"LI B ^AHY OF THE U N IVER.SITY OF ILLINOIS 510.84 Ufcr no. 2.50-256 cop. ^ Digitized by the Internet Archive in 2013 http://archive.org/details/designconsiderat254hors Tit"-, /h*M Report No. 25*+ DESIGN CONSIDERATIONS FOR MULTILAYER TRANSMISSION LINE CIRCUIT BOARDS by Larry Ray Horsman January 2, 1968 THE LIBRARY Of THE aug Id r::j UNIVERSITY Of ILLINOIS Report No. 25k DESIGN CONSIDERATIONS FOR MULTILAYER TRANSMISSION LINE CIRCUIT BOARDS* by Larry Ray Horsman January 2, 1968 Department of Computer Science University of Illinois Urbana, Illinois 618OI * This work was supported in part by the Advanced Research Projects Agency as administered by the Rome Air Development Center under Contract No. US AF 30(602)4lUU and submitted in partial fulfillment of the requirements for the degree of Master of Science in Electrical Engineering, January, I968 • ACKNOWLEDGMENT The author wishes to thank Professor Richard M. Brown for his guidance and counseling and for the opportunity to parti- cipate in the ILLIAC IV project- Thanks are also extended to Mrs. Anne Bruning who prepared the final manuscript. 111 TABLE OF CONTENTS Page 1. INTRODUCTION 1 1.1 High Speed Digital Systems 1 1.2 Statement of Problem 2 1.3 Method of Attack 3 2. TPANSMISSION LINES 5 2.1 Lossless Lines 5 2.2 Microstrip and Stripline 9 2.2.1 Fabrication Techniques 9 2.2.2 Advantages of Multilayer Boards 12 2-3 Mode of Operation 13 2.4 Time Domain Reflectometry .............. l4 3. CHARACTERISTIC IMPEDANCE ......... 19 3.1 Calculation of Impedance of Stripline 19 3.1.1 Conn's Equations ..... ..... 19 3-1.2 Bates' Equations 22 3.1.3 Field Mapping Techniques ...... 24 3«1.4 Comparison of Methods ....... 33 3.2 Calculation of Impedance of Microstrip 36 3.3 Circuit Board Manufacturing Tolerances 40 3.4 Experimental Determination of Characteristic Impedance ............ 52 4. CROSS COUPLING .............. 64 4.1 Analysis of Cross Coupled Lines ........... 6k 4.1.1 Directional Couplers ....... 6k 4.1.2 Coupling Mechanisms 68 4.1. 3 Classical Approach to Cross Coupling 69 4.2 Experimental Cross Coupling Measurements ...... 71 IV Page 5- SUMMARY AND CONCLUSIONS 75 BIBLIOGRAPHY ............ 77 APPENDIX I 80 APPENDIX II ........... . ..... 85 LIST OF FIGURES Figure Page 1. Approximate Representation of a Short Section of Transmission Line ........... 6 2- Microstrip and Stripline Geometries 10 3- Typical Multilayer Board Cross Section 10 k. Built Up Multilayer Board vith Drilled Hole for Mounting Component ........ 11 5. Plated-Through-Hole Circuit Board ........... 11 6. Typical TDR Arrangement ................ 16 7. TDR Oscilloscope Display l6 8. TDR Displays Related to Terminated Transmission jjiriGS • • o © © © * • • • • • • ■ • • • ® • • ♦ • • • -Lo 9« Parameters of Stripline 20 10. Field Plot Between Two Conductors 26 11. Two-Dimensional Potential Distribution . . 26 12. Potential Distribution of Stripline .......... 32 13« Comparison of Equations for Calculating Character- istic Impedance of Stripline ............. 35 Ik. Impedance Distribution Due to Variations in Line Width ...................... ^3 15* Impedance Distribution Due to Variations in Line Thickness .................... U5 l6. Impedance Distribution Due to Variations in Distance Between Ground Planes V7 17 • Impedance Distribution Due to Variations in Relative Dielectric Constant . . i+9 18. Impedance Distribution Due to Variations in All Parameters ..... 51 19* Microstrip Circuit Board .......... 53 VI Figure Page 20. TDR Display of Line 1 56 21. TDR Display of Line 2 . 58 22. TDR Display of Line 3 60 23 • Cross Coupling Circuit 65 2k. Oscilloscope Display of Cross Coupling 73 Vll 1. INTRODUCTION 1.1 High Speed Digital Systems During the past several years the speed or throughput of digital computer systems has increased by several orders of magnitude. This increase has been realized in most part by an increase in the speed of the logical circuit elements. Current state-of-the-art logical elements are capable of making switching decisions in less than 2 nanoseconds. Accompanying this increase in speed has been an increase in component density. Today's integrated circuits typically con- tain from two to ten logical elements per package. In terms of discrete components., this might represent as many as fifty transis- tors and seventy resistors interconnected within one fourteen pin package. The next step will be large scale integration with hun- dreds of logic functions in each package. The advances being made in digital circuits have made it necessary for the design engineer to revaluate his approach to compu- ter systems design.. It has been shown that 50 percent of the overall delay of a system is determined by the interconnection scheme. Thus> to take advantage of high speed circuits,- the design engineer must consider carefully signal propagation through the wiring. Signals with nanosecond rise times have frequency components in the hundreds of megahertz. In this range even short lengths of interconnecting wire begin to act as distributed transmission lines. There is a need to know the characteristics of the printed circuit board to a greater degree of accuracy than previously. The single sided printed circuit board is no longer adequate. Ground plane and stripline construction is necessary for undistorted transmission of high speed pulses. With increased speed; digital circuits have become more sensitive to noise. Again the design engineer must find new ap- proaches to insure tolerable noise margins. Transmission line theory offers an orderly approach to such problems as noise and cross talk. 1.2 Statement of Problem The ILLIAC IV computer will incorporate many state-of-the- art design concepts. This computer will use the fastest available logic with propagation delays of the order of 2 to 3 nanoseconds. The logic elements will be mounted on multilayer circuit boards. These circuit boards will replace much of the discrete backplane wiring commonly in use today. The purpose of this paper is to investigate the major design areas pertinent to multilayer transmission line board fabri- cation, characteristic impedance and cross coupling. The character- istic impedance of a line is determined by the geometry and material used to construct the circuit board. With increased clock frequencies and fast rise time pulses, control of the characteristic impedance of a printed circuit line becomes an important design objective since transmission line theory would be useless without controlled line impedance. Cross talk is defined, as the noise that is coupled to an inactive line when a signal propagates along a second line located in the vicinity of the inactive line. This coupling has two com- ponents, the electromagnetic coupling due to mutual inductance between the lines and electrostatic coupling due to the mutual capacitance between the lines. The amount of coupling between two printed lines depends upon factors such as the spacing between the lines, the dielectric material surrounding the lines, the termination impedances and the rise time of the signal on the active line. 1.3 Method of Attack The use of transmission line circuit boards is new to the digital field, but considerable theory has been written concerning microwave applications of stripline printed circuits. Much of this theory has been applied to the transmission line circuit boards now being used in the latest digital computers. The first part of this paper is devoted to a brief re- view of transmission line theory and a discussion of circuit boards in terms of existing theory. Wherever possible, analysis of typical circuit board geometries is made using this theory. It is the aim of the author to develop a general set of computer analysis programs which will be applicable to a wide variety of transmission line circuit board analysis and design problems. Experimental data have been obtained from a transmission line circuit board fabricated at Texas Instruments. This circuit board contains the basic geometries necessary for measuring impedance and cross coupling and is typical of the type of board that will be used in the ILLIAC IV computer. The experimental data are com- pared with theoretical results to determine the correlation between the two approaches . 2. TRANSMISSION LINES 2.1 Lossless Line s Transmission line theory is based on the distributed effect of a line. At higher frequencies any wiring system begins to exhibit distributed characteristics. The analysis of a transmission line is most easily carried cut by an extension of lumped- constant theory. Although the line constants are uniformly distributed along the line, the line can be treated as though it were made up of short sections of length Ax, as shewn in Figure 1. The parameters of the line are: R = series resistance per unit length L - series inductance per unit length G = shunt conductance per unit length C = shunt capacitance per unit length The difference between the instantaneous line-to-line voltages at de the two ends of the section will be v~Ax and is caused by the current i flowing through the resistance RAx and the rate of change ^r- in the inductance LAx- Summing voltages around the loop of length Ax yields: ||Ax = (RAx)i + (LAx)|i (2.1) The negative sign is used because positive values of i and of *rr- cause e to decrease with increasing x. o •H W W •H 6 w C! 03 Jl/c where f(\/LC x - t) represents any single- valued function of the argument vLC x - t- From Equation (2.1l) comes the second important result of lossless line theory. Since the function f represents a voltage, the quantity v L/C must have the dimensions of impedance. This quantity is known as the characteristic impedance of the lossless line: z =>Tl/c (2.12) 2.2 Microstrip and Stripline There are two basic geometries being used in the construction of transmission line circuit boards (Figure 2). The single ground plane configuration is known as microstrip and the double ground plane configuration is known as stripline. Stripline geometry gives better shielding but requires an additional ground plane. This is not a particular disadvantage in multilayer boards since one ground plane can serve two conductors., one on each side of the ground plane- Until recently the most popular geometry has been the micro- strip because of ease of layout and fabrication and circuit trace- ability. However, increased circuit speeds and component densities have made it desirable to use stripline techniques which allow the sandwiching of several layers to achieve the desired wiring density. A typical board cross section is shown in Figure 3° The intervening ground planes can be used as voltage distribution planes. The outer surfaces conveniently utilize the microstrip geometry. 2.2.1 Fabrication Techniq ues Each manufacturer has a different fabrication process for multilayer circuit boards., but two basic methods are currently in widespread use: the built-up process and the plated- through-hole process. The built-up process achieves interconnections between circuit layers by seqaential deposition of conductive patterns and dielectric material. Vertical interconnections are solid posts through which holes may be drilled for mounting components (Figure h) . 10 Ground Conductor Plane Dielectric fa — °° — h Microstrip > Dielectric Conductor -r>() ) > > > > i > > > i > > > > > >/> > ))>))) a ^ _^ ~S^ J L. Ground * \1 1 1 1\ <* L ^ Planes rr )))))) ) i > > > i > > > > i > i ) > ) ) > ) > > \ ^ Stripline Figure 2. Microstrip and Stripline Geometries. „ Dielectric ^y Conductors _< ' / i > / 1 1 i / ) 1 1 / C Ground I m — p"| q y-i p o / / / I 1 ) I / ) I I I ) I ) / I i i i i / i / / / I I ) ) ) ) 1 ) ) i / / ) v VI I l\ \> i > i\ \ ( HZZ2 XTTTT Figure 3" Typical Multilayer Board Cross Section. 11 Drilled Hole Dielectric Conductors Figure h> Built Up Multilayer Board with Drilled Hole for Mounting Component.. Plated- Through-Hole Dielectric on duct or s Figure 5° Pla' rough-Hole Circuit Board. 12 The built-up process allows internal layer to layer inter- connections that do not surface on either side of the board. This makes it possible to have more than one layer to layer interconnec- tion at each grid location. Built up boards are usually designed using buried circuit paths with only solder or welding pads on the surface. The plated- through-hole process achieves interconnections between circuit layers by means of either a through hole, plated with a conductive material, or a solid conductive post which can be drilled to accept circuit components (Figure 5)° The plated- through-hole process will allow only one layer to layer interconnec- tion at each grid location« The principle advantage of the plated- through-hole process over the built-up process is that it requires fewer fabrication steps and is therefore easier and more economical to produce. 2.2.2 Advantages of Multilayer Boards Multilayer circuit boards can reduce wiring space, cut weight, increase component density and assure higher reliability. In the ILI.IAC IV computer, multilayer circuit boards will replace wiring harnesses and paint to point backplane wiring typical of se- cond generation computers. The use of a ground plane in conjunction with closely controlled dielectric thicknesses and conductor width makes possible the production of multilayer stripline boards with constant impedance conductors. As discussed earlier, lossless trans- mission line theory can be used as an analysis tool on this type of 13 board. The use of wiring harnesses and point to point backplane wiring made the evaluation of noise., cross talk and reflections al- most impossible. The use of multilayer transmission line circuit boards should lead to better designs free from problems of noise and cross talk. 2.3 Mode of Operation The theory of the propagation of waves along a transmission line system, such as the shielded cable or wire over ground systems, is readily found in most transmission line and microwave textbooks. The microstrip and stripline geometries can be evolved from the famil- iar coaxial line geometry,, and the theory developed for the coaxial line is applicable to microstrip and stripline circuit boards. As with the coaxial cable., the microstrip and stripline systems operate in the TEM or transverse electromagnetic mode. At very high frequencies, when the wavelength of the signal approaches the size of the conductor to ground spacing, other modes of operation are possible, but these frequencies are much higher than any that are used in digital computers. There is a possibility that nonuniform TEM waves will be present in a microstrip system in which the two ground planes are not connected at regular intervals. W. T. Rhoades suggests in his paper that the ground planes should be connected together at distances less 2 than one-tenth tde wavelength of the highest frequency of interest. 14 2.4 Time Domain Reflectometry Time domain reflectometry is a relatively new technique used in the analysis of transmission lines. This technique was used in the analysis of the printed circuit board supplied by Texas Instruments. Since TDR has become an important analysis tool, the basic theory of this technique is presented belowo Time domain reflectometry gives quantitative information about mismatching, lcss^ reflection coefficients and other transmis- sion line parameters. A TDR system can measure lumped resistance and reactance as well as characteristic impedance. Measurement is by analysis of signals reflected from a step function signal that drives the line. TDR systems can display information as a function of distance, and in particular, show multiple discontinuities indivi- dually. A typical TDR arrangement is shown in Figure 6. The step function generator generates a series of step functions at the sweep rate of the oscilloscope so that the signals can be displayed con- tinuously on the oscilloscope. The generator drives a finite length of transmission line of characteristic impedance Z . The transmission line is usually terminated in some impedance Z . An oscilloscope dis- plays the voltage signals at the input end of the transmission line. A simple example will demonstrate the use of TDR. Assume an incident wave of voltage, denoted by e , traveling to the right on + + the line. The current associated with this voltage is i = e /Z . 15 - Z 7 (2.13) At the terminatior the relation Total e Total i "L must be satisfied . Unless Z T = Z , the wave traveling to the right L o e> e does not satisfy this relation and part of the incident wave will be reflected. This reflected voltage > denoted e t travels to the left along the line. The current associated with this voltage must be i = -e"/Z • At the termination. Equation (2.13) gives: + e + e 4—4 = z L ( 2 .i4) where the subscript t refers to the values at the termination. Equation (2.1.4) can be expressed in terms of Z as: + e + e t t e./Z - e./Z t' c V o Solving this equation for the ratio of reflected to incident voltage yields: e" Z. - Z (2.15) The ratio k is the reflection coefficient. Therefore a terminating impedance different from the characteristic impedance of the line will cause a reflected wave which travels back toward the generator. The reflection will again 16 Transmission Line R — c >— f Z o I ° 1 Oscilloscope \ % p s bej — i Funcbion Generabor ) Figure 6. Typical TDR Arrangemenb . Incidenb Step u Reflecbion ^- - +1 •R > Z , k < +1 L o ■\ = Z o> k = ° ^R L < Z Q , k > -1 R T = 0, k = -1 Time Required for Sbep bo Travel Down bhe Line and Back Figure 7° TDR Oscilloscope Display. 17 "be reflected upcn reaching the generator unless the impedance of the generator (R in Figure 6) is equal to Z • In all practical appli- cations of TDR, the generator impedance is made equal to the charac- teristic impedance of the transmission line- Figure 7 shows a typical oscilloscope display with various 3 values of resistive terminations . Tne horizontal axis of the oscil- loscope can be calibrated in either units of time or distance. The general theory of time domain reflectometry is based upon the reflection concept presented above. When a reactance is present along the transmission line., the oscilloscope displays a reflection with an exponential curve. The time constant of this curve can be used to calculate the magnitude of the reactance. Thus, any type of impedance mismatch can be detected at any point along the line and its magnitude determined from the oscilloscope display. Typical TDR displays cf a reactance along the line are shown in Figure 8. It should be remembered that the response to a resistance or reactance is a reflection and therefore multiple discontinuities can be displayed simultaneously on the oscilloscope. Footnotes 1. Johns on j W. C«, Transmission Lines and Networks , McGraw-Hill, New York (1950), pp. 3-22. 2. Rhodes, W. T., "Multilaminates for Nanosecond Circuitry," Nation al Ele ctronic Packaging and Production Conference 196^ Proceedings, Industrial and Scientific Conference Management, Inc., Chicago (Tune I96U), pp. 170-190. 3- "Tektronix Type 1S2 Sampling Unit Instruction Manual," Tektronix, Inc., Beaverton (1966). r 18 H ^ (a) Capacitor in Series with Line (b) Capacitor in Parallel with Line N^ T, -OP- T- * *. (c) Inductor in Series with Line (d) Inductor in Parallel with Line Figure 8. TDR Displays Related to Terminated Transmission Lines. 19 3- CHARACTERISTIC IMPEDANCE 3*1 Calculation of Impedance of Striplir.e The stripline transmission line geometry first came into widespread use in microwave circuits in the 1950' s. During this time several papers were written concerning the derivation of the characteristic impedance of the stripline geometry. 3«1«1 Cohn's Equations Cohn develops two equations for the characteristic impe- dance of stripline. The first equation concerns the wide- strip case where the width of the conductor is great enough so that the fringing fields at the ends of the conductor do not interact. This equation is: z - ■ --- 9l 4 1_, _ ohms J- f sZ* c f ^ (3a) V£ r rttTE + T6555T where w, b and t are the dimensions shown in Figure 9> e is the relative dielectric constant and C„ is the fringing capacitance in mmf/unit length from one corner of the strip to the adjacent ground plane. The exact formula for C„ has "been found by conformal mapping to be: • = 0.0885€r [2A ln(A + D _ ( A . D ln ( A 2 _ !)] ( 3 . 2 ) where 1 A ^ l-HE/b 20 U / / / / / / 1)1//) / / / f ; / \ ( i V- » H 1 1 1 \ V / / / /\ ) 1 ////////////// / / / / / /( Figure 9° Parameters of Stripline. 21 As the strip width decreases, Equations (3-1) and (3-2) become inaccurate due to the interaction of the fringing fields on the left and right of the conductor . Cohn states that these equa- tions are usable for w/(b - t) > 0.35^ with a maximum error of 1.2 percent at the lower limit of w. The second case considered by Cohn is the narrow- strip case, the approach being entirely different from the wide-strip case. This case assumes that at a sufficient distance from the rectangular conductor of the stripline, the magnetic field lines will be circular . With this assumption, an equivalent circular conductor can be found which will produce the same inductance per unit length as the rec- tangular conductor and therefore, the same characteristic impedance. 2 3 Flammer and Marcuvitz have both plotted graphs relating the equi- valent diameter, d , to the dimensions of a rectangular conductor. o' Knowing the equivalent diameter permits the use of the following approximate formula for the characteristic impedance of a circular conductor centered between two ground planes: z = 6o_ ln i+b ( } o J — nd v ' Cohn states that on the basis of flux distributions, Equation (3*3) is believed to be accurate to within 1.2 percent for w/(b - t) < .35 and t/b < .25- 22 A third formula is given by Cohn for the characteristic impedance of stripline. It is an exact formula when t = ' 0« z = ^SM (3 . U) ° K(k ) where K(k) and K(k ) are complete elliptic integrals of the first kind, and k = sech — , k = tan h tt- 2b 2b A family of characteristic impedance curves are plotted versus w/b, with t/b as a parameter, at the end of Cohn's paper. These curves are very helpful when designing stripline transmission lines. 3»1°2 Bates' Equations An exact solution for the characteristic impedance of the k stripline geometry has been determined by R. H. Bates. He uses conformal transformations to transform the stripline line into two coaxial cylinders for which the characteristic impedance is known- The problem involves the integration of an equation from a Schwarz- Christoffel transformation and another transformation involving Weirestrasse 1 s second order elliptic function. The mathematics of the transformations become quite involved. His final results are the following expressions: w _ 2K b " n \ 2 k sn a en a „/ \ j - Z(a) dn a - _j (3-5) 23 t a 2K ksnacna „ / \ / -, c\ b = K" ~ dnl Z(a) (3 ' 6) where : w = width of stripline conductor t = thickness of stripline conductor b = distance between ground planes K,K = complete elliptic integrals of the first kind with argument k k = modulus of the elliptic functions sn a ) en a • dn a - Jacobian elliptic functions with arguments a and k Z(a) = Jacobian zeta function with arguments a and k Bates has plotted i£ Z versus en a in his paper. He 2 suggests the following procedure for using Equations (3«5) and (3-6) : 1. Take a given value of Z . o 2. From the graph ie Z„ versus en a , find the correspond- X' o ing value of en a . 3" Assume several values of k and solve en a for a (en a is a function of a and k) . h> Substitute corresponding values of a and k into Equations (3-5) and (3-6) and calculate w/b and t/b. 5- Assume a value for w, b or t and solve for the other two dimensions • Bates includes in his paper a plot of t/b versus w/b with Jg Z as a parameter. This graph coders most of the practical stripline designs. 2k 3-l«3 Field Mapping Techniques A third method for calculating the characteristic impedance of a stripline transmission line employs field mapping techniques. Lossless line theory gives: \-7f (3 - 7) r where: V = velocity of propagation V = velocity of light Jj e = relative dielectric constant of medium r Two lossless line relations which were derived previously are: V. = ~= (2-9) P -/c~ Jj and substituting this equation into Equation (2.12) gives; r z ° ~ v L but V ^ 11.8 x 10 inches/ second, therefore. where C is measured in nanofarads per inch- Thus, the character- istic impedance can be calculated using Equation (3«8) if the capa- citance of the stripline is known. The capacitance of any geometry can be calculated from a field plot- Figure 10 shows a typical field plot in some arbitrary region between two conductors. The electric flux density at the midpoint between A and B may be approximated by: where : £¥ = flux in the tube m'b'b 1L - distance between adjacent flux tube boundaries The magnitude of E is then; E-i f- (3-9) The magnitude of the electric field intensity can also be found by the relation E = - gradient V. This relationship gives: *=§" (3-10) n 26 Equipotential Lines Flux Tube Boundaries Figure 10. Field Plot Between Two Conductors. h ^ » £ V V. V, Y -*-x Figure 11. Two- Dimensional Potential Distribution. 27 where : AV = potential difference between adjacent equipotential lines AL = distance between adjacent equipotential lines Equating Equations (3» 9) an d (3*10) gives: 1 AF_ AV t n or AV " G AL Assuming a constant increment of potential between equipotential lines and a constant amount of flux per tube gives: AL t -rr— = constant n and assuming this constant is unity (AL, = AL ) gives: AY AV £ (3-11) The capacitance is found from C = Q/V by replacing Q, by N AQ, and V by N /W where N is the number of flux tubes joining the two conductors and N is the number of equipotential increments be- tween conductors. Therefore: 28 but since AQ = 081, Equation (3-11) gives Afy/AV = e and N c == e IT (3-13) v The capacitance of any geometry can be calculated from its flux plot merely by counting the number of squares in two directions and using Equation (3°13)» Knowing the capacitance- the characteristic impedance can be found from Equation (3-8) . There are several techniques for finding the flux plot of a particular geometry. The relaxation method is particularly suited 5 for computer solution. This method assumes a two-dimensional po- tential distribution as shown in Figure 11. From LaPlace ' s equation : ox~ oy Approximate values for these partial derivatives are: 5E t 1 'a h cW 5x" V - v_ ■ . o 3 'a h from which o ^V I _ |Y I v-V-V+V o V ± ox 'a ox ' c 1 o o 3 3x 2 h h 2 and, similarly for — - >.2 V - V - V + Vi d V ^ 2 ) v ) V h Then from Equation (3-1 1 !) or ■x2 ^2 V + V + V + V - hV a v + o o o o o CJ o 1 o o o o O o a Oj c 1 C (•■ * V f- in CD IM m p- 1 r- m IM 00 PPI f» «- ♦1 1 - ^ - O * IM \ > IN m 1*1 •C * * •* ■j- •» en in M > o c •* m N *\ r- CM 1 1 m IM 0- 00 ►» »- in ■»■ : w* im m \ CM _- cs CM IM r- r- (M IM CC M • in fM M C fv. • C CM p- 1ft IT CD CC CM •c in CD . "" 0- o- r- ' O m - o e m o m « r4 s^ pH rn ■ pn M r r» *< in « CM c C <0 IM r - CM in, ^ vi y 1 1 =/ Vm IM \- •»■ ^"" w r- 00 in ■o c "■ CM "•b * J Tl ■* ___ *\ iml ~~»B] ■ • — -#M VLm c - — p ^* — o- •^ in « c !c ~]~ t op ( I* H i* Pi "1 c r° 1 lot I « CM in m eo 00 1 • • » !• 1 i *l le j col |e> 1 |m: ot _£ ... cm! » a ^^ m •o M ,! |cm m \ r ' t -^* r l C T* •"^ / •»• CM cv "* C 1 -o * 1 •* —1 im^^ "IS 'inl ' 1 • i *^ \ • V. J . u/"». > IN •«• s. f i> CO m ■0 1 ~* r% |cm , a\ % * "\f ' k ol Xi . ^s o V •*i m CM pM c . T" * \ -0 JM 1 'CmV. CI 1*^ ^h <^ CM <0 * i >>» r- C If — r» \ m •k^I y / ■o m ^ • v f P* J A M r- a LC -T *H \ IM m, yr, ;m in v" zS ^ m m CM " - C o "J m V • i~ • W ij o- it \^ Jf •«% L-J 49. f in - • - ITS c in o m o m o 9 ni op t re rn _ -:\ h rn o « r -r m L CO / / j: m m ID *^i c -r gd m p- IM * m ir o* ■o w r- m 00 >r -> -* IM y „ m en V CM N c « IN ^^ ^! r»- CO / • w r- CO « CM \ p* IM if <. m ^ <_> >r r~ Sn m m IM o IM IN ^IN 1" *T ^ P- jn c J- »■ p\ - QD CM n > ■t*. V 4. CM tr\ CO M m 2/ CD o «T i0 T> -4 B m < ■*i — — pH CM ;m P« — " — 1. ^* TN m # L \2 _ CM Fl IM H C CM ■* o 00 10 •r ■vj "" b "" "* ^ H •" "" c ■»■ 00 - -r « T « m o. v., 4 e >r 00 ff t C — CN » in -0 CD cr V F •o m ■T CM - ( c r» ■*• - r^ m /h CM p* m h b p» - * *- i - o ** CM CM m ' In 1 ■r <■ < •r 1 ,T1 J in •M rM *" 1 9 C c o » U o a k> o b o ! P a R 3 O 3 o g t. • • i n n o o « o o a 1" o o a u 3 o i P • o 9 o 33 The characteristic impedance is found from Equation (3*8) : Z o r r 11 ' 8C 11.8(-597 x 10" 3 £ ) z = i o 7.05 x 10" VJT J~e~ Z = 1U2 ohms r o 3-1.^ Comparison of Methods A computer program was written to compare the two equations for the characteristic impedance of a stripline derived "by Conn with the equation derived by Bates. The results are shown in Figure 13 ♦ The thickness of the line was assumed to be 2.8 mils and the distance between ground planes 40 mils. The ordinate is the line width since this is the parameter that is normally varied in actual board design to obtain a specific impedance. It is assumed that the equation given by Bates yields ac- curate values for impedance at all line widths. Based on this assumption, it is observed from Figure 13 that Conn' s Equation (3-l) yields similar results for line widths greater than about 12 mils. This corresponds to a value of w/(b - t) greater than 0-32 and is very close to Conn's value of 0«35 for the useful range of this equa- tion. Conn's Equation (3*3) yields similar results for line widths less than about 13 mils which agrees with Conn's range for this equation of w/(b - t) less than 0-35- 3^ •H H d< •H !h ■P CQ CH O d) o ro -d CD O ■H P w •H Jn QJ •P o CO fH CO -A o bO •H -P 03 H o H CO o Ph O

■H Pn 35 38 the velocity of propagation is related to the dielectric constant a s follows : P ^r Therefore, by measuring the velocity of propagation of a given micrcstrip geometry^ the effective dielectric constant can be com- puted by: e r (3.17) o H. R. Kaupp gives a linear approximation to the curve for effective relative dielectric constant versus relative dielectric constant of the circuit board material. This equation is: 0.1+75e + O.67 (3-18) A third method for calculating the effective dielectric 9 constant is based on mapping methods- The effective relative di- electric constant is defined as: t capacitance of line with nonhomogeneous dielectric r capacitance of line without dielectric or: t dielectric / -, n \ € r = c~ — ' ' — v3°i9; vacuum 39 From Equation (3*12), the capacitance without the dielectric is: C - rP AC (3-20) vacuum N It can be shown that the capacitance in a nonhomogeneous dielectric is C,. _ . . = AC Z 7— 7 (3-21) dielectric -, P + p TN - P m=l m e r L V m J where € = relative dielectric constant of base material r N = number of flux tube boundaries N = number of equipotential lines P = number of curvilinear squares in each individual m flux tube in the base dielectric Substituting Equations (3°2l) and (3-20) into Equation (3*19) gives: < - \ L p m ; ^ - y (3 - 22) The above three methods yield approximately the same re- sults. The first method is highly accurate but is an empirical approach due to the fact that the microstrip line must be built and the velocity of propagation measured before the effective relative dielectric constant can be determined. The linear approximation me- thod is less accurate but is simple to use. This is a good approxima- tion when the relative dielectric constant of the base material is in 40 the range of 2 to 5° The mapping method becomes exact as the distance between the equipotential lines and the distance between the flux tube boundaries approaches zero< It is apparent from the derivations of the characteristic impedance of stripline and microstrip geometries that impedance is a function of four parameters ■ These are- 1= relative dielectric constant 2° width of line 3« thickness of line h. distance between line and ground plane It is important tc know how these parameters vary on a typical circuit board and what effect these variations have on the characteristic im- pedance o 3= 3 Circuit Board Manufacturing Tolerances The Institute of Printed Circuits has issued design standards for multilayer circuit boards with recommended tolerances for various board parameters" A typical 50 ohm stripline circuit board might have the following parameter tolerances: conductor width = .008" t =002" conductor thickness - =0028" t -0007" distance between ground planes = .0311" - -002" relative dielectric constant = U«9 ~ °^ In order to investigate the effects of the parameter toler- ances on the characteristic impedance, a computer program was written 1+1 using a Monte Carlo technique to solve for the characteristic impe- dance of the above stripline. A listing of this program appears in Appendix II. The results of this analysis are shown in Figures 1-! through J Figures lh through 17 represent the characteristic impedance distribution as one parameter is -varied per graph- The impedance equation is solved 1000 times as the given parameter is varied ran- domly over a normal distribution specified by the design tolerance. The parameter tolerances listed above are taken as the ±3a points of a normal distribution. Figure 18 is a plot of the impedance with all parameters varying normally over their respective design tolerances. It is evident from Figure 14 that the variation in line width influences the variation in impedance to a greater extent than do variations in the other parameters. This is partly to be ex- pected because of the 25 percent tolerance in line width. The impedance is least affected by variations in the distance between ground planes (Figure l6). This is due to the fact that the distance between ground planes is relatively large, making the percent tolerance variation small. At higher impedances the distance between ground planes becomes less, thus increasing the influence of variations in this parameter. Figure 18 indicates that variations in the characteristic impedance due to variations in all parameters, can be as high as JT5 ohms. Although this represents a 10 percent variation in the impe- dance of the line, it should be remembered that because of the U2 3 ■H ^ 0) fl •H .J •H W ci o ■H •P CO •H £ O -P cu p a o •H a -p w •H P O g CO T) 01 £< •H ^3 ui 2 J b. I a 5 3 2 z i ol cr • , 1 UJ -H 00 - 1 I- OG I r5 io * UJ 1 z m 1 ii M ii j i < I* iy «C 1 (-» -o o w *-» |o 4J <~> O ICJ It) o ■» ! I - L« " r lh K |o io ! i kk o •H & CD •H •H w o •H -P a) •H ?H CO > o -p CD Q a o ■H •P -P W CJ CD ft •H Pn h5 (A (/> ui 2 H uj T 3 i h ui GO 3 z i ■H CO 3 ftf cr * ii ii ii £ h 00 oJ" Ho o u LJ bJ CJ t^ o ,u o o CJ t- o o <_} u o o o c L> CJ W o CJ o u O u u CJ u 1^ u (-> u o u w LJ <— 1 i w u w u \-- v_> ■w ,1— o (-J u- t-> u- *-/ W i u Sl u UI u w w ■ u W l«J Ui o y> i_; u> <_J u ^ ° w -*. *c U u. 1+6 ^7 5 Lu D o U8 > •H -P CO H 0) « a •H W o •H -P CO •H o p £ o •H o -P p 2 c fit CO •H p M w P a W o •H o o o a H rH 2 and 3 are shown in Figures 20^ 21 and 22., respectively. The first trace in each figure corresponds to the reflectometer placed on the left end of the line (see Figure 19) • The second 'race corresponds to the reflectometer placed on the right end of the line. Theoretically these two traces should be mirror images., but due to the degradation of the square wa/e pnl.se as it travels down the line,? the trace be- comes inaccurate toward the right hand eige of the display . Figure 20 shows the display of Line 1. The vertical scale is calibrated to measure distance. The step- function generator is connected to the board by coaxial cable.* part of which can be seen on the left side of all traces. The coaxial cable is soldered to the end of the printed line on the board, and there is a coaxial connector in the line approximately 6 inches from the board-. A coaxial cable is also soldered to the opposite end of the line;, and a standard 50 ohm microwave termination is used to terminate the line. The first discontinuity in the top trace of Figure 20 is due to the GR connector in the coaxial cable feeding the line. The 55 Figure 20. TDR Display of Line 1 56 A. Source on Left End HORZ. = 20 cm/DIV VERT. = .02'p/BIV B. Source on Right End HORZ. = 20 cm/DIV VERT. = .02p/DIV 57 Figure 21. TDR Display of Line 2 58 A. Source on Left End HORZ. = 20 cm/DIV VERT. = .02 p /DIV B. Source on Right End HORZ. = 20 cm/DIV VERT. = .02p/DIV 59 Figure 22. TDR Display of Line 3. 6o A. Source on Left End HORZ- = 20 cm/DIV VERT. = .02 p/DIV B- Source on Right End HORZ. = 20 cm/DIV VERT. = .02 p/DIV 61 magnitude of the discontinuity indicates a reflection coefficient equal to approximately . 00U or less than a 0.2 ohm discontinuity in impedance. The second discontinuity is indicated just before the beginning of the circuit board. This discontinuity is caused by the separation of the coaxial conductor and its shield at the point where the coaxial cable is soldered to the circuit board. There is a third mismatch as the signal enters the board. Here the abrupt drop indicates a lower impedance line on the circuit board. The reflection coefficient is approximately .02 correspond- ing to an impedance mismatch of about 2-5 ohms. The total variation in impedance along the entire line is less than 3 ohms. The right side of the upper trace of Figure 20 contains a short length of coaxial cable , a BNC coaxial connector and a 50 ohm termination. These features become less recognizable due to the degradation of the step- function as it passes the earlier discon- tinuities. The second trace of Figure 20 shows these features more clearly as the line is fed from the opposite end. The right angle corners in the line do not seem to cause any marked discontinuities. Figure 21 presents about the same information as Figure 20. The important feature on this line is the discontinuity as the line feeds through the plated- through-hole. This discontinuity can be explained by examining the cross section of a plated- through-hole , e.g. Figure 5» The increased area of the conductor as it goes through the hole and its close proximity to the ground plane has the same effect as a shunt capacitance to ground across the line. The effects of the right angle corners can be seen in the upper trace of Figure 21. 62 The effects of plated- through-holes can he seen plainly in Figure 22. Along the midsection of this line are six pairs of holes as shown in Figure 19° Again, these holes have the effect of shunt capacitors to ground- The combined effect of these holes is the lowering of the impedance of the midsection of the line by ap- proximately 3 ohms. This effect is similar to the loading effect obtained when several capacitive loads, such as the emitter coupled logic used in the ILLIAC IV, are placed along a transmission line which is terminated at its far end. All of the traces indicate a characteristic impedance for Lines 1, 2 and 3 of approximately UQ ohms. Using the actual dimensions of the board and Equation (3«l6), the characteristic impedance is found to be U7 . 5 ohms, indicating a close correlation between the theoretical and experimental results- Footnotes 1. Cohn, 8. B-, "Characteristic Impedance of the Shielded-Strip Transmission Line," IRE Transactions on Microwave Theory and Techniques , Vol. MTT-2, No. 2 (July 195*0; PP- 52-57- 2. Flammer, C, "Equivalent Radii of Thin Cylindrical Antennas with Arbitrary Cross Sections," Stanford Research Insti - tute Technical Report (March 15; 1950). 3. Marcuvitz, N», Waveguide Handbook, McGraw-Hill, New York (1951), pp. 2b3-2b5. ' ~ ho Bates, R. H., "The Characteristic Impedance of the Shielded Slab Line," IRE Transactions on Microwave Theory and Techniques , Vol. MTT-U (January 1956), pp. 28-33- 5. Hayte, W. H., Jr., Engineering Electromagnetics , McGraw-Hill, New York (1958), pp. 132-152- 63 6. "An Introduction to Engineering Analysis for Computers, " IBM Technical Publications Department, New York (1964), pp. 93-100. 7« Springfield, W. K-, "Designing Transmission Lines Into Multi- layer Circuit Boards," Electronics (November 1, 1965)^ pp. 90-96. 8- Kaupp, H- R., "Characteristics of Microstrip Transmission Lines," IEEE Transactions on Electronic Computers , Vol. EC-16, No. 2 (April 1967), pp. 185-193- 9. Cordi, V. A., Kingsley, S-, and Shah, A. M-, "Predicting Transmission-Line Properties of Printed-Circuit Conductor Geometry, " IBM Systems Development Division Technical Report No. TD OI.38O (June 25, 1965). ' 10. Messner, G., and Geshner, R., "Multilayer Printed Circuit Boards Technical Handbook," The Institute of Printed Circuits, Chicago (1966). Gk k. CROSS COUPLING l+.l Analysis of Cross Coupled Lines The problems of cross coupling first appeared when strip- lines were used for microwave applications . It was noted that signals from an active line were coupled into nearby parallel quiet lines. The phenomenon was studied exhaustively and a great deal of theory developed. Later the coupling problem again arose in the digital computer field. As the complexity of the computer increased, so did the maze of wiring carrying the signals throughout the machine. The backplane panel of typical computers became a particular source of cross coupling problems. Now the multilayer circuit board with its orderly layout seems to be the answer to many of these cross coupling problems. The existing theory on electromagnetic couplers has been used to design circuit boards free from cross coupling or cross talk problems. New approaches have also been taken which make the analysis of cross coupling more usable in terms of practical engin- eering problems. 4.1.1 Directional Couplers As cross coupling was being investigated in microwave applications, it was found that coupling was inherently directional. Referring to Figure 23; it can be shown that the following relations 65 V. K 25 2* 2V ( t 7 V, Figure 23- Cross Coupling Circuit exist: 66 -^=0 (k.l) V 2 . jk sin pi / 2 Vl - k cos pi + j sin pi (4.2) v ~ / — 2 \/l - k cos pi + j sin pi 0+-3) where k - maximum coupling coefficient ° Connolly has used these terminal equations and the equations for characteristic impedance of a stripline of zero thickness to de- velop two equations which are usable for the design of striplines hav- ing less than some predetermined valued of cross coupling. These equations are: w 2 + r = — tan h b it ■^v/kir) (k.k) \ e o , /(l - k ) -1 o f-f^^drfy^) C-5) where w and b are the parameters of the stripline as shown in Figure 9; s is the distance between the two parallel lines and k and k are given by: k = tan h e (2b) tan h i 2b ) 67 k o :; tanh - cot h 2b / With the above values for k and k , Equations (4.U) and e o (U.5) are transcendental and not easily solved. Several other re- lationships can be found that permit the solution of Equations (^-^) and (^■•5)- These relations are the following: where oe ( 1 + p Z ll - p OO \ o oe 00 K(>/l - k?) r ° K = complete elliptic integral Vl + k - v/l - k s/l + k + >/T Z = even mode impedance oe ^ Z = odd mode impedance 00 (U.6) z = z z (4.7) arw K(>/l - k 2 ) r c Z =30n J el (i+ } °° J7 K(kJ 68 An actual design would precede in the following manner: 1. Assume a characteristic impedance for the board. 2. Assume the maximum coefficient of coupling that is allowable. 3. Calculate Z and Z from Equations (4.6) and (4.7). 4. Solve for the ratios r- and — using Equations (4.8) ^ (4-9), (4.4) and (4-5). Any convenient value can then be chosen for w, b or s thus fixing the remaining two dimensions. 4.1.2 Coupling Mechanisms The coupling between any two circuits arises from two phenomena, magnetic induction due to mutual inductance between the lines and electrostatic induction due to mutual capacitance between 2 the lines. Greenstein and Tobin begin with these two parameters in their analysis of cross coupling. Their analysis is distinguished by the use of La Place transforms and the fact that they assume a general N- circuit complex rather than the two line circuit. The analysis assumes an output voltage V (s) of the p circuit that is the linear superposition of N source voltages: N V (s) = E G (s) V (s) (4.10) P ' -l np gn v where G (s) is the transfer function between the open circuit vol- np v ' tage source V (s) of the n circuit and the output voltage of the 69 p circuit. Equation (4.10) can be rewritten N V„(s) -- G_ V_(s) + Z G_(s) V_(s) (4-11) n=l rtfp P PP gP n=1 np^ gn where G is merely the voltage gain of the p circuit. The summation term of Equation (U.ll) therefore represents the cross coupled signal. The problem is to derive the general transfer func- tion G (s) in terms of the mutual coupling parameters. Greenstein and Tobin conclude that a rigorous solution of the transfer functions is prohibitive. The remainder of the paper discusses six approximation techniques that result from neglecting different coupling terms. The accuracy of each of these methods is discussed and a sample analysis is performed to illustrate some of the techniques. 4.1.3 Classical Approach to Cross Coupling The classical approach begins with the transmission line equations and coupling parameters and derives a general expression for the instantaneous voltage induced anywhere on the line. Again., the analysis indicates that there are two components to any coupled signal., a forward propagating wave and a backward propagating wave* For a homogeneous medium the resultant forward terminal voltage will be zero as proved by Connolly in Equation (4.1). If the medium is nonhomogeneous, as in the microstrip line, this forward voltage is not zero but becomes a function of the slope of the input signal and 70 the length of the line. The backward wave is an attenuated replica of the input voltage on the active line. With respect to the backward propagating wave there are two cases. The first case is the long line case in which the rise time of the signal on the active line is less than twice the propaga- tion time of the region of interaction between the quiet and active 3 lines. For this case the maximum amplitude of the coupled signal is: where T = propagation time of region of interaction L = mutual inductance m C = mutual capacitance m V,V - voltages from Figure 23 • The second case is the short line case in which the rise time is greater than twice the propagation time of the region of interaction. For this case the maximum amplitude of the backward coupled signal is: V = _*[ J2 + C Z 2 M Z o mo / where T is the rise time of the signal on the active line. The amplitude of the forward coupled signal is proportional to the slope (V/T ) of the driving signal and the length of the region of interaction. The pulse width is equal to the rise of the driving 71 signal. The polarity of the forward cross talk signal depends on 2 the relation between L , C and Z . If L > C Z the polarity is mm o m m o 2 opposite to that of the driving signal, and if L < C Z the po- * e m m o larities are the same- 4.2 Experimental Cross Coupling Measurements Several cross coupling measurements were made on the micro- strip board shown in Figure 19° The active line was driven by a fast rise time pulse generator and terminated in its characteristic impe- dance- The quiet line was terminated at both ends in its character- istic impedance. Figure 24 is typical of the voltage pulses measured at the terminals of the active and quiet lines. The top photograph compares the driving signal on Line 2 (bottom trace) with the forward terminal voltage (top trace) on Line 3> which is terminated at the first 90 degree bend. The frequency is 1 megahertz. If the medium had been homogeneous, the amplitude of the coupled signal would have been zero. But, because of the microstrip geometry, the medium is nonhomogeneous and the amplitude is a function of the slope of the input signal and the length of the line. The maximum coefficient of coupling is approximately =005° The bottom photograph of Figure 24 compares the driving signal on Line 2 with the near end terminal voltage on Line 3; again terminated at the first 90 degree bend. The rise time of the pulse generator is less than 1 nanosecond; therefore, the long line case 72 Figure 2k. Oscilloscope Display of Cross Coupling. 73 ■ 05V/DIV 2V/DIV • 2jUsec/DIV A. FAR END - V. . 05V/DIV 2V/DIV • 2iUsec/DIV Bo NEAR END - V, 7^ applies and the amplitude of the coupled signal is inversely propor- tional to the rise time of the signal on the active line. The maximum coefficient of coupling is approximately .0075- It is evident from these measurements that cross coupling on this board is insignificant" The maximum coefficient of coupling is approximately minus ^-0 dt>. This is due to the short length of interaction (about 6 inches) and the relatively large distance be- tween conductors (-05 inches) . Footnotes 1. Connolly, J. B-, "Cross Coupling in High Speed Digital Systems," IEEE Transactions on Electronic Computers , Vol. EC-15, No. 3 (June 1966), pp. 323-327- 2. Greenstein, L. J.., and Tobin, H. G«, "Analysis of Cable-Coupled Interference," IEEE Transactions on Radio Frequency Interference , Vol. RFI-5; No.l (March 1963), pp. i +3-55» 3- Feller, A., Kaupp, H. R., and Digiacomo, J. J., "Crosstalk and Reflections in High-Speed Digital Systems," American Federation of Information Processing Societies Conference Proceedi ngs . Vol. 27, Part I, Spartan Books, Washington (1965), pp. 511-525- 75 5- SUMMARY AND CONCLUSIONS The basic transmission line equations involving the propagation of a signal on a line were presented. The approximations leading to lossless line theory were introduced and found to be valid for most printed circuit board applications. Lossless line theory was used directly in the field mapping method for finding the charac- teristic impedance of any bounded geometry. The different circuit board geometries, microstrip and stripline., were described and discussed in detail. It was con- cluded that although the microstrip line is less expensive and easier to fabricate, the stripline geometry has the advantages of better shielding and the ability to be used in multilayer boards for in- creased circuit density. The built-up process and the plated- through-hole process were described as the two principle multilayer board fabrication techniques. Derivations for the characteristic impedance of the strip- line and microstrip geometries were discussed. Approximations used in the various derivations were indicated and ranges were given within which the approximations are valid. The equations for the character- istic impedance for stripline by Cohn and Bates were compared for a typical circuit board. The characteristic impedance of a stripline was also computed by the field plot method using the relaxation method. The effect of manufacturing tolerances on characteristic impedance was investigated using a Monte Carlo method to solve the impedance equation while varying the board parameters over a normal 76 distribution. It was shown that variations in the line width of currently available circuit boards have the greatest effect on im- pedance . It was also indicated that slowly varying changes in impedance are not as injurious to signal propagation and waveform as are sharp discontinuities. A printed circuit board with microstrip lines was analyzed using a time domain reflectometer. The experimental re- sults were found to be in close agreement with theoretical predic- tions. The problem of cross coupling was examined and several theoretical approaches to the calculation of cross coupling were presented. Typical oscilloscope displays of the terminal voltages of the active and quiet line were given. It was concluded that cross coupling is a minor factor in the design of current circuit boards, but as the region of interaction becomes smaller and the lines are placed closer together, cross coupling will become a problem that must be considered. In conclusion, the multilayer transmission line circuit board is well suited to current high speed digital applications. The impedance characteristics of these boards gives the designer better control over important factors such as velocity of propagation, cross talk and reflections. The multilayer transmission line board also increases component density for an effective increase in system speed. 77 BIBLIOGRAPHY Arvantakis, N. C, Kolias, J. T., and Radzelovage ; W<. , "Coupled Noise Prediction in Printed Circuit Boards for a High- Speed Computer System,. " IBM Systems Develop ment D ivis ion Technical Report No. TR 01.984 ,, November lb, 196T. Bates, R. H. T., "The Characteristic Impedance of the Shielded Slab Line," I RE Transactions on Microwave Theory and Techni ques , Vol. MTT-4, pp. 28-33; January 1956. Cohn, S. B-, "Characteristic Impedance of the Shielded-Strip Transmission Line," IRE Transactions on Microwave Theory and Techniques , Vol. MTT-2, No. 2, pp. 52-57/ July l^- Cohn, S. B., "Problems in Strip Transmission Lines," IRE Transactions on Microwave Theory and Techniques , Vol. MTT-3; PP° 119-126, March 1955= Cohn, S. B. "Chielded Coupled-Strip Transmission Line," IRE Trans- actions on Microw ave Theory and Techniques, Vol. MTT-3; pp. 29-38, October 1955. Connolly, J. B., "Cross Coupling in High Speed Digital Systems," IEEE Transactions on. Electronic _ jjOmp_ut_ers, Vol. EC- 15; No. 3, pp. 323- 3277~June 196 6 . " Cordi, V. A., Kingsley, S., and Shah, A. M., "Predicting Transmission- Line Properties of Printed- Circuit Conductor Geometry," IBM Systems Development Division Technical Report No . TD OI.38O, June "257^3^ 5- Feller, A., Kaupp, H. P., and Digiacomo, J. J., "Crosstalk and Reflec- tions in High-Speed Digital Systems," American Federati on of In formation Proce ssing Societies ^ Co nferer^e^Proce_e dings , Vol. 27, Part I, pp. 5II-525, Spartan Books, Washington, 1965. Firestone, W. L., "Analysis of Transmission Line Directional Couplers," Proceedings of the IRE, Vol. k2, pp. 1529-1538, October 195^- Flammer, C, "Equivalent Radii of Thin Cylindrical. Antennas with Arbitrary Cross Sections," S tanford Research I nstitute Technical Report , March 15/1950. Gray, H. J., Digital Computer Engineering , Prentice-Hall Inc., New York, 1963. ~* 78 Greenstein, L. J. and Tobin, H. G. ; "Analysis of Cable-Coupled Interference/' IEEE Transactions on Radio Frequency- Interference, Vol. EFI-5* No. 1, pp. ^3-55; March 1963. Hayte, Jr., W. H°, Engineering Elec tromagnetics, McGraw-Hill Book Company, Inc . , New York, 1958 » "An Introduction to Engineering Analysis for Computers," IBM Technical Publications Department, New York, 196U. Johnson, W. C, Transmission Lines an d Networks,? McGraw-Hill Book Company, Inc., New York, 1950c Jones, E. M. T., and Bolljahn, J. T=, "Coupled-Strip-Transmission- Line Filters and Directional Couplers," IRE Transactions on Microwa ve Theory and Techniques , Vol . MTT-4, pp. 75-81 , April 1956V ~ Kaupp, H. R., "Characteristics of Microstrip Transmission Lines," IEEE Transactions on Elec tr onic Computers , Vol . EC - 16 , No. 2, pp. 185-193V April ISJotT ~ ™~ Knechtli, R. C, "Further Analysis of Transmission-Line Directional Couplers/' Proceedings of the_J_RE, Vol. ^3; PP« 867-869, July 1955. Marcuvitz, N.., Waveguide Handbook, McGraw-Hill Book Company, Inc., New York, 1951° ~~ — Messner, G., and Geshner, R., "Multilayer Printed Circuit Boards Technical Handbook;" The Institute of Printed Circuits, Chicago, 1966. Oliver, B. M«, "Directional Electromagnetic Couplers," Proceedings of the IRE, Vol, k2, pp. 1686-I692, November,' 195'U. Polk, C, "Transient Response of a Transmission Line Containing an Arbitrary Number of Small Capacitive Discontinuities," IRE Transactio ns on Circuit Theory, Vol. CT-7, pp. 151-157; June I960. Rhodes, W. T., "Multilaminates for Nanosecond Circuitry," National Electronic Pack aging and Production Conference 19o4 Proceedings , pp. I7O-I9O, Industrial and Scientific Conference Management, Inc., Chicago,, June I96U. Rossing, J. F., and Walther, J. E., "Calculation of Cross-Coupled Noise in Digital Systems," I EEE Transactions on Electronic Computers, Vol. EC-16, No. 1, February" 1967 . " 79 Springfield, W. K., "Designing Transmission Lines Into Multilayer Circuit Boards," Electronics , pp. 90-9&, November 1, 1965' "Tektronix Type 1S2 Sampling Unit Instruction Manual," Tektronix, Inc., Beaverton, Oregon, 1966 . Wassel, G- N., "Multiple Reflections on Pulse Signal Transmission Lines - Model for Computer Solution," IEEE T ransactions on Electronic Computers , Vol. EC-1.6, No. 2, pp. 193-202, April 1967. Wested, J. H., "UHF Characteristics of Printed Circuits," IEEE Transactions on Component Parts , Vol. CP-11, pp. 11-19^ September 196^1. 80 APPENDIX I. POTENTIAL DISTRIBUTION PROGRAM DIMENSION V(200,24) »R(200,24) M = 20 N = 24 DEL=.01 ADD=10. <1 = 10 <2=n ti = n L2=14 M11=M-1 N11=N-1 L11=L1-1 L22=L?-1 K11=K1-1 K22=K2-1 K22?=K2+1 L222=L?+1 DO 2 I=1»M V( I »1)=0. 81 2 V( I »N)=0. DO 4 J=1»N V( 1 »J)=0. 4 V(M»J)=0 # MIDN=N/2 DO 6 !=2»M11 DO 6 J=2»MIDN V( I »J)=V( 1 »J-1 )+ADD JJ=N-J+1 6 V( I »JJ)=V( I »J) DO 8 J=L1.L2 V(K1»J)=100. 8 V(K2tJ)=l00. DO 9 T=K1»K2 V( I »L1 )=100. 9 V( I »L2)=lOO. DO 10 1=1. M DO 10 J=1»N 10 R( I »J)=0. DO 12 I =2 »<1 1 DO 12 J=2»N11 12 R(I»J)=V(I»J-l)+V(I»J+l>+V(In}tJ)+V(I+l»J)-4.0*V(I.J) DO 14 I=K1»K2 DO 14 J=2»L11 jj=J-2+L222 82 r( i »jj)=v( i ,jj-d + v( i »jj+i>+v< i-i,jj)+v( i+i»jj)-4-o«v( i »jj) 14 R< I »J)=V( I »J-1 )+V< I »J+1)+V< 1-1»J)+V( I+1»J)-4.0*V( I, J) DO 16 I=K222»M11 DO 16 J = 2»NH 16 R( I »J)=V( I ,J-1 )+V( I .J+l )+V( 1-1 »J)+V( 1 + 1 »J)-4.0*V( I ,J) 18 KK = 2 DO 22 I=2»<11 DO 22 J=2»N11 IF(ABS(R( I »J) ) -DEL ) 22 ♦ 22*20 20 KK=1 RDEL=R( I»J)/4.0 V( I »J)=V( I »J)+RDEU R( I ♦ J)=0. R( I »J-1 )=R( I »J-1)+RDEL R( I »J+1 )=R(I »J+1 )+RDEL R( 1-1 , J)=R( 1-1 ♦JJ+RDEL R( 1+1 »J)=R( 1*1 »J)+RDEL 22 CONTINUE DO 26 I=K1,K2 DO 26 J=2.L11 IF(ABS(R( U J) ) -DEL) 26*26.24 24 KK=1 RDEL=R( I»J)/4.0 V( I »J)=V( I »J)+RDEL R( I »J)=0. 83 R( I »J-1)=R( I » J-D+RDEL R( I »J+1)=R( I , J+D+RDEL R< I-lt J)=R( T-l *J)+RDEL R(I+1»J)=R( !*1 »J)+RDEL 26 CONTINUE DO 27 I=K1»K2 DO 27 J=L222,N11 IF( ABSCRt I »J) ) -DEL ) 27 » 27*25 25 KK=1 RDEL=R( I »J) /4.0 V( I »J)=V( I ,J)+RDEL R( I ♦ J)=0. R« I *J-1 )=R( I ♦ J-l )+RDEL R( I »J + 1 )=R( I . J+D+RDEL R( 1-1 »J)=R( I-1»J)+RDFL R( 1+1 »J)=R( 1+1 *J)+RDEL 27 CONTINUE DO 30 I=K222»M11 DO 30 J=2»N11 IF( ABS(R( It J! ) -DEL) 30 » 30, 28 28 KK=1 RDEL=R( I »J)/4,0 V( I »J)=V( I ,J)+RDEL Rf I »J)=0. p( I ,j-l )=R( T » J-l )+RDEL 84 R( I »J+1)=R( I»J+1)+RDEL R( 1-1 ♦ J)=R( T-l ♦J)+RDEL R( 1+1 t J)=R( 1+1 »J)+RDEL 30 CONTINUE GO TO ( 18,32) »KK ->>? WRITE(6»500) WRITE (6 ,600) ((V(I,J),J=1,N)»I=1»M) 500 FORMAT( 1H1 ) 600 FORMAT( ///21F5.1) END 85 APPENDIX II. IMPEDANCE DISTRIBUTION PROGRAM C C COMPUTE IMPEDANCE OF STRIPLINE BY COKN APPROXIMATION C DIMENSION A(l000»5)tD(25»6)»E(25»2)»F(25»2)»G(25»2)»H(25»2) DIMENSION R(25«2) NZ=lOOO IX=463 AMW=8. AMT=2.8 AMB=31.1 AME=4.9 SW=2./3. ST=.7/3. SB=2./3. SE=.4/3. SORSF=SORT( AME) DO 50 1=1, NZ CALL GAUSS( IX»SWtAMW»W) CALL GAUSS( IX»ST»AMT,T) 86 CALL GAUSSt IX*SB»AMB»B) CALL 6AUSS(IX»SEtAMEtE) SQRTE=SQRT(E) TBYB=T/B TBYBl=AMT/AMB TBYB?=T/AMB TBYB3=AMT/B C=l./( l.-TBYB) Cl=l./( 1.-TBYB1 ) C2 = U/(1.-TBYB2) C3=1./(1.-TBYB3) CFBYE=(2«*C*AL0G(C+1. )-(C-l. )*ALOG (C**2-l . ) )/ 3. 14159265 CFBYl=(2,*Cl*AL0G(Cl+l. )-(Cl-l. ) »ALOG( C 1**2-1 . )) /3 • 14159265 CFBY2=(2.*C2*ALOG(C2+l.)-(C2-l.)*ALOG(C2**2-l. ) )/3. 141 59265 CFBY?=(2,*C^*ALOG(C3+l. )-(C3-l. ) *ALOG ( C3**2~l . ) ) /3 • 14159265 WBYB=W/B WBYBl=W/AMB WBYB2=AMW/AMB WBYB3=AMW/B A( I »1 )=94.15/(SQRSE*(Cl*WBYBl+CFBYl) ) A( I »2)=94.15/(SQRSE*(C2*WBYB2+CFBY2) ) A( I ,3)=94.15/(SQRSE*(C3*WBYB3+CFBY3) ) A( I »4)=94.15/(SQRTE*(Cl*WBYB2+CFBYl) ) 5 A( I »5)=94.15/(SQRTE*(C*WBYB+CFBYE> ) 87 C C SORT A MATRIX IN ASCENDING ORDER C DO 6 K=l,5 DO 6 I=1*NZ DO 6 J=I»N2 IF( A( I»K)-A( J»K) )6»6*4 4 Z=A( I ,<) A( I *<)=A( J»K) A( J»K)=Z 6 CONTINUE C C GENERATE D MATRIX FOR PLOT SUBROUTINE C BEGIN=44. DEt_TA = O.S M = 25 J=l K = l KK=1 D(l .1 )=BEGIN DELL=DELTA/2. DDEL=REGIN+DELL DO 9 J=2,6 DO 8 T=1»M 88 8 D( I »J)=0. DO }0 I=2»W XI=I-1 10 D( I »1)=D(1»1 )+XI*DELTA DO 24 JJ=2»6 12 DO 16 II=KK,NZ IF(A( II ♦JJ-D-DDEL) 14 ♦ 14 » 18 14 K=K+1 1ft D( J»JJ)=D( J»JJ)+1. GO TO 22 18 J=J+1 DDEL=D(J.1)+DELL KK = K IF( J-M) 12.20*20 20 D( J»JJ)=NZ-II 22 CONTINUE J = l < = 1 DDEL=BEGIN+DELL 24 KK=1 WRITE(6»400) 400 FORMAT(lHl) WRITE(6»200) ( (D< I »J) *J=1»6>»I=1»M) 200 FORMAT(6F20.5) DO 26 L=1»M 8 9 F(L»1)=D(L»1 ) F(L»2)=D(L,2) FtL»l)=0(L»l ) F(L,2)=D(L,3) G(L»1)=D(L»1 ) G(L»2)=D(L»4) H(L»l)=0(Ltl ) H(L»2)=D(L»5) R(L»1 )=D(Ltl ) 76 R(L»?) = D(L»f>) WRITF(6»400) CALL PLOKE»25»2t25»OtO.,AOO.) WRITE(6»400) CALL PLOT(F,25»2»25»OtO.,400.) WRITE(6,400) CALL PLOT(G»25»2»25t0t0,»400.) WRITE(6.400) CALL PLOT(H,25»2»25»0,0.,400.) VRITF(6,400) CALL PLOT(R»25»2»25»0»0.»400.) FND 90 Plot Subroutine Adapted from IBM 360 Scientific Subroutine Package SUBROUTINE PLOT ( A »N »M tNL »NS t YMI N t YMAX ) DIMENSION OUT(lOl) »YPR( 11) »ANG<9) »A( 1) 1 FORMAT! 1H1,60X»7H CHART »I3»//) 2 FORMAT! 1H ♦ Fl 1 . 4 » 5X » 1 01 Al / ) 3 FORMAT (1H ) 4 FORMATdOH *+0 = 56789) 5 FORMAT! 10A1) 7 FORMATC1H ,16X»l0lH. . 1 • • • • 8 FORMAT! 1H0,9X»11F10. A) NLL=NL IF(NS) 16» 16» 10 10 00 IS I=1,N DO 1A J=I,N IF(A( I )-A( J) ) 14. 14* 11 11 L=I-N LL=J-N DO 12 K»1»M L = L + N 91 LL=LL+N F=A(L) A(L)=A(LL) 12 A(LU=F 14 CONTINUE 15 CONTINUE 16 IF(NLL) 20, 18* 20 18 NLL=50 20 CONTINUE REWIND 13 WRITE (13»4) REWIND 13 READ (13»5) BLANKt (ANG( I)»I«1»9) REWIND 13 XSCAL=( A(N)-A(1 ) ) /(FLOAT (NLL-1) ) Y5CAL=(YMAX-YMIN) / 100,0 XB=A( 1) L = l MY=M-1 1 = 1 45 F=I-1 XPR=XB+F*XSCAL 50 DO 55 IX=ltl0l 55 OUT( IX)=BLANK DO 60 J=1,MY 92 LL=L+J*N JP=( (A(LL)-YMIN)/YSCAL)+1.0 OUT( JP)=ANG( J) 60 CONTINUE WRITE(6*2)XPR,(0UT( I Z ) » IZ=1 » 101 > L = L + 1 80 1=1+1 IF(I-NLL) 45» 84t 86 34 XPR=A(N) GO TO 50 86 WRlTE(6t7) YPR(1)=YMIN DO 90 KN=ltO QO YPR(KN + 1 )=YPR(KN)+YSCAL*10.0 YPR( 11 )=YMAX WRlTE(6t8) (YPR( IP)»IP«ltll) RETURN END 93 Gauss Subroutine from IBM 360 Scientific Subroutine Package SUBROUTINE GAUSS ( I X tS . AM »V ) A = 0.0 DO 50 1=1.1? CALL RANDU( IX»IY»Y) TX = IY 50 A**A+Y V=< A-6.0\*S+AM RETURN END Randu Subroutine from IBM 360 Scientific Subroutine Package SUBROUTINE R ANDU ( I X » I Y » YFL) IY=IX*65539 IF( IY\5»6»6 5 IY=IY+21474&3647+1 6 YFL=IY YF L=YPL», 465661 3E-9 RETURN, END AUG 1 6 1SB8