ECE 451 Circuit Synthesisemlab.uiuc.edu/ece451/notes/synth.pdfECE 451 –Jose Schutt‐Aine 18 In...
Transcript of ECE 451 Circuit Synthesisemlab.uiuc.edu/ece451/notes/synth.pdfECE 451 –Jose Schutt‐Aine 18 In...
ECE 451 – Jose Schutt‐Aine 1
ECE 451Circuit Synthesis
Jose E. Schutt-AineElectrical & Computer Engineering
University of [email protected]
ECE 546 – Jose Schutt‐Aine 2
MOR via Vector Fitting
• Rational function approximation:
• Introduce an unknown function σ(s) that satisfies:
• Poles of f(s) = zeros of σ(s):
• Flip unstable poles into the left half plane.
1
( )N
n
n n
cf s d sh
s a
1
1
( ) ( )
( )1
Nn
n n
Nn
n n
cd sh
s as f ss c
s a
1
1 1
1 1
( )1
N Nn
nn n n
N Nn
nn nn
cd sh s z
s af s
cs z
s a
s-domain
Re
Im
ECE 546 – Jose Schutt‐Aine 3
Passivity Enforcement
• State‐space form:
• Hamiltonian matrix:
• Passive if M has no imaginary eigenvalues.
• Sweep:
• Quadratic programming:– Minimize (change in response) subject to (passivity compensation).
x Ax Buy Cx Du
T T
T T T T
A BKD C BKBM
C LC A C DKB
1 1T TK I D D L I DD
Δ Δ Δmin subject to ( ).vec vec G vec T( C) H ( C) = C
( ) ( )Heig I S j S j
3/28
ECE 451 – Jose Schutt‐Aine 4
• Time-Domain simulation using recursive convolution
• Frequency-domain circuit synthesis for SPICE netlist
Use of Macromodel
Macromodel Circuit Synthesis
ECE 451 – Jose Schutt‐Aine 5
• Circuit can be used in SPICE
Objective: Determine equivalent circuit from macromodel representation*
Motivation
Macromodel Circuit Synthesis
*Giulio Antonini "SPICE Equivalent Circuits of Frequency-Domain Responses", IEEE Transactions on Electromagnetic Compatibility, pp 502-512, Vol. 45, No. 3, August 2003.
• Generate a netlist of circuit elements
Goal
ECE 451 – Jose Schutt‐Aine 6
Circuit RealizationCircuit realization consists of interfacing the reduced model with a general circuit simulator such as SPICE
11 1
1
N
N NN
s s s sS s
s s s s
Model order reduction gives a transfer function that can be presented in matrix form as
or
11 1
1
N
N NN
y s y sY s
y s y s
ECE 451 – Jose Schutt‐Aine 7
Y-Parameter - Circuit Realization
Each of the Y-parameters can be represented asL
kij
k 1 k
ay ( s ) ds p
where the ak’s are the residues and the pk’s are the poles. d is a constant
ECE 451 – Jose Schutt‐Aine 8
Y-Parameter - Circuit Realization
We need to determine the circuit elements within yijk
The realized circuit will have the following topology:
ECE 451 – Jose Schutt‐Aine 9
Y-Parameter - Circuit Realization
We try to find the circuit associated with each term:
Lk
ijk 1 k
ay ( s ) ds p
1. Constant term d
ijdy ( s ) d
2. Each pole-residue pair
kijk
k
ay ( s )s p
ECE 451 – Jose Schutt‐Aine 10
Y-Parameter - Circuit Realization
In the pole-residue case, we must distinguish two cases
k k k kijk
k k k k
j jy ( s )s j s j
(a) Pole is real kijk
k
ay ( s )s p
(b) Complex conjugate pair of poles
In all cases, we must find an equivalent circuit consisting of lumped elements that will exhibit the same behavior
ECE 451 – Jose Schutt‐Aine 11
Circuit Realization – Constant Term
1Rd
ECE 451 – Jose Schutt‐Aine 12
Z sL R
Consider the circuit shown above. The input impedance Z as a function of the complex frequency s can be expressed as:
Circuit Realization - Real Pole
1//
LY ss R L
kijk
k
ay ( s )s p
kL 1 / a k kR p / a
ECE 451 – Jose Schutt‐Aine 13
21 1
2 2
11/ 1
RZ sL R sL RR sC sCR
1 2 2
2
11
R sL sCR RZ
sCR
Consider the circuit shown above. The input impedance Z as a function of the complex frequency s can be expressed as:
Circuit Realization - Complex Poles
ECE 451 – Jose Schutt‐Aine 14
2
1 22 1 2
2 2
1/1 s CRY
L R RL CR Rs sLR C LR C
2 2
1 22 1 22
2 2
1/CR s CRY
R RL CR RLCR s sLR C LR C
Circuit Realization - Complex Poles
ECE 451 – Jose Schutt‐Aine 15
1 2
1 2
ˆ r rYs p s p
Circuit Realization - Complex Poles Each term associated with a complex pole pair in the expansion gives:
Where r1, r2, p1 and p2 are the complex residues and poles. They satisfy: r1=r2* and p1=p2*
It can be re-arranged as:
1 2 2 1 1 2
1 2 21 2 1 2
/ˆ s r p r p r rY r r
s s p p p p
ECE 451 – Jose Schutt‐Aine 16
2
1 22 1 2
2 2
1/1 s CRY
L R RL CR Rs sLR C LR C
1 2 2 1 1 2
1 2 21 2 1 2
/ˆ s r p r p r rY r r
s s p p p p
1 2p p p
1 2g p p
1 2a r r
1 2 2 1x r p r p
product of poles sum of residues
sum of poles cross product
DEFINE
Circuit Realization - Complex Poles We next compare
and
ECE 451 – Jose Schutt‐Aine 17
1/L a1 2
x gRa a
2 2
p x gRx a a
2
1pa gCx a x
Circuit Realization - Complex Poles
We can identify the circuit elements
ECE 451 – Jose Schutt‐Aine 18
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
1 2
1 2
ˆ r rYs p s p
1 2
1 2 1 2
ˆ
Augmented Circuit Compensation Circuit
r rYs p s p s p s p
Circuit Realization - Complex Poles
Can show that both augmented and compensation circuits will have positive elements
ECE 451 – Jose Schutt‐Aine 19
S-Parameter - Circuit Realization
Each of the S-parameters can be represented asL
kij
k 1 k
as ( s ) ds p
where the ak’s are the residues and the pk’s are the poles. d is a constant
ECE 451 – Jose Schutt‐Aine 20
Realization from S-Parameters
We need to determine the circuit elements within sijk
The realized circuit will have the following topology:
ECE 451 – Jose Schutt‐Aine 21
S-Parameter - Circuit Realization
We try to find the circuit associated with each term:
Lk
ijk 1 k
as ( s ) ds p
1. Constant term d
ijds ( s ) d
2. Each pole and residue pair
kijk
k
as ( s )s p
ECE 451 – Jose Schutt‐Aine 22
S-Parameter - Circuit Realization
In the pole-residue case, we must distinguish two cases
k k k kijk
k k k k
j js ( s )s j s j
(a) Pole is real kijk
k
as ( s )s p
(b) Complex conjugate pair of poles
In all cases, we must find an equivalent circuit consisting of lumped elements that will exhibit the same behavior
ECE 451 – Jose Schutt‐Aine 23
S- Circuit Realization – Constant Term
o1 dR Y1 d
ECE 451 – Jose Schutt‐Aine 24
S-Realization – Real Poles
ECE 451 – Jose Schutt‐Aine 25
1
1 2 1 21
1 22
sR R R R C
YR R s
R C
S-Realization – Real Poles
Admittance of proposed model is given by:
ECE 451 – Jose Schutt‐Aine 26
S-Realization – Real Poles
ˆo
s aY Ys b
, andk k k ka p r b p r
1 2
1 2o
R RYR R
2o
b aC
b Z
21R
bC
kijk
k
rs ( s )s p
1 21R R
aC
From S-parameter expansion we have:
which corresponds to:
where
from which
ECE 451 – Jose Schutt‐Aine 27
Realization – Complex Poles
2
22 1 2 1 2
11 1a
o o
sCRY YR s LCR s L R R C R R R
2 1 2 2 1 2
2 2
2 1 2 1 2
2 2
1o o
o
o
L R R C R R C R R Rs sLCR LCR
YR L R R C R Rs s
LCR LCR
Proposed model
which can be re-arranged as:
ECE 451 – Jose Schutt‐Aine 28
1 2 1 2 2 11 22
1 2 1 2 1 2
ˆ s r r r p r pr rSs p s p s s p p p p
2
2
1ˆ1ˆˆ1 1
o o
sa xS s sg pY Y Ysa xS
s sg p
which corresponds to an admittance of:
From the S-parameter expansion, the complex pole pair gives:
Realization – Complex Poles
ECE 451 – Jose Schutt‐Aine 29
22
2 2ˆ
o o
s s g a p xs sg p sa xY Y Ys sg p sa x s s g a p x
1 2p p p
1 2g p p
1 2a r r
1 2 2 1x r p r p
product of poles sum of residues
sum of poles cross product
WE HAD DEFINED
Realization – Complex Poles
The admittance expression can be re-arranged as
ECE 451 – Jose Schutt‐Aine 30
1o
o
RY
1 2
2
oR R Rp xLCR
1 2
2
R Rp xLCR
1 2
2
2 22 oR R RpLCR
2
2 oRxLCR
Realization – Complex Poles
Matching the terms with like coefficients gives
ECE 451 – Jose Schutt‐Aine 31
Realization from S-Parameters
2oRLa
11 22
oo
gR LxR Ra a
2 1
1
2
opRxR R
2
aCR x
Solving gives
ECE 451 – Jose Schutt‐Aine 32
Typical SPICE Netlist* 32 -pole approximation *This subcircuit has 16 pairs of complex poles and 0 real poles .subckt sample 8000 9000 vsens8001 8000 8001 0.0 vsens9001 9000 9001 0.0 *subcircuit for s[1][1] *complex residue-pole pairs for k= 1 residue: -6.4662e-002 8.1147e-002 pole: -4.4593e-001 -2.4048e+001 elc1 1 0 8001 0 1.0 hc2 2 1 vsens8001 50.0 rtersc3 2 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 r2cd6 6 0 -9.682e+003 :*constant term 2 2 -6.192e-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 0.020 fs10 0 8001 vp10 -1.0 gs10 0 8001 10 0 0.020 fs16 0 8001 vp16 -1.0 gs16 0 8001 16 0 0.020 fs22 0 8001 vp22 -1.0 gs22 0 8001 22 0 0.020 fs28 0 8001 vp28 -1.0 gs28 0 8001 28 0 0.020
ECE 451 – Jose Schutt‐Aine 33
Realization from Y-Parameters
Recursive convolution SPICE realization
ECE 451 – Jose Schutt‐Aine 34
Realization from S-Parameters
Recursive convolution SPICE realization