ECE 451 Black Box Modeling

Transcription

1 Black Box Modeling Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois

2 Simulation for Digital Design Nonlinear Network 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 2

3 Macromodel Implementation Frequency-Domain Data IFFT MOR Discrete Convolution Recursive Convolution STAMP Circuit Simulator Output 3

4 Black Box 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 4

5 IFFT/Convolution Approach to Macromodels V I V g Z g2 V gi Z gi : : I i Blackbox [S] V i In frequency domain B=SA In time domain b(t) = s(t)*a(t) Convolution: s(t)*a(t) s(t )a( )d 5

6 Discrete Convolution When time is discretized the convolution becomes t s(t)*a (t) s(t )a( ) Isolating a(t) t s( t )* a( t ) s( )a( t ) s( t )a( ) Since a() is known for < t, we have: t H(t) s(t )a( ) : History 6

7 Termination Conditions Defining s'() =s()we finally obtain b(t) = s'()a (t) + H (t) a(t) = (t)b(t) + T(t)g(t) By combining these equations, the stamp can be derived 7

8 Stamp Generation from Convolution i( t ) Y v( t ) I stamp o Y Z s'() s'() stamp stamp o stamp I 2Z s'( ) H(t ) Termination i Black Box + I g Y g v Y stamp Istamp - ( Y Y ) v( t) I I g stamp g stamp 8

9 Inverse FFT Procedure. The sampled data for the blackbox will come with the following set of parameters: number of points M, start frequency f, stop frequency f 2 (and frequency step f). 2. We wish to process the data. For this purpose we will define our own set of parameters which may or may not be the same as those of the original data. Number of points N, start frequency f start, stop frequency f stop. From the discussion above we learned that the start frequency is best set to zero; therefore, all of the parameters will be set by the choice of two quantities: N and f stop. 3. Perform extrapolation and interpolation of raw data and map the M points from the raw data into the N points of the processed data 4. Determine the frequency step, f step =f stop /N. From this, the time step (t step =/(2Nf step )), the total duration of the simulation (t stop =Nt step ) are determined; the frequency-domain data points can now be be arranged as follows a b c d e f 5. Set the imaginary parts of the first and last points to zero: Imag{a}=, and Imag{g}= 9

10 Inverse FFT Procedure 6. Fold the data with respect to its conjugate mirror image in order to insure that the time-domain response will be pure real; this looks as follows (* indicates complex conjugate): g f* e* d* c* b* a b c d e f where g and a are now real quantities. The total number of points is N 2 =2N 7. Feed those N 2 points to IFFT routine to obtain inverse FFT. After the IFFT call, scale all the points in the returned array by dividing them by (N 2 t step ). The data now looks like o p q r s t u v w x y z 8. These points should all be real. They are the time-domain impulse response. 9. If everything is fine, proceed by keeping only the first N points of the sequence (o to t). They represent the impulse response of the data and will be used for the time-domain convolution

11 Importance of Low-Frequency Calculating inverse Fourier Transform of: V( f) 2sin 2 ft 2 ft No DC component (lowest freq: MHz) With DC component Volts - Volts Time (ns) Time (ns) Left: IFFT of a sinc pulse sampled from MHz to GHz. Right: IFFT of the same sinc pulse with frequency data ranging from - GHz. In both cases points are used

12 Recursive Convolution Limitations Convolution is slow Linear network block can be large Motivation for Recursive Convolution Faster Requires transfer function to have the form: H( ) A L a i j/ i ci 2

13 Time Domain Solution Using Recursive Convolution Given the frequency-domain relation: If the transfer function is written as H ( ) Y( ) H( ) X( ) A L i a i j / ci Then, the time domain relationship (for step invariant model) is: where L y( nt ) Ax( n K) T y pi ( nt ) i cit cit y ( nt) a ( e ) x ( nk ) T e y ( n) T pi i pi This is also called recursive convolution 3

14 Macromodel Approximation Objective: Approximate frequency-domain transfer function to take the form: H( ) A L a i j/ i ci Methods AWE Pade Pade via Lanczos (Krylov methods) Rational Function Chebyshev-Rational function Vector Fitting Method 4

15 Attributes for Macromodel Numerical Robustness Accuracy Passivity Causality User Flexibility Automatic selection of starting poles Automatic determination of order Compensation Features Frequency truncation No DC information Measurement noise 5

16 Causality - Hilbert Transform Analysis Enforce causality ht ( ), t Every function can be considered as the sum of an even function and an odd function ht () h() t h() t e o he () t h() t h( t) 2 ho () t h() t h( t) 2 Even function Odd function h () t o he (), t t he (), t t h ( t) sgn( t) h ( t) o e 6

17 Causality - Hilbert Transform Analysis ht () h() t sgn() th() t e e In frequency domain this becomes H ( f) He( f) * He( f) j f H( f) H ( f) jhˆ ( f ) e e Imaginary part of transfer function is related to the real part through the Hilbert transform Hˆ ( f ) is the Hilbert transform of H ( f ) e x( ) xˆ( t) x()* t d t t e Discrete Hilbert Transform HT ( h ) hˆ n k 2 2 n odd n even hn, k even k n hn, k odd k n 7

18 HT for Coupled Lines: 3 KHz 6 GHz Actual is red, HT is blue Actual Transform Real Part of S Actual Transform Imaginary Part of S Real(S) Imag(S) Actual Transform Actual Transform A A Imag(S2) Real(S2).5.5 Real(S2) Imag(S2) Observation: Good agreement A A 8

19 HT for Via: MHz 2 GHz Actual Transform Actual is red, HT is blue Actual Transform Real(S) - Via Imag(S) - Via Actual -. Actual Actual Transform A -.5 Actual Transform Real(S2) - Via A Imag(S2) - Via.5.5 Real(S2) Imag(S2) A A Observation: Poor agreement (because frequency range is limited) 9

20 Orders of Approximation S-parameter Approximation function Measurement Data S-parameter Approximation function Measurement Data S-parameter Approximation function Frequency Frequency Measurement Data Frequency Low order Medium order Higher order L T L T L T LT C T C T C T 2

21 Measurement Data.- 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 /2 inches Frequency sweep: 3 KHz 6 GHz 2

22 DISC: Approximation Results DISC: Approximation order 9 22

23 COUP: Approximation Results COUP: Approximation order 75 Before Passivity Enforcement 23

24 Results After Passivity Check Magnitude plot of Y, measured data and the 3-th order rational approximation with passivity check TL with capacitive discontinuity Phase plot of Y, measured data and the 3-th order rational approximation with passivity check 24

25 Coupled Lines Measured on ANA Port : a Port 2: d Data from 3 KHz to 6 GHz a b x d x y c d.6 Exact.6 Convolution B.2 B A Observation: Good agreement A 25

26 Microstrip with Steps Measured on ANA Microstrip line with discontinuities Data from 3 KHz to 6 GHz 2.2 Exact.2 Exact B B A Observation: Good agreement A 26