight microstripstu b 7.1 Microstrip circuit used in the example - top v iew ............................. 39 7.2 Program execution time comparison, the approximate vs. full-wave M oM .................................................................................................... 40 7.3 Coordinate system used for radiation pattern calculation.................. 41 7.4 Radiation pattern for the circuit shown in Fig. 7.1, <|>=0o ................... 42 7.5 Radiation pattern for the circuit shown in Fig. 7.1, <j)=90o ................ 42 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 X 7.6 Radiation pattern accuracy over the frequency ran g e ........................ 43 7.7 Simplified model of PCB with attached power c o rd ........................ 44 7.8 Symmetric and anti-symmetric current modes on P C B .................... 45 7.9 Short stub attached to power cord ...................................................... 46 I I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER I INTRODUCTION 1.1 Research goals The goal of this research is to produce a method for fast computation of current distributions on Microwave Integrated Circuits (MICs) and Printed Circuit Boards (PCBs) in cases when the circuit size is of the order of a few wavelengths or less. Such MICs and PCBs are commonly found in personal computers and commercial mobile hardware (such as mobile phones) which operate at RF frequencies or lower. MIC and PCB circuits of any size can be analyzed using a full-wave Method of Moments (MoM) using one of several available commercial software packages. However, there are two issues related to commercially available MoM CAD software that need to be addressed. Firstly, full-wave MoM is a computationally intensive task, with most of the computing time used to calculate and invert an impedance matrix. Because the fullwave Green’s function is frequency dependent in a complex manner, the computation of the impedance matrix needs to be repeated for each different frequency point even if shape or physical length of the circuit has not changed. Circuit analysis over a frequency range is quite common in research and development design process, in particular in the circuit optimization and design verification stage. A method that would analyze the circuit once, and then use the information to compute more quickly the impedance matrix and current distribution at all frequencies within the frequency i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 range would reduce circuit analysis software run time, and would therefore be beneficial to RF design engineers. The second issue comes from the fact that full-wave MoM is known to be less accurate if a microwave circuit has electrically thin substrate. The way to overcome this problem is to refine the MoM mesh geometry, i.e. use smaller subdomain cells. This approach, however, increases the size of the impedance matrix [Z], which in turns increases CPU time needed to evaluate current [I]. The core of this problem is in the way full-wave Green’s function is computed; it involves curve-fitting of the Green’s function into a polynomial, and then using the polynomial coefficients to evaluate surface integrals. If, for the class of problems described in the previous paragraph, we eliminate the need for curve-fitting, and further evaluate the surface integrals analytically, we would improve stability of the current distribution. We propose to speed up the computation of impedance matrix over a frequency range by making modified quasistatic approximations to the full-wave impedance matrix. The approximation has three stages. First, the full-wave Green’s function is approximated with a static Green’s function. Second, the infinite summation in the Green’s function is approximated with a single term. The constitute static elements Lnn and On- are then calculated analytically, using the approximate static Green’s function. Analytic computation of static elements Lnn- and Cm,- and the fact that we do not curve-fit the Green’s function into a polynomial increases the accuracy of the current distribution when the circuit substrate is electrically thin. The third and final approximation stage is to multiply L^- and Cnn- with phase shifts <j>to get quasistatic inductance and capacitance matrices [L«,] and [C*,]. Phase shifts are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 chosen to be a linear function of wave number ko. Finally, modified quasistatic impedance matrix [Z] is assembled from j©[L«,] and l/jcofCeq]. The resulting impedance matrix is a function of frequency in a simple manner. To calculate the impedance matrix [Z] at different frequency, we just need to recompute phase shifts using appropriate wave number at that frequency, and multiply constitute static elements L™' and Cnn' with jo) and appropriate phase shifts. Therefore, repeated evaluation of double integrals at new frequencies is not required using our approximate method; however, impedance matrix inversion is still required to solve for the unknown currents. The approximate MoM algorithm was implemented by modifying Pmesh, a full-wave MoM code developed by Zheng and others [5]. The unmodified version of Pmesh was used to run full-wave circuit analysis, and to compare results. The choice of the code was based on the access to the software and the intimate understanding of the inner workings of the Pmesh model by professors and students affiliated with Electromagnetic Laboratory at University o f Colorado at Boulder. It should be pointed out that the basic approach and results are not unique to Pmesh, and can be implemented, with appropriate modifications, in other RF or microwave MoM codes. 1.2 Thesis Format The reminder of the thesis consists of seven chapters. Chapter 2 derives Mixed Potential Integral Equation (MPIE) which is the principal equation from which unknown currents on a MIC or PCB is calculated. Chapter 3 outlines MoM and describes in some detail some basic elements of MoM - test functions, mesh grid. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 subdomain cells, and calculation of unknown current coefficients. It also gives details on impedance matrix calculation and describes different elements that constitute an impedance matrix term. Chapter 2 and 3 are preliminary chapters that review the material necessary to understand the new work that follows in Chapter 4 through 7. From now on, we focus on the new work that is the core of the research presented in this thesis. Chapter 4 gives a brief background of the Green’s function, and continues with description of four terms that constitute the quasi-static Green’s function. It then proceeds to outline the static approximation, and in particular the approximation of an infinite summation in electric part of the Green’s function with a single term. Details on how this approximation is carried out are given. The chapter closes with an example that computes the error in static Green’s function that has been committed by making such approximation. Chapter 3 presents static impedance matrix calculation, and outlines the transformation of double surface integrals used to calculate impedance matrix terms into double line integrals. This transformation allows analytic computations of constitute static elements Lnn' and Cnn' .which improves the current calculation accuracy when the circuit substrate is electrically thin. Lnn' and Cnn' are then multiplied with various phase shifts to yield quasistatic inductance and capacitance matrices [L*,] and [0*,]. These phase shifts are a function of wave number ko, circuit geometry, substrate thickness, and relative permittivity £r- The modified quasistatic impedance matrix [Z] is then assembled from jtofLcq] and l/jo)[Ceq]. Chapter 6 describes an error bound used in estimating error in current that results from making approximations in Chapter 4 and 5. The error bound we used is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 less known in the numerical electromagnetic literature, but proved to give more accurate error estimates than some other better known error bounds [15]. Further in this chapter, we present examples of the numerical calculation of impedance matrix and current distribution on three different microstrip circuits, using full-wave and approximate MoM analysis. The accuracy between those two methods is compared, as well as the accuracy of the current error estimate with an actual current error. Chapter 7 presents possible means o f implementing the approximate method we outlined in Chapter 4 and 5. The first example shows improved accuracy of the current calculation on MIC when the substrate thickness is electrically thin. The next example compares the CPU time needed to calculate current distribution on MIC over frequency range using the approximate method and full-wave MoM. The third example is MIC radiation pattern estimate using the approximate MoM method. The last section in this chapter opens with background and literature overview on unwanted radiation from PCB. Paul and others [12] claimed that radiation pattern from a PCB circuit may be approximated with a radiation pattern from a short antenna. We attempted to verify this claim using both holi-wave and approximate method. Neither approach, however, could verify what has been claimed in literature. Finally, Chapter 8 summarizes the findings in this research effort and gives recommendation for future research efforts. I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER n MIXED POTENTIAL INTEGRAL EQUATION Let us consider a circuit given in Figure 2.1. The surface S is the surface of metalization along the dielectric/air interface. The metalization, with thickness t and width w, is the circuit. The dielectric slab, with permittivity Er and thickness h, and ground plane are considered to extend to infinity to simplify the problem. Let us impress a time harmonic electric field £ '(r)o n surface S. z Metalization > w 1 Figure 2 .1 : Microstrip printed circuit geometry A current density, J(?) with r e S will be induced on the metalization. Induced current causes scaterred electric field £*(r), where £ T(F)can be expressed as: £ ' (F) = jG (r, r ') •J(r')dS s 2. 1 where G(F, ?') is a dyadic Green’s function. In order to solve for the unknown current density, we need to enforce the appropriate boundary condition on the tangential electric field on the metalization surface S. Assuming perfectly conducting metalization, the total tangential field on surface S is: £,' (?) + E\ (r) = 0 reS i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 .2 At higher frequencies, metalization is not completely lossless, so in order to account for those losses, we can introduce equivalent surface impedance, Z s(r)and modify Eq. 2.2 to become: E\ (F) + E* (F) = Z s (F)7 (F); reS 2 .3 Z s(r) is a function of position, frequency, metalization edge effects, etc. A detailed discussion of metallization effects and surface impedance is given in [1]. Taking into account the planar geometry of the circuit shown in Figure 2.1, we can simplify dyadic Green’s function into two scalar Green’s functions once we let the metalization thickness approach zero. As we let t-*0, we take the surface S to have two sides, top (S*) and bottom (S')- We modify Eq. 2.3 to become: £/+(?) + £ ~ (F) = Z s (F; r)7* (F) r e S+ £;-(F) + E ;-(r) = Z ,(r;f)7 -(r) reS' 2. 4 We then simplify Eq. 2.4 as: £ /( r ) + £ ; ( r ) = - i z i (r;r)7 (r) reS 2. 5 The current 7(F) is the total current density on surface S. Once we split the dyadic Green’s function into two scalar Green’s functions [2], [3], we can rewrite Eq. 2.5 as a Mixed Potential Integral Equation (MPIE) [4]: E't ( r ) = ^ Lj Gn (p ) J ( r ') + -V V, Ge{ p ) T • 7(F') d5' + i z j (F;r)7(F) Ajc j ko | Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2. 6 CHAPTER HI THE METHOD OF MOMENTS To solve the MPIE (Eq. 2.6), it is typical to expand current density 7(F) into a complete set of basis functions: 7(?) = £ / „ « „ ( ? ) 11=1 3 .1 where Bn(r) is the basis function and /„ are unknown current coefficients. When we substitute Eq. 3.1 into Eq. 2.6 we get: S; (?) = 4sr j r / . J [G . (/> )« . ( 7-) + - J V,Ge( p ) V fl.(r W + Z.Cr-.Ofl.Cr)|3. 2 n= l From Eq. 3.2 we get a matrix equation when we multiply Eq. 3.2 by a set o f test functions f n(r) , n=l,2,3..., integrate them over the surface of the structure and truncate the infinite number of basis and test functions into finite number of them. A technique where the test functions are chosen to be the same as basis functions, Ta(?) = Bn(?), is known as Galerkin’s method. Furthermore, we define each basis function Bn(?) to be a “rooftop” function A„ (r) that exists only on a subdomain of S. Taking these assumptions into account, we write a matrix equation [Z][I]=[V] as: £ z „ J ..= V ,; h = UZ3..JV <1 =1 where the impedances Z nn- and the voltages V'can be written as [5] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 .3 9 +zsjK c*. y)• a . (*« 5 K, 3 .4 = J A„ (x, y) •£/ (x , y)dS s Function n„ (x, y ) , defined as n „ (x, y ) = V, • An(x, y ) , is commonly referred to as a “pulse” function. The next step is to define subdomains S, S ' over which the integration Eq. 3.4 is performed. The original circuit is replaced by rectangular and/or triangular subdomain cells over which basis functions exist, as shown in Figure 3.1. Figure 3.1: Subdomain rectangular and triangular cells “Rooftop” and “pulse” basis functions are always defined on a pair of neighboring sub-domain cells, which we label (oti, Cb) and (a /.a iO as shown in Figure 3.2 and 3.3. The normal component to the cell edge of the “rooftop” test function A„(x, y) is continuous across cell boundaries and vanishes on the boundary of the structure. Each impedance term Znn' in Eq. 3.4, is defined between a pair of cell edges n and n ' and has four integrals that contribute to it. The integrals are defined on subdomains Sa and Sa-; using shorthand notation, we identify them as I t Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 JJ .11.11 - / / 0 t\O \ . Each of these four integrals is, in fact, impendance defined CtiCty over subdomain cells; this is “local” impedance Zcaaj • Each “global” impendance term Z„n- defined between a pair of cell edges n and n' has four “local” impedance terms defined on a pair of subdomain cells a* and a'j. cell a t n cell a 2 cell o'i n' cell afz Figure 3.2: Roof top functions defined over two pairs of cells ( a t, 0 C2 ) and (a 'i, o ' 2 ) cell o'- cell a 2 I M cell a t cell o ' 1 Figure 3 3 : Pulse functions defined over two pairs of cells (cti, 0 2 ) and (a ' 1, oc'2 ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 Now we can rewrite the impedance term in equation Eq. 3.4 as w ,-=I j = I JJ (G. 1 n„ ( P ) A „ , ( * . V )• A „ . ( X , y ) - z a„' * J K ( x , y ) n „ - ( x . y ') V S « J S a. *0 Equation 3.5 establishes the relationship between “local” impedance elements, Zoa' defined on a pair of cells otj and a 'j and “global” impedance terms Zmr defined between a pair of cell edges n and n'. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER IV GREEN’S FUNCTION APPROXIMATION AND ERROR 4.1 Green’s function approxim ation The exact expressions for the Green’s functions are [6]: 4. 1 ua +u \anh(uh)ka p dk C ,(P ) = T ^ - - p „ ( ^ P ) -----------D tm D te 9 2^o o 4.2 where D te = u q * u coth(w/i) 4.3 Dm = e ru0 * u coth(«/i) |io is the free space permeability and £o is the free space permittivity. The quantity er is the relative permittivity of the substrate region, Jo is a Bessel function of order zero, and uo and u are two functions of the integration variable kp: “o -* o 4 .4 where ko = W / ^ o 4.5 There are known quasi-static approximation for Gmand Ge that are valid for lk o p l« l, as given in [6]. The quasi-static approximations for Gra and Ge are: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 // -y io /io 4* /^> „ - jk o R i ) = g: ( / » + g; ( p ) 4 .6 4^0 [ i=i = G?{p) + G‘e(p) where 6 —1 4 .7 R 2 = p 2 +(2ih)2 Gmand Ge each can be broken down into 2 terms; a direct term, Gd that contains Ro term, and image term G1that contains Rj terms. If we further assume that the IkoRikcl, we get a static approximation for Green’s function: a - ° (e )=Jr i 4 k Ro »».o, x (P) = ~ Mo I — 4 k Ri 4 .8 G ^°(p) = ^ — S l — 4 K£o Ro G '» { p ) = - (1-/72) 4KEo .=i Zero indices in superscript in Eq. 4.8 indicate static approximation. We would like to further simplify G ^°(p), the last term in Eq. 4.8, to avoid having to compute infinite series for different p’s. Mosig [7] suggested approximating the infinite series with a simple ground image; following his suggestion we introduce a single term function F(p) that has a behavior similar to the biggest term in the summation ( 1/R|)» with different amplitude A and thickness coefficient hcff: i i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 4.9 F (p )= n - i J Mosig suggested p —»0 and p — as choice matching points where F(p) =G'°(/>) .Our approach is to use different matching points p max and p mm as function o f a distance p. p max and p rain are defined as: o' r 'm in = rO + 0v *25 'm i n —^522. ^ 4. 10 PLx=Pn»n+l-25 Pm ax P tm n where pmax and p„un are the maximum and minimum distances between a pair of cells a and a ' over which p is defined, as illustrated in Figure 4.1. mm Cell a Cell a ' Figure 4 .1 Definition of maximum and minimum distances between cells a and a . By letting p —» p min and p -» p max and matching F(p) and G '° O ) , we get llim T 1 = — = = A.= = = )T _1/(-TJ) 1— *-/>„*, I=l R . 4. 11 N lin» Z W 'i r T -" I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 tf < 1-6. we Set: h 2 _ p 'L g ° 2( p p - p ! < 2 °2( P L ) ' c ; o! ( p 1 )- g : ° 2(p L ) 4. 12 A = V ( P l + a tf)G i° 2( P l.) If, however, *0(P™n + p ^ ) > 1.6, then we let p'™, -» 0 and p',** -» °°, to get: l.2 _ A “ *i 2l 2 * * -(— >A 1 ,24 ln2( - = ^ - ) er +l 4.13 A =■ 2 *rr + l The scaling coefficients in Eq. 4.10 (0.25 and 1.2 respectively) are chosen by trial and error to yield the least error in current. The “cutoff’ distance kofp'min+p'max) that defines the domain over which Eq. 4.12-13 are valid is also chosen by trial and error to yield the minimum error in current distribution. N in Eq. 4.11 is taken to be big, but finite; we take the first N terms in the summation that would yield error less than 1%. Taking into account that for large N terms in infinite summation are approximately we need N=100 terms to get a 1% accuracy. It should be pointed out that A and hefr are a function of permittivity Erand substrate thickness h only; therefore, we need to calculate A and heff only once for a particular circuit and then reuse the coefficients in a subsequent analysis o f the same circuit. A further simplification of the outlined method for simplifying the Green’s function would be to approximate coefficients A and heff valid over a pair o f cells as given per Eq. 4.12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 ,p ) valid over ko(p'„ui.+ p'm ax) ^1-6. This with a single curve fitting function /( A , is a topic for further study, and is not included in this thesis. 4.2 G reen’s function e rro r As a conclusion to this chapter, we give an example that shows how much error is introduced as a result of approximating G' °(p) with F(p). In this example we choose an open ended microstrip stub with relative permittivity Er = 2.25. The length of the circuit is p = 1.1A., the thickness of the substrate is h = A/25. Approximation function F(p) as given per Eq. 4.9 is compared with G'e°(p) as given per Eq. 4.8. The resulting error in Green’s function is given in Figure 4.2 Error in image term of electric part of Green’a function 3 ---------------------------------------------------------------------------------------- — ------------ 2.5 * 0.5 0.0 0.5 0.9 1.4 1.9 2.4 2.B 3.3 3.8 4.2 4.7 5.2 5.7 6.2 6.6 Circuit length kp, normalized Figure 4 .2 Error in image term of electric part of the Green’s function G'e°(p) It should be mentioned that the shape of the circuit did not affect the error distribution significantly. While we have shown the data for a simple open-end stub, i ! Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 we ran simulations on two more complicated open-end circuits that will be used as examples in Chapter 6. Resulting differences in Green’s function error distribution were negligible, and are not shown here. The conclusion we make is that, for practical purposes, Green’s function approximation as outlined in this chapter, is independent of circuit shape. The sharp increase in the error at kp = 0.8 is due to the fact that our Green’s function approximation is not continuous at that point. While Eq. 4.13 is valid for k o (p ,min+p,max) > 1-6, we did not match Eq. 4.12 and 4.13 at kp = 0.8; instead we used kp = Oas a matching point in Eq. 4.13. This choice ( kp = 0)of matching point does not appear to be particularly intuitive; in particular it increases the overall error in Green’s function. However, it yields less current error than the other ( kp = 0.8) choice, and was therefore used in all examples following this chapter. Another rather intuitive approach as to how to reduce error in Green’s function is to apply Eq. 4.12 throughout the whole circuit, and completely abandon Eq. 4.13. The advantage o f it seems rather obvious: because Eq. 4.12 is defined over a pair of cells, it is a better approximation to G '°(p) than Eq. 4.13. We decided not to do that for two reasons. First, using many “local” Green’s function approximations, i.e. approximations valid over only a pair o f cells, we lose simplicity and increase computation time, because we need to calculate finite series for every pair of cells over which Eq. 4.12 is valid. Second, we ran a simulation when we used Eq. 4.12 over the whole domain over which gI °( p ) is valid, and while the error in Green’s function indeed decreased, the current error slightly increased, which made us abandon this approach. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We also tried a completely different approach [8] that suggests making an approximation in spectral domain Green’s function, and then transforming the approximation from spectral into spatial domain. This approach yielded inferior results to the spatial domain Green’s function approximation. In closing, we should point out that we did extensive simulation on Green’s function approximation throughout the research effort; the approach outlined in Eq. 4.9-13 consistently gave the best overall current approximation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER V IMPEDANCE MATRIX APPROXIMATION In the light of the static approximation in Eq. 4.8 we see that electric and magnetic impedance terms can be written as: C : = Z "J'° + Z w = J" JJ<c " <P>+ G “ U y) • /)< « < « SnSn 5. 1 + Z " '“ = - £ J J ( 0 " ( p ) + - — = = ) n . n . ^ s i s ' We take advantage of the divergence theorem to transform double surface magnetic and electric image term integrals into double line integrals, using the following identity: JJ f{R )d S d S '= - $ g ( R ) d [ • dl 5. 2 3sds" s s" where g(R) is a solution of V fg(R) = /( R ) 5 .3 and B x2 By1 «=V pr +*T p = J (x - x 'f+ ( y - y ') 2 The surfaces S and S' are defined as illustrated in Figure 5.1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5- 4 20 dS Figure 5.1 Definition of a distance p between surfaces S and S ' and vectors associated with Eq. 5.2 Applying Eq. 5.2 in cylindrical coordinates to electric image impedance Z":° we get: JJ A dSdS ss-^p +heff - A § ( J p : +h^, asar .d P 5.5 Line integrals associated with Eq. 5.5 are derived in Appendix A. It should be pointed out that if the cells a and a ' are rectangles, then right hand side of Eq. 5.5 consists of integrals that have the form of JJg(/?)<£cdlr’or j j g(R)dydy\ Since p is symmetric with respect to (x-x7) and (y-y7), once solved, these two integrals are easy to implement, because many corresponding terms are dual, i.e once we solve the first one, we can implement the second one by replacing x with y and x' with y'. Applying Eq. 5.2 in cylindrical coordinates to the magnetic image impedance Z ”J.'° is a bit more complex, since it involves test functions that are linear functions of x,y and x',y'. We are giving a simple example here that involves square cells (ct|,a 2 ) and (a 7i.aS) that are collinear with x and x ' axis (Figure 5.2), so that A„ (x, y) = anxax + bn and A„.(jc\ y ')= an.x'at. + bn-, where a„. bn, an*. b„' are constants. i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 ” x Cell ct| Cell o£\ Cell a 2 Cell aS Figure 5.2: Collinear pair of cells over which test functions A„ and A,,.are defined Since there is no dielectric present, the coefficients A and heff are now A= L and hen=h. As a result, we get the following set of equations: JJ . . xxi —dSdS' = - ^ g { R ) x x ' d l • £ + ss-Jp2 + h2 a— sas- •dr (/?)(«„• • ax )(an • dx )dldV + asas- + fy x f 2(R)(an. • a x.){an •a x-)dldl' asas- I S d S '= - $ g (R ) x d I • d T + $ M R ) ( a a. • d x)(dn . a x)dldl’ s s j p 2 +h2 asas- asas- f f ■ T x '-r -= dSdS ' = - t i g ss j p 2 + h2 f[ r_ L - = f / M ’= ll-Jp 2* ^ (R)x'dl *d?+ & f^ R ) { a n. * a x.){an • ax.)dldl’ Msaa' g(R )df • d/~ aS- where k(R), fi(R) and f2 (R) obey the following equations: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5. 6 22 V 2k(R) = g(R) # ,( * ) dx g(R ) 5. 7 and consequently: g(R) = i j p 2 + h 2 - h ln(— ^ 2h !fD\ h t P 2 ^ 1.2 v I IT 3 ^ P 2 \ l / ^ + V / ?2 + /t(/?) = —p -1 + (— ----------h )Jp~’ +h~ +/(^------— ) In(----- —---------) 4 9 36 6 4 2/2 /,(/?) = ^ Y ~ 4 hl + P 2 + +'(y2—— ln(* ~ x ’+ Jh 2 + P 2)~ h{-{x - x ’) - , ,. _|. h ( x - x ') , . h + ijh 2 + p 2 - i y - y ’) tan '( ------/ -) + ( * - * )ln(----- -- ) + ( y - y ’)4 h + P~ + h ln(.r —x'+ ^h2 + p 2) } f 2( R ) = - ± - ± J h 2 + p 2 + h 2 + i y 2 2 + , ,. _|. h (x-x') . ( y - y ) tan ' ( ------------— j = = = = = ( y - y ' ) y h + P~ ln(-(.r - JC’) + 4 h 2+ p 2) - h { x - x'+ . )-(x -x )ln( h + J h 2 + p~ ---------) + h ln(-(.r - x ’) + J h 2 + p 2) } Line integrals given in Eq. 5.6 are solved analytically in Appendix A. Here, we have four double surface integrals that we transform into double line integrals. In Appendix A, we also include a more general case with test functions over rectangles and triangles. In the general case, we have nine double surface integrals that we Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ® 23 transform into double line integrals, which are then solved analytically. We end up computing image electric and magnetic impedance terms using analytical expressions, which increases the accuracy of impedance matrix computation when the circuit substrate is electrically thin. Once all components of impedance matrix terms Z^, . are computed, we need to include phase shifts to our quasistatic impedance matrix that would make for a good approximation to a full-wave impedance matrix. A good choice for phase shift is <|>=koRnn' because it is frequency dependent in a linear manner. Unknown Rnn- is chosen to minimize the error that we get by approximating the full-wave impedance matrix terms with static impedance matrix terms. For magnetic part of impedance, we have: "An(P) • A,' (p)dSdS = \\\^ .< .P ) * A P ) d S d S - 5 .9 CC' t\ I f k o (R -R n n ') « 1. t h e n e s 1- jk 0( R - R ^ ) , and we get: jjA'(p)'A„-(p)dSdS' 5. 10 CC* ** Similarly, for electric part of impedance, we get: JJna(p)nff.(p)dSds' cc* 'CUt I 5. 11 \ \ ± n a(p)Ua.(P )dSdS’ It should be pointed out that Eq. S. 11 should be applied to each of the four terms in the “local” impedance matrix as defined per Eq. 3.5. In other words, we have Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 four phase shifts in electric part o f impedance for each impedance matrix term. However, we have a unique phase shift for the magnetic part of the impedance terms, so Eq. S. 10 is calculated only once for each pair of subdomain cells (a i,a 2 ) and (ot i,a 2 ). We now rewrite impedance matrix term Znn- as: =Z J °e m *jm,i 2 2 IaA-A 2 2 :Ake-‘ + Z ^° e J +2 f Z £ V +£ f Z £V 5. 12 ar=i a =1 ar=i a =1 j Where various phase shifts <j>are defined as: JJ A„ (/?) • A„- (p)dSdS' a ? J J - A „ ( p ) .A „ ( p ) r f .S < iS ' t f A .( p ) 'K A p ) < tS d S ' 0«m' —^O^nn' ~ ^0 sr_______________________ JjiA.(p).A„.(pWS<tS' JJn „ (p )n ^ (p )< /s< /s' J J - n „ ( p ) n „ .< p ) d M s ' SS- P \ \ n a{p)Yl a.(P)dSdS' Qcaz' = ^O^atr' = ^0 “ j J J ^ n a ( p ) n a. ( p ) ^ 5 ' 5 J3 55' From Eq. 5.12-13 we see that the only frequency dependance in impedance matrix [Z] comes from wave number ko in phase shifts 0mand 0e, and from jco in amplitudes Z° and Zm. This fact makes impedance matrix elements frequency dependant in simple fashion. To emphasize this fact, we rewrite Eq. 5.12 as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 ^nn ~ J0*'™' + + 5. 14 where j< P nn + [ ft-ie ' L'q. nn = L'qd nn e~ J'i'nn' + Ltqie nn eM 1 1 .. nn ■- j * z + ‘ * nn e jif j waa 5. 15 nn where CJ = £- Jf 4* s J « Ro i f f = - | 2- IT J - A .< - t.y )-A ,- (/.y V S d S ' 4* s iw Ri i 2 2 —!— = y Y CZJ ff I n i 2-— unnadsds' J J a - ^ o Ro — = y tmd y JJff I. i A—, 2 n „" n nadsds' f'tq j nn « = l« = l SaSa’y jP ~ + The equivalent inductance U ^ f, and capacitance C ^ f , C%:‘ elements are functions of cell geometry, substrate thickness h and, for the capacitance, permittivity £r, but are frequency independent. In order to compute the impedance matrix [Z] over a frequency range they need to be calculated only once. Matrix computation at multiple frequencies co* is done by first multiplying these elements with phase shifts <|W and (jw (with appropriate wave number kj) to get U^. and C% , which are then multiplied with jo* and l/jo* to yield matrix elements Znn'. Impedance matrix computation using the method outlined above is far less CPU intensive than the fullwave analysis over a frequency range. It is also more accurate for electrically thin substrates. This topic will be further explored in Chapter 7. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER VI ERROR ESTIMATE AND COMPUTATION 6.1 Error estimate In Chapter 4 and 5 we introduced the approximation to Green’s function and impedance matrix elements and gave examples how big of an error in Green’s function these approximations cause on simple microstrip circuits. In this chapter we discuss further the relation between impedance and current error and give guidelines on how to estimate current error once impedance error is known. A conventional way [15] to bound error in matrix equation ([A]+[E])[Y]=[B], where [A][X]=[B], and [E] is a perturbation of [A], is to apply the Banach Lemma [9]. In our matrix equation, we identify [A] as the full-wave impedance matrix [Z] as defined in Eq. 3.4 and [E] as an “error” impedance matrix [5Z] resulting from introducing approximations, as previously described, into the impedance matrix [Z]. [B] is equivalent to the excitation column vector [V] as defined per Eq. 3.4. [X] is equivalent to the current column vector [I] resulting from solving the full-wave matrix equation [Z][I]=[V]. Y is the current column vector [Iapp] resulting from solving the matrix equation ([Z]+ [5Z]) [Iapp]=[V]. Setting [Iapp] = [I]+[8I], where [81] is current error matrix, we get: m _ gg) i Im + fld l tz i 6 . i where m ) = l z - % z t i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 2 27 is one definition of the condition number. An alternative definition of condition number in Euclidian L2 norm K(Z)=lXtnaxl/lA,mml is also commonly used. We note that K(Z) > iX^n.l/lXminl in any norm. Equation 6.1 can be used to estimate current error, once the impedance matrix [Z] and error impedance matrix [8Z] are known. However, our experience with Eq. 6.1 indicates that it tends to give a rather crude estimate of the current error, much larger than the actual error. A less well known error bound has proved to give a more accurate estimate of the current error [10]. Using the same notation as we did in Eq. 6.1 we write the bound as: 6.3 We can further bound the right hand side of Eq. 6.2 as: I z - ][SZ\1£ p - ' $ ffi(|= |Z - 6 .4 From Eq. 6.3 - 4 we get: 6.5 Indeed, Eq. 6.5. clearly indicates that Eq. 6.3 represents a closer bound to current error than Eq. 6.1. However, we have yet to see how much more accurate the bound in Eq. 6.3 really is. In the next two sections we calculate condition number and error bounds for different microstrip circuits, and compare the estimates with the real current error we get from solving the full-wave matrix equations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 6.2 Impedance m atrix e rro r In this chapter we give a comparison between the true impedance matrix [Z] (as, for example, calculated by Pmesh, a MoM algorithm developed by Zheng and others [5]), and the approximate impedance matrix [Zapp], a calculation of which was outlined in Chapters 4 and 5. The circuit used in this example was an open ended microstrip stub with relative permittivity £r = 2.25. Substrate thickness was taken to be h = 7J25 while circuit length was left as a variable parameter, we ran simulation for circuit lengths that varied from 1= 0.7A. to I = 1.2X. Three different versions of the open-ended microstrip stub that were used in this simulation are shown in Figure 6.1. Circuit length I is taken to be along the middle of the circuit, as shown in the figure. L shaped Straight U shaped Figure 6.1 Microstrip circuits used in examples - top view The impedance error is taken in Li„f norm, i.e. the maximum error norm: \ Z " S Z „ X = ||Z -C Z -Z _ > |_ - m a j X K 6.6 7=1 Linf error norm in this case is the maximum error among matrix columns of the matrix [A], where matrix terms Ay are defined as: Ay — *=i Aty(upp)) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 .7 29 where Z "1are matrix terms for inverse o f the true impedance matrix [Z], and Z mapp) are matrix terms for approximation matrix [Zopp], Maximum impadanca error for various circuit lengths 1000 * - Straight stub I — L shaped stub | — li shaped stub tubir 100 ^ w 2 0.7 0.75 0.8 0.85 0.9 0.9S 1 1.05 1.1 1.15 1.2 Circuit Length I, normalized Figure 6.2: Impedance error distribution for various lengths of the three circuits shown in Figure 6.1. The error shown in Figure 6.2 comes from three staged approximation we have made so far. The first step is to approximate full-wave Green’s function with its static version. The second step was to approximate the infinite summation in the image term of electric part of Green’s function, G'e’°(p) with a single term function F(p) as given per Eq. 4.9. Finally, in order to re-introduce frequency dependence, but in much simpler fashion, we make a final adjustment to impedance matrix by multiplying its components with phase shifts, as given per Eq. 5.12-13. In Chapter 4, we saw that the shape of circuit did not have a significant effect on the Green’s function error distribution. However, Figure 6.2 indicates that shape of the circuit, as well as the total length of the circuit, impacts the maximum impedance error. We have to keep in mind that impedance error as defined per Eq. 6.6. is a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 function of the inverse matrix [Z]'*. If the smallest eigenvalue of matrix [Z] is very small, then the largest eigenvalue of matrix [Z]'1will be very large, thus making the product [Z] *[Z-Zapp] potentially very large. In Table 6.1 we list the smallest and the largest eigenvalues (L in . Amax) and condition number K(Z) for the three circuits when their length is near resonance (p=A). From the table we see that the increase in condition number for all 3 circuits is indeed due to a drop in the minimum eigenvalue, Lnin. Because Lm„ is so small, small errors in matrix [Z] cause large errors in the inverse matrix [Z]**, therefore causing large overall error. Table 6.1: Eigenvalues and condition number K(Z) for circuits in Fig. 6.1 Straight stub L shaped stub U shaped stub Length p Amin Amax K(Z) Amin Amax K(Z) Amin Amax K(Z) 0.95 A. 3.5E-3 0.78 226 6.1E-3 0.79 131 6.4E-4 0.8 208 1.0 A. 1.6E-3 0.78 480 1.6E-3 0.79 505 3.8E-3 0.8 1250 1.05 A. 7.8E-3 0.78 100 9.8E-3 0.78 79 7.5E-3 0.8 106 Finally, we look into how condition number affects impedance error. The magnitude of condition number for each of the three circuits is plotted against impedance error in Figure 6.3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 Im padanc* error a s a function of condition num ber ioo-t- ►straight stub * | ML shaped stub j o I A ll shaped stub |_ 5 so J E 1 40 a ^ b ** 100 C ondition N um ber, K(Z) Figure 6.3 The maximum impedance error as a function of condition number for microstrip circuits shown in Figure 6.1 It is evident that the general trend is that higher condition number yields higher error in matrix. However, different circuits having the same condition number can have substantially different impedance error. Upon further inspection, it is clear that more complex circuits have higher impedance error for similar condition number value, i.e. straight stub and L shaped stub clearly have lower impedance error than U shaped stub. Im pedance error for the first two circuits is less than 22% when K(Z) < 100; impedance error for U shaped stub was 22% - 60% when K(Z) < 100. 6.3 Current error To give numerical examples and compare how close the two bounds to the actual error are, we ran computer simulation on the three circuits shown in Figure 6.1. The length of each circuit varies from 1= 0.7 k to 1 = 0.7k. The results are shown in Table 6.2. For reference purposes, the right hand sides of Eq. 6.1 and Eq. 6.3 are labeled as “Est. A”, and “Est. B” respectively. The actual current error is labeled as ! Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 “error”. The actual current error calculated is the maximum current error of the current column vector ||/|| maxi/I “ IS/SJV1 1 maxl/l I S i iN ' 1 All error values in Table 6.2 are given in percentage (%). Table 6.2: Comparison between the bounds in Eq. 6.1 and Eq. 6.3 and the real current error from full-wave MoM solution Stub Length est. A U shaped stub L shaped stub Straight stub est. B error est. A est. B error est. A est. B error 0.70 X 26.32 2.94 1.76 27.6 3.18 1.6 110.2 27.1 15.4 0.75 X 33.73 3.46 2.01 25.11 3.5 1.7 97.8 21.4 15.2 0.80 X 34.08 4.72 2.86 30.27 4.4 2.1 99.2 22 12.9 0.85 X 53.18 7.55 5.18 45.51 6.1 3.4 148.5 26.5 8.6 0.90 X 103 18.3 12.1 79.01 10.5 7.2 236.8 38 16.8 0.95 X 768.4 53.5 36.3 209.52 30.3 17.8 489.6 72.5 28.9 1.0 X 931.2 108 37.1 494.9 84.7 113 4063 615 93.5 1.05 X 430 23.3 13.5 161.2 23.7 18.5 676 113 74 1.10X 294.8 16.1 8.5 107.44 15.2 11.3 349.8 60.4 19 1.15 X 254.21 15 6.4 79.7 12 8.16 252.4 42 9.2 1.20 X 240.8 15.5 5.4 71.07 13 5.8 199 33.3 13.6 It is quite obvious from Table 6.2 that “Est. B”, i.e. error bound given in Eq. 6.3. is at least of the order of magnitude more accurate than the error bound given in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 Eq. 6.1. From the data it can also be inferred that the bound estimate is more accurate when the circuit is not near resonance (l-X.). The data also indicates that error bound in Eq. 6.3 is typically 1.5 —3 times higher than the actual current error, depending on the length of the circuit. This is still a rather coarse estimate, but is clearly superior to the one given in Eq. 6.1. Current distribution on a circuit is a function of voltage source position; therefore the actual current error is also a function of voltage source position. In the previous example, the voltage source was placed at the end of each circuit. To see how the current error changes when the source is placed in a different position in a circuit, we run another set o f simulations, with a simple stub as a test circuit, with two different circuit lengths, 1|=0.75 A. and h= 1.0 X. The results are shown in Figure 6.4. Current error as a function of source position 3.0 2.5 - 35 2.0 - 20 - 10 0.75 X 0.5 0.0 O' O Or 0 C0 > O O O' O' O' O' O' O' O' O' O' O' O' O' Source position from circuit edge, normalized Figure 6.4: Current error variation as a function of voltage source placement It should be noted that the vastly different scales are used in this figure to show error for these two circuits. The data from Figure 6.4 indicates that relative I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 voltage source position in the circuit affects the current error distribution the most when the circuit length is at resonance (1=X). This is an expected result, since a circuit in resonance is quite sensitive to impedance perturbation, as seen from Figure 6.2; consequently this sensitivity propagates to current error. In particular, when the voltage generator is placed near a point in the circuit where the impedance is big (l=0.5A.), the current error is small. Since impedance at the point where generator is placed is big, a given absolute impedance variation (error) at that point is relatively small, therefore causing a small perturbation in the current. A circuit that is not near resonance (l2=0.75A.) is far less sensitive to generator placement when it comes to current error distribution. Predictably, a generator placed in the middle of the circuit (bigger impedance) yields smaller current error, but the overall error variation is nowhere as dramatic as it is when a circuit is in resonance. The maximum current error from Figure 6.4 is 2.4% for 1=0.75A. case, and 38.8% for l=A.. Error bound from Table 6.2 for the corresponding two cases is 3.46% and 53.5% respectively. Therefore, we confirmed our earlier observation that error bound in Eq. 6.3 gives error estimate between 50-200% higher than the actual current error. The low number corresponds to a circuit that is not near resonance; circuits in resonance get poorer error bounds. As a final note in this chapter, we present the actual current distribution along a microstrip circuit in Figure 6.5. We chose a simple open-ended microstrip stub with length 1=1.25 A., 6r=2.25 and substrate thickness h=A/25. The voltage generator is placed in the middle of the circuit. In this example, the maximum current error is 3.6%. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 Currant distribution along microatrip circuit Pmesh. amplitude 0.0035 150 approx. amplitude Pmesh, phase 0.003 - - approx. phase 100 0.0025 0.002 3 ta O 0.0015 -50 0.001 -100 0.0005 -150 ■v V Distance from circuit adga, normalizad Figure 6.5: Current distribution along open-ended straight microstrip stub As expected, the maximum current error is located at the maximum peak amplitude, which is the first peak in the figure. Overall agreement in both amplitude and phase is quite good, as expected for a current distribution with maximum error of 3.6% Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER VH APPLICATIONS OF THE PROPOSED METHOD 7.1 Current computation for electrically thin substrates In Chapter 1 we mentioned that full-wave MoM algorithm suffers from insufficient accuracy if the MIC substrate thickness is electrically thin. The principal reason for this deficiency is the fact that Green’s function is first computed N numerically from (4.1-2) and then curve fitted into polynomial of the form ^ a tp ‘ . /= -1 Curve-fitting number N is chosen as a function of p; bigger values of p imply smaller N. Consequently, coefficients a/ are also a function of p [4]. Green’s function is then approximated with the polynomial, and inserted back into Eq. 3.4. This effectively splits the equation into N double surface integrals that are computed to evaluate Znn'However, this approach does not yield accurate results when the MIC substrate is electrically thin. This is the consequence of the fact that for electrically this substrate the direct and image terms of the Green’s function have almost equal amplitude, but opposite signs. The result is a very small number that is difficult to curve-fit accurately. One way to circumvent this problem is to increase the number of subdomain cells in mesh grid, and/or increase the curve-fitting order N. The former method is CPU intensive at all frequencies; the latter method is cumbersome, and as suggested in [4], CPU intensive as well, especially at higher frequencies. We suggested in Chapter 1 that using approximation method facilitates analytic computation of impedance matrix terms (as given per Eq 5.12), which improves accuracy of the current computation when the substrate is electrically thin. We now show an example that supports our claim. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 We use straight stub as shown at Figure 6.1 with length l=0.45A. and permittivity £r = 2.25. The voltage source is placed at the beginning of the circuit, and the thickness of the substrate is left as variable. In Table 7.1 we show the maximum current error between the approximate and full-wave solution (Li„f) in percentage as a function of substrate thickness and subdomain cell length. Subdomain cell width was kept to A/20 at all times. Table 7.1: Current error (%) as a function of substrate thickness and mesh grid size Substrate thickness h Subdomain cell length A/25 A/50 A/100 A/150 A/200 A/20 20 9 37 136 23 A/40 8 5 11 6 22 From Table 7.1 we see that that the current error is of the reasonable order of magnitude when the subdomain cell length is A/20 if the substrate thickness h is less than A/100. If A/100 < h < A/200 we have to decrease subdomain cell length to A/40 to achieve reasonable accuracy. However, if h > A/200, even subdomain cell length of A/40 is not sufficient to improve current accuracy. We now have to show that a dramatic improvement in current error for the substrate thickness between A/100 and A/200 is largely due to a dramatic improvement in full-wave current computation. This would prove the point that the most of the error when substrate thickness is smaller than A/100 comes from the inaccuracy in full-wave current distribution when subdomain cell size is A/20 by A/20. We calculate current error L'jnfas a difference between full-wave current Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. distribution when subdomain cell length is A/20 and current distribution when it is reduced to A/40. We find this difference to be L'i„f= 43%. We then calculate a difference between the approximate current distribution for the same two cases, when subdomain cell length is A/20 and when it is A/40. We find that difference to be only L " nf= 4.6%. Clearly, the changes in the full-wave current distribution when the finer mesh grid is used far outweigh the change in the approximate current distribution. Therefore, most of the improvement gained by increasing the circuit mesh refinement comes from the improved accuracy of the full-wave current distribution. Looking at Table 7.1, one may come to a conclusion that for h > A/200, we need to refine mesh grid even further, possibly to cell length of A/80 to achieve better accuracy for fiill-wave MoM. However, we have to keep in mind that every time we reduce the length of grid size, we increase the size of the impedance matrix by the factor of four. In the next section, we will show how computationally expensive refining the mesh grid (i.e. increasing the impedance matrix size) really is. 7.2 Current computation over frequency range In Chapter 5 we mentioned that although computing the double line integrals outlined in Appendix A requires somewhat more CPU time than the Gaussian integration of double surface integrals for a single frequency point, the proposed approximation is far superior than full-wave MoM analysis over a frequency range. We now give an example that supports such claim using test circuit shown in Figure 7.1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 The circuit was analyzed over 50 frequency points, with center frequency fo=30 MHz. At frequency f0 the length of each arm in the loop is 1=0.25X, while the length of the stub attached to the loop is L=3.lX. Initially, the circuit mesh consisted of rectangular equilateral subdomain cells of length X/20. The total number of unknowns in this circuit is 85 for both approximate and full-wave MoM. The thickness of the substrate was h=A/25. +------ I ► Figure 7.1 Microstrip circuit used in the example - top view In the next simulation run, the same circuit was analyzed with substrate thickness h=X / 100. As we have seen in the previous section, it is necessary to reduce subdomain cell length to X /40 to achieve acceptable current computation accuracy when using full-wave MoM on a circuit with this thickness. However, it is not necessary to reduce subdomain cell length in the approximate MoM analysis of a MIC circuit regardless of the thickness size, so we still use the same equilateral X /20 by X /20 subdomain cells as mesh grid for the approximate MoM analysis. Therefore, full-wave MoM analysis of the circuit with thickness t=X /100 has 170 unknowns, while the approximate MoM analysis still has only 85 unknowns, the same number as it had in the first simulation run. The results are shown in Figure 7.1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 Program execution time 1200 Full-wave, hslambda/25 ■Approximate MoM ' Full-wave. h=lambda/1001 1000 1 o 800 § % 600 S 400 1 5 10 15 20 25 30 35 40 45 50 Number of frequency points, N Figure 7.2: Program execution time comparison, the approximate vs. fullwave MoM Even for MICs that are not electrically very thin, such as the circuit with thickness h=A/25, the approximate MoM is faster than the full-wave MoM after 4 frequency points. While the full-wave MoM needs 11 seconds to compute one frequency point, the approximate method needs only 2 seconds. Both methods have the same number of unknowns (85) to solve for when the substrate thickness is h=A/25 and both require [Z] matrix inversion to solve for the unknown current [I]. Therefore, 80% of the CPU time per frequency point is used to compute the impedance matrix, and only 20% to invert it and solve for [I]. The advantage of the approximate MoM is even more clear for electrically thin substrate (h=X/100). We need to double the number of unknowns (170) for the full-wave MoM, while the approximate MoM still has the same number of unknowns (85). As a result, the approximate MoM is faster even if there are onlv 2 frequency Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 points that need to be analyzed. It takes 18.5 minutes to analyze all 50 frequency points using the full-wave MoM, and just over 2 minutes for the approximate MoM with substrate thickness h=A/100, making the approximate MoM 9 times faster. 7 3 Radiation p a ttern prediction and accuracy In Chapter 6 we gave a comparison between the current distribution of the approximate and full-wave MoM. However, in PCB and MIC analysis, current distribution is only an intermediate step needed to find radiation pattern or S parameters or some other quantity that is commonly used to qualitatively describe the circuit. In the next example, the radiation pattern of the microstrip circuit illustrated in Figure 7.1 is calculated using the approximate and fiill-wave MoM. If the substrate thickness is h = A/25, and all other parameters the same as in section 7.2, the maximum current error is computed to be 7.8%. The coordinate system used for the radiation pattern calculation is shown in Figure 7.3. The comparison between radiation patterns calculated using the approximate and the full-wave MoM is shown in Figure 7.4-5. z Figure 7 3 : Coordinate system used for radiation pattern calculation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 Radiation Pattern, <j>=0o 5? jS* jP Jp .*£ -50 -100 o '-150 j — fu ll-w a v e M o M 2*-200 ------ . - - ap p rox M oM L N -250 -300 -350 0, d eg rees Figure 7.4: Radiation pattern for the circuit shown in Fig. 7.1, <j>=0o plane cut Radiation Pattern, 4^90° -50 -100 i — f u ll- w a v e M o M -150 * * approx M oM -250 -300 -350 8 , d eg rees Figure 7.5: Radiation pattern for the circuit shown in Fig. 7.1, <|>=90o. The maximum error for <J)=0o cut is 3.6 dB, and it is at one of the radiation pattern nulls. However, we have to keep in mind that for certain application, such as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 unwanted radiation from PCB, it is only important how accurately the overall maxium radiation is estimated, not overall radiation pattern. Since the error in the direction o f the maximum radiation (broadside) is only 0.8 dB, and since the FCC [18] allows the maximum radiation to exceed up to 6 dB of the specified maximum radiated electric field, we conclude that the computed accuracy is acceptable for 0=0° plane cut. Likewise, the radiation pattern error in the direction o f maximum radiation for 0=90° plane cut is 0.6 dB, we see that computed radiation pattern accuracy is acceptable by the FCC standards for this plane cut as well. We now need to see how the accuracy holds up over the frequency range. The circuit shown in Figure 7.1 was used in simulations, with h = A/25 with the other parameters the same as in Section 7.2. The number of unknowns is 85. The results are shown in Figure 7.6. Radiation pattern difference 12 p h i= 0 10 - > - p h i= 9 0 e f us- w eM 6 4 2 0 0 .9 7 5 0 .9 8 4 0 .9 9 4 1 .0 0 4 1 .0 1 4 1 .0 2 5 OTo Figure 7.6: Radiation pattern accuracy over the frequency range Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The frequency sweep range was 5% of the central frequency f0. Taking into account that the maximum tolerable error is 6 dB, we see that we have one frequency point that was above the FCC error tolerance range. Given that the circuit size was over 3X, which is somewhat more than what we would consider an optimum size of the circuit for this type of approximations, we conclude that results are reasonably accurate. For better accuracy, circuits of the smaller size should be used. 7.4. Unwanted radiation from PCB A very simple model of a PCB is illustrated in Figure 7.7. It consists of a rectangular loop circuit with a resistance R, a voltage generator V, and a power cable attached to the loop. PCB JT L /m etallic rrarp Power cable a Figure 7.7: Simplified model of PCB with attached power cord A common approach to analyze the circuit is to disconnect the cable and determine the current on the rest of the circuit, and then calculate the radiation pattern from it. However, it has been reported in literature [11] that such an approach yields results ihat are off by an order of magnitude or more. It has been suggested in [ 12] that by removing the cable, we calculate only anti-symmetric currents in the loop. Anti-symmetric currents flow in the opposite direction in loop arms, and as such, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i 45 largely cancel out in the far field pattern if the loop is electrically small. However, the presence of the attached power cable induces symmetric currents; these currents flow in the same direction, i.e. are in phase (Figure 7.8), and as such, they add up in the far field radiation pattern. (a) (b) Figure 7.8: Symmetric (a) and anti-symmetric (b) current modes on PCB Although their amplitude is a few orders less than the amplitude o f the anti symmetric currents, the radiation pattern resulting from the symmetric currents is significant, even bigger than the pattern of anti-symmetric currents. Because of that, the overall radiation pattern from those two current components is much bigger than what the circuit analysis that takes into account only anti-symmetric currents predicts. This is unwanted radiation from PCB and needs to be suppressed. Filtering these currents from the power cable does not alleviate the problem, because even the small amount of symmetric current that passes the filter, such as higher order harmonics of the symmetric mode current, causes significant radiation. To minimize the symmetric currents, extensive simulations need to be performed to determine optimum cable attachment position. In the next paragraph, we outline a procedure that we had hoped would reduce the number of simulations needed by introducing an equivalent circuit that can be used instead of the original PCB. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It has been suggested in literature [13] that equivalence may be established between a printed circuit board and a short dipole antenna. To expand on that idea, we establish equivalence between a circuit on Figure 7.7 and a short dipole attached to a power cable, as illustrated in Figure 7.9. The length L^, and the tilt angle 5 of the dipole are determined from the condition that both circuits have approximately the same far field radiation pattern. The equivalent voltage source Veq that is driving the current in the Figure 7.9 is determined from the condition that the current amplitude at the cable junction is the same on both circuits. Figure 7.9: Short stub attached to power cord The motivation behind this equivalence is the hope that the parameters of the equivalent circuit in Figure 7.9 are not very sensitive to the position of the voltage source V and the length L and orientation of the cable of the circuit illustrated in Figure 7.7. If that is the case, we only need to know the amplitude V and the position of the cable attachment to analyze the circuit in Figure 7.7. We did simulations on a circuit illustrated in Figure 7.1 to see if we can establish a proof of concept. While the straight stub of the length L attached to the loop is not an exact model of a power cable of the length L attached to the PCB as shown in Figure 7.7, the presence of the straight stub introduces the symmetric ti Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 currents in the circuit. Therefore, we can assume that the circuit in Figure 7.1 is a first order approximation of the circuit in Figure 7.7. We experimented with various PCB loop arm length 1, and various stub lengths L. As an example, we list the results we got for the circuit with I=0.2A., £f=0.2A. and h=A/25. The length of the stub L varied, and the voltage position was the same as in Figure 7.1. The results are shown in Table 7.2. Table 7.2: Tilt angle 8 and length Leq as a function of stub length L L 8 (degrees) U, 3X/20 -24 0.18 A. 2A. -18 0.15 A. 3A. -16 0.13 A. 4X -14 0.11 A. We see that the tilt angle 8 and the length of the short stub Lcq depend on the length of the attachment L. When we moved the voltage source to a different position in the circuit, we got different values for 8 and L«, than the ones listed in Table 7.2. This implies that we need to know the exact position of the voltage source V and length L and orientation of the power cable to be able to characterize the equivalent circuit in Figure 7.9. These findings deny any practical use of the equivalence between the two circuits, because it does not make analysis of the original circuit any simpler or faster. Therefore, we conclude that we can not use the idea of establishing the equivalence between a PCB circuit and a short stub to simplify analysis of unwanted radiation from PCB. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER VHI CONCLUSION The thesis research has introduced a modification of the Moment Method based algorithm for the analysis of Printed Circuit Boards and Microwave Integrated Circuits. The primary goal is to develop a CAD tool that can efficiently analyze arbitrarily shaped PCBs and MICs that are no bigger than few wavelengths with only a modest loss in accuracy compared to full-wave MoM. The algorithm began by approximating the full-wave microstrip Green’s function with its static form. The approximation was then further simplified by using a single term instead o f infinite summation in the image part of the Green’s function. In full-wave analysis, double surface integrals that are used to determine impedance matrix are computed numerically. Here, we transform double surface integrals into double line integrals. The point of this transformation is to increase the accuracy of the solution when the circuit substrate is electrically thin. The transformation allows for analytic computation of constitute static elements Lnn' and Cnn*; however it should be pointed out that this transformation is possible for any Green’s function that is a simple function of p. Static elements L,^ and C„n'are then multiplied with phase shifts <tw and <J>aa' that are a linear function of wave number ko to yield quasistatic inductance and capacitance matrices [Leq] and [Ceq]. Modified quasistatic impedance matrix [Z] is assembled from jcofLcq] and l/jcDfC,*,]. Therefore, for each new frequency, we just need to calculate new phase shifts (jw and <|w to get [Leq] and [Ceq] in order to determine impedance matrix [Z] at that frequency, which is a very simple and CPU inexpensive task. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 Further, we use a less well known error bound to estimate current error based on the know led ge of the impedance matrix and its inverse. This bound gives more accurate current error than the one that is widely used in numerical electromagnetic literature. We support this claim by comparing the bounds that these two estimates give on three microstrip circuits of different shape. We continue with examples of possible implementation of our method. It was shown that the approximate method is clearly superior in terms of computation time over full-wave analysis when a range of frequencies is covered. Depending on the number of subdomain cells, the approximate method may be faster than the full-wave approach if number of frequency points is as low as 2. Since incremental run time per frequency point is much lower for the approximate method, clearly as the number o f frequency points increases, so does the approximate method efficiency compared to the full-wave approach. To summarize, we demonstrated: • Accurate computation of current for electrically thin MICs • Fast computation of current over frequency range. This method was shown to be up to 9 times faster than full-wave MoM over frequency range (50 frequency points) • Prediction of maximum radiated power within accuracy required by the Federal Communication Commission • Reasonably accurate prediction of radiation pattern of MICs We made an effort to use this approach to facilitate estimation of unwanted radiation from PCB with a power cord attached to it, but the results we got imply that the equivalence between radiation pattern coming from PCB and the one coming Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 from a short dipole antenna is not straightforward. A possible way forward for the work in this direction may be to use a piecewise approximation to a short dipole using diacoptic theory, as given in [16] and [17]. We need to bear in mind that this approach has its limitations; namely it should be used only for circuits that are at most a few wavelengths in size. The accuracy decreases rapidly if larger circuits are used. It also needs to be pointed out that applications that require high current estimate accuracy, such as S parameters, may need more accurate Green’s function approximation. A possible way to improve Green’s function approximation outlined in Chapter 4 is to use more image terms, as shown, for example, in [14]. It should be pointed out that, in all examples given in this thesis, only circuits with rectangular subdomain cells are used. In Appendix A, we derived the whole set of analytical expressions needed to implement double line integration over either rectangular or triangular cells. While the source code that calculates all integrals included in appendix A has been implemented in the software, additional code adjustment unrelated to integrals in the appendix needs to be done, before examples that include triangular subdomain cells could be presented. This task is left for future generations of Pmesh aficionados. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. BIBLIOGRAPHY [1] C. L. Holloway, E. F. Kuester, “Edge shape effects and quasi-closed form expression for the conductor loss of microstrip lines”, Radio Science, Volume 29, Number 3, pp 539-559, May-June 1994 [2] J. X. Zheng, D.C. Chang, “Numerical modeling of chamfered bends and other microstrip junctions of general shape for MIMICs”, 1990 EEEE MTT-S International Microwave Symposium Digest, Volume H, pp 709-712, May 1990 [3] R. Delyser, “Homogenization Analysis of Electromagnetic Strip Grating Antennas”, Ph.D. Thesis, University of Colorado at Boulder, December 1991 [4] J. C. Moore, “A Pertubational Solution Methodology for Solving Integral Equations of Microstrip Circuit Discontinuities”, Ph.D. Thesis, University of Colorado at Boulder, May 1992 [5] J. X. Zheng, “Electromagnetic Modeling of Microstrip Circuit Discontinuities and Antennas of Arbitrary Shape”, PhD. Thesis, University of Colorado at Boulder (December 1990) [6] J. M. Dunn, “A uniform Asymptotic Expansion for the Green’s Functions Used in Microstrip Calculation”, IEEE Transactions on Microwave Theory and Techniques, Vol. 39, No. 7, pp. 1223-1226, July 1991 [7] J. R. Mosig, “Closed Formula for Static Three-Dimensional Green Function in Microstrip Structures” [8] V. I. Sementsov, V. B. Golovchenko, “Calculation of the Partial Capacitances in Multilayer Thin-Film Printed Circuits”, Radio Eng. Electron. Phys. Vol. 17, pp. 103-19, 1972 [9] J. M. Ortega, “Numerical Analysis; a second course”, Academic Press, NY, 1972 [10] Brother K. E. Fitzgerald, “Error Estimates for the Solution of Linear Algebraic Systems”, Journal of Research of the National Bureau o f Standards - B. Mathematical Sciences, Vol. 748, No. 4, pp. 251-310, October-December 1970 [11] C. R. Paul, “A comparison of the Contributions of Common-Mode and Differential-Mode Currents in Radiated Emissions”, IF.EE Transactions on Electromagnetic Compatibility, Vol. 31, No. 2, pp. 189-193, May 1989 [12] K. B. Hardin, C. R. Paul, K. Naishadham, “Direct Prediction o f Common-mode currents”, 1991 International Symposium on Electromagnetic Compatibility, Cherry Hill NJ, pp 67-71, August 12-16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 [13] R. F. German, H. W. Ott, C. R. Paul, “Effect of an Image Plane on Printed Circuit Board Radiation”, 1990 International Symposium on Electromagnetic Compatibility, Washington DC, pp. 284-291, August 21-23 [14] M. I. Aksun, R. Mittra, “Derivation of Closed-Form Green’s Functions for a General Microstrip Geometry”, IEEE Transactions on Microwave Theory and Techniques, vol. 40, No.l 1, pp. 2055-2062, November 1992. [15] C. Klein, R. Mittra, “Stability of Matrix Equations Arising in Electromagnetics”, IEEE Transactions on Antennas and Propagation, vol. 21, pp. 902-905, November 1973. [16] F. Schwering, N. Puri, C. Butler, “Modified Diakoptic Theory o f Antenna”, IEEE Transactions on Antennas and Propagation, vol. 34, pp. 1273-1281, November 1986. [17] E. Niver, H. Smith, G. Whitman, “Frequency Chacterization o f a Thin Linear Antenna Using Diakoptic Antenna Theory”, IEEE Transactions on Antennas and Propagation, vol. 40, pp. 245-250, November 1992. [18] The Federal Communications Commission, Title 47 Code of Federal Regulation, Part 15.32, U.S. Government printing office, Washington D.C., 1997 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX A In this appendix, we do a detailed derivation of double line integrals that are used in Chapter 5 for quadruple surface integral transformation. A- 1 Snow White (Integral 1) /, = JJTv ---2- = ss'ijp' +h f tg W i asar •<"'= tf(Vp2+ h 2 - h l n ( h + ^ asar = JJ^/72 + h2dxdx'—Ajjln(---- ^2h l * h )<U .<//■' 2“ ))dxdx'+ JJ-^p2 + hrdydy A.. I .. /i + J fl" + h~ - hJJln( ------ ))4yd[y' 2h = / ,, (x , y , x \ / ) + / , , (x , y , x , y ) + / l3(x , y, x \ y ) + IU(x , y , x \ y ') 9 5" h + J p 2 +h2 -------- ) g(R) = j p 2 + h 2 - h ln( A. 2 p = J ( x - x ' ) 2 + ( y - y ’) 2 A. 3 /? = -yjp2 + h 2 A. 4 / , , (x ,y ,x ',y ') = JJ J p 2 + h 2dxdx’= ^ { 2 h 2 + ( y - y ')2 - (x - x')2)<Jp2 + h 2 + + y ( / i 2 + ( y - y ') 2) ( x 'ln ( x - x '+ >/ p 2 + A2 A. 5 + x ln (-(x - x') + i ] p 2 + h 2)) /1 2 (x, y, x’y ’) = -/iJJln ( - +- ~ ^ +— )dxdx' = — A 6 121 + A 22 + A 23 + A 24 + A 25 ) / 12I (x, y, x', y') = J (x - x v ( x - x') = ^-(x - x ')2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A. 7 54 7122(■*. y. ■*’. / ) = (y - y')j ta n “l 7 _ y) d (x - x ') A. 8 7x —xy)+-(y - y ’) 2 In p = (x - x’)(y - y’) tan"1 A23(X, y, X, y') = ( y - y ') J ta n ■*( ^ - J = = ) d (x - x') ( y - y h j p ’ +h = (x - x')[(y - y ) tan"1(------A(*- . - —) + (y-ytyp + h A. 9 „ . , « y - y ' ) 2+ h 2)(2h2 + p 2 +2hTlp2 + h 2 1, + - (y - y) in(- - - - - - - - - - - - - - - - - - - - - - - - - - - —)] —2 7124(x, y, X , y ) = - J ( x - x') ln(-------^ ------ )d (x - x') _ , U --Q 2 h n m j ^{y-yf A. 10 \n(h + J p 2 + h Z ) _ 2 - 2h - 7 ~ y . )2. in[(/t2 + (y - y') 2)(2A2 + p 2 + 2 h j p 2 + h 2)] 4 7125(■*. y,x ',y ' ) = h j In(-(x- x') + j p 2 + h 2 )d(x- x') A. 11 = h i jp 2 + h 2 + h ( x - x ')ln (-(x - x') + J p 2 + h 2) / l3(x, y, x , / ) = JJ *Jp2 + h 2dydy' =7,, (y, x, y , x ) A. 12 /,4(x, y , x , y ) = - h J J ln(/>+ A. 13 ~ —-)dydy = / 12(y, x, / , x') Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 A- 2 G rum py (Integral 2) I 2 = JJ i x -dSdS' = - § x g ( R ) d I . d? + ^ f l{R)(an. * a I )(an . a x)dldl' ss' -yP + h = 121 + 3s3s' asas- A . 14 ^22 ^ 9 - = g(R) A . 15 ox / , (R) = /i(x ,y ,x \y ') = X yjh2 + p 2 + h In(x- x + J h 2 + p 2 ) - h [ - ( x - x') - ( y - y' ) tan~l ( .X \ = ) A . 16 (y - y')-/h2 + p ‘ t” - / h +\ h 2 + p 2 , I 9 + (x - x ) ln ( ) + A ln(x-x + -)]h + p )] 2h I 2l( x , y , x \ y ,) = -jjx g (R)d xd x, - x j j g ( R ) d y d y ' = I 2n + / 212 + / 2I3 / 21, (x, y, x', y') = -J J x(i] p2 + h2 )dxdx = + / 2,,) A. 17 A. 18 / 21, (x, y, x \ y' ) = - J ((x - x') + x') - - ~ - ^ p 2 + h 2d (x - x ) = - ^ r ( h 2 + ( y - y ' ) 2) l n ( x - x ' + J p 2 + h 2 ) 16 ~ tV p 2 2 + /l2 A. 19 +K 7 U --C ' ) 2 + ^ x '( x - x ' ) 2 4 3 + i-(x - x')(/z2 + (y - y')2+Ix'(/z2 + (y - y')2)] o / 2I, (x, y, x', y') = 3 J ((X_ x ') + x') ln(-(x - x') + + -yjp2 + h 2) d (x - x') = = h ^ i y - y V ^ — 7 4 + ^ ^ r ^ T ( h 2 + ( y - y Y ) l n ( x - x ' + J p 2~+h2 ) + 1 (x - x ' ) 2 ln(-(x - x') + Vp 2 +A2)] I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A 20 56 / 2 1 2 (x ,y,x',y') = —h j j x I ( +H ~)dxdx 2h —1Ja -L. /* + \ C +1* Jr l t JL.lf 212 T 1 212 T 1 212 ^ I 2 l 2 ^ t 2 l 2 ^ t 212 n / 212(x. y*x'»/ ) = J (x - x * ) ( x - * ) = - a A. 21 (x -x Y A. 22 / 1212 (x, y. x', / ) = - h ( y - y ')f (x - x') tan (—— ^j) d (x - x') J x-x I A. 23 / = —h(y - y')[(x- x')(y - y ' ) + p 2 tan -1 ( y y, )] 2 x —x /m (x, y, x , y') = —h(y —/ ) J (x - x') tan -1 (------h^X j . f - ~ =)d (x - x') (y - y y p 2 + h 2 - A. 24 K y - y\p 'x zn -'(r hiX - X,) ' 2 ( y - y ' ) ^ P 2+ h2 - h ( y - y ' ) ln(x - x + <Jp2 + h i )] 12,2 (x, y.x', y') = Af ( x - x ' ) 2 ln(— V^ ^ — )d (x - x') J 2h = - j { - y ( x - x ' ) 3 + ( x - x ') ( y - y ' ) 2 + ^ h ( x - x ' ) i j p 2 + h2 A. 25 /. 3 rt h + J p 2 + h 2 + ( y ~ y ) [*n(-------2A h{x-x) . ) + tan ‘(------ v , / ) (y - y ) J p 2 +it2 - tan ( X -;)] - ^-h(h2 + 3(y - y ') 2) ln(x - x + J p 2 + h2)) y-y 2 1212 (x»y>x', v') = —h 2j ( x —x') ln(-(x - x') + ^ p 2 + h 2)d (x - x') = —- / i 2 [ ( x - x ' ) 2 ln(-(x- x') + - J p 2 + h 2) + —( x - x ') ijp 2 + h 2 A. 26 2 ~ ^ { h 2 + ( y - y ') 2 )ln ( x -x ' + 2 A/ / ? 2 + h 2)] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 / 12 1 2 (•*>y.*'i y') = -Ar* f L* - x '+ (y - y') tan -1 ( - - - ,) + J X —X , *. + (y -/> ta n h ( x —x') ^ h +J p 2+h2 (------) - ( x - x ' ) l n ( ---------) ( y - y ) ^ P +h2 2n A 27 + h In(—(jc - x') + -yjp2 + h 2)]d ( x —x') = = x'In (x,y,x',y') l2i3(x*y’x,’y') = - x j j 8 ( R)‘ty‘ty, =-x(.Ii3( x , y , x \ y ' ) + I u (x,y,x',y')) A. 28 In (x,y,x',y') = $ f x{R)(,an.»ax){dn • ar )dldl' = J J / , (R)dydy dSdS* A. 29 1 222 = ^221 1 223 ^ 224 ^ 225 / ^ ^221 C*. y .* y') = X- * JJV p 2 Jr h 2dydy' = / / I3 (x, y ,x \ y') A+ 0 2 + h 2 1222 ('r’ y» ■*'. y#) = -A JJ(x - x') ln(----- 3L_ -------)</y</y' = (x - x ')/,4(x, y, x', y') 2h 12 2 3 (x,y,x ', y ') = JJ A 2 ^ ln(x- x ' + p z +h 2 )dydy' A. 30 A. 31 A. 32 — ^2231 + 1 2232 7223i y. ■*'. y') = “ h 2JJ ln(* - ■*'+4 P 2 + h 2)dydy' A. 33 ~ 2 ^ ^2231 + ^2231 + ^2231 + ^2231 + ^2231 ) /^ ,i (x, y , x , y') = J (y - y V (y - y') = ^ (y - y')2 A. 34 / 4 ji (■*, y.-t'. / ) = -A J tan ' 1( y J )d(y- / ) A. 35 = - A ( y - / ) « a n - 'r i + i A = l n ( ^ 5 ^ - ) n 2 h Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 ,.r tan -i,(y * -* ')w ) d, ( y - y ^ ) (------y7 ')(* /miC x ,y,x',y') = «J 3 h-Jp +h s^ ( y - y ^ i an-| (^ y ] (x ~ x^ )+ lA l» i( ^ t .l £ Z 5 ) ') h j p 2+h2 2 /i2 + ( y - ) 0 2 A. 36 In(A2 + 2( x - x ' ) 2 + (y - y ')2 + l \ x - x ^ p z + h 2)} i^3i{x,y,x',y') = (x - x')J ln(-(y - y') + *Jp2+ h 2)d (.y - / ) ( x - x ') . , l m ( y , x , y ,x ) h A * 37 I$23i(x,y,x',y') = - f (y - y') ln(x - x '+ i ] p 2 + h 2)d ( y - y ' ) = -^ (y - / ) 2 l2 ~ < J p 2 + h 2 - ^ ( y - y ' ) 2l n ( x - x ' + y l p 2 + h 2) * A 38 + (* -* V 4 |x - x ] ) ( '* 2 + 2 (x -x ') 2 + ( y - y ') 2 - 2 |x - x y p 2 +A2)) + ( 1 - f ^ r ) In( ^ 2 + (y - y ')2)] |x - x ] | j j ( y - y ') 2 ln ( x - x '+ y jp 2 + h 2 )dydy A. 39 2 ^ 2 2 3 2 + ^2232 iam i i x , y , x , y ' ) =- 7 2232 {x,y,x,y) l zczsi(x>yix\y') g * )^ 2 2 3 2 + ^ 2232 + ^2232 + ^2232 + ^2232 ) - ( y- yV( y- y' ) = ~ ( y - y V J J A. 40 O = ^ J ( y - y ') 3^ ( y - y ,) = ^ - ( y - y /) 4 =J(y - y'Up1 + h 2d ( y - a .4 1 y) A. 42 - ( x - x ) ( p 2 + h 2) i j p 2 + h 2 3 t m i (x,y,x',y')- 5r r J tan "1 (— ~ )d(y - / ) = - - (x, y, x', y') Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A. 43 59 rm i { x , y , x , y ) = - 3 i $ m ( x , y , x \ y ' ) = —j J tan -1 ( (* j )(y ~ p )d(y - y ) = h i jp + h 3 J (y ~ y ' f (x, y, x7, y7) A. 44 In(x - x + <Jp2+ h 2)d (y - y ) = ^ h 2( y - y ' ) z + - ^ ( y - y V +^ ln^h ~ + ( ^ - / ) 2) - — ~- - - - - l n ( x - x + i J p 2 + h z ) 12 A. 45 ^ — ( x —x')(5h2 - ( y - y ' ) 2 + 2 ( x - x ' ) 2)ylp2 + h 2) 36 +T7 24 |x - x ] + ( x - x 7) 2 )(/i2 + ( y - y 7) 2 + 2 ( x - x 7) 2 + <- 2 | x - x y p 2 + / z 2"))+ (i ^ 2( x ,y , x f,y') —x + x |x - x | W 2+ tv - y')2)] [(x - x ' ) z +3h2]J In(-(y - y') + J p z + h 2) d ( y - / ) 6 A. 46 = —j Q h 2 + ( x - x ) 2)/^ ,, (x, y, x7, y7) 6 iru (x ’ y * x \ y r) : h{x - x7)J f dydy = h ( x - x') yy' A. 47 h{x - x7) I 725(x,y,x',y') : /»J J ( y - y 7)ta n " '(------n^ ~ X) )dydy' ( y - y ) ^ P +h a . 48 h U ^+ l^+ l^+ lis) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 I (x, y, x , / ) = ~ U y ~ y Y tan ' 1 (------(y - / ) 21 ( y - y ' ) J p 2+ h 2 = - | [ | / i ( x - x /)V/P2 +h2 + " X)6*+ k 3 In(ft2 - K x - x 7) 2) ft3 + h 2 + ( y - y Y + 2( x - x Y + 2{ x - x ) J p z + h 2 ft2 + ( y - 6 ( x - x 7)3 , + — <— 6 / ) 2 } A. 49 2hz + { y - y Y + ( x - x Y + 2 h j p 2 + h 2 In(-------------- ; ~ ( x - x r) r + ( j - y )^ ----------------- ) •tan ( h(X - x ' ) (y-y')Jp2+h 0 ] rb , / /x h e - i X x - x ) ( y - y ) , / ^ ( x . y . x . y ) = — I tan ( r )d(y-y) 2 h^jp +h A. 50 = ^ ICzui i x , y , x \ y ' ) Tc / / /x ( x - x 7) 2 r -i, h (y-y ') ... ,. Iz2s ( < x , y , x , y ) = J tan (------- - T = = = ) d ( y - y ) (x - x')-yjp2 + h 2 A. 51 x-x I $2 5 (x»y • / ) = ft(* - X7)J ln(-( y - .y7) + y j p 2 + h zd ( y - y ' ) = 25 = (x -x ')/| (3',J:,/,Jr/) j Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A. 52 61 A- 3 B ashful (Integral 3) = \ \ - i= = d S d S , = - ^ g { R ) x d i •<//'+ g / j C K K a . . . *az.)dldl' A. 53 ssryp +h asar asar 73 — 7 3l + 7 32 M .= * (* ) 9xr A. 54 /i^ + f v — / 2 (/?) = / 2 (x. y. x \ y') = +------ , - In(-(JC - x7) + h2 + p 2 ) A. 55 (y - y ' h l h * ’ + P 4 T" - / h +J h 2 + , I o - ( x - x ))In( In() + / i l n ( - ( x - x ) + y A 2 + / ? 2 )} 2n 73i =-JJx'^(/?)d[rd^, - x 'J J g ( /? ) J y f //= /3 1I + / 3I2 + / 3l3 A. 56 / 31, (x, y, x \ y ) = - JJ x ' J p 2 + h 2dxdx = / 2II (x7, y, x, y') A. 57 i (x, y , x \ y') = /iJ J x ln(h + ^ 73 2 2 A ^ —)dxdx' = I 2n(x7, y,x, y ') 7313(x. y,x', y') = - x 'J J (ylp2+ h 2 - hln(H+ ^ *— ))</y</y7 A. 58 A. 59 = - x ' ( / 13 (x, y, x7, y7) + / , 4 (x, y, x7, y7)) 732 = J ^ / 2 ( 7^)(^n*• «x. )(«„ • a x*)<//<// = / 32, + / 322 + / 323 + / 324 + / A. 60 asar 7 321 (x, y , x'y') = - ^ J J 4 p 2 + h 2dydy' = 7 322 (x, y. x \ y ’) = /tJJ(x - x’) In(-----3L_ 2h /l j/I ^ ^ ^ 2 (x, y,x7, y7) A. 61 )dydy'= (x - x ’) / l4 (x, y, x \ y ’) A. 62 7 ,3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I m (x,y,x',y') = JJ k + ( *2 y ) ln(~(x ~ x ' ) + t ] p 2 + h 2 )dydy' A ^ = I 2a{ x \ y , x , y ) I 324(x,y,x',y') = - h ( x - x ' ) j j dydy = I m ( x \ y, x, y') A. 64 I 37S( x ,y , x \ y ') = ~ h \ \ (y - y') tan (----- ^ - = = = = ) d y d y ' = l m ( x \ y \ x , y ) (y-y'> Jp A .65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 A- 4 /4= Evil Stepm other (Integral 4) JJ -HSdS' = asar ■Tjp2 +fi2 ss~ .J? JJx'/i (/?)(«„• • az )(d„ • a x)<//<//' isa r • a^JCa. • a I.)dldl' A. 66 \S3S- $k(R)dI •dl' asas- = / 4l + / 42 + / 43 + /+; 7 2k ( R ) = g ( R ) A. 67 k{ R ) A. 68 / 4 l ( x , y , / ) = - J f g ( R ) x x ' d ! • d l ' = - j j x x ' y j p 2 + h 2d x d x ' asas- ^ Jdto/r'- x r'J JJ p 2 + h 2dydy + fax 'JJ ln(~ ) ^ y ' = / 4| I + h\2 + ^413 + ^414 - + /i JJx c'ln (h + A. 69 / 4,, (x, y, x , / ) = - J J x x ' J p 2 + h 2 dxdx' = - J ( ( x - x) + x')2[^Y~^p2 +h2 _ h 2 + ( y ^ j )_ ln(_(jc _ xf) + ^ 2 +A2)Jrf(jt _ ^ A. 70 r '2 — T“ _ r 7 *411——~ I 411 c + I 411 J + / 411 f + / 411 r — 7 411 - i J(x - x ' ) 3^ p 2 + h 2d ( x - x ) 2 r( x - x ')4 + ( x - x V ! ? + ( y - y V ( 30 10 (h2+ ( y - / ) 2)2 ) j p 2+ h7 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A. 71 / 4II( x ,y ,x ',/ ) = j ( x - x ' ) 2^Jp2+ h zd ( x - x ' ) ( x - x ')3 4 .2 ^ j ^ U 8 ,.h 2+ ( y - y ') 2 A. 72 8 '\ 2 x2 - u , (x - y + V ^ F ) / 4„ (x, y , x \ y') = J (x - x')tJpz + h 2d ( x - x') = j ( p 2 + h 2)<Jp2 + h2 A. 73 Ifu (x, y, x , / ) = ^ L ± £ L _ Z 2 _ J (x _ x') 2 in(-(x - x ') + ^ p 2 + h 2)d ( x - x ') 2 _ h_+(y y) ^ _xy in(_ (x_ x') + ^/p 2 + /j2) o ( ( x - x ') 2 - 2 (ftz + ( y - / ) 2) A. 74 Vp 2 + /»2 i / 4M(x,;y,x',;y') = (/i 2 + ( y - y') 2)x'J ( x - x ' ) ln (-(x - x ' ) + y jp 2 + h2) d { x - x ) A. 75 - - ^ +(~f2 J/) x 7 2,2 (x, y,x', y') •2 1 ” '' 2 / iiI = h 'r ' 2 J ln (-(x - X ) + -yjp2 + h 2) d ( x - x') 2 h 2 + ( y - y ' ) 2 ,2l -x'2/ 125(x ,y ,x ',y ') 2h ^ IM2= h j j x x ' l n ( H+ i f/ / ^121 / 2 'w r = J(X + ( x - x ) ) [ h +h )dxdx' = I M2l + (x -x ')2 h A. 77 / 4122 r ~ i — TT + —y p A. 76 C * - * 7) 2 , +h -•------ h +J p 2+ h2 ln(-------— )+ - y- y -- ln((/i2 + ( y - y')2)(2h2 + p 2 + 2 h j p 2 + h 2 ))]d ( x - x ) A. 78 = / u + / * + i c + rd 14121 ^ 1412! T 1412! T 14121 7 4I2I = Y j (C*- x ) + x ' ) ( h j p 2+ h 2 - i ( x - x ' ) 2)d ( x - x') 1, /\4 . 1 /, M . I rr^ /i 2 . -2\ 1 •/ = - — ( x - x ')4 + — x '( x - x ')2 + - M - ( / i 2 + P 2) ---- x \ x - x ' ) ] J p 2 + h : 16 12 2 3 2 ^ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A. 79 65 14.21 = \ \ ( x - x ) 3 ]n(h + J ? - — ) d (x - x ) 2J 2h = L (x- xy ( y - y y - ± ix - xy + L x - xy ^ * ± V g l ± £ ) 8 32 8 2A + ^ - M U - .r ' ) 2 24 16 f 4.21 (*. >~ / ) 4 /) =^ 2 A2 - 5 a . 80 (y-y')2hlP2+/»2 ln[(A2 + (.y - / ) 2 )( 2 A2 + p 2 + 2 hj p 2 + h 2)] x ' j ( x - x ' ) 2in(—- +— — ) d u - x ') A. 81 = t t x'/ 2 .2 (x , y , x ',y') 2h l M2\ =^-(3,- / ) 2J((x - x,) + x')ln[(/i2+ ( y - / ) 2) (2 h 2 + p 2 + 2 h j p 2 + h 1)\d(X - X') = 7 ( y ~ y ' ) { h j p 2 + h 2 +2 x \ y - /)[tan"' ( 4 h--X -.-XJ ) ( y -y ') < J p + h 2 A. 82 . tan- \ I Z ± . )] + ii p 2 - ( x - x ')x ')in[(h2 + ( y - y ' ) 2) y -y 2 (2h2 + p 2 + 2h^[p2 + h 2) ] - ^ x - x ' ) 2 + 2 x '(x - x ' - ( y - y ' ) ) - 2hx'ln(x - x ' + y j p 2 + h 2) } h m = f((x - x ) + -0 J 2 [ U - x ') + ( y - / ) t a n - 1( - ^ 7) x —x + ( y - / ) t a n _,(------, x ) ) - ( x - x ') l n ( ^ +" y , + **- ) ( y - y ) i j p +h 2h A 33 A’83 + h ln (-(x - x') + -Jp2 + h 2 )]d(x —x') = ^4122 + ^4122 + ^4122 + ^4122 + ^4122 + ^4122 + ^ 4 .2 2 K m = \ ( x - x ' ) { { x - x ' ) + x')2d ( x - x ) (j* X*) ) = t( x - x *y\ 2(s -^ x ' 2 +, 2- x /(, x - x /), + ----2 3 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A. 84 66 K m = ( y - / ) f ((x - x ) + x ')2 tan ' 1(~ — ^ r ) d ( x - x ' ) J x —x = ( y ~ y ' ) ^ & x ' p 2 + (x - x')((x- x ' ) 2 + 3x'z)] tan ' 1(^— ^ ) 6 x —x + ( y - y'X(x - x 'X x +5 x') - ((y - / ) 2 - 3x'2) In p 2 )] ^4122 =(y- / ) f ((* - X)+x ') 2 tan ( A. 85 hSx-*i- -)d(x- x) (y-yWP +h = ~ ( y ~ y '){ -^ (y ~ /W P 2 + h 2 - h ( y - y')x’ln(x- x ' + J p 2 + h 2) A. 8 6 - i k y - y Y - x p 2+ ( x - x V 2] tan'* (----- ^ x -r * ] 3 ( y-y'wp +h 0 - r'x* '2 2 (y-yym ^ 6 ) + < - y - y ^ 2h2+p 2+2h^ p 2+l' \ p K m = ^ J ((x ~ x') + x ')2 In(-(x - x') + ^ p 2 + h 2 )d (x - x') A. 87 = — ~T--- m K\ I ix, y, x , y') + 2 x 7 2I2 (x, y, x \ y') + I l25(x, y, x', / ) ) h + (y -y ) K m = - \ ( x - x ') 3 In( - + ^ 2 h ^4122 = ~2x' f (X - x ')2 ln( hJ K m = ~x '2 J x )\n ( fl + ^ ~ 2h 2h b = ~21 4121 —— ) d (x - x') = “ h A. x 7 212(x, y, x \ y') +h )d ( x - x') = x ' 2/ l24(x, y , x , y ' ) 14i3 = - x x ' j j yjp 2 + h 2dydy' = -xx'1 13(x, y, x , y') / 4I4 = hxx'jj ln(— +— )dydy' = - x x ' I u (x, y, x', y ) 88 A. 89 A. 90 A. 91 A. 92 Ki = $ x'fx(RXd„- • axXan • ax )dldl' = x 'J J / , (R)dydy'= x l n (x, y, x , y') asas- A. 93 143 = $ x f 2(R Xan- • ax.)(an • ax.)dldl' = x J J / 2(R)dydy‘ = xln (x, y,x', y') asas- A. 94 I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 I M = § k { R ) d l • d l ' = §k{R)dxdx’+ ^ k ( K ) d y d y ' = I w + l w asas- A .95 ^441 —^k(R )dxdx —^4411 + ^4412 A . 96 ^4413 ^4414 ^4415 I Mn( x , y , x ' , y ' ) = ^ h j f p 2dxdx' = ^ h x x \ { y - y ' ) 2 + ^ x 2 —^xr'+^x'2) A . 97 ^44 n ( ^ y . * » / ) = ^ f f p 2J p 2 + h 2)dxdx' = .(- ~ -xi ( 3 ( y - / ) 4 + 2 h 2{ y - y ' ) 2 - h * ) l n ( - x + x ' + ^ p 2 + h 2 ) 7 2 ______________________________ - ~ ^ J p 2 + / , 2 ((-c - x ')4 + ^ ( x - x ' ) 2C7hz + 2 7 ( y - y ' ) 2) + U h 4 - 2 2 h 2( y - y ' ) 2 - 2 K y - y Y ) 6 l m i { x , y , x ' , y ' ) = - ^ h 2j j J p 2 + h 2dxdx' = - ^ h 2I lt( x yy , x ' , y ' ) U. ».-t'. / ) = / . ( i / i 2 - 1 -(y - / ) 2 )j} ln ('l + ^ 6 4 JJ A . 99 + — )A A ' 2A a . 100 = -(-^/r " ( y - / ) 2)/,2(x ,y ,* ',/) 6 fwisC*.? ’*'’ / ) 4 JJ 4 ln(“ -----TZ------- )rfwfc' 2h A. io i “ ~ ~ ^ ( ^ 4 4 1 5 + ^4415 + ^441S + ^4415 + ^4415 + ^4415 ) 4 C 15 (x, y,x', / ) = \ J(x - x ) [ U x - x ' ) 2 - { y - y ') 2]d(x - x') A. 102 = ^ [ - ! r ( x - x ) 4 - i ( x - x ,) 2 ( y - / )2] 3 12 2 /«i sU .y.Jf'.y') = —7 J (* “ X'W P 2 + h 2d ( x - x ' ) =- j l c4u(x,y, x', / ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A. 103 68 /« ,5 {x, y, x , / ) = (y ■ - f tan _l ( x X,)d(x - x') = J y -y 3 = .^ yjLJ 0 ~ —tan -1 (——^j))d(x —x') J 2 x -x 3 A. 104 A2 = O * ~ ” 3 / l22 (X’ • / ) = ~ (y / )- J lan' (— ^ 3 =^ J X’y')] j d( x- x' ) ( y - y )Vp +/l A. 105 /|23 (■*. y*■*'. y') 1+415(•*. y. X, y') = - f (x - x ' ) 3 ln( A+ ^ +h )d(x - x') = - 21 *,, (x, y, x', y') A. 106 J 2h l{4l5( x , y , x \ y ' ) = - ^ ( h z + 3(y - y ')2) f ln(-(x- x') + J p 2 + h 2 )d{ x - x’) 6 A. 107 = ^ ( h 2 + 3 ( y - y ' ) 2)Il25( x , y , x ,y ') 6 7442 = JJ k(R)dydy = 7^, (y,x, y',x') i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A. 108 69 A* 5 /5 = H appy (Integral 5) JJ i * - - 2 JSdS' = - f j y g ( R ) d T •<£'+ § M R ) ( a n. . a r )(an *dy)dldV ss-'Jp + h asasasasA. 109 — ^51 + ^52 ^4p-= s(R ) dy A. 110 M R ) = f 3(x ,y ,x , y ) = ? + h2 + ( x - x ') 2 2 ^h2 +p 2 , [~2 2 \ ln ( y - y + ^ h + p ) ^ / 2 2" - h[-{y - y ' ) + h ln(y - y '+yjh2 + p 2 ) + (y - /)ln (-^ + *** + P ) A. 111 2h - ( x - x ' ) t a n ( x - x ') - J h 2 + )} / 5I (x, y, j :', y' ) = - j j yg(R)dl • d/' = - J J yg(R)dydy' - yjjg(R)dxdx' asas= J5i1 (*. y> ls\i (Jf. y . /) / ) + 1512 (*. y. /) = -J J y y j p z + h 2dydy' + /iJJ y ln ( ^ ^ - ^ - ^ - ) < /y < /y ' = ( > • / • -r ') (>. y'. *') ^211 A. 112 A. 11 3 + 1 212 / 5 l2 (x,y,x,t y#) = -yJJ^(/?)d! 2aZr, = / 2 l3 (y ,x ,y ',x /) / S2(x, y, x \ y') = J f / 3 (^)(a„- • dy ){dn • 5 y)dldV = J J / 3 (R)dydy = I 12(y,x,y',x') Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A. 114 A. 115 70 A- 6 Dopey (Integral 6) / h = JJ j y 2 dSdS' = - $ y ' g ( R ) d T . d ? + § U R X a n. . a y.)(an *ay.)dldl' ss- J p + h 2 asasasasA . 116 = ^61 + ^62 A. 117 —r— = g(R) ov’ / , ( ^ ) = / 4 U, / ) = - y 2 ' " V^ +h 2 +p2 **-■■ ln (-(y - y') + ^ h 2 + p 2 ) - M y - / + ( * - * ' ) tan "'( , —- — ) (x-x ')y h 2 +p 2 h +■Jh2 -\lh + + p* p2 , 118 f~n —------- ( y —y )in(— —■*—2h H ) + /jln(—(y —y ) + y h +p 5 ” )} =f i { y , x , y \ x ) 161 (*. y. X, y') = - JJ y'g(R)dI • d? = - J J y'g{R)dydy - y 'J J g(R)dxdx' asas- A. 119 = / 6u (jt. y. jr. y') + 1612(jt. y. jr/. y) 1 6 ii (Jr. y. Jr'. y') = / 6m (jr. y. y’)+hin(jr. y. Jr'. y') ^ iii (Jr. y , x , y') = - J J y ' J p 2 + h 2dydy' =/3ll (y,x, y ',x ) ^6112{x , y , x \y ') = A. 120 A. 121 122 ^^(jr.y.jr'.y') = - y 'J J s ( ^ ) ^ r ^ '= ^3i3(y.jr.y,.jr') A. 123 /62 (jr. y. jr'. y') = JJ /» (R)(an-• a.v-)(«„ • a y.)dldl' = J J /4 (R)dydy' =i&(y. jr. y#. jr#) A. 124 I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A- 7 Doc (In teg ral 7) = J J - r ^ r f dM $ ' = - § V g (W Z ss' tjP +n asas- + §y'MR)&n- * 5 x)(5» • a z)dldl' A. 125 asas- + jjx f 4 (fl)(a„' • «,.)(«. • a y.)dldl'+ fyk(R)dl *di' asasasas= / 7I + / t2 + / t3 + 7?4 / 7I (x, y, x', y') = - j j g(R)xy'dl • dl' = - y 'J J xg(R)dxdx' - x JJy'g(R)dydy' asas= ^711 + ^7I2 A. 126 77I, (x, y,x', y') = - y 'J J xg{R)dxdx = y'(72,, (x, y,x', y') + / A. 127 2 I2 (x, y, x', y')) / 7 I 2 (x, y, x', y') = xJJ y'g(R)dydy' = x l 6l, (x, y, x', y') 172 (x, y, x', y ') = A. 128 J f yft (/?)(«„• • ax ){dn • a x)dldl' = JJ y f x(R)dydy ’ A. 129 = I ni + lj22 + fj23 "*■^724 + ^725 9 7721(■*. y . y ') = — JJ y ' J p 2 +h 2dydy' = (x ,y ,x ',y /) = -/> (x -x ')JJy 'ln ^ + ^ / 16 i11 (■*. y . x* y ') + ^ My^y’ A - 130 A 1 3 1 = ( x - x ') / 6 1 l2 (x,y,x',y') 77 2 3 (x, y, x', y ') = h ( x - x') JJ ydtydy = ^- /i(x - x') yy ' 2 77 2 4 (*• y > y ' ) = J 77241C*.y. J ^ A. 132 y 'ln (x - x '+ J p 2 + h 2)dydy' = 1 ^ + 1 n42A. 133 y') = - ~ r J J y ln(* - x ' + 4 p 2 + h 2)dydy A. 134 _ _h*_n — 2 ^ T24** + 72412) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 Jtmu = j f y \ n ( x - x ' + y l p 2 + h 2) d y d ( y - y ' ) A. 135 = 7 7 2 4 ll + 7 7241l + 7 7 2 4 I1 + 7 724l1 ■*" 7 7 2 4 l1 + ^7 2 4 1 1 + 7 724t I + 7 724l I + 7 7241l ^ 4 1 1 (x, y.x', y') = /(O ' - / ) + y'Xy - y ) d { y - y ) A. 136 = ~ ( y ~ y )3 + 4 y'(y - y V 7£ui I (Jr. y. x', y') = -h y 'j tan _l (^ - ~ ) d ( y - y') = y ' l (x, y, x', y') ^ 4 1 1 (x, y,x\ y') = ~hj (y - y') tan - 1 ( l Z 2 _)</(y - y') J n A. 137 A. 138 = —^-[(A2 + (y - y V ) tan - 1 ( ^ ^ - ) - My - / ) ] 2 A 7™.i (x,y .x ',y ') = hy'j tan -1((X 'p ^ y 7 . V -)d(y- y') = y 7 ^ ,( x , y,x,y') J h-\jp +h 7 A. 139 w n (x , y,x ,y') = h j ( y - y') tan - 1 ( (x ~ x ) ( y - y ) ) J ( y _ ^ J hyjp2 + h 2 =V ta n A. 140 2 f ( y - y V t a n - '^ T - ^ - J L l ) h ^ p 2 +h2 - A(x - x') In(y - y ' + J p 2 + h2)] 7 m \ i(x .y .x \ y ' ) = * '(x - x ' ) j ln (-(y - y') + <Jp2 + h 2)d(y - y') A. 141 = ~ x'(x - x')1 125 (y, X, y \ x ' ) 7724ii (x, y, x , y') = ( x - x ' ) j (y - y') ln(-(y - y') + J p 2 + h 2)d(y - y') A. 142 = “ TT (x ■ ~ x') I e2ll (y, x, y x ') h 7724i 1 (x, y, y') = - y ' j (y - y') ln(x - x + y lp 2 + h 2)d(y - y') = - y /7223i(x.y,x',y') i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A. 143 73 1 724ii (*. y**'* y) = - J ( y ~ y )2 ln(* -*'+Jp2 +hz)d(y - y7) = ^ [ h z (y - y7) - y (y - y ' ) 3 + (y - y 7) 3 ln(x - x + *Jpz + h z ) + \ ( x - x ' ) ( y - y ' ) i j p 2 + h2 + h3 tan~‘( ('r .x ^ y - ^ ) 3 h^Jp2 + h z yi. 144 .^ ta n - ^ Z lJ L ) /i " ( x - x 'X C x - x 7) 2 + S‘ hz)\n(y —y ' + -Jp z + h 2)] 172412(-c*y *•*'. / ) = JJ(y- y')ln(* - x ' + i ] p z + h z )dyd(y - y) i. 145 = — ( / “ 4- I b J- I c A- l d 'I 72412 72412 72412 ^ 1 72412' 1 £412 (*. y**• y') = - J ( y - y ') 2< /(y-y7) = - y ( y - y Y f T2412(■*. y* y') = 2Cjt - x')j ylp2 +hzd(y- / ) = (x - x 7X(y - y')Jp2 +h +(hz + ( x - x 7) 2) In(y-y'+Jp2 +h 2)) *£»«(■*. y.Jf'.y') = 2 f ( y - y Y ln ( x - x ' + ^ p 2+hz ) d ( y - y ') =- 2 I^u(.x,y,x',y') l. 146 l. 147 / u 148 ^7 2 * 1 2 (Jf’y -r,»y#) = A2Jln[(/i2 + ( x - x 7)2)(/i2 + 2 (x —x7)2 + ( y - y 7)2 + + 2(x - x)ylpz +hz )]d(y - y7) / = /i2[-2(y - / ) + 2A tan '1( ^ L ) h + 2 ( x - x 7) I n ( y - y 7+ -^ p 2 + /i2) hjp2 ^ / .. 149 2 + (y - y7) ln[(/i2 + (x - x7) 2)(/i2 + (y - y7)2 + 2(x- x7)2 + 2(x- x’) 4 p z +h z )] ^7242^* y»x'» y') = T J/Cy ' y ') 2y'Wx-x' +Jp2 +hZ)dydy = / 72421+ /t 2422 ^ . 150 i i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 ^72421 (xyy , x \ / ) = | j j ( y - y ') 2y l n ( x - x ' + J p 2 + h 2)dyd(y - y ) —j a + rb + fc 4. id +. j c _i_ i f A. 151 4. / * 1 72421 T 1 72421 T 1 72421 T 1 72421 T * 72421 ^ 1 72421 T 1 72421 ^421 (Jf» y.*'. / ) = j J ( / + (y - y'))(^(y - / ) 3— h2(y—y*y)d(y- = T3 [ /( “4 (y - y ')4 ——2 (y ~y')2) /) = A. 152 + ^ ( y - / ) 5- - y 0 ' - / ) !] ^72421 (Jf. y.*'. y') = —“— / J (y - / W p 2 +h2d{y- y) X X lc-nxi\ (*• y. x \ y ) = ,rc A. 153 t , J (y - y')2-\jp2 + h zd (y - y ) A. 154 X X f9 4bll(y,x,y / /,x) /\ f 72421(*. y. Jf'. y^ = -Y /( y '+ ( y - y')) tan ( l - J L ) d { y - y') A. 155 = -■“t 3 I (*. y. / / 72421(■*. y. "1 y') + / ^ 41 , (j:, y, x \ y')) ^ y ') = ~ J (y - y f in(* - ■*'+V p 2 +h2)d(y - y') Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A. 156 75 1 4*21 (* .y » / ) = - j J ( y - y =“ [- ^ ') 4 (5 ^ 4 In(.r—x + i j p 2 + h 2 )d( y - y') + D (y - / ) 5 + - ^ h 2( y - y ' ) 3 - ( y ~ y '*X~ X ) 0 h 2 - 2 ( y - yO 2 + 3 { x - x ) 2) J f 7 t f + ] : ( y - y ' ) l n ( x - x ' + y j p 2 + h 2) +1 ^ 5 A. 157 tan- ( f - ^ ) ( y - / ) h 5 h ffT i? + - ( x - x ' ) ( l 5 h 4 + l0h2( x - x ' ) 2 40 + 3(.r - x Y ) ln(y - y '+ J p 2 + h 2)] l nn\ix,y,x\y') = ~ { x ~ x')((x - x ) 2 + 3h2)J ((y - y') + y') ln(-( y - y ^ + 6 + j p 2 + h 2) d ( y - y ' ) A. 158 ( x - x ' ) ( ( x - x ' ) 2 +3h2) 6h V&»i i (■*. y. y ')+ y ^125 (y» ■*. y'. *')) J72422(■*»y.^ y') = T JJ(y ” y")3,n(*~x'+4 p 2 + h 2)dyd(y - y') — 4. /* J- Ic J. A. 159 Jd — 1 12422 T 1 72422 T i 72422 T 1 72422 ^422 y.Jf'. y7) = j J(y ~y 7) 2^ 2 —- ( y - y ') 2W(y - y') A. 160 = 8- ( -3/ i 2( y - y 7) 3 - —10 ( v - y ') 5) / M ^ y ^ ' - y 7) = ~ 5/* X) J V p 2 + ^ 2 ^ ( y - y 7) /v 2 5/T + ( x - x ') 12 «y-y')Jp2 +h2 + + (/t2 + ( x - x ') 2)In (y -y '+ < s/ p 2 + h 2)) j Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a. 161 76 I ^ a 2 ( x , y , x ' , y ) = - ^ j ( y - y ' ) 2J p 2 + h 2d ( y - y ' ) A. 162 Wft ^ = TI f f u /\ ’y ’x t*. y»*'. / ) =jf(y~ yY^P2 +hZd(y- y')=j ^ n (■*. y^'*/ ) A. 163 I h m u . v .j : ',/ ) = - - ^ -Jln [(A 2 + ( x - j t ') 2 )(/i2 + 2( x - x ')2 + ( y - y ' ) 2 + + 2(x - x ' ) i j p 2 + h 2 )]d{y - y') A. 164 1 = —J’/l 2 / n4i2 U y --1'- / ) 8 ^ = /iff/(y -/)ta n J h (x-x') ’(--- * , * )dxdx ( y - y y j p +h2 A. 165 = ^*(^7251 + ^7252 ) 1 72si = JJy (y ~ / ) tan (------ - (X~ X) ( y - y ') V P + * )</y</(y - y') A. 166 — ^7251 + ^7251 + ' ^7251 + ^7251 + ^7251 + ^T251 + ^ T U l + ^7251 I™1(*.y . /) = -T f(y- y'Ytan’ 1(-----—c~ x ) )</(y- y') (.y - y ' ) y l p 2 + h 2 2 = j ( x - Jt')[(y- y')^jp2 + h 2 - 2(h2 o 7 + (x -x ')2)In (y -y '+ p 2+/t2)] . i r/ /> +-l(y-y) 4 8 . -i, h(x-x') tan l(------ — p - - _ . ) ( y - A /P ^ + (jr-a :') 4 tan-l( jj ( x - x ' ) t J p 2+ h 2 h* - i f {x —x')(y —y') + — tan l (---■ 8 h^jp2 + h 2 =~V J ') Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A. 167 77 l^,(x,y.x',y') = “ / J ( y - / ) ! tan ' 1( 2 x1 )dyd(y - / ) O -y W ^ T * A . 168 =^ r l ^ x , y , x \ y ) h I$2 5 i ( x , y , x \ y ' ) = ^ y ' j x a n ' \ ~ - J ^ y J ^ - ) d ( y - y ' ) = 2hI^4U( x , y , x \ y ' ) A. 169 ^72sit*.y*x* y) = 2 J ( y - / ) tan 1 ( (x ? ^ y-- ^ ) d ( y - / ) h^Jp Jch A. 170 = 2hlen4U( x , y , x , y ') , / A. Insi(.x ’y>x ’y ) = ( jt - x ' ) ‘ 1 2 ,r - i , h (y-y ) ... , )d(y-y ) y J tan (--------- , J ( x - x y p 2+ h 2 x') , —( x --— y/,»(y.jr.y ,, 4,(*,y.t , / ) 1 = 2 A. 171 , jt) J(y - y')tan-1(— ~\ y r y- - - ) d ( y - y ) ( x - x y p 2+h2 *— J A. 172 x - x rc , - r 7 - ^ i2 ( y * jf.y •■*) 2h I rai U. y. Jf'. y') = - *V j ln(-(y - y') + J p 2 + h 2)d(y - y ) A. 173 = y'( x - x')I l25(y, x, y x ) Irai U. y. y') = - *') J (y - yO ln(-(y - y') + 4 p2 +h2 )d(y- y') A. 174 = -/i(x - x ')/ 2,2 (•*»y>Jr', y') / 7252 (Jr.y.Jr'.y') = JT C y -y V tan -1 (----- h(<X * 1 _ . A d y d ( y - y ') (y-y Up +h A. 175 = ^7252 + ^7252 + ^7252 + ^7252 + ^7252 + ^7252 / 7252(-r.y.Jr’.y ’)= ^ -A (jr-x ’) |V ^ r +Ar < /(y -y ’) = ^ A /^ 4l2(x ,y ,x >,y ’) I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A. 176 78 lT25i(•*»y*x'*/ ) = J lnK^ 2 + (* - x/)2)(^2 + (y - / ) 2 + 2<x ~ xV + + 2 ( x - x ' ) ^ J p 2 + h 2) ] d ( y - y ' ) A. i l l = 7^412 ( x , y , x ' , y ) o I ^ 2( x , y , x , y ' ) = ^ - p ^ - j l n [ ( h 2 + ( x - x Y ) ( 2h 2 + ( y - y ')2 + ( x - x )7 + 2h*Jp2 + h 2)]d (y - y') = iX - - -3 [-2(y - y') + 2/j ln(y - y ' + J p 2 + h 2) A . 178 - 2 ( x - x ') t a n - ( - *( y ~ A . - ) (x - x ')tJ p +h + 2 (x - x')tan_l( y- y, ) - 2(x - x')tan_1(----- ^ —=====) x-x ( y - y ' ) ^ p +h + ( y -y ') ln (( /i2 + ( x - x ') 2)(p 2 +2/I2 + 2 h^Jp2 + h 2))] / T2S2(x x',y - y'./i) = --— Jln( 6 2 + (y - y V ) d ( y - y) f »3 = - — [2/itan_,( ^ =^ - ) + 6 h + (y - y') ln (/r + ( y - y ) 2) - 2(y - y')] ( x - x ’Y 77 2 5 2 (X - x ' , y - y \ h ) = - J ln((y - y')2 + ( x - x ')2) d ( y - y ) 6 J A . 179 A 180 = ^ 2i h , y - y ' , x - x ) 7ra i(x, y , x \ y ) = ^ J (y - y ')3 tan 1(----- h(x 3 x ) )d(y - y') (y - y U p 2 + h 2 A . 181 = ~ I$a i(x,y,x',y') I n (x,y,x',y') = ^ x f 4 (#)(£„• • d y )(an • ay.)dldl' = JJxf 4 (R)dxdx' asas— 7731 + 7732 + 7^3 + A . 182 + ^735 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 / 7 3 I(x ,y ,x ',y ') = ~~~2 J /W ^ 2 +h2dxdx' = Im ( / , i , y.x) = A(y - y')JJxln(-^— ^ j - ^ —)dxdx' = / ra (y ',x ', y.x) / 7 3 3 (.r, y .x '.y ') = -A (y - y')JJ**fo&' = Im {y',x, y.x) ^ 734 (*’ y*x\y*) _ JJ h +^*- xln(—( y - y') + <Jp2 + h 2)dxdx' A. 183 A. 184 A. 185 A. 186 = i t » { y W ,y ,x ) / 7 M(x ,y ,x ',y /) = -h JJ x (x - x ')tan - 1 (----- — y ■ )dydy' JJ ( x - x ) J p 2+h2 A. 187 = / 7 2 5 (y,,x ,,y ,x ) l u ( x , y , x , y ' ) ■= • <//' = JJ*(/?)dtax'+JJ*(fl)</ydy' asas- /^ (x , y,x',y') Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A. 188 80 A- 8 Sneezy (Integral 8) = L = d S d S ’= - § x y g ( R ) d l • dl' ss-Jp'+ h asas+ j jx '/ 3 (fl)(a„. • a y )(a„ • d j d l d l ' asas- A. 189 + $ y f2 ( R ) ( “n •5 ,0 (5 , . a x.)</W/'+ $ k ( R ) d I • dl' asasasas= ^81 + ^82 + ^83 + ^84 / 81(x, y, x', y ') = - £ j y x ' g { R ) d l • d f = - y J J x'g(R)dxdx'~ x 'J J yg(R)dydy’ asas= ^811 + ^812 A. 190 1811 t*. y. A. 191 ^12 1 82 y') = - y JJ x'g(R)dxdx' = y (/ 3I, (x, y, x \ y ) + / 312(x, y, x \ y')) (■*. y»■*'. y') = ~ x ' j j yg(R)dydy' = x751, (x, y, x', y') (*. y. A. 192 y') = <£f^ 3 (R)(an- • « v)(5« • 5 »)^W// = fJ*’/» (R)dxdx' a&- A. 193 = ^821 + ^822 + ^823 + ^824 + ^825 h n (■*.y.Jf',y') = y y JJx'V p 2 + h zdxdx' = / 731(x',y',x,y) h + -Jo2 + h 2 )dxdx' = / l « 2 2 (*• y. Jf#. y') = -A(y - y') JJ * ln(------- — ■/t- + J ) = ^734(Jf'.y'.x, y) J8 1 2 5 (x', y', X , y) A. 195 y) A. 196 x ' \ n ( - ( y - y ' ) + ylp2 + h 2)dxdx' A J97 *823 (x’ y > y') = h ( y - y')JJxdxdx = / 733 (x', y ^ ( x yy,x',y') = J J 732 A. 194 X, y . y ' ) = hJ J x '(x - x ) tan_l(----------- — ^ = )d xdx = / 735(x ',y ',x ,y ) A. 198 ( x - x ')tJ p +h2 83(■*. y. Jf', y') = £f y f2(R)(dn. • ax )(a„ • dx.)dldl' = JJyfA( R)dydy a5ar = ^831 + ^832 + ^833 + ^834 + ^835 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A. 199 81 x —x / J J y yjp 2 + h2dydy’= I n i (x ',y '.x ,y) l « n ^ , y , x \ y ) = h ( x - x ' ) J J y ln( h J ^ ~ + /i ^ + = ^7 2 A. 2CX) >) 2 A- 201 l m ^ , y , x \ y ) = —A(-c - x')JJ yrfwfy' = /^ (x', y', x, y) A. 202 t ) L ^y ...v ^ ( - (v„j c - x') + J p 2 + h 2)dydyf / 834(x ,y ,x \ / ) _ j j __h_ +{y___y_) 2 A. 203 = / 724(x/,y ',x ,y ) ^35 =)dydy' = (xt y,x',y') = h \ \ y (y - y ')tan"1(------ ~ ~~ 7 ^ JJ ( y - y ') J p +h2 (x ',y \x ,y ) A. 204 /g4(x,y,x',y') == $ * (* > // • d? = JJ£(/?)<£a£c'+J J k(R)dydy' asaslu {x,y,x,y) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A. 205 82 A- 9 Sleepy (In teg ral 9) ss--\Jpz + h z iSdS' = - &yy'g{R)dl . dl' asas- + asas8 y%ms.- -s,K S ..sy)didi’ A. 206 + f t y f A W G n ' •*?■)&„ *ay.)dldl'+ § k { K ) d i . d l ' asas= / 9I + l n + / 93 + / w asas- t 9l( x , y ,x \ y ') = -j$g (R )y y'd l . d l ' = - j f y y 'j p ^ + f S d y d y ' asas+ * J J y y ' I n C^ — - yy'JJV P2 + h2dxdx' + Z A 2Q7 i y y ' J J )dxdx' = I u ( y ,x ,y \ x ') l n ( x , y , x , / ) = £ f y f t (/?)(£,,> • d y)(5n • a v)dldl' = y 'J J / 3(R)dxdx' asar A. 208 = / 42(.y l 93(x,y ,x',y') = y/4(#)(«„. • a v.)(«n • d y.)dldl' = y J J / 4(R)dxdx' a&= / 43(y ,x ,y ',x ') A. 209 fc(R)dl • d? = J J k(R)dxdx'+ J J k(R)dydy' asas= I M( x , y , x \ y ') A. 210 / w (j:, y, x', y') = Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. IMAGE EVALUATION TEST TARGET (Q A -3 ) 4 8* ^ T c r '(■ 150mm 6 " IM/4GE.Inc 1653 East Main Street Rochester, NY 14609 USA Phone: 716/482-0300 Fax: 716/288-5989 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i A *>

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