Numerical Method for Calculating the Terms of the
Closed System Determinant in Two-Graph Representation
of Electronic Circuits
Marian Pierzchała1 and Benedykt Rodanski2
Abstract − This paper describes a cancellation-free
method for obtaining symbolic transmittances of
electronic circuits. The method is numerical and is
based on the original modification of the Binet-Cauchy
theorem, applied to the closed system determinant in
two-graph representation of a circuit. We also
introduce a new compact notation of symbolic
expressions, called the road map.
1 INTRODUCTION
It is a widely held view, that the two-graph tree
enumeration technique is the most suitable method
for symbolic analysis of large analog circuits that
cannot be partitioned hierarchically in an efficient
way, such as complex operational amplifiers [1].
This technique operates on two weighted graphs, the
voltage graph GV and the current graph GI. Usually,
the two graphs are constructed over the so-called
‘closed system’ (or ‘augmented circuit’), i.e., a
circuit in which the independent signal source was
replaced by a dependent source, controlled by the
output variable. This allows us to obtain the desired
transmittance by calculating a single determinant [2].
It is well known that the determinant of the Node
Admittance Matrix of the closed system is equal to
the algebraic sum of the weights of all common
spanning trees of GV and GI. The original method
works only for networks containing resistors,
capacitors, inductors and voltage-controlled current
sources (RLC-gm networks). When other elements
are
present,
some
preliminary
network
transformations might be required [2].
Clearly, the original two-graph method requires an
algorithm that enumerates common spanning trees.
Several tree enumeration algorithms have been
reported in literature [2-5]. The two graphs are
constructed by inspection of the circuit. The weight
of each edge in these graphs is equal to the
admittance of the corresponding element in the
circuit. The absolute value of a term is equal to the
product of the weights of the edges of a common
spanning tree. The sign of a term is determined using
topological information [5].
1
Another approach is also possible. If the
augmented circuit is described by a hybrid equation,
the system determinant is equal to unity plus the
algebraic sum of terms that are always products of
equal number of admittances and impedances (so the
determinant is unitless). This approach was first
suggested in [6]. A similar method was later
described in [2].
In this paper we describe a method for determining
the terms of the closed system determinant of the
R,L,C,E,I,DS networks (R,L,C,E,I,DS stand for
resistors, inductors, capacitors, independent voltage
and current sources and all types of dependent
sources). Our technique is numerical and is based on
the original modification of the Binet-Cauchy
theorem [6]. It requires only a topological procedure
for finding out the fundamental loop and the
fundamental cut-set matrices. The method operates
on the symbols of newly defined matrices ( PB TV and
PTCI ) whose elements are the products of impedance
and admittance corresponding to each non-zero
element of the fundamental loop and the fundamental
cut-set matrices. The sign of each determinant term
is obtained directly by calculating the value of easily
formulated, unimodular determinants. Efficiency is
further improved by the fact that for product terms
involving only passive elements we calculate only
one unimodular determinant. If dependent sources
are present in a product term, we must calculate two
determinants, one from the loop matrix and the other
from the cut-set matrix.
The algorithm consists of a generation procedure in
which terms are “grown” without duplication by
adding elements of the product matrix ( PB TV or
PTCI ), one element at a time. Every time an element
is added, it is tested whether it causes the appropriate
submatrices of the fundamental loop and
fundamental cut-set matrices to be singular. If any
submatrix is singular, then the most recently added
element of the product matrix is removed and
another element is examined.
Institute of Telecommunication, Teleinformatics and Acoustics, Wrocław University of Technology (WUT), Wrocław,
Poland. E-mail: [email protected]
2
Faculty of Engineering, University of Technology, Sydney (UTS), P.O. Box 123, Broadway, NSW 2007, Australia.
E-mail: [email protected]
In addition, we introduce a new notation of
symbolic expressions called the road map. This
notation has the advantage that in the process of
repetitive evaluation of transfer function for the
analysed network we do not have to calculate the
whole expression but only to update the terms which
are changed. Moreover, the method generates the
symbolic network function in a cancellation-free
form.
2 SYMBOLIC FORMULAE FOR RLC-gm
NETWORKS
The networks to be considered here are connected,
linear, time-invariant active or passive networks
modeled with immittances and voltage-controlled
current sources only. (If other types of controlled
sources are present, an extension of the two-graph
method, proposed in [7], should be used.) It is
assumed that any independent current source has been
replaced by a dependent current source controlled by
the desired output variable, thus forming a closed
system. Such network will be represented
topologically by a linear graph G consisting of a
voltage graph GV and a current graph GI. The hybrid
systems of equations for this class of networks for a
complete (common) tree T is
⎡ −I
⎢B V
Hx= ⎢ T
⎢ 0
⎢
⎣ 0
0
−I
0
YC
ZT
0
−I
0
0 ⎤
0 ⎥⎥
Q CI ⎥
⎥
−I ⎦
⎡ vT ⎤
⎢v ⎥
⎢ C ⎥ =0
⎢ iT ⎥
⎢ ⎥
⎣ iC ⎦
(1)
where the first and the last rows consist of tree (T)
and cotree (C) element functions and the second and
third rows consist of fundamental circuit and
fundamental cut-set equations for GV and GI,
respectively. The matrices ZT and YC are diagonal, I
is an identity matrix of an appropriate order.
The associated primitive signal-flow graph F(H) of
the hybrid systems of equations (1) is shown in Fig.
1.
B TV
vC
vT
Q
iT
)
) Y ⎤⎦
(
= det ⎡ I + B TV Z T −Q CI
⎣
(2)
C
For a given matrix B TV ( QCI ) of order m×n (n×m) let
B TV ( I u , J v ) [ Q CI ( J v , I u ) ] be the submatrix of B TV
( Q CI ) consisting of the rows and columns
corresponding to the integers in the sets Iu and Jv,
respectively, where Iu = {i1, i2, …, iu} and Jv = {j1, j2,
…, jv} are subsets of Sm = {1, 2, …, m} and Sn = {1, 2,
…, n}.
Using a modification of the Binet-Cauchy theorem
[6], we have
det(H ) = 1 +
∑∑ Z
( I1 ) ( J1 )
Y
Ti1 C j1
det[B TV ( I1 , J1 )]det[TCI ( I1 , J1 )]
2
+ ∑ ∑ ∏ ( ZT YC ) det[B TV ( I 2 , J 2 )]det[TCI ( I 2 , J 2 )]
ik
( I 2 ) ( J 2 ) k =1
jk
s
+ … + ∑ ∑∏ ( ZT YC ) det[B TV ( I s , J s )]det[TCI ( I s , J s )]
( I s ) ( J s ) k =1
ik
jk
(3)
where: s = min(m,n),
TCI ( I u , J v ) = ⎡⎣ −Q CI ( J v , I u ) ⎤⎦
t
and the summations are taken over all
possible subsets Iu, and Jv.
Although the formula (2) is obtained topologically,
the procedure for its evaluation (3) is purely
B TV ( I x , J x )
numerical [all submatrices
and
TCI ( J x , I x ) are unimodular] and no sign rule is
required. The network in this method is described by
a closed signal-flow graph, thus the numerator and
denominator of the desired transfer function are
calculated in a single process.
For the analysis of large electronic circuits, the
formula (3) should be expressed in another form.
Namely, we can group the terms related to non-zero
elements of the matrix B TV ( −Q CI ). In order to do
this, the new matrices - the product matrices PB TV
and PTCI - must be defined first.
Definition 1:
The product matrices PB TV and PTCI are obtained
YC
ZT
(
det(H ) = det I − B TV Z T Q CI YC
I
C
iC
Fig. 1. A primitive signal-flow graph of (1).
Thus, from Fig. 1, we can write the expressions for the
graph determinant:
from matrices B TV and TCI as follows:
i) Give the names Y(i) [Z(i)] and Z(j) [Y(j)] to the
ith row and jth column of B TV ( TCI ),
ii) Substitute symbol bij ← sgn Y(i) Z(j) [tij ← sgn
Z(i) Y(j)] for each non-zero element of the matrix
B TV ( TCI ), where ‘sgn’ is equal to the sign of this
non-zero element.
□
From Definition 1, using the matrix TCI , we can
obtain a simpler form of (3)
n1
2
2
n2
det(H ) =1 + ∑ sgn(i ) ti + ∑ sgn(i ) ∏ ti + …
i =1
i =1
ns
s
i =1
k =1
k =1
k
8
+ ∑ sgn(i ) ∏ ti
3
5
4
7
8
1
1
(a)
k
(4)
where:
n1 – number of non-zero elements in TCI ,
n2 – number of representative combinations of
two non-zero elements,
ns – number of representative combinations of s
non-zero elements,
s – min(m,n); m,n – number of rows and
columns in PTCI , respectively,
B TV
Y(1)
Z(1)
Z(2)
Z(3)
0
−1
−1
Y(2)
Y(3)
Y(4)
Y(5)
−1
−1
0
−1
−1
−1
−1
−1
0
0
−1
−1
−Q CI
Y(1)
Y(2)
Y(3)
Y(4)
Y(5)
Z(1)
Z(2)
Z(3)
−1
−1
−1
0
−1
0
−1
−1
0
0
−1
−1
−1
−1
−1
Using Definition 1 we obtain the product matrices
PB TV and PTCI (Table II)
TABLE II
Y6
Y8
+
V6
- Y 5V 6
Z3
Y7
Z4
+
V7
-
Fig. 2. Example circuit.
The linear voltage and current graphs representing
the closed system are shown in Fig. 3.
7
(b)
Example 1
Z2
5
4
TABLE I
elements,
sgn(i) = det[B TV ( I l , J l )]det[TCI ( I l , J l )] .
Consider a simple circuit shown in Fig. 2. The
independent current source I1 has been replaced by
the current source controlled by the output variable
V7 (I1 = Y1V7), thus forming the closed system.
2
The matrices B TV and −Q CI for these graphs, based
on a tree T = {2,3,4}, are shown in Table I.
ti – ith non-zero element in PT ,
tik – kth component of the product of non-zero
By representative combination we mean a single
combination of non-zero elements in a submatrix
PTCI ( I u , J v ) which represents the determinant of this
submatrix (in general, we can have a number of such
combinations).
Formula (4) is associated with the non-zero
elements of the fundamental cut-set matrix Q CI . A
similar formula can be derived, based on the
fundamental loop matrix B TV . As a rule, we choose
this matrix which has fewer nonzero elements.
3
Fig. 3. Linear voltage graph GV (a) and current graph
GI (b) of the closed system in Fig. 2.
I
C
I1 = Y1V7
6
6
PB TV
Y(1)
Y(2)
Y(3)
Y(4)
Y(5)
PTCI
Y(1)
Y(2)
Y(3)
Y(4)
Y(5)
Z(1)
Z(2)
Z(3)
0
−b12
−b13
−b21
−b31
0
−b51
−b22
−b32
−b42
−b52
0
0
−b43
−b53
Z(2)
Z(3)
−t11
−t12
−t13
0
−t31
0
−t51
−t22
−t32
−t42
−t52
0
0
−t43
−t53
Z(1)
where:
bij = Y(i)Z(j), tij = Y(i)Z(j) and
Z(1) = Z2, Z(2) = Z3, Z(3) = Z4, Y(1) = Y1,
Y(2) = Y5, Y(3) = Y6, Y(4) = Y7, Y(5) = Y8.
Consider now the matrices shown in Table II.
Calculating the determinant terms according to (4)
we obtain (terms associated with the numerator of
the transfer function, T = V7/I1, are marked with bold
letters):
det(H)=1+(+1)×t12+(+1)×t13+(+1)×t22+(+1)×t31+
(+1)×t32+(+1)×t42+(+1)×t43+(+1)×t51+(+1)×t52+
(+1)×t53+(−1)×t11t22+(+1)×t13t22+(+1)×t13t31+
(+1)×t13t32+(+1)×t22t43+(+1)×t22t53+(+1)×t31t42+
(+1)×t31t43+(+1)×t31t53+(+1)×t32t43+(+1)×t32t53+
(+1)×t42t51+(+1)×t43t51+(+1)×t22t43t51+(+1)×t31t42t53
□
The number of product terms, even in this simple
matrix determinant, is large. We need therefore a
compact method to represent (and store) the
determinants of network functions. Before we
introduce one such method, let us describe the
algorithm for generating all valid terms in (4).
3 ALGORITHM FOR GENERATING VALID
SYMBOL COMBINATIONS
In order to implement (4) we need an algorithm to
generate all symbol combinations, represented by the
product terms
∏t
ik
. The algorithm is based on the
k
closely related matrices TCI and PΤCI ; for active
circuits the fundamental loop matrix B TV is also
needed (in the remainder we assume dealing with an
active circuit).
A. Basic Algorithm
We start by choosing an element of PΤCI that has a
corresponding nonzero in B TV . We mark its column
and row as ‘used’. It becomes the first (besides the
unity) term in (4). Then, we try to extend the length
of the term by appending another element of PΤCI
(from an ‘unused’ row and column). The new
element is accepted if the sign of the extended term,
as defined in (4), is ±1. The extension phase
continues until the length of the current term reaches
s = min(m,n). Then the algorithm switches to the
exchange phase. In this phase we try to exchange
the last element of the current term with another
unused element of PΤCI . When there are no more
elements to exchange, the sequence is shortened by
removing the last element and the extension phase
resumes. The algorithm alternates between extension
and exchange, until a single term is generated that
cannot be extended nor exchanged.
B. Avoiding Spurious Terms
The basic algorithm, described so far, may generate
spurious terms that correspond to a valid symbol
combination already obtained (our experiments
suggest that for large circuits, whose B TV and Q CI
matrices are very sparse, such spurious terms are
rare).
To illustrate the possibility of generating unwanted
terms, let us consider the matrix PTCI from Table II.
It can be easily verified that three different terms:
t31t42t53, t31t43t51 and t32t43t51 correspond to the same
symbol combination: Y(3)Z(1)Y(4)Z(2)Y(5)Z(3).
This is because all three terms come from the same
major of PTCI , made of its last three rows. Clearly,
we need to consider only one of these terms (the
representative combination).
To detect such situations, our algorithm calculates
the permanent of the TCI submatrix, corresponding to
each generated term. (The permanent of a square
matrix is an analogue of a determinant where all the
signs in the expansion by minors are taken as positive.
Therefore, the permanent of a unimodal matrix is
equal to the number of nonzero terms in the
determinant expansion.) If the permanent is greater
than one, we check the list of permanents of this order
and either reject the term or accept the term as a
representative combination and append its generating
minor to the list (validation phase).
The pseudocode of our algorithm is presented
below.
procedure validate_term
calculate the permanent of the submatrix
corresponding to the term;
if (permanent > 1) {
if (submatrix already used)
reject the term;
else {
accept the term;
record the submatrix as used;
}
}
else
accept the term;
procedure extension
while (length of term < s) and (element available) {
extend the term by adding an unused element;
if (sign ≠ 0)
validate_term;
else
remove the last element from the term;
}
procedure exchange
sign = 0;
while (element available) and (sign = 0) {
exchange the last element in the term with the next
unused element;
if (sign ≠ 0)
validate_term;
}
procedure term_generation
term = NULL;
do {
extension;
exchange;
} while length of term > 0
C. Relationship to Common Spanning Tree
Generation
Although our algorithm and the common spanning
tree generation algorithm are not related, the final
results of both algorithms are related very closely.
The result of applying the common tree
enumeration algorithm to graphs GV and GI is a
sequence of unique tree admittance product terms.
Now, choose any common tree and divide every term
of the sequence by the admittance product of the
chosen tree. The resulting sequence will have
identical terms (generally, in different order) as the
sequence obtained by our method applied to matrices
B TV and −Q CI , based on the same tree.
have to calculate the whole determinants but only to
update the terms which are affected by the change.
Moreover, our method generates the symbolic
network functions in a cancellation-free form.
To illustrate the method we shall introduce an
implicit representation for the product terms in the
matrix determinants, called a road map. The det(H)
of the circuit in Fig. 2 can be represented by the road
map, shown in Fig. 4. In this map each term of the
determinant is represented by the road. Each road
has the number and its unique label. Formally, a road
map is a signed, directed, acyclic graph with paralleledges. The graph contains n vertices and the root,
and the end vertices. The number of vertices n is
equal to the number of nonzero elements in PTCI or
PB TV (we choose the matrix with fewer nonzeros).
The road map is represented by a collection of
linked lists of destinations (Table III). Each road
number is complemented by a symbol: N for the
roads related to the numerator and D for the roads
related to the denominator of the transfer function.
Each road label L represents a symbolic expression
La;Lb, defined as follows:
1) If X is a terminal vertex of the road, then La is
denoted by E and Lb is equal to the sign of the
determinant term (±1),
2) If X is a non-terminal vertex of the road, then La
represents the next vertex and Lb is denoted by P.
4 THE ROAD MAP REPRESENTATION OF A
DETERMINANT
As the number of terms in network determinants
grows exponentially with the network size, it is
essential that a compact scheme is employed to
efficiently store and retrieve the terms.
Several methods for compact representation of
network determinants have been reported in the
literature. One such scheme generates a nested
expression, based on the sparse recursive Laplace
expansion of determinant with minor storage [8].
More recently a method was proposed that consists
of representing the symbolic determinant of a circuit
by a graph – called a determinant decision diagram
(DDD) – and performing symbolic analysis by graph
manipulation [9]. The DDD method offers
significant improvement over previously used
techniques, utilising fully expanded (as opposed to
nested or hierarchical) formulae.
In existing methods the process of generation of a
symbolic network function is very complicated and
one must calculate the entire network function if a
single circuit element is changed. In this paper we
propose a method of generation of symbolic network
functions which is very simple and has the advantage
that in the process of the repetitive evaluation of
transfer function for the analysed network we do not
2D;L
11D;L
12D;L
3N;L
13D;L
19D;L
1D;L
START
1N;L
9D;L
t11
10D;L
t12
2N;L
3D;L
14D;L
15D;L
5N;L
t13
4N;L
t22
6N;L
10D;L
4D;L
5D;L
17D;L
9D;L
18D;L
16D;L
12D;L
11D;L
19D;L
13D;L
6D;L
4N;L
5N;L
6N;L
18D;L
3N;L
t31
8D;L
t32
14D;L
15D;L
t42
t43
17D;L 18D;L
t51
19D;L
16D;L
t53
t52
7D;L
Fig. 4. The road map of the determinant of the
example circuit in Fig. 2 (without the end vertex).
TABLE III
Start
RN
L
1N
1,1;P
2N
1,2;P
3N÷6N 1,3;P
1D÷4D 2,2;P
5D÷9D 3,1;P
10D÷12D 3,2;P
13D÷14D 4,2;P
15D÷16D 4,3;P
17D
5,1;P
18D
5,2;P
1.1
1.2
RN L RN L
1N 2,2;P 2N E;1
5.3
RN L
4D E;1
7D E;1
9D E;1
12D E;1
1.3
2.2
3.1
3.2
RN L
RN
L
RN
L RN L
3N E;1
1N
E;-1
5N
E;1 6N E;1
4N 2,2;P 4N
E;1
5D
E;1 10D E;1
5N 3,1;P 1D
E;1 6D÷7D 4,2;P 11D 4,3;P
6N 3,2;P 2D÷3D 4,3;P 8D 4,3;P 12D 5,3;P
4D 5,3;P 9D 5,3;P
4.3
5.1
4.2
RN L RN L
5.2
RN L
RN
L
2D E;1 3D E,1
6D E;1
3D 5,1;P 14D E;1
18D
E;1
7D 5,3;P
8D E;1 16D E;1
15D
E;1
13D E;1
Each road between the start vertex and the terminal
vertex represents one term of the determinant det(N)
or det(D) and is equal to a product of symbols tmn
(bmn) of those labels that are lying along this road.
For example, the road 3D has the following labels:
(2,2;P→4,3;P→5,1→P;E;1), and the corresponding
determinant term is equal to +t22 t43 t51.
The above list of destinations (Table III) has the
advantage that in the process of repetitive evaluation
of transfer function for the analyzed network, we do
not have to calculate the whole determinants but only
to update the terms which are changed. For example,
if we change the value of Y5 (only element t22 will be
affected), we must recalculate only the terms related
to the following roads:
- for the numerator: 1N, 4N,
- for the denominator: 1D, 2D, 3D, 4D.
With a view to simplicity of calculations, the
immitances are best represented in the exponential
form: Ya = |Ya|·exp(jφa), Zb = |Zb|·exp(jφb). In this
case calculation of both individual symbols tij = YaZb
and the product of symbols tij…tkl is very simple.
5 CONCLUSION
This paper introduced a new graph-based
representation of exact symbolic network functions
for analogue networks, called a road map. Although
the formulae are topological, the procedures for their
generation are purely numerical. For the sake of
notational simplicity the road map is represented by
a collection of linked lists of destinations. This
notation has the advantage that in the process of
repetitive evaluation of transfer function for the
analyzed network we do not have to calculate the
whole expression but only to update the terms which
are changed.
The method is based on the two-graph
representation of electronic circuits and a
modification of the Binet-Chauchy theorem. Thus,
this method can guarantee that cancellation of terms
does not occur.
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