ECE 451 Circuit Synthesis Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois jesa@illinois.edu ECE 451 Jose Schutt Aine 1
MOR via Vector Fitting Rational function approximation: N cn f ( s) d sh s a n 1 n Introduce an unnown 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
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). 3/8 ECE 546 Jose Schutt Aine 3
Macromodel Circuit Synthesis Use of Macromodel Time-Domain simulation using recursive convolution Frequency-domain circuit synthesis for SPICE netlist ECE 451 Jose Schutt Aine 4
Macromodel Circuit Synthesis Objective: Determine equivalent circuit from macromodel representation* Motivation Circuit can be used in SPICE Goal Generate a netlist of circuit elements *Giulio Antonini "SPICE Equivalent Circuits of Frequency- Domain Responses", IEEE Transactions on Electromagnetic Compatibility, pp 50-51, Vol. 45, No. 3, August 003. ECE 451 Jose Schutt Aine 5
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 451 Jose Schutt Aine 6
Y-Parameter - Circuit Realization Each of the Y-parameters can be represented as y(s) ij d L 1 a s p where the a s are the residues and the p s are the poles. d is a constant ECE 451 Jose Schutt Aine 7
Y-Parameter - Circuit Realization The realized circuit will have the following topology: We need to determine the circuit elements within y ij ECE 451 Jose Schutt Aine 8
Y-Parameter - Circuit Realization We try to find the circuit associated with each term: y(s) 1. Constant term d ij d L 1 s a p y ijd(s) d. Each pole-residue pair y (s) ij a s p ECE 451 Jose Schutt Aine 9
Y-Parameter - Circuit Realization In the pole-residue case, we must distinguish two cases (a) Pole is real y (s) ij s a p (b) Complex conjugate pair of poles y (s) ij j j s j s j In all cases, we must find an equivalent circuit consisting of lumped elements that will exhibit the same behavior ECE 451 Jose Schutt Aine 10
Circuit Realization Constant Term R 1 d ECE 451 Jose Schutt Aine 11
Circuit Realization - 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 a s p 1/ L y s ij(s) R/ L L 1/ a R p / a ECE 451 Jose Schutt Aine 1
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 451 Jose Schutt Aine 13
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 451 Jose Schutt Aine 14
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 1 1 / 1 s r p r p r r s s p p p p 1 1 ECE 451 Jose Schutt Aine 15
Circuit Realization - Complex Poles We next compare p Y and pp 1 1 s1/ CR L L CRR R R 1 s s LRC LRC Yˆ r r 1 product of poles 1 1 1/ 1 s r p r p r r s s p p p p DEFINE 1 1 a r r 1 sum of residues g p p 1 sum of poles x rp rp 1 1 cross product ECE 451 Jose Schutt Aine 16
Circuit Realization - Complex Poles We can identify the circuit elements L 1/ a R x a 1 g a R p x g pa 1 g C x a a x a x ECE 451 Jose Schutt Aine 17
Circuit Realization - Complex Poles 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 451 Jose Schutt Aine 18
S-Parameter - Circuit Realization Each of the S-parameters can be represented as s(s) ij d L 1 a s p where the a s are the residues and the p s are the poles. d is a constant ECE 451 Jose Schutt Aine 19
Realization from S-Parameters The realized circuit will have the following topology: We need to determine the circuit elements within s ij ECE 451 Jose Schutt Aine 0
S-Parameter - Circuit Realization We try to find the circuit associated with each term: s(s) 1. Constant term d ij d L 1 s a p s ijd(s) d. Each pole and residue pair s (s) ij a s p ECE 451 Jose Schutt Aine 1
S-Parameter - Circuit Realization In the pole-residue case, we must distinguish two cases (a) Pole is real s (s) ij s a p (b) Complex conjugate pair of poles s (s) ij j j s j s j In all cases, we must find an equivalent circuit consisting of lumped elements that will exhibit the same behavior ECE 451 Jose Schutt Aine
S- Circuit Realization Constant Term 1 d R Yo 1 d ECE 451 Jose Schutt Aine 3
S-Realization Real Poles ECE 451 Jose Schutt Aine 4
S-Realization Real Poles Admittance of proposed model is given by: 1 s R R R R C Y 1 1 RR 1 1 s RC ECE 451 Jose Schutt Aine 5
S-Realization Real Poles From S-parameter expansion we have: r s ij(s) s p which corresponds to: ˆ s a Y Yo s b from which Y o where R R RR 1 1 a p r, and b p r R 1 bc C b a bzo R R 1 1 ac ECE 451 Jose Schutt Aine 6
Realization Complex Poles Proposed model Y Y 1 1 scr 1 R slcr s L RRC R R R a o 1 1 o which can be re-arranged as: 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 ECE 451 Jose Schutt Aine 7
Realization Complex Poles From the S-parameter expansion, the complex pole pair gives: Sˆ 1 1 1 1 r r s r r r p r p s p s p s s p p p p 1 1 1 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 451 Jose Schutt Aine 8
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 451 Jose Schutt Aine 9
Realization Complex Poles Matching the terms with lie 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 451 Jose Schutt Aine 30
Realization from S-Parameters 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 451 Jose Schutt Aine 31
Typical SPICE Netlist * 3 -pole approximation *This subcircuit has 16 pairs of complex poles and 0 real poles.subct sample 8000 9000 vsens8001 8000 8001 0.0 vsens9001 9000 9001 0.0 *subcircuit for s[1][1] *complex residue-pole pairs for = 1 residue: -6.466e-00 8.1147e-00 pole: -4.4593e-001 -.4048e+001 elc1 1 0 8001 0 1.0 hc 1 vsens8001 50.0 rtersc3 3 50.0 vp4 3 4 0.0 l1cd5 4 5 1.933e-007 rocd5 4 0 5.000e+001 r1cd6 5 6 5.895e+003 c1cd6 6 0 3.474e-015 rcd6 6 0-9.68e+003 : *constant term -6.19e-003 edee397 397 0 9001 0 1.0e+000 hdee398 398 397 vsens9001 50.0 rterdee399 398 399 50.0 vp400 399 400 0.0 rdee400 400 0 49.4 *current sources fs4 0 8001 vp4-1.0 gs4 0 8001 4 0 0.00 fs10 0 8001 vp10-1.0 gs10 0 8001 10 0 0.00 fs16 0 8001 vp16-1.0 gs16 0 8001 16 0 0.00 fs 0 8001 vp -1.0 gs 0 8001 0 0.00 fs8 0 8001 vp8-1.0 gs8 0 8001 8 0 0.00 ECE 451 Jose Schutt Aine 3
Realization from Y-Parameters Recursive convolution SPICE realization ECE 451 Jose Schutt Aine 33
Realization from S-Parameters Recursive convolution SPICE realization ECE 451 Jose Schutt Aine 34