ECE 546 Lecture 15 Circuit Synthesis

Transcription

1 ECE 546 Lecture 15 Circuit Synthesis Spring 018 Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois ECE 546 Jose Schutt Aine 1

2 MOR via Vector Fitting Rational function approximation: N cn f ( s) d sh s a n 1 n Introduce an unknown function σ(s) that satisfies: Poles of f(s) = zeros of σ(s): ( sf ) ( s) N cn d sh s a n 1 n N ( s) c n f ( s) N n 1 1 s a n 1 n n1 N N cn n 1 n n 1 n N 1 cn d sh s z s a 1 s z s a n n Flip unstable poles into the left half plane. Im s-domain Re ECE 546 Jose Schutt Aine

3 Blackbox Formulation Transfer function is approximated H( ) d L k1 ck 1 j / Using curve fitting technique (e.g. vector fitting) In the time domain, recursive convolution is used ck 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 Recursive convolution is fast ECE 546 Jose Schutt Aine 3

4 Passivity Enforcement State space form: Hamiltonian matrix: x Ax Bu y Cx Du M T T T T C LC A C DKB T T A BKD C BKB T T 1 1 K I D D L I DD Passive if M has no imaginary eigenvalues. Sweep: H eig I S( j) S( j) Quadratic programming: Minimize (change in response) subject to (passivity compensation). T min vec( ΔC) H vec( ΔC) subject to = G vec( ΔC). ECE 546 Jose Schutt Aine 4

5 Macromodel Circuit Synthesis Use of Macromodel Time-Domain simulation using recursive convolution Frequency-domain circuit synthesis for SPICE netlist ECE 546 Jose Schutt Aine 5

6 Macromodel Circuit Synthesis Objective: Determine equivalent circuit from macromodel representation Motivation Circuit can be used in SPICE Goal Generate a netlist of circuit elements ECE 546 Jose Schutt Aine 6

7 Circuit Realization Circuit realization consists of interfacing the reduced model with a general circuit simulator such as SPICE Model order reduction gives a transfer function that can be presented in matrix form as or S s Y s s11 s s1 N s sn1 s snn s y11 s y1 N s yn1 s ynn s ECE 546 Jose Schutt Aine 7

8 Method 1: Y-Parameter/MOR* Each of the Y-parameters can be represented as y(s) ij d L k1 ak s p where the a k s are the residues and the p k s are the poles. d is a constant k *Giulio Antonini "SPICE Equivalent Circuits of Frequency- Domain Responses", IEEE Transactions on Electromagnetic Compatibility, pp 50-51, Vol. 45, No. 3, August 003. ECE 546 Jose Schutt Aine 8

9 Equivalent-Circuit Extraction Macromodel is curve-fit to take the form Constant term d Real Poles Complex Poles Y(s) d L k1 rk s p Need to find equivalent circuit associated with k ECE 546 Jose Schutt Aine 9

10 Y-Parameter - Circuit Realization Each of the I ij can be realized with a circuit having the following topology: The resulting current sources can then be superposed for the total current I i leaving port i All Y parameters are treated as if they were one-port Y parameters ECE 546 Jose Schutt Aine 10

11 Y-Parameter - Circuit Realization We try to find the circuit associated with each term: L aijk y(s) ij dij s p k1 For a given port i, the total current due to all ports with voltagesv j (j=1,,p) is given by P P P L aijk Ii yijvj dijvj V j s p j1 j1 j1 k1 ijk For each contributing port, with voltage i, the total current due to a voltage V j at port j is given by L aijk Iij dijvj V j s p k1 ijk ijk ECE 546 Jose Schutt Aine 11

12 Y-Parameter - Circuit Realization We try to find the circuit associated with each term: y(s) 1. Constant term d ij d L k1 s a k p k y ijd(s) d. Each pole-residue pair y (s) ijk a s k p k ECE 546 Jose Schutt Aine 1

13 Circuit Realization Constant Term R 1 d ECE 546 Jose Schutt Aine 13

14 Circuit Realization Pole/Residue In the pole-residue case, we must distinguish two cases (a) Pole is real y (s) ijk s a k p k (b) Complex conjugate pair of poles y (s) ijk k jk k jk s j s j k k k k In all cases, we must find an equivalent circuit consisting of lumped elements that will exhibit the same behavior ECE 546 Jose Schutt Aine 14

15 Case (a) - Real Pole Consider the circuit shown above. The input impedance Z as a function of the complex frequency s can be expressed as: Z sl R Y s L 1/ a k from circuit a s p 1/ L Ŷ(s) k s R/ L Comparing Y(s) and Ŷ(s) yields the solution R p k / ak from pole and residue k ECE 546 Jose Schutt Aine 15

16 Circuit Realization - Complex Poles Each term associated with a complex pole pair in the expansion gives: r1 r Yˆ s p s p 1 Where r 1, r, p 1 and p are the complex residues and poles. They satisfy: r 1 =r * and p 1 =p * It can be re-arranged as: Yˆ r r / 1 s r p r p r r s s p p p p 1 1 ECE 546 Jose Schutt Aine 16

17 Circuit Realization - Complex Poles The pole/residue representation of the Y parameters is given by: Yˆ r r 1 1 1/ 1 s r p r p r r s s p p p p 1 1 p pp 1 product of poles DEFINE a r r 1 sum of residues g p p 1 sum of poles x rp rp 1 1 cross product ˆ s x/ a Y a s sg p ECE 546 Jose Schutt Aine 17

18 Circuit Realization - Complex Poles Consider the circuit shown above. The input impedance Z as a function of the complex frequency s can be expressed as: 1 Z slr slr 1 1 1/ R sc 1sCR R Z 1 R sl scr R 1 1 scr ECE 546 Jose Schutt Aine 18

19 Circuit Realization - Complex Poles Y LCR s s CR s 1/ CR L CRR LR C 1 R 1 R LR C Y 1 s1/ CR L L CRR R R 1 s s LRC LRC 1 ECE 546 Jose Schutt Aine 19

20 Circuit Realization - Complex Poles Comparing Y 1 s1/ CR L L CRR R R 1 s s LRC LRC 1 We can identify the circuit elements with ˆ s x/ a Y a s sg p L 1/ a R x a 1 g a R p x g pa 1 g C x a a x a x ECE 546 Jose Schutt Aine 0

21 Circuit Realization - Complex Poles L 1/ a R x a 1 g a R p x g pa 1 g C x a a x a x ECE 546 Jose Schutt Aine 1

22 Negative Elements In the circuit synthesis process, it is possible that some circuit elements come as negative. To prevent this situation, we add a contribution to the real parts of the residues of the system. In the case of a complex residue, for instance, assume that r1 r Yˆ s p s p 1 Yˆ r r s p1 s p s p1 s p 1 Augmented Circuit Compensation Circuit Can show that both augmented and compensation circuits will have positive elements ECE 546 Jose Schutt Aine

23 Why S Parameters? Y-Parameter Test line: Zc, l Y 1 e 11 l Z (1 e ) c Z c : microstrip characteristic impedance : complex propagation constant l : length of microstrip Y 11 can be unstable short Reference Line Zo S-Parameter Test Line Test line: Zc, l ( 1 e ) S 11 l 1 e Z c Z c Z o Z o Reference Line S 11 is always stable Zo ECE 546 Jose Schutt Aine 3

24 Y Versus S Parameters S Y Impulse Responses Observation: S-parameters decay rapidly; Y parameters do not. ECE 546 Jose Schutt Aine 4

25 Optimizing Reference System o o S ZZ I ZZ I 1 o using Z I S I S Z 1.5 S11 - Linear Magnitude Z o = as reference S11 1 using 0.5 S11-50 Ohm S11 - Zref Z o = as reference Frequency (GHz) ECE 546 Jose Schutt Aine 5

26 Method : S-Parameter /MOR 1 A V Z I i i o i Need equivalent circuit for S ijk 1 Bi ViZoIi All S parameters are treated as if they were one-port S parameters ECE 546 Jose Schutt Aine 6

27 Strategy For a given circuit, a relationship between the input admittance Y ijk (s) of the circuit and the associated one-port S-parameter representation S ijk (s) can be described by S (s) ijk Y Y o o Y (s) ijk Y (s) ijk 1 S (s) ijk Y ijk(s) Yo 1 S ijk (s) Y o is the reference admittance ECE 546 Jose Schutt Aine 7

28 Equivalent-Circuit Extraction Macromodel is curve-fit to take the form S( s ) d L k1 s r k p k Need to find equivalent circuit associated with Constant term d Real Poles Complex Poles ECE 546 Jose Schutt Aine 8

29 Constant Term 1 S ijk 1 d R Yo Y 1 S ijk 1d o S ijk R=Y o (1-d)/(1+d) ECE 546 Jose Schutt Aine 9

30 S- Circuit Realization Constant Term 1 d R Yo 1 d ECE 546 Jose Schutt Aine 30

31 S-Parameters - Poles and Residues In the pole-residue case, we must distinguish two cases (a) Pole is real s (s) ijk s a k p k (b) Complex conjugate pair of poles s (s) ijk k jk k jk s j s j k k k k In all cases, we must find an equivalent circuit consisting of lumped elements that will exhibit the same behavior ECE 546 Jose Schutt Aine 31

32 S-Realization Real Poles Proposed Circuit Model Admittance of proposed model is given by: 1 s R R R R C Y 1 1 RR 1 1 s RC ECE 546 Jose Schutt Aine 3

33 Real Poles Y from circuit 1 s R R ak 1 R R C 1 S ijk(s) RR 1 s pk 1 s RC r 1 ˆ 1 S ijk s p s a Y Yo Yo Y o 1 S r ijk 1 s a s p from pole and residue ECE 546 Jose Schutt Aine 33

34 Solution for Real Poles Comparing Y ˆ( s) with Y () s gives C bzo b a R 1 bc R R 1 1 ac where a p r, and b p r k k k k ECE 546 Jose Schutt Aine 34

35 Realization Complex Poles From the S-parameter expansion, the complex pole pair gives: Sˆ r r s r r r p r p s p s p s s p p p p which corresponds to an admittance of: sa x ˆ 1 ˆ 1 S s sg p Y Yo Y 1 Sˆ sa x 1 s sg p o ECE 546 Jose Schutt Aine 35

36 Realization Complex Poles The admittance expression can be re-arranged as ˆ s sg psax s s ga px Y Y o Y s sg psax s s ga px o p pp 1 product of poles WE HAD DEFINED a r r 1 sum of residues g p p 1 sum of poles x rp rp 1 1 cross product ECE 546 Jose Schutt Aine 36

37 Realization Complex Poles ˆ s sg psax s s ga px Y Y o Y s sg psax s s ga px This can be further rearranged as o in which ˆ s sab Y s sdf E A ( ga), B px, D ( ga), F px and E Y o 1 H ECE 546 Jose Schutt Aine 37

38 Realization Complex Poles There are several circuit topologies that will work Model 1 Model 9 Model 13 Model 1 ECE 546 Jose Schutt Aine 38

39 Realization Complex Poles More circuit topologies that will work Model 10 Model 11 Model 8 Model 1 ECE 546 Jose Schutt Aine 39

40 Complex Poles Model 1 In particular, if we choose model 1 Y LRR CR R C R R R s s 1 LCR LCR R o LRR 1 C R1R s s LCR LCR o 1 o o 1 Must be matched with sa x 1 1 ˆ S ijk s sg p Y Yo Y 1 S sa x ijk 1 s sg p o ECE 546 Jose Schutt Aine 40

41 Complex Poles Model 1 Matching the terms with like coefficients gives p x R R R o LCR 1 R o 1 Y o p x R R LCR 1 o p R R R LCR 1 R o x LCR ECE 546 Jose Schutt Aine 41

42 Complex Poles Model 1 Solving gives R R o p x 1 R 1 L R o a R 1 1 gro Lx a a R o C a Rx ECE 546 Jose Schutt Aine 4

43 Complex Poles Model 9 If we choose Model 9 Y 9 A B E L CRR3 CR1R R3 R R 3 s s R1 R3 LC R1R3 LC R1R3 RR 1 3 CR R 1RR3 LR1 1R R3 s s LCR1R3 LCR1R3 D F from which the circuit elements can be extracted ECE 546 Jose Schutt Aine 43

44 Complex Poles Model 9 F R 1 BH C BD AF H F R BD BF ADF F B ABD BD A F BF ADF F H R 3 F L B D A F B FH B ABD BD A F BF ADF F H ECE 546 Jose Schutt Aine 44

45 Special Case Model 4 A special case exists when x=0 or x rp 1 rp 1 0 ˆ s s ga px s s ga p Y Y o Y s s ga px s s ga p o in which ˆ s sa B Y s sdb E A ( ga), B px, D ( ga), F p and E Y o 1 H ECE 546 Jose Schutt Aine 45

46 Special Case Model 4 The proposed circuit model is Model 4 This model has one inductor, one capacitor and two resistors. ECE 546 Jose Schutt Aine 46

47 Special Case Model 4 The admittance seen at the input is: R 1 DE A Y 4 A B F E 1 1 s s R1 R C R 1 R LC RR s s CR LC D FB The element are extracted as R R E R 1 1 E DR 1 C 1 L FC ECE 546 Jose Schutt Aine 47

48 Method 3 - Convolution with S Parameters In frequency domain B=SA In time domain Convolution: b(t) = s(t)*a(t) s(t)*a(t) s(t )a( )d ECE 546 Jose Schutt Aine 48

49 SPICE Macromodel STAMP stamp 1 o 1 Y Z 1s'(0 ) 1 s'(0 ) stamp o 1 1 I Z 1s'(0) H(t) Most of the computational effort is in the convolution calculation of the history H(t) at each time step. t1 Ht () s( ) at ( ) We make the convolution faster using a -function expansion for s(t) 1 This operation can be accelerated ECE 546 Jose Schutt Aine 49

50 In the frequency domain, assume that S parameter can be expanded in the form: Sl () M s( p) c ( pk) k 1 S-Parameter Expansion M j lk ce k k 1 In the time domain, this corresponds to: k Which is an impulse train of order M. Convolution will then give: M M y( p) s( p)* x( p) ck ( pk) x( p) ckx( pk) k1 k1 If we truncate the summation to the Q largest impulses: Q Q y( p) ck ( pk) x( p) ckx( pk) k1 k1 If the reference system is optimized, most of the coefficients c k will be negligibly small; therefore Q will be a relatively small number Fast Convolution is achieved by making Q small ECE 546 Jose Schutt Aine 50

51 Strategy s ( u) c ( u k) ijk ijk Since the S-parameter is a simple delay, the oneport circuit representation is a transmission line with characteristic impedance Z o and delay k ECE 546 Jose Schutt Aine 51

52 Circuit Interpretation s ( u) c ( u k) ijk ijk kt t time step d ECE 546 Jose Schutt Aine 5

53 Method 1: Y-Parameters/MOR Recursive convolution SPICE realization ECE 546 Jose Schutt Aine 53

54 Method : S-Parameters/MOR Recursive convolution SPICE realization ECE 546 Jose Schutt Aine 54

55 Comparisons SPICE simulation from MOR generated Netlist (Method ) Direct Convolution ECE 546 Jose Schutt Aine 55

56 Coupled Lines (4-port) Direct SPICE simulation Using generated netlist (Method ) ECE 546 Jose Schutt Aine 56

57 Comparison of S-methods SPICE Netlist from MOR with S (Method ) SPICE Netlist from Fast convolution with S (Method 3) Convolution (no SPICE netlist) ECE 546 Jose Schutt Aine 57