Structural analysis of electric circuits and consequences for MNA

TLDR: In this paper, the authors analyse electric circuits with respect to their structural properties in order to give circuit designers some help for fixing modelling problems if the numerical simulation fails, and discuss the index of the differential algebraic equations obtained by this kind of modelling.

Abstract: The development of integrated circuits requires powerful numerical simulation programs. Naturally, there is no method that treats all the different kinds of circuits successfully. The numerical simulation tools provide reliable results only if the circuit model meets the assumptions that guarantee a successful application of the integration software. Owing to the large dimension of many circuits (about 107 circuit elements) it is often difficult to find the circuit configurations that lead to numerical difficulties. In this paper, we analyse electric circuits with respect to their structural properties in order to give circuit designers some help for fixing modelling problems if the numerical simulation fails. We consider one of the most frequently used modelling techniques, the modified nodal analysis (MNA), and discuss the index of the differential algebraic equations (DAEs) obtained by this kind of modelling. Copyright © 2000 John Wiley & Sons, Ltd.

Figures

Table VCCS condition a

Table VCCS condition a

Full text

Structural analysis for electric circuits and
consequences for MNA
D. Estevez Schwarz

and C. Tischendorf
y
Abstract
The development of integrated circuits requires powerful numerical
simulation programs. Of course, there is no method that treats all the
dierent kinds of circuits successfully. The numerical simulation to ols pro-
vide reliable results only if the circuit mo del meets the assumptions that
guarantee the successful application of the integration software. Because
of the large dimension of many circuits (about 10
7
circuit elements) it
is often dicult to nd the circuit congurations that lead to numerical
diculties. In this paper, we analyze electric circuits with respect to their
structural properties in order to give circuit designers some help for xing
modelling problems if the numerical simulation fails. We consider one of
the most frequently used modelling technique, the modied nodal anal-
ysis (MNA), and discuss the index of the dierential algebraic equations
(DAEs) obtained by this kind of mo delling.
Key words:
Circuit simulation dierential-algebraic
equation DAE index
modied no dal analysis MNA structural properties modelling.
AMS Sub ject Classication:
94C05, 65L05.
1 Structural analysis
In the following we discuss lump ed electric circuits containing nonlinear and p os-
sibly time-variant resistances, capacitances, inductances, voltage sources and
current sources. Usually circuit simulation to ols are based on these kinds of
network elements. For two-terminal (one-port) lumped elements, the current
through the element and the voltage across it are well-dened quantities. For
lumped elements with more than two terminals, the currententering anyter-
minal and the voltage across any pair of terminals are well dened at all times
(cf. 3]). Hence, general n-terminal elements are completely describ ed by(
n
;
1)
currents entering the (
n
;
1) terminals and the (
n
;
1) branchvoltages across
each of these (
n
;
1) terminals and the reference terminal
n
.

Humboldt University of Berlin, Germany
y
At present at Lunds University,Sweden
1

12345
6 7 ...
n (reference terminal)
Figure 1.1:
n
-terminal circuit element
In particular,
n
-terminal resistances can be modeled by an equation system of
the form
j
k
=
r
e
k
(
u
1
 ::: u
n
;
1
t
) for
k
=1
:::n
;
1
if
j
k
represents the currententering the terminal
k
and
u
l
describes the voltage
across the pair of terminals
f
l n
g
(for
k l
=1
:::n
;
1). The Kirchho 's Current
Law implies the currententering the terminal
n
to b e given by
j
n
=
;
P
n
;
1
k
=1
j
k
.
The conductance matrix
G
e
(
u
1
:::u
n
;
1
t
) is then dened by the Jacobian
G
e
(
u
1
 ::: u
n
;
1
t
):=
0
B
B
@
@r
e
1
@u
1
:::
@r
e
1
@u
n
;
1
.
.
.
.
.
.
.
.
.
@r
e
n
;
1
@u
1
:::
@r
e
n
;
1
@u
n
;
1
1
C
C
A
:
The index
e
shall specify the correlation to a special element of a circuit. Later
on wewillintroduce the conductance matrix
G
(
u t
) describing all resistances
of a circuit. Correspondingly, the capacitance matrix
C
e
(
v
1
:::v
n
;
1
t
) of a
general
n
-terminal capacitance is given by
C
e
(
u
1
:::u
n
;
1
t
):=
0
B
B
@
@q
e
1
@u
1
:::
@q
e
1
@u
n
;
1
.
.
.
.
.
.
.
.
.
@q
e
n
;
1
@u
1
:::
@q
e
n
;
1
@u
n
;
1
1
C
C
A
if the voltage-current relation is dened by means of charges by
j
k
=
d
dt
q
e
k
(
u
1
:::u
n
;
1
t
) for
k
=1
:::n
;
1
:
In order to illustrate what the matrices
C
e
may look like, let us consider a
MOSFET-model as an example of a common
n
-terminal element.
d(u )
Gate
Bulk
DrainSource
G
S
B
D
R
C
BS BD
C
C
GS GD
C
BS
DS GS BS
r(u ,u ,u )
BD
d(u )
Figure 1.2: MOSFET-mo del
2

Choosing the source node S as the reference node, wehave the reference voltages
u
GS
,
u
DS
, and
u
BS
. For the currents we obtain
j
G
=
C
GS
_
u
GS
+
C
GD
(_
u
GS
;
_
u
DS
)

j
D
=
;
C
GD
(_
u
GS
;
_
u
DS
)
;
C
BD
(_
u
BS
;
_
u
DS
)
+
d
(
u
BS
;
u
DS
)+
g
(
u
GS
u
DS
u
BS
)

j
B
=
C
BS
_
u
BS
+
C
BD
(_
u
BS
;
_
u
DS
)
;
d
(
u
BS
)
;
d
(
u
BS
;
u
DS
)
:
Note that
j
S
is given by the formula
j
S
=
;
j
G
;
j
D
;
j
B
due to Kircho 's
CurrentLaw. Nowitiseasytoverify that
C
e
(
u
GS
u
DS
u
BS
)=
0
@
C
GS
+
C
GD
;
C
GD
0
;
C
GD
C
GD
+
C
BD
;
C
BD
0
;
C
BD
C
BS
+
C
BD
1
A
for the MOSFET-mo del from 14].
Inductances can b e modeled by means of uxes by
u
k
=
d
dt

e
k
(
j
1
:::j
n
;
1
t
) for
k
=1
:::n
;
1
:
Then, the inductance matrix
L
e
(
j
1
:::j
n
;
1
t
) of a general
n
-terminal induc-
tance is given by the Jacobian
L
e
(
j
1
 ::: j
n
;
1
t
):=
0
B
B
@
@
e
1
@j
1
:::
@
e
1
@j
n
;
1
.
.
.
.
.
.
.
.
.
@
e
n
;
1
@j
1
:::
@
e
n
;
1
@j
n
;
1
1
C
C
A
:
A commonly used method for network analysis in circuit simulation packages
like TITAN
1
and SPICE is the Mo died No dal Analysis (MNA).
It represents a systematic treatment of general circuits and is imp ortantwhen
computers perform the analysis of networks automatically. The scheme to set
up the MNA equations is:
1. Write no de equations by applying KCL (Kirchho 's CurrentLaw) to each
node except for the datum node:
Aj
=0
:
(1.1)
The vector
j
represents the branch current vector. The matrix
A
is
called the (reduced) incidence matrix and describes the network graph,
the branch-node relations.
2. Replace the currents
j
k
of voltage-controlled elements by the
voltage-
current relations of these elements in equation (1.1).
3. Add the current-voltage relations for all current-controlled elements.
1
SIEMENS AG.
3

Note that, in case of multi-terminal elements with
n
terminals, we speak of
branches if they represent a pair of terminals
f
l n
g
with 1

l

n
;
1.
In general, the MNA leads to a coupled system of implicit dierential equations
and nonlinear equations, i.e., to a dierential-algebraic equation (DAE)
f
(_
x
(
t
)
x
(
t
)
t
)=0

(1.2)
where the partial derivative
f
0
_
x
(_
x
(
t
)
x
(
t
)
t
) is singular. The analytical and nu-
merical solutions of (1.2) dep end strongly on its structure and index. For a
detailed discussion of this fact we refer to 7], 9], 12], and 13]. Let us note
that numerical methods can fail in higher index cases, particularly if the index
is greater than 2. Therefore, we are lo oking for conditions (depending on the
network topology) that guarantee a lower index (

2).
In order to obtain more detailed information ab out the structure of (1.2) wesplit
the (reduced) incidence matrix
A
into the element-related incidence matrices
A
=(
A
C
A
L
A
R
A
V
A
I
)

where
A
C
,
A
L
,
A
R
,
A
V
, and
A
I
describe the branch-current relations for ca-
pacitive branches, inductive branches, resistive branches, branches of voltage
sources and branches of current sources, respectively. Denote by
e
the no de
potentials (excepting the datum no de) and by
j
L
and
j
V
the currentvectors of
inductances and voltage sources. Dening the vector of functions for current
and voltage sources by
i
and
v
, resp ectively,we obtain the following quasi-linear
DAE-system from the MNA:
A
C
dq
(
A
T
C
e t
)
dt
+
A
R
r
(
A
T
R
e t
)+
A
L
j
L
+
A
V
j
V
+
A
I
i
(
A
T
e
dq
(
A
T
C
e t
)
dt
j
L
j
V
t
) = 0

(1.3)
d
(
j
L
t
)
dt
;
A
T
L
e
= 0

(1.4)
A
T
V
e
;
v
(
A
T
e
dq
(
A
T
C
e t
)
dt
j
L
j
V
t
) = 0
:
(1.5)
Note that the vectors
A
T
C
e
,
A
T
L
e
,
A
T
R
e
and
A
T
V
e
describe the branchvoltages for
the capacitive, inductive, resistiveandvoltage source branches, resp ectively.
Remark:
Due to the fact that the currents through resistances are functions
of the branch p otentials, we do not include them separately as controlling func-
tions. Of course, if the network does not contain controlled sources, then the
source functions reduce to functions
i
(
t
)and
v
(
t
)which dep end on time only.
Nowadays circuit simulation packages use two dierent approaches for solving
(1.3)-(1.5), the conventional and the charge-oriented one.
The conventional MNA
For the con
ventional MNA the vector of unknowns consists of all no de voltages
and all branch currents of current-controlled elements.
4

Dening
C
(
u t
):=
@q
(
u t
)
@u
 q
0
t
(
u t
):=
@q
(
u t
)
@t
 L
(
j t
):=
@
(
j t
)
@j
 
0
t
(
j t
):=
@
(
j t
)
@t
we obtain
A
C
C
(
A
T
C
e t
)
A
T
C
de
dt
+
A
C
q
0
t
(
A
T
C
e t
)+
A
R
r
(
A
T
R
e t
)
+
A
L
j
L
+
A
V
j
V
+
A
I
i
(
A
T
e C
(
A
T
C
e t
)
A
T
C
de
dt
j
L
j
V
t
) = 0

(1.6)
L
(
j
L
t
)
dj
L
dt
+

0
t
(
j
L
t
)
;
A
T
L
e
= 0

(1.7)
A
T
V
e
;
v
(
A
T
e C
(
A
T
C
e t
)
A
T
C
de
dt
j
L
j
V
t
) = 0
:
(1.8)
Later on we will also need
G
(
u t
):=
@r
(
u t
)
@u
 r
0
t
(
u t
):=
@r
(
u t
)
@t
:
The charge-oriented MNA
In comparison with the conventional MNA, the vector of unknowns consists
additionally of the charge of capacitances and the ux of inductances. Moreover,
the original voltage-charge and current-ux equations are added to the system.
The resulting system is then of the form (cf. 8])
A
C
dq
dt
+
A
R
r
(
A
T
R
e t
)+
A
L
j
L
+
A
V
j
V
+
A
I
i
(
A
T
e
dq
dt
j
L
j
V
t
) = 0

(1.9)
d
dt
;
A
T
L
e
= 0

(1.10)
A
T
V
e
;
v
(
A
T
e
dq
dt
j
L
j
V
t
) = 0

(1.11)
q
;
q
C
(
A
T
C
e t
) = 0

(1.12)

;

L
(
j
L
t
) = 0
:
(1.13)
Topological characterization of the splitted incidence ma-
trix
The splitting of the incidence matrix
A
= (
A
C
A
L
A
R
A
V
A
I
) corresp ond-
ing to certain branches leads to the following useful structural information for
lumped circuits:
Theorem 1.1
Given a lumped circuit with capacitances, inductances, resis-
tances, voltage sources and curr
ent sources. Then, the fol lowing relations are
satised for the (reduced) incidence matrix
A
=(
A
C
A
L
A
R
A
V
A
I
)
.
1. Then matrix
(
A
C
A
L
A
R
A
V
)
has ful l row rank, because cutsets of current
sources are forbidden.
5

Frequently asked questions

Q1. What are the contributions in this paper?

In this paper the authors analyze electric circuits with respect to their structural properties in order to give circuit designers some help for xing modelling problems if the numerical simulation fails The authors consider one of the most frequently used modelling technique the modi ed nodal anal ysis MNA and discuss the index of the di erential algebraic equations DAEs obtained by this kind of modelling 

Q2. What are the future works in this paper?

The presented results provide the possibility to obtain information about the index of the systems and by topological analysis of the network 

Q3. what is the vector of unknowns for the conventional MNA?

Of course if the network does not contain controlled sources then the source functions reduce to functions i t and v t which depend on time onlyNowadays circuit simulation packages use two di erent approaches for solving the conventional and the charge oriented oneFor the conventional MNA the vector of unknowns consists of all node voltages and all branch currents of current controlled elements 

Q4. What is the simplest way to set up the MNA equations?

The scheme to set up the MNA equations isWrite node equations by applying KCL Kirchho s Current Law to each node except for the datum nodeAjThe vector j represents the branch current vector 

Q5. what is the condition that controlled voltage sources do not form a part of a C V?

The condition that controlled voltage sources do not form a partof a C V loop is equivalent to QV CQV C tHere QV C t denotesthe upper part of QV C corresponding to AV tProof A controlled voltage source forms a part of a C V loop if and only if the column as of AV co corresponding to this source depends linearly on the columns of AC AV where AV denotes the matrix AV reduced by the column as i e there is a vector v such thatACAV v and vsfor the s th component of v corresponding to the controlled source considered 

Q6. What is the simplest way to denote a DAE?

This index concept requires only weak smoothness conditions Furthermore solvabil ity and stability results exist for index tractable and index tractable DAEs see e gThe authors consider nonlinear DAEsf x x tfor which N ker f x x x t is constant and f is continuously di erentiable 

Q7. What is the condition that a controlled current source does not form a part of an L?

The condition that controlled current sources do not form a part of an L The authorcutset is equivalent to the relation QTCRVAI Q T CRV AItProof A controlled current source forms a part of an L The authorcutset if and only if the column as of AIa AIb AIc corresponding to this controlled source isThe controlling currents of a CCCS can be currents ofinductancesindependent current sourcesresistances or VCCSs for which the controlling nodes are connected bya capacitancesb voltage sourcesc paths containing only the elements described in a and bbranches that form a cutset with the elements described in orTable CCCS condition aThe controlling current of a CCCS can be the current ofinductancesindependent current sourcesresistancesvoltage sources that do not form a part of a C V loopVCCSa branch that forms a cutset with the elements described in andTable CCCS condition blinearly independent of the columns belonging to AC AR AV i eas im ACARAV and therefore Q T CRV asBut this is equivalent to the condition that QTCRV AIa AIb AIc q e dThus assumption a of Theorem implies thatQTCRVAI i A T edq ATCe tdt jL jV t QT CRVAItiti AT e dq ATCe tdt jL jV t ia AT Ce A T V e jL tfor a suitable function iaFurthermore assumption b of Theorem implies by de nition thatQTCAIbi AT e dq ATCe tdt jL jV t ib AT e jL PV CjV tThe controlling current of a CCCS can be the current ofinductancesresistancesindependent current sourcesVCCSa branch that forms a cutset with the elements described in andTable CCCS condition cfor a suitable function ibFinally assumption c of Theorem implies thatQTV CQ T CAIci AT e dq ATCe tdt jL jV t ic AT e jL tfor a suitable function icRegarding and the assumptions imply thatQTCRVAI i A T edq ATCe tdt jL jV t QT CRVAItitis always ful lled 

Q8. What is the di erential index of the network?

If the di erential index is then the network con tains at least a C V loop or an L The authorcutsetProof Let us now suppose that the di erential index is 

Q9. how to obtain a qcv expression when mul tiplying by H?

Therefore the authors considerde dt PC de dt QCPV C de dt QCQV CPR CV de dt QCRV de dtand observe that the authors obtain the needed expression for QCRV de dt when mul tiplying by H after substituting the expressions for PC de dt QCPV C de dt QCQV CPR CV de dt and djL dtIf the network does not contain C V loops then QTCAV has full column rank cf point in Theorem Therefore PV C The authorand the authors obtain an expression for djVdt when multiplying by H A T VQC after substituting the obtained expressions for de dt and djL dt This transformation is reversible as well as can be seen by multiplication by H Q T CAVH 

Q10. what is the simplest way to determine the di erentiation of the network?

At this point it is recognizable that N S represents those components for which the di erential index de nition requires two di erentiations to obtain the rep resentation of their derivative as a continuous function of the variables