ECE 546 Lecture 04 Resistance, Capacitance, Inductance Spring 2020 Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois [email protected] ECE 546 –Jose Schutt‐Aine 1 What is Extraction? - + - + - - - + + + - + - + Process in which a complex arrangement of conductors and dielectric is converted into a netlist of elements in a form that is amenable to circuit simulation. Need Field Solvers ECE 546 –Jose Schutt‐Aine 2 Electromagnetic Modeling Tools We need electromagnetic modeling tools to analyze: Transmission line propagation Reflections from discontinuities Crosstalk between interconnects Simultaneous switching noise So we can provide: Improved design of interconnects Robust design guidelines Faster, more cost effective design cycles ECE 546 –Jose Schutt‐Aine 3 Field Solvers – History 1960s 1990s Conformal mapping techniques Adapting methods to parallel computers Finite difference methods (2-D Laplace eq.) Including methods in CAD tools Variational methods 1970s 2000s Boundary element method Incorporation of Passivity Finite element method (2-D) Incorporation of Causality Partial element equivalent circuit (3-D) 1980s Time domain methods (3-D) 2010s Finite element method (3-D) Stochastic Techniques Moment method (3-D) Multiphysics Tools rPEEC method (3-D) ECE 546 –Jose Schutt‐Aine 4 Categories of Field Solvers • Method of Moments (MOM) • Application to 2-D Interconnects • Closed-Form Green’s Function • Full-Wave and FDTD • Parallel FDTD • Applications ECE 546 –Jose Schutt‐Aine 5 Capacitance Relation: Q = Cv Q: charge stored by capacitor C: capacitance v: voltage across capacitor i: current into capacitor dv dQ it() C dt dt 1 t vt() i ( ) d C 0 ECE 546 –Jose Schutt‐Aine 6 Capacitance d + V A : area oA C = d o : permittivity For more complex capacitance geometries, need to use numerical methods ECE 546 –Jose Schutt‐Aine 7 Capacitance Calculation ()r g(,rr ') (')r dr ' (rkn) potential ( own) grr( , ') Green' s function () known (r') charge distribution ( unknown) Once the charge distribution is known, the total charge Q can be determined. If the potential =V, we have Q=CV To determine the charge distribution, use the moment method ECE 546 –Jose Schutt‐Aine 8 METHOD OF MOMENTS Operator equation L(f) = g L = integral or differential operator f = unknown function g = known function Expand unknown function f f nfn n ECE 546 –Jose Schutt‐Aine 9 METHOD OF MOMENTS in terms of basis functions fn, with unknown coefficients n to get L n ()fn g n Finally, take the scalar or inner product with testing of weighting functions wm: L n wmm,,fn w g n Matrix equation lmn n gm ECE 546 –Jose Schutt‐Aine 10 METHOD OF MOMENTS wL11, ff1 wL, 2 ... l wL,,ff wL ... mn 221 2 .......... ............. ... 1 w1,g g, w g n 2 m2 . . Solution for weight coefficients 1 n lnm gm ECE 546 –Jose Schutt‐Aine 11 Moment Method Solution D xyz',' , ' xyz,, dx''' dy dz E 4 R 2 222 R xx''' yy zz L Green’s function G: LG = ECE 546 –Jose Schutt‐Aine 12 Basis Functions Subdomain bases 1 xxxnn Pxn 22 0 otherwise x1 x2 xn 1 xxnn xx Txn 22 0 otherwise x1 x2 xn Testing functions often (not always) chosen same as basis function. ECE 546 –Jose Schutt‐Aine 13 Conducting Plate Z Y 2b 2b 2a X Sn 2a conducting plate aa x',y ', z ' x,,yz dx'' dy aa 4 R = charge density on plate ECE 546 –Jose Schutt‐Aine 14 Conducting Plate Setting = V on plate 22 R xx'' yy aa xz',y ', ' V dx'' dy 22 aa4'' x xyy for |x| < a; |y| < a q 1 aa Capacitance of plate: Cdxdy'' x', yz', ' VVaa ECE 546 –Jose Schutt‐Aine 15 Conducting Plate Basis function Pn ss 1 xxx mm22 ss Pxnmn,1 y y n y y n 22 0 otherwise Representation of unknown charge N xy, nfn n1 ECE 546 –Jose Schutt‐Aine 16 Conducting Plate Matrix equation: N Vlf mn n n1 Matrix element: 1 ldxdy '' mn 22 xy mn4'' xmnxyx N 1 1 Cslsnn mnn V nmn1 ECE 546 –Jose Schutt‐Aine 17 Parallel Plates z y 2a x d 2a Using N unknowns per plate, we get 2N 2N matrix equation: [ltt ][ltb ] [l] [lbt ][lbb ] Subscript ‘t’ for top and ‘b’ for bottom plate, respectively. ECE 546 –Jose Schutt‐Aine 18 Parallel Plates Matrix equation becomes tt tb t t llmn mn n g m 1 Solution: ttttbt mnll g mn charge on top plate Capacitance C 2V 1 t nns 2V top Using s=4b2 and all elements of gt V 1 C 2b2 ltt ltb mn mn ECE 546 –Jose Schutt‐Aine 19 Inductance Relation: = Li : flux stored by inductor L: inductance i: current through inductor v: voltage across inductor di d vt() L dt dt 1 t it() v ( ) d L 0 ECE 546 –Jose Schutt‐Aine 20 Inductance Magnetic Flux z I r H I R Idl Total flux linked Inductance = Current BdS S ECE 546 –Jose Schutt‐Aine 21 2‐D Isomorphism Electrostatics Magnetostatics znˆ ˆ A 0 znˆ ˆ tVi 0 t zi 1 qs nˆ V nˆ t Azi o J z ri t i o ri CV Q LI Consequence: 2D inductance can be calculated from 2D capacitance formulas ECE 546 –Jose Schutt‐Aine 22 2-D N-line LC Extractor using MOM s w t h r • Symmetric signal traces • Uniform spacing • Lossless lines • Uses MOM for solution ECE 546 –Jose Schutt‐Aine 23 Output from MoM Extractor Capacitance (pF/m) 118.02299 -8.86533 -0.03030 -0.00011 -0.00000 -8.86533 119.04875 -8.86185 -0.03029 -0.00011 -0.03030 -8.86185 119.04876 -8.86185 -0.03030 -0.00011 -0.03029 -8.86185 119.04875 -8.86533 -0.00000 -0.00011 -0.03030 -8.86533 118.02299 Inductance (nH/m) 312.71680 23.42397 1.83394 0.14361 0.01128 23.42397 311.76042 23.34917 1.82812 0.14361 1.83394 23.34917 311.75461 23.34917 1.83394 0.14361 1.82812 23.34917 311.76042 23.42397 0.01128 0.14361 1.83394 23.42397 312.71680 ECE 546 –Jose Schutt‐Aine 24 RLGC: Formulation Method Electrical and Computer Engineering y x z 3 (x,y,z) 350 m =1 y=d r n t 70 m n 2 2 y=dn-1 n-1 =3.2 100 m y=dn-2 t r 1 1 r=4.3 200 m (x o,y o,z o) 150 m y=d P.E.C. m P.E.C. m y=dm-1 m-1 y=dm-2 Closed-Form Spatial Green's Function * Computes 2-D and 3-D capacitance matrix in multilayered dielectric * Method is applicable to arbitrary polygon-shaped conductors * Computationally efficient • Reference – K. S. Oh, D. B. Kuznetsov and J. E. Schutt-Aine, "Capacitance Computations in a Multilayered Dielectric Medium Using Closed-Form Spatial Green's Functions," IEEE Trans. Microwave Theory Tech., vol. MTT- 42, pp. 1443-1453, August 1994. ECE 546 –Jose Schutt‐Aine 25 Multilayer Green's Function Optional Top Ground Plane y=dNd Nd, o Nd y=dNd-1 y=d2 y 2, o 2 y=d1 1, o 1 x y=0 Bottom Ground Plane ECE 546 –Jose Schutt‐Aine 26 Extraction Program: RLGC RLGC computes the four transmission line parameters, viz., the capacitance matrix C, the inductance matrix L, the conductance matrix G, and the resistance matrix R, of a multiconductor transmission line in a multilayered dielectric medium. RLGC features the following list of functions: h ECE 546 –Jose Schutt‐Aine 27 RLGC – Multilayer Extractor • Features – Handling of dielectric layers with no ground plane, either top or bottom ground plane (microstrip cases), or both top and bottom ground planes (stripline cases) – Static solutions are obtained using the Method of Moment (MoM) in conjunction with closed-form Green’s functions: one of the most accurate and efficient methods for static analysis – Modeling of dielectric losses as well as conductor losses (including ground plane losses – The resistance matrix R is computed based on the current distribution - more accurate than the use of any closed-form formulae – Both the proximity effect and the skin effect are modeled in the resistance matrix R. – Computes the potential distribution – Handling of an arbitrary number of dielectric layers as well as an arbitrary number of conductors. – The cross section of a conductor can be arbitrary or even be infinitely thin • Reference – K. S. Oh, D. B. Kuznetsov and J. E. Schutt-Aine, "Capacitance Computations in a Multilayered Dielectric Medium Using Closed-Form Spatial Green's Functions," IEEE Trans. Microwave Theory Tech., vol. MTT-42, pp. 1443-1453, August 1994. ECE 546 –Jose Schutt‐Aine 28 RLGC – General Topology 350 m r=1 70 m r=3.2 100 m r=4.3 200 m 150 m P.E.C. Three conductors in a layered medium. All conductor dimensions and spacing are identical. The loss tangents of the lower and upper dielectric layers are 0.004 and 0.001 respectively, the conductivity of each line is 5.8e7 S/m, and the operating frequency is 1 GHz ECE 546 –Jose Schutt‐Aine 29 3-Line Capacitance Results 3 t 2 2 t 1 1 P.E.C. 350 m r=1 70 m r=3.2 100 m =4.3 200 m 150 m r P.E.C. Capacitance Matrix (pF/m) 142.09 21.765 0.8920 145.33 23.630 1.4124 21.733 93.529 18.098 22.512 93.774 17.870 0.8900 18.097 87.962 1.3244 17.876 87.876 Delabare et al. RLGC Method ECE 546 –Jose Schutt‐Aine 30 Modeling Vias l1 Possible P.E.C.