ECE 546 Lecture 04 Resistance, Capacitance, Inductance

Home , Capacitance

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   Pxn   22 0 otherwise 

x1 x2 xn    1 xxnn xx Txn   22 0 otherwise

x1 x2 xn Testing functions often (not always) chosen same as basis function.

ECE 546 –Jose Schutt‐Aine 13 Conducting Plate

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', ' VVaa

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 n1

ECE 546 –Jose Schutt‐Aine 16 Conducting Plate

Matrix equation:

N Vlf  mn n n1 Matrix element:

1 ldxdy '' mn  22 xy mn4'' xmnxyx

N 1 1 Cslsnn mnn V nmn1

ECE 546 –Jose Schutt‐Aine 17 Parallel Plates

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ll g  mn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

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

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:

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

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

Possible P.E.C. y=dn Nd y=dn-1

2 y y=d1 1 x l l2 3 y=0 Possible P.E.C. Medium Via

1.6 2

4 l2 4.02 50 l1 3.2

l 1 50 3 4 4.02

Via in multilayer medium Equivalent circuit

ECE 546 –Jose Schutt‐Aine 31 Modeling Discontinuities

l1 l l l1 l2

w1 w

e ce c l2 w2 Open Bend

l1 l3 l3 l1 l2 l l2 1 l1 l2

w1 w1 w2 e c l ce w2 2

Step T-Junction

ECE 546 –Jose Schutt‐Aine 32 3D Inductance Calculation Loop Inductance

I I

 a I ak I

 11    Lloop Bda   A da II I aa QUESTION: Can we associate inductance with piece of conductor rather than a loop? PEEC Method

ECE 546 –Jose Schutt‐Aine 33 Partial Inductance (PEEC) Approach QUESTION: Can we associate inductance with piece of conductor rather than a loop?

I I

 a I ak I   4 4 1  dli dl j Lloop          daidaj i1 j1aiaj 4 r  r  aia j li l j i j DEFINITION OF PARTIAL INDUCTANCE   1  dli dl j Lpij        daidaj aiaj 4 r  r  aia j li lj i j 4 4 Lloop   sij Lpij i1 j1

ECE 546 –Jose Schutt‐Aine 34 Circuit Element K [K]=[L]-1

• Better locality property

• 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 Locality of K Matrix

h 1 2345

11.4 4.26 2.54 1.79 1.38  103 34.1 7.80 4.31 3.76 4.26 11.4 4.26 2.54 1.79 34.1 114 31.6 6.67 4.31   []L 2.54 4.26 11.4 4.26 2.54  []K  7.80 31.6 115  31.6 7.80   1.79 2.54 4.26 11.4 4.26 4.31 6.67 31.6 114  34.1 1.38 1.79 2.54 4.26 11.4 3.76 4.31 7.80 34.1 103 

ECE 546 –Jose Schutt‐Aine 36 Package Inductance & Capacitance

Component Capacitance Inductance (pF) (nH) 68 pin plastic DIP pin† 435 68 pin ceramic DIP pin†† 720 68 pin SMT chip carrier † 27

† 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 Package Inductance & Capacitance

Component Capacitance Inductance (pF) (nH) 68 pin PGA pin†† 2 7 256 pin PGA pin†† 5 15 Wire bond 1 1 Solder bump 0.5 0.1

† 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 38 Metallic Conductors

 Area h Lengt

Re sist an ce : R Length R    Area Package level: Submicron level: W=3 mils W=0.25 microns R=0.0045 /mm R=422 /mm

ECE 546 –Jose Schutt‐Aine 39 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 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 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 Dielectric Materials

1 v  LC

Material Conductivity (-1-m-1) Germanium 2.2 Silicon 0.0016 Glass 10-10-10-14 Quartz 0.510-17

 tan  

ECE 546 –Jose Schutt‐Aine 43 Dielectric Materials

r

1 v  LC

Material r v(cm/s) Polyimide 2.5-3.5 16-19 Silicon dioxide 3.9 15 Epoxy glass (FR4) 5.0 13 Alumina (ceramic) 9.5 10

ECE 546 –Jose Schutt‐Aine 44 Conductivity of Dielectric Materials

r j i

Material Conductivity ( -1 m-1) Germanium 2.2 Silicon 0.0016 Glass 10-10 -10-14 Quartz 0.5 x 10-17

 Loss TANGENT : tan 

ECE 546 –Jose Schutt‐Aine 45 Combining Field and Circuit Solutions

Network Macromodel Field Solution Description Generation

Circuit  Bypass extraction Simulation procedure through the use of Y, Z, or S parameters (frequency domain)

ECE 546 –Jose Schutt‐Aine 46 Full‐Wave Methods   B E  Faraday’s Law of Induction t   D H  J  Ampère’s Law t  D   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 FDTD ‐ Formulation

 FDTD solves Maxwell’s equations in time-domain E 1  H t 0 H 1  E t 0 • Problem space is discretized • Derivatives are approximated as u uv()()  v uv  v  00 vv2 • Time stepping algorithm • Field values at all points of the grid are updated at each time step

ECE 546 –Jose Schutt‐Aine 48 Finite Difference Time Domain (FDTD)

z Ey

Hz y

x Ex Ex

Ez Ez

Ey Hx

Hy Hy

Ey Hx Ez

Ex Ex Hz

Ey Space Discretization

ECE 546 –Jose Schutt‐Aine 49 FDTD –Yee Algorithm

ct Enn ijk,, E11/21/2 H n ijk ,,  H n  ij ,   1, k  xx y z z z Ey ct nn1/2 1/2 y Hz HijkHijk,, ,,  1  x yy Ex Ex  z Ez Ez Ey Hx Hy Hy Ey Hx Ez Ex Ex Hz ct Ey H nnnn1/2ijkH,, 1/2  Eij ,  1, k Eijk ,, xxzz y ct Enn1/2  i,, jk 1 E i ,, jk z yy

ECE 546 –Jose Schutt‐Aine 50 2D-FDTD

Hz y

x 1 1 t 1 1 1 1 En i  ,j  En1i  ,j  Hn 1/ 2 i  ,j   H n1/2 i  ,j  x   x   z   z   2 2  oy  2 2 2 2  1 1 t 1 1 1 1 En  i, j    En1i, j    Hn1/ 2 i  ,j   H n1/2 i  ,j  y   y   z   z   2 2 ox  2 2 2 2 

n1/2  1 1 n 1/2  1 1 t  n  1  n  1   Hz i  ,j   Hz i  ,j   Ex i  ,j1  Ex i  ,j 2 2 2 2 oy  2 2  t  n  1 n  1 - Ey i 1, j    Ex i, j   ox  2 2 

ECE 546 –Jose Schutt‐Aine 51 Absorbing Boundary Condition: 2D-PML Formulation

Simulation Medium PML Medium

E H Ex Hz  x  z o Ex  o t y t y Ey H Ey Hz z   o E y  o t x t x E H E Ey Hz Ex y  z *H  x  o   o t z y x t y x y x   *  o o No reflection from PML interface

ECE 546 –Jose Schutt‐Aine 52 Importance of the PML  Example: Simulation of the sinusoidal point source

PML is “on”

PML is “off”

ECE 546 –Jose Schutt‐Aine 53 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 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 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 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 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): 2 tT0   Rectangular mesh pulse e spread Plane wave source with Gaussian distribution Perfectly matched layer (PML) for the ABC

ECE 546 –Jose Schutt‐Aine 58 3D FDTD for Single Microstrip Line

Computational domain size: 90x130x20 cells

(in x, y, and z directions, respectively) T=100

* Cell size 0.026 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 T=300 * Microstrip 9 cells wide

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. 3D FDTD for Patch Antenna

Microstrip antenna at Microstrip antenna at T=300 T=400

Patch dimensions 47 x 60 cells

ECE 546 –Jose Schutt‐Aine 60 Simulation of the Microstrip Antenna Frequency‐Dependent Parameters

fft(inc ) S ( ) 20 log abs • S for the patch antenna 11  11 fft(ref )

Our simulation By D. Sheen et. al

ECE 546 –Jose Schutt‐Aine 62 Simulation of Microstrip Structures

• Source setup:

• Microstrip Patch Antenna

ECE 546 –Jose Schutt‐Aine 63 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 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 Single Straight Microstrip • Simulation with length doubled (example of what happens when the mesh is bad)

ECE 546 –Jose Schutt‐Aine 66 Single Straight Microstrip • Simulation with the adjusted mesh

ECE 546 –Jose Schutt‐Aine 67 Meandered Microstrip Lines • Test boards were manufactured

ECE 546 –Jose Schutt‐Aine 68 Simulation and Measurements

• Scattering parameters for the m‐line #3

ECE 546 –Jose Schutt‐Aine 69 Comparison with ADS Momentum • The line was also simulated with Agilent ADS Momentum EM simulator

ECE 546 –Jose Schutt‐Aine 70 Comparison with ADS Momentum

• S21 parameters

ECE 546 –Jose Schutt‐Aine 71 References

• A. Taflove, S.C. Hagness, Computational Electrodinamics: The Finite – Difference Time‐Domain Method. 3‐d edition. Artech House Publishers, 2005. • D. Sullivan, Electromagnetic simulation using the FDTD method, IEEE Press series on RF and microwave technology, 2000. • 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 1990.

ECE 546 –Jose Schutt‐Aine 72