Nonideal Conductor Models

Transcription

1 Nonideal Conductor Models 吳瑞北 Rm. 340, Department of Electrical Engineering url: cc.ee.ntu.edu.tw/~rbwu S. H. Hall et al., High-Speed Digital Designs, Chap.5 1

2 What will You Learn How to model lossy tx-lines? How to deal with tx-line loss due to finite conductivity of metal? How to calculate dc and skin-effect loss? How to deal with tx-line loss due to conductor surface roughness? How to model surface roughness loss?

3 Tx-Line Losses Conductor Losses DC losses in the conductor Frequency dependent conductor losses Lossy TEM Theory & Computations Surface Roughness 3

4 Driving forces by IC Nonideal Effects Higher speeds, with higher frequency content Smaller form factors, with shrinking dimensions High-speed impacts on design Some high-speed characteristics largely ignored in designs of the past becomes critical in modern times. Deal with technically difficult issues Contend with a greater number of variables 4

5 Transmission Line Losses (cont d) Three categories of losses Metal losses: normal metals not infinitely conductive Scattering losses: surface of metal not perfectly smooth Dielectric losses: Dipoles oscillating with applied time varying field takes energy Effects of losses Degrade amplitude, severe problems for long buses Degrade edge rates, significant timing push-outs Degrade waveform, severe ISI due to dispersive loss Ultimately a primary speed limiter of current technology 5

6 Incorporate Losses Into Circuit Model A series resistor, R, to account for conductor losses in both power and ground planes. A shunt resistor, G, to account for dielectric losses z R z L z C z G z 6

7 Conductor Losses 7

8 DC Resistive Losses Low freq. current spreads out as much as possible. DC losses dominated by cross sectional area & inverse of conductivity of signal conductor R DC A cross-section w wt t Current flows through entire cross section of signal conductor and ground plane Reference Plane DC loss by ground plane is negligible. DC losses of FR4 are very negligible 8

9 Typical DC Losses As dimensions shrinking, losses are often a first-order effect which degrades SI and deserves rigorous analysis. DC losses are of particular concern in small geometry conductors, very long lines, and multiload buses. Ex. : Cu 5810 For PCB line 6 Sm R dc 1 t L W t 0.05, W 1.5mm R 0.3 Ω/m For IC line t 5, W 15. m R.3 Ω/mm 9

10 AC Resistive Losses High-freq. current migrates towards periphery, skin effect. Coaxial Cable Cross Section at High Frequency Inner (signal) conductor Outer (Ground) conductor Areas of high current density Current flows in a smaller area than DC case. As such, the resistance will increase over DC 10

11 Line Impedance R A high f ( D) L( ) 1 f for Cu f m R( ) R&L of RG-58/U (AWG0) coax D 10 R 10 ( AWG 10)/ 0 ( AWG 10)/10 inch m/ft AWG0 D 0.8 mm D/ f 7 KHz 11

12 Propagation Constant RC region : Low - loss region : flat Skin effect f region : f j ( R jl) jc 1

13 Wave in Conductive Media Maxwell equations: E jh H Ji E j E E good conductor Wave eq.: Propagation in x direciton: d dx E j H j E; E E E j E 0; E 0; = j air 0 Conductor (,, ) 0 y z x 13

14 Electric filed E : E( x) E e E e e s Plane Wave Field x x j x s Propagation constant : 1 j j x Skin depth: e e x f Magnetic filed H : 1 xˆ 1 H ( x ) E E x x ˆ E j EEse j Z E j j Wave impedance Z H H E x 14

15 Amplitude Skin Depth, microns Skin Effect When field impinges upon conductor, field will penetrate conductor and be attenuated Remember signal travels between conductors Field amplitude decreases exponentially into a skin depth of conductor, defined as the penetration depth at a freq. where amplitude is attenuated 63% (e -1 ) of initial value Penetration into conductor 10 9 Skin Depth In Copper 8 Electromagnetic Wave X f 0 0.E+00 1.E+09.E+09 3.E+09 4.E+09 5.E+09 6.E+09 Frequency, Hz 15

16 Current Skin Effect Spatial View Fields induce currents flowing in the metal Total area of current flow can be approximated to be in one skin depth because total area below exponential curve can be equated to area of a square Area w J( y, z) J ( y) e Skin Depths 0 (1 j) z/ t w 16

17 Current Calculation of Skin-Effect Resistance Area w Skin Depths J ( y, z) J ( y) e 0 (1 j) z/ w w (1 j) z/ j I J ( y) dy e dz J ( y) dy w w w z/ (, ) 0( ) 0( ) Pd dy J y z dz J y dy e dz J y dy 4 w w w RAC 0( ) AC 0( ) 0( ) RAC Pd I J y dy R J y dy J y dy 4 t w Assume uniform current: J0( y) J0 RAC w J0 wj 0 w 17

18 Microstrip R due to Signal Conductor Assumptions on current flow: confined to on skin depth; while return path neglected concentrated in lower portion of conductor due to local fields w t E-fields R AC f f A w w w current_flow 18

19 Resistance, Ohms Microstrip Freq-Dependent R Estimates Resistance will stay at approximately the DC value until skin depth is less than conductor thickness, then it will vary with f Example of frequency dependent resistance R tot R DC R AC Frequency Tline parameter terms R R0 Rs f 0 0.E+00 1.E+09.E+09 3.E+09 4.E+09 5.E+09 6.E+09 Frequency, Hz Unphysical, since not an analytic function R0 ~ resistance/unit length Rs ~ resistance/sqrt(freq)/unit length 19

20 Microstrip R due to Return Path Return current in reference plane also contributes t w h I( y) I O 1 h 1 ( y / h) Effective width estimated for area of return current flow. R I ( y) dy ground 0 Aeffective I 0 y (Current Density in plane) h 0

21 Total Microstrip AC Resistance Total resistance is approximately sum of signal and ground path resistance R AC _ total R signal R ground R total 1 1 F /m w h This is an excellent back of the envelope formula for microstrip AC resistance 1

22 Empirical Formula Microstrip (by Collins) Derived using conformal mapping techniques Being not exact, it should only be used for estimates R R signal ground LR LR w 1 ln f t w w w w for h h h 1 for f w for w h ( h w) h h w h

23 Stripline Losses In stripline, fields are referenced to two planes. Total current distributes in both planes, and in both upper and lower portions of signal conductor I ( y ) 1 ( 1 y / h ) For example: in symmetrical stripline, area in which current travels increases by and R decreases by. This inspires the parallel microstrip model 3

24 Calculating Stripline Losses R ac _ strip R R R ( h1_ micro) ( h _ micro) R ( h1_ micro) ( h _ micro) t h1 w h where the resistances are calculated from the microstrip formulae at appropriate heights. 4

25 Surface Resistance for Microstrip Surface resistance (Rs) used to evaluate resistive properties of a metal AC loss R is proportional to square root of freq. 1 1 RAC f RS f w h 1 1 RS w h sec Rs is a constant that scales square root behavior Is caused by the skin loss phenomena Used in specialized T-line models (i.e., W-Element) 5

26 Lossy TEM Theory & Computations 6

27 Mechanism in Conductor Near conductor boundary ( x 0) H yh ˆ ( y) e 1 * 4 Lint I Wm 4 H H dxdy 4 H0( y) dy 0 = W stored in 0 x when PEC is placed at x m 0 0 J zj ˆ ( y) e E J (1 j) x/ (1 j) x/ J ( y) J ( y) e dx s 0 0 (1 j) x/ 0 as, J ( y) H ( y) s 1 j J 0( y) H0( y) Stored magnetic energy inside conductor Dissipated power inside conductor 1 1 * 1 ac 0 R I J J dxdy J ( y) dy 1 J ( y) 1 j 0 0( ) 0( ) int int H y dy H y dy L I R L 0 H E E y H 7 x

28 Incremental Inductance Rule (Wheeler, 194) At high frequencies, say, < 0.1t, current crowds to conductor surface within a skin depth of. L R R s(f) n L L L ( f ) L ext int ext L n δ/ h w h +δ w-δ t t-δ h 1 h 1 +δ δ/ 8

29 Lossy-TEM Modal Field Theory 1 H A; A za z E j A ; Total current: I E da z 1 Ez ds j n Parameters: d dz d dz M n1 m gnd ( R jl ) I mn mn n ˆ Free space: TEM ( E E ) ( x, y) e jkz E j A jk A z 0 z 0 t z 0 H H A zˆ 1 t t z Conductor: TM/LE ( j ) R.C.: z H J E E j E 0 E t z t t z z 0 const. in (x,y) t E j A d dz B.C. at Az z 1 ˆ 1 t t z t z H A z E zˆ j : (tang. H continuous) m j A E d dz z Az j n z Ez n const. at m Ref. and J. C. Yang, Boundary integral equation formulation of skin effect problems in multiconductor transmission lines, IEEE Trans. Magn., vol. 5, pp , July

30 Applications to Parallel Plates Solution of PDE s: E j E 0 E a e ( yh/) z z z Matching B.C. s: z 0 z A j A b y h j Az Ez d dz b 1 a V V a jaz n Ez n b a 1 d dz V h 1 h w y x d dz V I-V relation: 1 E 1 w/ z dez w w I ds dx a V j n j w/ dy j j 1 d dz d dz V ( R jl) I 1 h h V j (1 ) (1 ) h R jl j I w w w w w R R ; L Lint Lext ; Lint w 30 h

31 Parallel Plates (low-freq. limit) Solution of PDE s: for y d de dy 0 at y d E a d y h h h z h z cosh Matching B.C. s: I-V relation: for y : ja b y h h b a cosh d V V a h b a sinhd cosh d sinh d z d dz V 1 h w y x d dz V 1 Ez w dez dez w sinhd I ds V h j n j dy h dy h j cosh d sinh d y d y V h j d R jl j R I wtanhd w dc 1 (1 3 d ) d R j L int j wtanh d wd wd 3w low freq. limit Validate L int by solving static magnetic field and finding stored energy? 31 tanh j d d jl ac

32 Validation of Low-Freq. Inductance Current flow I J ( x, y) zˆ for h h y d wd Magnetic field h y d x H J I dh in conductor x wd H xh ˆ x( y); J z dy 0 in free space I h wd ( d - y) in conductor H x I w in free space Stored magnetic energy I wh I wd 1 Wm dx H dy LI h 3 d 0 x w w d 3 h L Lext Lint w d 3w w H x 3

33 Proximity Effect ( 4.7mm) X (cm) Rem: Proximity effect causes a slight increase in attenuation X (cm) 33

34 Example - Lossy Tx-Line 1 V 0. y Current distribution 1 z x unit: mm 34

35 Current Distribution vs. Frequency 35

36 Internal inductance Three-conductor Example 8,4, 8,4, 36

37 Surface Roughness 37

38 Surface Roughness alters Rs The formulae presented assumes a perfectly smooth surface The copper must be rough so it will adhere to the laminate Surface roughness can increase the calculated resistance 10-50% as well as frequency dependence proportions Increase the effective path length and decreases the area Tooth height is typically m in RMS value, peak with 11 m. Trace Skin-Depth Tooth structure (4-7 microns) Plane 38

39 Frequency Dependence Surface roughness is not a significant factor until skin depth approaches the tooth size (typically 100 MHz 300 MHz) At high freq., loss becomes unpredictable from regular geometric object because of heavy dependence on a random tooth structure. No longer varies with the root of freq. something else Measurement S 1 P P 1 P( f) P( f) 0log ; : received power : injected power 1 39

40 Example of Surface Roughness Measurements indicate that the surface roughness may cause the AC resistance to deviate from f 0.5 PCB Modeling Tooth Structure PCB X-section POOL Stackup PCB Performance right turns do not equal a right and a left. Fiberglass Bundles 40

41 Hammerstad Model [1980] Rac KHRs f ; measurement Hammerstad h RMS hrms 1.m Hammerstad coeff.: K H 1 h RMS 1 tan Skin-effect R & L R H ( f) KHRs f if t Rdc if t hrms 5.8m RH ( f) if t f LH ( f ) Lext RH( f t) if t f t 41

42 Hammerstad model breaks down in case of very rough copper foil. Other models needed. Surface Profile Measurement 4

43 Hemispherical Model The rough surface is characterized as random protrusions. Assume a TEM incident on the hemisphere at a grazing angle. Calculate total cross section tot of a sphere, then divide it by two gives power loss of hemisphere. Power loss absorbed by flat conducting plane is also considered. K hemi 1 1 tot A base 1 ; 4 Atile Atile f ( kr, r) tot 43

44 Implementation for Rough Surface Prolate spheroid model Surface model 1 r 3 e htooth bbase A A base tile d 1 b peaks base 44

45 Huray Model [009] K Huary N 1 n 1 4 tot 1 ; A tile Fig. 5-3 N is determined s.t. total surface area equals that of the hemispheroid constructed before. 1 bbaseh tooth sin 1 Alat ; 1 bbase htooth Alat N ; a sphere radius, e.g., 0.8m 4 a 45

46 Results & Comparisons Huray model predicts I.L. with error < 1.5dB up to 30GHz. Hammerstad model saturates at, not enough loss for rough copper. It should be used only for h RMS < m Hemisphere model overpredicts loss at middle freq. and slightly under-predicts at high freq. Fig

47 Did You Learn How to incorporate loss into tx-line model? How conductor loss will affect propagation constant and attenuation of tx-lines? How to calculate dc and skin-effect resistance for microstrip and striplines? How to describe change in em-field of tx-line due to conductor finite conductivity? How to model tx-line resistance due to conductor surface roughness? Can you distinguish Hammerstad and Huray model? 47

48 Further Reading -1 W. T. Weeks, L. L. Wu, M. F. McAllister, and A. Singh, Resistive and inductive skin effect in rectangular conductors, IBM J. Res. Develop., vol. 3, pp , Nov and J. C. Yang, Boundary integral equation formulation of skin effect problems in multiconductor transmission lines, IEEE Trans. Magn., vol. 5, pp , July A. J. Gruodis and C. S. Chang, Coupled lossy transmission line characterization and simulation, IBM J. Res. Develop., vol. 5, pp. 5-41, Jan R. Ding, L. Tsang, and H. Braunisch, Random rough surface effects in interconnects studied by small perturbation theory in waveguide model, Proc. IEEE EPEPS, 011, pp R. Ding, L. Tsang, and H. Braunisch, "Wave propagation in a randomly rough parallel plate waveguide," IEEE T-MTT, May

49 Further Reading - X. C. Guo, et al., "An analysis of conductor surface roughness effects on signal propagation for stripline interconnects, T-EMC, pp , Jun B. Curran, et al., "On the modeling, characterization, and analysis of the current distribution in PCB transmission lines with surface finishes, T-MTT, pp , Aug M. Y. Koledintseva, et al., "Method of effective roughness dielectric in a PCB: measurement and full-wave simulation verification, T-EMC, pp , Aug F. Bertazzi, et al., "Modeling the conductor losses of thick multiconductor coplanar waveguides and striplines: A conformal mapping approach," T-MTT, pp , April 016. A. V. Rakov, et al., "Quantification of conductor surface roughness profiles in printed circuit boards," T-EMC, pp , Apr