ECE 546 Lecture 04 Resistance, Capacitance, Inductance

Transcription

1 ECE 546 Lecture 04 Resistance, Capacitance, Inductance Spring 2018 Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois ECE 546 Jose Schutt Aine 1

2 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

3 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

4 Field Solvers History 1960s Conformal mapping techniques Finite difference methods (2-D Laplace eq.) Variational methods 1970s Boundary element method Finite element method (2-D) Partial element equivalent circuit (3-D) 1980s Time domain methods (3-D) Finite element method (3-D) Moment method (3-D) rpeec method (3-D) 1990s Adapting methods to parallel computers Including methods in CAD tools 2000s Incorporation of Passivity Incorporation of Causality 2010s Stochastic Techniques Multiphysics Tools ECE 546 Jose Schutt Aine 4

5 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

6 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 () ( ) Capacitance vt i d C 0 ECE 546 Jose Schutt Aine 6

7 Capacitance V + d C = oa d A : area o : permittivity For more complex capacitance geometries, need to use numerical methods ECE 546 Jose Schutt Aine 7

8 Capacitance Calculation () r g(, rr') ( r ') dr ' ( r) potential ( known) 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

9 METHOD OF MOMENTS Operator equation L(f) = g L = integral or differential operator f = unknown function g = known function Expand unknown function f f f n n n ECE 546 Jose Schutt Aine 9

10 n Matrix equation METHOD OF MOMENTS in terms of basis functions f n, with unknown coefficients n to get nl( fn ) n Finally, take the scalar or inner product with testing of weighting functions w m : g n wm, Lfn wm, g l g mn n m ECE 546 Jose Schutt Aine 10

11 METHOD OF MOMENTS l mn n 1 2. Solution for weight coefficients w1, Lf1 w1, Lf2... w2, Lf1 w2, Lf l g n nm w1, g g m w2, g. m ECE 546 Jose Schutt Aine 11

12 Moment Method Solution D E xyz,, x', y', z' dx ' dy ' dz ' 4 R 2 L Green s function G: LG = ' ' ' R xx y y zz ECE 546 Jose Schutt Aine 12

13 Subdomain bases 1 Pxn 0 Basis Functions xn x xn 2 2 otherwise T x x 1 x 2 x n n 1 x xn x xn otherwise x 1 x 2 x n Testing functions often (not always) chosen same as basis function. ECE 546 Jose Schutt Aine 13

14 Conducting Plate Z Y 2b 2a S n 2b X 2a conducting plate x, yz, a dx' a a a dy ' x', y ', z ' 4 R = charge density on plate ECE 546 Jose Schutt Aine 14

15 Setting = V on plate Conducting Plate ' ' 2 2 R xx y y V a a dx' a a x', y ', z' dy ' 4 ' ' x x y y 2 2 for x < a; y < a Capacitance of plate: a q 1 C dx' dy' x', y', z' V V a a a ECE 546 Jose Schutt Aine 15

16 Conducting Plate Basis function P n s s 1 xm x xm 2 2 s s Pnxm, yn1 yn y yn otherwise Representation of unknown charge N xy, f n n n1 ECE 546 Jose Schutt Aine 16

17 Conducting Plate Matrix equation: Matrix element: N V lmn f n1 n 1 lmn dx' dy' xm yn 4 ' ' x x y x m 2 2 n 1 N 1 n n mn n n1 mn C s l s V ECE 546 Jose Schutt Aine 17

18 Parallel Plates z y 2a d x 2a Using N unknowns per plate, we get 2N 2N matrix equation: [l] [ltt ] [l tb ] [l bt ] [l bb ] Subscript t for top and b for bottom plate, respectively. ECE 546 Jose Schutt Aine 18

19 Matrix equation becomes Parallel Plates l tt l tb t t mn mn n g m t tt tb t m l l g mn n Solution: 1 Capacitance Using s=4b 2 and all elements of C 2b 2 mn C charge on top plate 2V 1 t ns 2V top g t V l tt l tb 1 mn n ECE 546 Jose Schutt Aine 19

20 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 Inductance ECE 546 Jose Schutt Aine 20

21 Inductance Magnetic Flux z I I Idl r R H S BdS Inductance = Total flux linked Current ECE 546 Jose Schutt Aine 21

22 2 D Isomorphism Electrostatics Magnetostatics zˆ nˆ V 0 t i zˆ nˆ A 0 t zi nˆ q s ritv i o nˆ 1 t Azi oj ri z CV Q LI Consequence: 2D inductance can be calculated from 2D capacitance formulas ECE 546 Jose Schutt Aine 22

23 2-D N-line LC Extractor using MOM s t w h r Symmetric signal traces Uniform spacing Lossless lines Uses MOM for solution ECE 546 Jose Schutt Aine 23

24 Output from MoM Extractor Capacitance (pf/m) Inductance (nh/m) ECE 546 Jose Schutt Aine 24

25 RLGC: Formulation Method Electrical and Computer Engineering y z x y=d n y=d n-1 y=d n-2 n n-1 (x,y,z) t 2 t 1 70 m r =1 r = m 100 m y=d m y=d m-1 y=d m-2 m m-1 (x o,y o,z o ) P.E.C. 150 m r =4.3 P.E.C. 200 m 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 , August ECE 546 Jose Schutt Aine 25

26 Multilayer Green's Function Optional Top Ground Plane y=d Nd Nd, o N d y=d Nd-1 2, o 2 y=d 2 y 1, o 1 y=d 1 y=0 x Bottom Ground Plane ECE 546 Jose Schutt Aine 26

27 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

28 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 , August ECE 546 Jose Schutt Aine 28

29 RLGC General Topology 70 m r =1 350 m r = m 150 m r =4.3 P.E.C. 200 m Three conductors in a layered medium. All conductor dimensions and spacing are identical. The loss tangents of the lower and upper dielectric layers are and respectively, the conductivity of each line is 5.8e7 S/m, and the operating frequency is 1 GHz ECE 546 Jose Schutt Aine 29

30 3-Line Capacitance Results 3 2 t 2 1 t 1 P.E.C. 70 m r =1 r = m 100 m 150 m r =4.3 P.E.C. 200 m Capacitance Matrix (pf/m) Delabare et al. RLGC Method ECE 546 Jose Schutt Aine 30

31 Modeling Vias l 1 Possible P.E.C. N d y=dn y=dn-1 l 2 l 3 Possible P.E.C. 2 1 y=d1 y=0 y x Via Medium l l l 3 Via in multilayer medium Equivalent circuit ECE 546 Jose Schutt Aine 31

32 Modeling Discontinuities l l l 1 l 1 l 2 w w 1 c e w 2 l 2 c e Open Bend l 1 l2 l 1 l 2 l 1 l 3 l 3 l 1 l 2 w 1 w 2 w 1 c e w2 l 2 c e Step T-Junction ECE 546 Jose Schutt Aine 32

33 3D Inductance Calculation Loop Inductance I I I a I a k 1 1 L B da A da loop I I I a QUESTION: Can we associate inductance with piece of conductor rather than a loop? PEEC Method a ECE 546 Jose Schutt Aine 33

34 Partial Inductance (PEEC) Approach QUESTION: Can we associate inductance with piece of conductor rather than a loop? I I L loop 4 4 i1 j1 I 1 a i a j a a i a j l i L pij 1 a i a j 4 L loop a i l j I a k d l i dl j 4 a j l i l j 4 4 s ij L pij i1 j1 r i r da i da j j DEFINITION OF PARTIAL INDUCTANCE dl i dl j r i r da i da j j ECE 546 Jose Schutt Aine 34

35 Better locality property Circuit Element K [K]=[L] -1 Leads to sparser matrix Diagonally dominant Allows truncation of far off-diagonal elements Better suited for on-chip inductance analysis ECE 546 Jose Schutt Aine 35

36 Locality of K Matrix l h w s [ L] [ K] ECE 546 Jose Schutt Aine 36

37 Package Inductance & Capacitance Component Capacitance Inductance (pf) (nh) 68 pin plastic DIP pin pin ceramic DIP pin pin SMT chip carrier 2 7 No ground plane; capacitance is dominated by wire-to-wire component. With ground plane; capacitance and inductance are determined by the distance between the lead frame and the ground plane, and the lead length. ECE 546 Jose Schutt Aine 37

38 Package Inductance & Capacitance Component Capacitance Inductance (pf) (nh) 68 pin PGA pin pin PGA pin 5 15 Wire bond 1 1 Solder bump No ground plane; capacitance is dominated by wire-towire component. With ground plane; capacitance and inductance are determined by the distance between the lead frame and the ground plane, and the lead length. ECE 546 Jose Schutt Aine 38

39 Metallic Conductors Area Length Re sist an ce : R Length R Area Package level: W=3 mils R= /mm Submicron level: W=0.25 microns R=422 /mm ECE 546 Jose Schutt Aine 39

40 Metallic Conductors Metal Conductivity -1 m 10-7 ) Silver 6.1 Copper 5.8 Gold 3.5 Aluminum 1.8 Tungsten 1.8 Brass 1.5 Solder 0.7 Lead 0.5 Mercury 0.1 ECE 546 Jose Schutt Aine 40

41 Dielectrics Dielectrics contain charges that are tightly bound to the nuclei Charges can move a fraction of an atomic distance away from equilibrium position Electron orbits can be distorted when an electric field is applied E ECE 546 Jose Schutt Aine 41

42 Dielectrics Charge density within volume is zero Surface charge density is nonzero sp D= o (1+ e )E=E ECE 546 Jose Schutt Aine 42

43 Dielectric Materials v 1 LC Material Conductivity ( -1 -m -1 ) Germanium 2.2 Silicon Glass Quartz tan ECE 546 Jose Schutt Aine 43

44 Dielectric Materials r v 1 LC Material r v(cm/s) Polyimide Silicon dioxide Epoxy glass (FR4) Alumina (ceramic) ECE 546 Jose Schutt Aine 44

45 Conductivity of Dielectric Materials r j i Material Conductivity ( -1 m -1 ) Germanium 2.2 Silicon Glass Quartz 0.5 x Loss TANGENT : tan ECE 546 Jose Schutt Aine 45

46 Combining Field and Circuit Solutions Field Solution Network Description Macromodel Generation Bypass extraction procedure through the use of Y, Z, or S parameters (frequency domain) Circuit Simulation ECE 546 Jose Schutt Aine 46

47 Full Wave Methods B E t H J D t D Faraday s Law of Induction Ampère s Law Gauss Law for electric field B 0 Gauss Law for magnetic field FDTD: Discretize equations and solve with appropriate boundary conditions ECE 546 Jose Schutt Aine 47

48 FDTD Formulation FDTD solves Maxwell s equations in time-domain E 1 H t H t Problem space is discretized Derivatives are approximated as u v 0 1 E 0 uv ( 0 v) uv ( 0 v) 2v Time stepping algorithm Field values at all points of the grid are updated at each time step ECE 546 Jose Schutt Aine 48

49 Finite Difference Time Domain (FDTD) z E y y H z x E x E x Ez Ez E y H x H y H y E y H x Ez Ex H z E x E y Space Discretization ECE 546 Jose Schutt Aine 49

50 FDTD Yee Algorithm x z Ex Ey y Hy Hz Ez Hx Ey n n c t n n Ex i j k Ex Hz i j k Hz y i j k c t n1/2 1/2 Hy i j k Hy z i j k Ex Hy 1 1/2 1/2,,,,, 1, Ez n,,,, 1 Ex Hx Ey Hz Ez Ey Ex n1/2 n 1/2 c t n n H x i j k Hx Ez i j k Ez i j k y c t n1/2 n E,, 1,, y i jk Ey i jk z,,, 1,,, ECE 546 Jose Schutt Aine 50

51 2D-FDTD E y E x H z y x E n x i 1 2,j E x n1 i 1 2,j t o y H z n 1/2 i 1 2,j 1 2 H z n1/2 i 1 2,j 1 2 E n y i, j 1 2 E y n1 i, j 1 2 t o x H z n1/2 i 1 2,j 1 2 H z n1/2 i 1 2,j 1 2 n1/2 H 1 z i 2,j 1 2 H z n 1/2 i 1 2,j 1 2 t o y E x n i 1 2,j1 E x n i 1 2,j t - o x E n 1 y i 1, j 2 E n 1 x i, j 2 ECE 546 Jose Schutt Aine 51

52 Absorbing Boundary Condition: 2D-PML Formulation Simulation Medium PML Medium o E x t H z y o E x t E x H z y o E y t H z x o E y t E y H z x o H z t E x y E y x y o H z t *H z E x y E y x No reflection from PML interface x o * o ECE 546 Jose Schutt Aine 52

53 Importance of the PML Example: Simulation of the sinusoidal point source PML is on PML is off ECE 546 Jose Schutt Aine 53

54 Some Features of the FDTD Advantages FDTD is straightforward (fully explicit) Versatile (universal formulation) Time domain (response at all frequencies can be obtained from a single simulation) EM fields can be easily visualized Issues Resource hungry (fields through the whole problem space are updated at each step) Discretization errors Time domain data is not immediately useful Problem space has to be truncated ECE 546 Jose Schutt Aine 54

55 Pros of The FDTD Method FDTD directly solves Maxwell s equations providing all information about the EM field at each of the space sells at every time step Being a time domain technique, FDTD directly calculates the impulse response of an electromagnetic system. Therefore? A single FDTD simulation can provide either ultrawideband temporal waveforms or the sinusoidal steady state response at any frequency within the excitation spectrum FDTD uses no linear algebra Being a time domain technique, FDTD directly calculates the nonlinear response of an electromagnetic system ECE 546 Jose Schutt Aine 55

56 Cons of The FDTD Method Computationally expensive, requires large random access memory. At each time step values of the fields at each point in space are updated using values from the previous step FDTD works well with regular uniform meshes but the use of regular uniform meshes leads to staircasing. Implementation of nonuniform meshes, on the other hand, requires special mesh generation software and can lead to additional computer operations and instabilities Requires truncation of the problem space in a way that does not create reflection errors ECE 546 Jose Schutt Aine 56

57 Numerical Dispersion Occurs because of the difference between the phase speed of the wave in the real world and the speed of propagation of the numerical wave along the grid Distortion of the pulse propagating over the grid (time domain data is recorded at different reference points) ECE 546 Jose Schutt Aine 57

58 Setting Up a Simulation Main steps: Discretize the problem space create a mesh Set up the source of the incident field Truncate the problem space create the absorbing boundary conditions (ABC) We are using (mainly): Rectangular mesh Plane wave source with Gaussian distribution Perfectly matched layer (PML) for the ABC pulse e t0 T spread 2 ECE 546 Jose Schutt Aine 58

59 3D FDTD for Single Microstrip Line Computational domain size: 90x130x20 cells (in x, y, and z directions, respectively) T=100 * Cell size cm * Source plane at y = 0 * Ground plane at z = 0 * Duroid substrate with relative permittivity 2.2. Electric field nodes on interface between duroid and free space use average permittivity of media to either side. * Substrate 3 cells thick * Microstrip 9 cells wide T=300 Figures on the left show a pulse propagating along the microstrip line. A Gaussian pulse is used for excitation. A voltage source is simulated by imposing the vertical Ez field in the area underneath the strip.

60 3D FDTD for Patch Antenna Microstrip antenna at T=300 Microstrip antenna at T=400 Patch dimensions 47 x 60 cells ECE 546 Jose Schutt Aine 60

61 Simulation of the Microstrip Antenna

62 Frequency Dependent Parameters fft( inc) S 11 for the patch antenna S 11 ( ) 20 logabs fft( ref ) Our simulation By D. Sheen et. al ECE 546 Jose Schutt Aine 62

63 Simulation of Microstrip Structures Source setup: Microstrip Patch Antenna ECE 546 Jose Schutt Aine 63

64 Microstrip Coupler Branch line coupler Scattering parameters of the branch line coupler Our simulation By D. Sheen et. al ECE 546 Jose Schutt Aine 64

65 Single Straight Microstrip Comparison with measured data Comparison is only qualitative, since parameters used correspond to the line with (length/2) ECE 546 Jose Schutt Aine 65

66 Single Straight Microstrip Simulation with length doubled (example of what happens when the mesh is bad) ECE 546 Jose Schutt Aine 66

67 Single Straight Microstrip Simulation with the adjusted mesh ECE 546 Jose Schutt Aine 67

68 Meandered Microstrip Lines Test boards were manufactured ECE 546 Jose Schutt Aine 68

69 Simulation and Measurements Scattering parameters for the m line #3 ECE 546 Jose Schutt Aine 69

70 Comparison with ADS Momentum The line was also simulated with Agilent ADS Momentum EM simulator ECE 546 Jose Schutt Aine 70

71 Comparison with ADS Momentum S 21 parameters ECE 546 Jose Schutt Aine 71

72 References A. Taflove, S.C. Hagness, Computational Electrodinamics: The Finite Difference Time Domain Method. 3 d edition. Artech House Publishers, D. Sullivan, Electromagnetic simulation using the FDTD method, IEEE Press series on RF and microwave technology, D.M. Sheen, S.M. Ali, M.D. Abouzahra, J.A. Kong, Application of the Three Dimensional Finite Difference Time Domain Method to the Analysis of Planar Microstrip Circuits, IEEE Trans. Microwave Theory Tech., vol. 38, no 7, July ECE 546 Jose Schutt Aine 72