Automatic circuit equation formulation for non-smooth electrical circuits
Vincent Acary, Olivier Bonnefon, Pascal Denoyelle
INRIA Rhône-Alpes
February 07, 2008
Automatic circuit equation formulation for non-smooth electrical circuits
February 07, 2008
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1
Basics on circuit topology
2
Modified Nodal Analysis
3
MNA and DAE
4
Piecewise Linear modeling
5
Extend MNA for non-smooth components
6
MLCS and MLCP
7
Cost of the automatic equation formulation
8
Simulation
9
Conclusion
Automatic circuit equation formulation for non-smooth electrical circuits
February 07, 2008
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Basics on circuit topology
Circuit branch
Vk
Ia
Ua
Vj
Figure: Basic branch
A branch is described by a pair of branch variables: the tension Ua and the current Ia . Moreover,
the Branch Constitutive Equation can be expressed in a general implicit form:
F (Ua , Ia ,
dUa dIa
,
, ...) = 0
dt dt
(1)
Some examples with a resistor, a capacitor and a current source:
Ua = RIa
Ia = C
dUa
dt
Ia = αUb
We will see the different forms of this relation.
Automatic circuit equation formulation for non-smooth electrical circuits
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Basics on circuit topology
The Kirchhoff Current Law (KCL)
At any node in an electrical circuit where charge density is not changing in time, the sum of
currents flowing towards that node is equal to the sum of currents flowing away from that node.
I3
1
V1
2
V2
L3
I2
I1
I4
R4
C2
I1
V0
0
With this example
−I1 + I2 + I3 = 0
−I3 + I4 = 0
I1 − I2 − I4 = 0
(KCL1)
(KCL2)
(KCL0)
or the matrix formulation :
AI = 0
where A is known as the incidence matrix and I is the vector of branch currents.
Automatic circuit equation formulation for non-smooth electrical circuits
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Basics on circuit topology
The Kirchhoff Voltage Law (KVL)
The directed sum of the electrical differences around a closed circuit must be zero.
I3
1
V1
2
V2
L3
I2
I1
I4
R4
C2
I1
V0
0
With this example
U1 + U2 = 0
−U2 + U3 + U4 = 0
U1 + U3 + U4 = 0
or the matrix formulation :
BU = 0
where B is known as the loop matrix and U is the vector of branch tensions.
Automatic circuit equation formulation for non-smooth electrical circuits
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Basics on circuit topology
Relation between A and B
BAt = 0
Automatic circuit equation formulation for non-smooth electrical circuits
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Basics on circuit topology
Relation between A and B
BAt = 0
Proof:
0
B
B =B
@
0
1
b1
. C
C
. A
bnl
B
A=B
@
1
KCL1
C
.
C
A
.
KCLnn
Let p ∈ {1, nl } and q ∈ {1, nn }. We proof that bp .KCLtq = 0.
q
q
j
1
2
The node
The node
q
q
k
j
k
is not on the loop p , then bp and KCLq have no common non null coordinate.
is on the loop p :
bp = (...1... − 1...)
KCLq = (...1...1...)
The other coordinates are not simultaneous not null. Therefore :
bp .KCLtq = 0
Automatic circuit equation formulation for non-smooth electrical circuits
February 07, 2008
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Basics on circuit topology
An other KVL formulation
kp
jp
Up
It consists in writing :
∀p ∈ {1, nb }
Up = Vjp − Vkp
This system of equation implies BU=0.
The matrix formulation is:
U − At V = 0
where V is the vector of nodes potential and U the vector of branches tensions.
⇒ In further consideration, BU=0 is eliminated.
Example:
U1 = V1 − V0
U3 = V2 − V1
Automatic circuit equation formulation for non-smooth electrical circuits
U2 = V0 − V1
U4 = V0 − V2
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Basics on circuit topology
The Sparse Tableau Approach STA leads to the following system:
AI = 0
(KCL)
t
U−A V =0
(KVL)
F (Ua , Ia , ...) = 0
For all branches :
(BCE )
Example:
UL
UC
V1
IL
L
C
IC
The vector of unknowns: (IL , IC , UL , UC , V1 )t
−IL + IC = 0
UL + V1 = 0
CUC′
(KCL1)
UC − V1 = 0
(KVL)
LIL′
(BCE )
− IC = 0
Automatic circuit equation formulation for non-smooth electrical circuits
− UL = 0
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Modified Nodal Analysis
A Current-Defined Branch (CD)
The branch is current-defined if its current is a function of its own voltage, controlling variable or
their time–derivatives:
dUa dUb dIc
Ia = Fi (Ua , Ub , Ic ,
,
,
)
(2)
dt
dt dt
A Voltage-Defined Branch (VD)
The branch is voltage-defined if its voltage is a function of its own current, controlling variable or
their derivatives:
dIa dUb dIc
Ua = Fu (Ia , Ub , Ic ,
,
,
)
(3)
dt dt dt
Examples :
A resistor is a voltage-defined branch because Ua = RIa .
A inductor is a voltage-defined branch because Ua = L dIdta .
a
.
A capacitor is a current-defined branch because Ia = C dU
dt
MNA Hypothesis:
The M.N.A. assumes that smooth branches are explicit functions of current or voltage. It means
each branch is either Voltage Defined or Current Defined.
Automatic circuit equation formulation for non-smooth electrical circuits
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Modified Nodal Analysis
MNA unknowns
The vector of unknowns contains:
All node potentials Vi
Currents through all voltage defined branches
The following equations are written:
The KCL is applied to every node.
The Branch Constitutive Equation for all voltage defined branches.
Example:
UL
IL
UC
V1
L
C
IC
The vector of unknowns is :(V1 , IL )t
CV1′ − IL = 0
Automatic circuit equation formulation for non-smooth electrical circuits
LIL′ + V1 = 0
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MNA and DAE
MNA and DAE
MNA leads to a DAE of the form:
C (X )X ′ + J(X ) − S(t) = 0
With
C(X) describing the dynamic elements.
J(X) describing the static ones.
S(t) describing the independent sources.
Index of the DAE : 0,1,2.
In SPICE,the discretization of this equation is linearized and solved using the Newton-Raphson
iterations.
But, the non-smooth components such as diodes or transistors, result in troubles in this iterative
method. A novel approach is to model this components with piecewise linear function and to
formulate the problem as a Complementary Problem.
Automatic circuit equation formulation for non-smooth electrical circuits
February 07, 2008
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Piecewise Linear modeling
Ideal Diode
Branch Constitutive Equation of the diode:
Id = Is (exp (−UD /C ) − 1)
The complementary formulation:
0 6 Id ⊥ −UD > 0
Automatic circuit equation formulation for non-smooth electrical circuits
February 07, 2008
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Piecewise Linear modeling
Ideal Transistor
The complementary formulation:
Id =
`
...
0
´
λ
Id = −Is
1
Vd
Y = A @ Vg A + I λ + C
Vs
c1
c10
0 6Y ⊥ λ >0
Automatic circuit equation formulation for non-smooth electrical circuits
February 07, 2008
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Extend MNA for non-smooth components
Extended MNA
It consists in replacing the Constitutive Equation of the non-smooth branch with the
complementary formulation.
Unknowns
Before go ahead, the unknowns vector X is decomposed in:
x which contains only the dynamic unknowns(currents in inductor and tensions from
capacitor branches)
Zs which contains only the non dynamic unknowns(Voltage nodes,... ).
A good choice of unknowns yields the following system:
x ′ = A2x x + A2zs Zs + Rλ + A2s
(4)
0 = B2x x + B2zs Zs + B2λ λ + B2s
(5)
Y = D2x x + D2zs Zs + D2λ λ + D2s
(6)
06Y ⊥ λ >0
(7)
Automatic circuit equation formulation for non-smooth electrical circuits
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MLCS and MLCP
Mixed Linear Complementary System
Given the matrices Ax ∈ Rn×n , Az ∈ Rn×p , Av ∈ Rn×m , Bx ∈ Rp×n , Bz ∈ Rp×p , Bv ∈ Rp×m ,
Cx ∈ Rm×n , Cz ∈ Rm×p ,Cv ∈ Rm×m , and the vectors a ∈ Rn ,b ∈ Rp ,c ∈ Rm , the explicit MLCS
denoted by EMLCS(Ax , Az , Av , Bx , Bz , Bv , Cx , Cz , Cv , a, b, c) consists in finding three vectors
x ∈ Rn , z ∈ Rp and v ∈ Rm such that
8 ′
x = Ax x + Az z + Av v + a
>
>
>
<0 = B x + B z + B v + b
x
z
v
.
(8)
>
>
>
:
0 ≤ v ⊥ Cx x + Cz z + Cv v + c ≥ 0
The time discretization of a MLCS leads to :
Mixed Linear Complementary Problem
Given the matrices A ∈ Rn×n , B ∈ Rm×m , C ∈ Rn×m , D ∈ Rm×n , and the vectors
a ∈ Rn , b ∈ Rm , the MLCP denoted by MLCP(A, B, C , D, a, b) consists in finding two vectors
u ∈ Rn and v ∈ Rm such that
8
>
<Au + Cv + a = 0
.
(9)
>
:
0 ≤ v ⊥ Du + Bv + b ≥ 0
At each step, SICONOS solves a MLCP.
Automatic circuit equation formulation for non-smooth electrical circuits
February 07, 2008
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Cost of the automatic equation formulation
Cost evaluation
Let
Nb the number of branches.
Nn the number of nodes.
N = max{Nb , Nn }
The costs of the automatic circuit equation formulation:
To parse the Netlist : It consists in reading and storing the Netlist. O(N).
A topological analysis to build the unknowns vector: The Minimum Spanning Tree algorithm
complexity is O(Nlog(N))
Stamping method: Each component writes its contribution in the table equation. O(N).
Matrix product: Multiplied two dense matrices costs O(N 3 ).
Automatic circuit equation formulation for non-smooth electrical circuits
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Simulation
Diodes Bridge
N1
4
5
7
6
N2
N3
2
3
N0
10
’DiodeBridge.traj’
8
U23
6
4
2
0
0
50
100
150
200
250
300
350
400
450
time e-5 s
Automatic circuit equation formulation for non-smooth electrical circuits
February 07, 2008
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Simulation
Buck converter
3
Vramp
9
comparator
Verror
−
4
i2
10
$A$
C21
R21
−
6
7
5
C11
8
R11
L
i1
VI
R12
+
1
i3
Vref
ampli
2
+
Vgate
C
i4
Rload
0
0.1
0
Inductor current
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
0
2e-5
4e-5
6e-5
8e-5
1e-4
1.2e-4 1.4e-4 1.6e-4 1.8e-4
2e-4
time s
Automatic circuit equation formulation for non-smooth electrical circuits
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Conclusion
Conclusions
Software aspects
The actual software version is able to parse and simulate a Netlist composed of:
Non constant sources.
Linear components : resistor, capacitor, inductance.
Non smooth components : diode, MOS transistor, comparator.
Automatic circuit equation formulation for non-smooth electrical circuits
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Conclusion
Conclusions
Software aspects
The actual software version is able to parse and simulate a Netlist composed of:
Non constant sources.
Linear components : resistor, capacitor, inductance.
Non smooth components : diode, MOS transistor, comparator.
Theoritical aspects
Formulation of nonsmooth circuits in MLCS
New algorithms in Siconos/Numerics for MLCP
Automatic circuit equation formulation for non-smooth electrical circuits
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Conclusion
Perspectives
Software Aspects
Forecast active set: use a local piecewise linear approximation to solve a smaller MLCP.
Integration of the nonlinear SPICE component: MNCP solvers.
Integration of piecewise smooth model of MOS transistor
Specific electrical heuristics.
Improve the component model.
Automatic circuit equation formulation for non-smooth electrical circuits
February 07, 2008
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Conclusion
Perspectives
Software Aspects
Forecast active set: use a local piecewise linear approximation to solve a smaller MLCP.
Integration of the nonlinear SPICE component: MNCP solvers.
Integration of piecewise smooth model of MOS transistor
Specific electrical heuristics.
Improve the component model.
Theoritical Aspects
Topological analysis for the index reduction of the DAE and the relative degree of the EMLCS
Topological analysis to reduce the number of unknowns.
Multi–grid algorithm: use a set of embedded piecewise linear approximation (from coarser to
finer).
Other Algorithms for MLCP and MNCP (Nonsmooth Newton methods and Projection
methods for VI)
Adaptive time–stepping scheme
Automatic circuit equation formulation for non-smooth electrical circuits
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