ECE 451 Macromodeling

Transcription

1 ECE 451 Macromodeling Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois ECE 451 Jose Schutt Aine 1

2 Blackbox Macromodeling Nonlinear Network 1 Nonlinear Network 3 Linear N-Port with Measured S-Parameters Nonlinear Network 4 Objective: Perform timedomain simulation of composite network to determine timing waveforms, noise response or eye diagrams Nonlinear Network 2 ECE 451 Jose Schutt Aine 2

3 Macromodel Implementation Frequency-Domain Data IFFT MOR Discrete Convolution Recursive Convolution STAMP Circuit Simulator Output ECE 451 Jose Schutt Aine 3

4 Blackbox Synthesis Motivations Only measurement data is available Actual circuit model is too complex Methods Inverse-Transform & Convolution IFFT from frequency domain data Convolution in time domain Macromodel Approach Curve fitting Recursive convolution

5 Blackbox Synthesis Terminations are described by a source vector G() and an impedance matrix Z Blackbox is described by its scattering parameter matrix S

6 Blackbox - Method 1 Scattering Parameters B( ) S( ) A( ) Terminal conditions A( ) B( ) TG( ) (1) (2) where and o o U ZZ U ZZ T U ZZ o 1 1 U is the unit matrix, Z is the termination impedance matrix and Z o is the reference impedance matrix ECE 451 Jose Schutt Aine 6

7 Blackbox - Method 1 Combining (1) and (2) 1 A( ) U S( ) TG( ) and 1 B( ) S( ) A( ) S( ) U S( ) TG( ) 1 V( ) A( ) B( ) U S( ) U S( ) TG( ) 1 I Z A B Z U S U S TG 1 1 ( ) o ( ) ( ) o ( ) ( ) ( ) v() t IFFT V ( ) i() t IFFT I( ) ECE 451 Jose Schutt Aine 7

8 Method 1 - Limitations No Frequency Dependence for Terminations Reactive terminations cannot be simulated Only Linear Terminations Transistors and active nonlinear terminations cannot be described Standalone This approach cannot be implemented in a simulator ECE 451 Jose Schutt Aine 8

9 Blackbox - Method 2 In frequency domain B=SA In time domain Convolution: b(t) = s(t)*a(t) s(t)*a(t) s(t )a( )d ECE 451 Jose Schutt Aine 9

10 Isolating a(t) Discrete Convolution When time is discretized the convolution becomes t1 s( t )* a( t ) s( 0 )a( t ) s( t )a( ) 1 Since a(t) is known for t < t, we have: t s(t)*a (t) s(t )a( ) 1 t1 H(t) s(t )a( ) : History 1 ECE 451 Jose Schutt Aine 10

11 Terminal Conditions Defining s'(0) =s(0)we finally obtain b(t) = s'(0)a (t) + H (t) a(t) = (t)b(t) + T(t)g(t) By combining these equations, the stamp can be derived ECE 451 Jose Schutt Aine 11

12 Stamp Equation Derivation The solutions for the incident and reflected wave vectors are given by: 1 a(t) 1 (t)s'(0) T(t)g(t) (t)h(t) b(t) = s'(0)a (t) + H (t) The voltage wave vectors can be related to the voltage and current vectors at the terminals 1 a( t ) v( t ) Zoi( t ) 2 1 b(t) v(t) Zoi(t) 2 ECE 451 Jose Schutt Aine 12

13 Stamp Equation Derivation From which we get 1 s'(0) v(t) Zoi(t) v(t) Zoi(t) H(t) 2 2 or Zoi( t ) s'(0 )Zoi( t ) 2H( t ) 1s'(0 ) v( t ) or 1s'(0) Z i(t) 1s'(0) v(t) 2H(t) which leads to o o o i(t) Z 1s'(0) 1s'(0) v(t) 2Z 1s'(0) H(t) ECE 451 Jose Schutt Aine 13

14 Stamp Equation Derivation i(t) can be written to take the form i( t ) Y v( t ) I stamp stamp in which stamp 1 o 1 Y Z 1s'(0) 1s'(0) and stamp o 1 1 I 2Z 1s'(0 ) H(t ) ECE 451 Jose Schutt Aine 14

15 Stamp Equations i( t ) Y v( t ) I stamp 1 o 1 Y Z 1s'(0) 1s'(0) stamp stamp 1 o stamp 1 I 2Z 1 s'(0 ) H(t ) ( Y Y ) v( t) I I g stamp g stamp ECE 451 Jose Schutt Aine 15

16 Effects of DC Data No DC Data Point With DC Data Point If low frequency data points are not available, extrapolation must be performed down to DC. ECE 451 Jose Schutt Aine 16

17 Effect of Low-Frequency Data 1 Port 1 - No DC Extrapolation Port 1 - With DC Extrapolation Port Port 2 - No DC Extrapolation Port 2 - With DC Extrapolation Port volts 0.4 Volts Time (ns) Time (ns) ECE 451 Jose Schutt Aine 17

18 Effect of Low-Frequency Data Calculating inverse Fourier Transform of: V( f) 2sin 2 ft 2 ft 0 No DC component (lowest freq: 10 MHz) 1.2 With DC component Volts -1 Volts Time (ns) Time (ns) Left: IFFT of a sinc pulse sampled from 10 MHz to 10 GHz. Right: IFFT of the same sinc pulse with frequency data ranging from 0-10 GHz. In both cases 1000 points are used ECE 451 Jose Schutt Aine 18

19 Convolution Limitations Frequency-Domain Formulation Y( ) H( )X( ) Time-Domain Formulation y(t) h(t)*x(t) Convolution Discrete Convolution t y(t) h(t)*y(t) h(t )y( )d 0 t1 1 t h( t )* x( t ) h( t )x( ) 1 H(t ) h(t )x( ) : History Computing History is computationally expensive Use FD rational approximation and TD recursive convolution ECE 451 Jose Schutt Aine 19

20 Frequency and Time Domains 1. For negative frequencies use conjugate relation V(-)= V*() 2. DC value: use lower frequency measurement 3. Rise time is determined by frequency range or bandwidth 4. Time step is determined by frequency range 5. Duration of simulation is determined by frequency step ECE 451 Jose Schutt Aine 20

21 Problems and Issues Discretization: (not a continuous spectrum) Truncation: frequency range is band limited F: frequency range N: number of points f = F/N: frequency step t = time step ECE 451 Jose Schutt Aine 21

22 Problems and Issues Problems & Limitations (in frequency domain) Discretization Consequences (in time domain) Time domain response will repeat itself periodically (Fourier series) Aliasing effects Solution Take small frequency steps. Minimum sampling rate must be the Nyquist rate Truncation in Frequency Time domain response will have finite time resolution (Gibbs effect) Take maximum frequency as high as possible No negative frequency values Time domain response will be complex Define negative frequency values and use V( f)=v*(f) which forces v(t) to be real No DC value Offset in time domain response, ringing in base line Use measurement at the lowest frequency as the DC value ECE 451 Jose Schutt Aine 22

23 Complex Plane An arbitrary network s transfer function can be described in terms of its s-domain representation s is a complex number s = + j The impedance (or admittance) or transfer function of networks can be described in the s domain as T() s m as m a s... asa n s b s bsb m1 m1 1 0 n1 n ECE 451 Jose Schutt Aine 23

24 T() s Transfer Functions m as m a s... asa n s b s bsb m1 m1 1 0 n1 n The coefficients a and b are real and the order m of the numerator is smaller than or equal to the order n of the denominator A stable system is one that does not generate signal on its own. For a stable network, the roots of the denominator should have negative real parts ECE 451 Jose Schutt Aine 24

25 Transfer Functions The transfer function can also be written in the form T() s a m sz sz... sz 1 2 sp sp... sp 1 2 m m Z 1, Z 2, Z m are the zeros of the transfer function P 1, P 2, P m are the poles of the transfer function For a stable network, the poles should lie on the left half of the complex plane ECE 451 Jose Schutt Aine 25

26 Model Order Reduction Large Network (>1,000 nodes) Reduced Order Model (< 30 poles) SPICE Y(t) v(t) = i(t) Y ( ) A1 L a1 i 1 j/ i1 ci 1 Y() V() = I() Order Reduction Y() = ~ Y() ~ Recursive Convolution ~ Y(t) v(t) = i(t) ECE 451 Jose Schutt Aine 26

27 Objective: Approximate frequency-domain transfer function to take the form: Methods Model Order Reduction H( ) A AWE Pade Pade via Lanczos (Krylov methods) Rational Function Chebyshev-Rational function Vector Fitting Method 1 L a1 i 1 j/ i1 ci 1 ECE 451 Jose Schutt Aine 27

28 Model Order Reduction (MOR) Question: Why use a rational function approximation? Answer: because the frequency domain relation L c k Y( ) H( )X( ) d X( ) k11 j / ck will lead to a time domain recursive convolution: where y() t dx( t T) y () t L pk k 1 () ( )1 ckt ckt y t a x tt e e y ( tt) pk k pk which is very fast! ECE 451 Jose Schutt Aine 28

29 Model Order Reduction Transfer function is approximated as H( ) d L k1 ck 1 j / ck In order to convert data into rational function form, we need a curve fitting scheme Use Vector Fitting ECE 451 Jose Schutt Aine 29

30 History of Vector Fitting (VF) Original VF formulated by Bjorn Gustavsen and Adam Semlyen* Time-domain VF (TDVF) by S. Grivet-Talocia Orthonormal VF (OVF) by Dirk Deschrijver, Tom Dhaene, et al Relaxed VF by Bjorn Gustavsen VF re-formulated as Sanathanan-Koerner (SK) iteration by W. Hendrickx, Dirk Deschrijver and Tom Dhaene, et al. * B. Gustavsen and A. Semlyen, Rational approximation of frequency responses by vector fitting, IEEE Trans. Power Del., vol. 14, no. 3, pp , Jul ECE 451 Jose Schutt Aine 30

31 Vector Fitting (VF) Vector fitting algorithm ( s) f ( s) ( s) N n 1 N n 1 c n s c s a n a n n d sh 1 N N c n c n d sh f ( s) f ( s). s a s a n1 n n1 n Avoid ill-conditioned matrix Guarantee stability Converge, accurate Solve for c, c, d, h n n With Good Initial Poles Can show* that the zeros of (s) are the poles of f(s) for the next iteration * B. Gustavsen and A. Semlyen, Rational approximation of frequency responses by vector fitting, IEEE Trans. Power Del., vol. 14, no. 3, pp , Jul ECE 451 Jose Schutt Aine 31

32 Examples 1.- DISC: Transmission line with discontinuities Length = 7 inches 2.- COUP: Coupled transmission line2 Line A u v Line B w x d x y z d x = 5 1/2 inches Frequency sweep: 300 KHz 6 GHz ECE 451 Jose Schutt Aine 32

33 DISC: Approximation Results DISC: Approximation order 90 ECE 451 Jose Schutt Aine 33

34 DISC: Simulations Microstrip line with discontinuities Data from 300 KHz to 6 GHz Exact 1.2 Exact B B A Observation: Good agreement A ECE 451 Jose Schutt Aine 34

35 COUP: Approximation Results COUP: Approximation order 75 Before Passivity Enforcement 35

36 Port 1: a Port 2: d Data from 300 KHz to 6 GHz COUP: Simulations a c b x d x y d 0.6 Exact 0.6 Convolution B 0.2 B A Observation: Good agreement A ECE 451 Jose Schutt Aine 36

37 Orders of Approximation S-parameter Approximation function Measurement Data S-parameter Approximation function Measurement Data S-parameter Approximation function Measurement Data Frequency Frequency Frequency Low order Medium order Higher order L T L T L T LT C T C T C T ECE 451 Jose Schutt Aine 37

38 Accurate:- over wide frequency range. Stable:- All poles must be in the left-hand side in s-plane or inside in the unit-circle in z-plane. Causal:- Hilbert transform needs to be satisfied. Passive:- H(s) is analytic MOR Attributes hn [ ] h[ n] h[ n] H( j) H ( j) jh ( j) e o R I H s H s () ( ), [ ( ) ( )] 0, [ ] 0 T T * z H s H s z s,for Y or Z-parameters. T * IH ( s ) H( s) 0, [ s ] 0,for S-parameters.

39 MOR Issues Bandwidth Low frequency data must be added Passivity Passivity enforcement Causality Causality enforcement High Order of Approximation Orders > 800 for some serial links Delay need to be extracted ECE 451 Jose Schutt Aine 39