IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 53, NO. 9, SEPTEMBER 2006 2011
Symbolic Framework for Linear Active Circuits
Based on Port Equivalence Using Limit Variables
David G. Haigh, Fellow, IEEE, Thomas J. W. Clarke, and Paul M. Radmore
Abstract—This paper proposes a new framework for linear active
circuits that can encompass both circuit analysis and synthesis. The
framework is based on a definition of port equivalence for admit-
tance matrices. This is extended to cover circuits with ideal active el-
ements through the introduction of a special type of limit-variable
called the infinity-variable (
-variable). A theorem is developed for
matrices containing
-variables that may be utilized in both circuit
analysis and synthesis. The notation developed in this framework
can describe nonideal elements as well as ideal elements and there-
fore the framework encompasses systematic circuit modeling.
Index Terms—Active circuit analysis, active circuit synthesis,
active circuits, circuit modeling, admittance matrix, nullor.
I. INTRODUCTION
O
NE of the difficulties in the field of linear active circuits
has been the lack of general mathematical techniques to
underpin the analysis and synthesis of practical designs. The
critical factor that prevents this possibility is the nonexistence
of a simple basis for which the descriptions of active circuit ele-
ments and the corresponding circuit functions exist. The pur-
pose of this paper is to show how an admittance description
can fulfill this requirement, providing that certain limiting cases
are allowed. This is made possible through the concept of port-
equivalence and a novel notation based on limit-variables.
Co-ordinate-free descriptions for nondegenerate linear active
circuits already exist, since it is recognized [1] that the behaviour
of such a circuit corresponds to an
dimensional subspace of
the
dimensional vector space spanned by the voltage and
current unit vectors. Linear active circuits may thus be char-
acterized as points in a Grassmannian [2]. Intrinsically nonsin-
gular co-ordinate systems exist in formalisms due to Belevitch
[1],
, and Youla [3], . These
elegant approaches have been used to prove important results in
circuit theory. However, they do suffer from a lack of economy
in that the number of coordinates is greater than the dimension-
ality of the space and the nonuniqueness of the equations means
that they have to be normalized to yield canonical forms. Hence,
these descriptions for circuits have not been widely accepted for
use in design and synthesis of active circuits.
The admittance basis for describing circuits, where the node
voltages are independent variables and the node currents are
Manuscript received June 6, 2005; revised November 30, 2005. This paper
was recommended by Associate Editor P. K. Rajan.
D. G. Haigh and T. J. W. Clarke are with the Department of Electrical and
Electronic Engineering, Imperial College London, London SW7 2BT, U.K.
(e-mail: dhaigh@ee.ic.ac.uk; e-mail: t.clarke@imperial.ac.uk).
P. M. Radmore is with the Department of Electronic and Electrical En-
gineering, University College London, London WC1E 7JE, U.K. (e-mail:
pradmore@ee.ucl.ac.uk).
Digital Object Identifier 10.1109/TCSI.2006.882815
dependent variables, is potentially an attractive candidate for a
general framework [4]. In order that a framework can accommo-
date circuit synthesis, it must be able to describe ideal circuit el-
ements, because synthesis using nonideal elements is usually in-
tractable. However, the admittance matrix representation suffers
from the problem that key ideal circuit elements, including the
nullor, which can represent the ideal transistor and op-amp, most
dependent sources, and the impedance converter, require infi-
nite matrix elements. This problem has been overcome by the
modified nodal approach (MNA) in which additional columns
and rows are incorporated into the standard admittance matrix
[4], [5]. As a consequence, the MNA has become the industry
standard for both numerical and symbolic circuit analysis. An-
other approach is to associate a series combination of a positive
resistor
and a negative resistor with each problem ele-
ment and then convert the combination of the problem element
and one of the resistors into an acceptable form using source
transformations [6]. For the MNA and
approaches, the di-
mensions of the matrix and the basis for the representation are
dependent on the type of elements contained in the circuit. This
does not greatly obstruct circuit analysis where the circuit el-
ements are known in advance and the matrix dimensions and
basis can be set up accordingly. However, this is a problem for
circuit synthesis where the types of elements needed in the cir-
cuit are not known
a priori. For a coordinate framework encom-
passing both circuit analysis and synthesis it is necessary for the
dimensionality, and the chosen basis, to be independent of the
type of element. This condition is satisfied by the admittance
basis, as the dimensions of the matrix are determined by the
number of nodes in the circuit, but the problem of infinite ma-
trix elements remains.
The use of a variable that is initially treated as a finite variable
(such as the gain of an op-amp,
) and then at some point al-
lowed to tend to infinity in order to define a limit is well known
[7]. Talbot in 1965 used an infinite parameter
in
order to describe a number of elements in admittance matrix
form, including the op-amp and the transformer [8]. Piercey
(working with Talbot) and then Sewell extended the work of
Talbot to the realm of circuit synthesis [9]–[11]. Using an infi-
nite parameter
and by deploying node introduction
matrices and sometimes transformation matrices, Sewell syn-
thesized the negative impedance converter and the circulator
[10] and various single amplifier Sallen and Key-type circuits
from their admittance matrices [11]. In the field of calculus, the
hyper-real numbers (
or ) have been identified, which are
greater than any real number but less than infinity and may be
used like finite variables [12].
This paper extends previous work on the use of matrices with
infinite elements. We will study their properties systematically
1057-7122/$20.00 © 2006 IEEE
2012 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 53, NO. 9, SEPTEMBER 2006
and show that they imply the existence of matrix equivalences.
We will use those equivalences to derive a theorem and use that
to provide a framework for analysis and synthesis of active cir-
cuits. The paper is based on preliminary work in [13] and [14].
We begin by considering a mathematical equivalence for admit-
tance matrices of passive circuits.
II. P
ORT
EQUIVALENCE FOR
PASSIVE
NETWORKS
A. Case Where All Circuit Nodes Are Accessible
Consider a circuit with
nodes, apart from the reference
node. At this stage, let the circuit consist entirely of 2-terminal,
linear, passive elements. Such an element has the admittance
matrix stamp
(1)
where
is the element admittance which is connected between
nodes
and ; node names in (1) act as labels for the rows
and columns the matrix elements occupy. The nodal admittance
equations for the circuit may be expressed in the (homogeneous)
form
(2)
where
is a column vector of node currents, ’,
is a column vector of node voltages, , and
is the nodal admittance matrix (NAM), which con-
sists of a superposition of stamps of the form of (1). The matrix
(2) defines a set of linear equations between components of
and . For a given circuit, is unique, and therefore there is a
one-to-one correspondence between the circuit and
.Wenow
consider the case where only some of the nodes are port nodes
and the remainder are inaccessible, or internal, nodes for which
the node current is zero.
B. Case Where Internal Nodes Exist
Consider a circuit with
nodes and ports .At
this stage, we still assume that the circuit consists entirely of
2-terminal, linear, passive elements. The nodal equations may
be expressed in the form
(3)
The partitioning separates rows and columns corresponding to
the port nodes,
, from those corresponding to the in-
ternal nodes,
. The second subscript as-
sociated with some of the sub-matrices denotes that the dimen-
sions of the complete admittance matrix are
. Kirchhoff’s
current law (KCL) implies that the dependent current elements
, in rows corresponding to the internal nodes,
are zero.
We can apply row operations iteratively in (3) in order to
perform Gaussian elimination, subtracting the bottom row from
each of the other rows after scaling it by a factor which reduces
Fig. 1. The nullor. (a) Nullator. (b) Norator.
the element in the last column to zero. The bottom row and last
column may then be discarded to yield
(4)
Application of
such reductions will lead to the port matrix
(5)
The reduced matrices could equally well have been obtained
by performing column operations with scaling to make the ele-
ments in the last row zero. In order to ensure that when the ma-
trices are reduced the port variables
are preserved,
the source row or column for each operation must always cor-
respond to an internal node.
Since each matrix in the series that starts with the NAM in
(3) and ends with the port matrix in (5) reduces to the same port
matrix, we may state that the matrices are equivalent in the sense
that they describe the same port behaviour
(6)
The Gaussian elimination process, moving from left to right in
(6) is one of circuit analysis. A reversal of this process, pivotal
expansion, would correspond to circuit synthesis. For passive
circuits in general, the synthesis problem posed this way may
be intractable [15]. However, we have shown that the removal
or introduction of internal nodes in a circuit permits the deriva-
tion of equivalent matrices that preserve the port behaviour. We
refer to such equivalence between admittance matrices as port
equivalence.
III. P
ORT EQUIVALENCE FOR ACTIVE NETWORKS
A. Limit Description for the Nullor
It has been shown that a sufficient set of elements to con-
struct any active network consists of a number of passive el-
ement types and a single type of active element, the universal
active element [16]. The universal active element is also known
as the nullor and consists of a pair of 2-terminal elements called
the nullator and norator, the symbols for which are shown in
Fig. 1. The nullator imposes two constraints on its voltage and
current,
and ; the norator imposes no constraint on
its voltage and current. The nullor may represent a small-signal
HAIGH et al.: SYMBOLIC FRAMEWORK FOR LINEAR ACTIVE CIRCUITS 2013
Fig. 2. Nullor equivalents. (a) Ideal op-amp. (b) Ideal FET and bipolar junction
transistor.
model for the ideal op-amp and transistor as shown in Fig. 2,
[17], [18]. The nullor may also be used, by itself or in con-
junction with passive elements, to realise higher level active el-
ements such as dependent sources [17], [19] or the complete
family of current, voltage and hybrid types of op-amp [16], [20].
It is known that the nullor can be derived as a limit of any
of the four types of dependent source when its gain tends to in-
finity. Since we are working with admittance matrices and the
voltage-controlled current source (VCCS) is the only dependent
source that possesses an admittance matrix, here we shall con-
sider the nullor as a VCCS for which the transconductance gain
tends to infinity. The admittance matrix stamp for the represen-
tation of a nullor with nullator connected between nodes
and
and norator connected between nodes and , as in Fig. 1, can
be considered as that for a VCCS with transconductance
(7)
where
is taken to a limit of infinity. One way to preserve
finiteness in an equation containing a parameter that tends to
infinity is to divide the relationship by that parameter. Let us
apply this approach in a set of NAM equations containing pas-
sive and active element stamps as in (1) and (7). At this stage,
we assume that different nullor representations never co-exist
in the same row or column of the NAM; this restriction will be
removed later. For a nullor whose norator is connected to nodes
and ,rows and of the NAM equation set have the form
(8)
where
(If or are internal nodes, then or
, respectively). Now consider dividing rows and of
the matrix equation by
(9)
where
. Dependent current variable terms on the LHS
and finite terms on the RHS vanish when the limit is taken
(10)
We are left with a relationship involving independent variables
only. Both rows corresponding to the norator nodes in the NAM
set of equations yield the same relationship between the inde-
pendent variables, namely
(11)
Since the nullor description in (7) has no entries in row
or row
, we also have
(12)
Hence the nullor description in (7) with
imposes fi-
nite relationships between the nodal voltages and currents which
correctly describe the nullator. The symmetry of the coefficients
in (7) imposes the constraint that the current entering the norator
is equal to that leaving it and (8) imposes KCL at nodes
and
; however, the norator voltage and current are otherwise un-
constrained.
Thus, infinite limits of elements in the NAM (2) may be used
providing we understand that the limit applies to the NAM equa-
tion rather than the NAM elements in isolation. Formally, the
NAM port equivalence class, which defines circuit behaviour,
has a well-defined limit in this case even though the NAM itself
does not. One advantage of this formulation is that a single ex-
pression may be used to represent both a real circuit, for which
the element has a finite value, and an idealized circuit. Taking
the limit may thus be viewed as an abstraction, or approxima-
tion, operation that converts real circuits into the related ideal
circuits. We shall see that this duality allows the framework we
are developing to handle not only circuit analysis and synthesis,
but also circuit modeling.
In order to make working with limits more practical, we now
introduce a special notation.
B. Limit Variables
In the case where a matrix contains a variable
that ap-
proaches a limit
, we replace each instance of the variable
in the matrix by the limit-variable . In the limit variable no-
tation
, the subscript denotes the variable that is involved
in the limit and
denotes its limit value. The limit variable for
may be abbreviated to , providing refers unambigu-
ously to the circuit variable
.
Where the limit is a limit to infinity, the limit variable is
called an infinity-variable, or
-variable, and written , where
refers to the circuit element whose parameter is being taken to
infinity. Using
-variables, the nullor description in (7) takes
the form
(13)
The ideal short circuit is equivalent to the parallel connection of
a nullator and norator; thus it has an admittance matrix descrip-
tion similar to that for the nullor in (13) except that the elements
are arranged symmetrically
(14)
2014 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 53, NO. 9, SEPTEMBER 2006
Replacement of regular variables in an NAM by limit-vari-
ables implies that these variables can no longer be given nu-
merical values and have to be treated as symbolic variables that
may be manipulated by hand or by using symbolic computa-
tion. Since
-variables are shorthand for finite variables with
infinite limits, and algebraic transformations may be applied be-
fore taking the limit, it follows that
-variables must conform
to the rules of algebra for regular variables, including Gaussian
elimination and pivotal expansion of admittance matrices.
1
The
concept of port-equivalence, developed in Section II-B for pas-
sive networks, is therefore equally applicable to matrices con-
taining
-variables. Using the -variable as a placeholder for a
variable that can tend to infinity at some point is just a notational
convenience. The real advantage comes from special operations
that are possible only for matrices containing
-variables.
C. Operations for Matrices Containing
-Variables
We begin by considering the general form that admittance
matrix elements may take during a process of Gaussian elimina-
tion. A circuit consisting of passive elements and nullors can be
described by an NAM consisting of a superposition of stamps as
in (1) and (13). Each element of the NAM must consist, in gen-
eral, of a signed sum of passive element admittances
and
-variables . It may be shown that, provided the admit-
tance of each circuit element is represented by a unique vari-
able, reduction of the NAM to the port matrix leads to matrix
elements at every stage which are bilinear functions of each cir-
cuit variable [17], [21]. Thus, in terms of an
-variable for a
particular nullor (nullor
), each matrix element at every stage
of the reduction has the form
(15)
where
and denote the row and column the element is in and
denotes the stage of the reduction process .We
now consider taking limits in respect of such a typical matrix
element.
Provided the coefficients in (15) are finite, then taking the
limit in respect of
can yield only three possible limiting
values, which are
or , where and are finite.
2
Hence, in the case where the element survives the limit, it
may be multiplied by a finite quantity.
In the case where the coefficients in (15) contain other
-vari-
ables, a number of cases arise: 1) a finite limit is obtainable; 2)
a function of two or more
-variables may be set equal to a
composite
-variable (this case applies if a differential pair of
field-effect transistors (FETs) are represented and the node the
sources are connected to is eliminated [22]); and 3) known re-
lationships between
-variables may be introduced in order to
reduce the number of
-variables, to one (as when the geome-
tries and bias conditions of FETs have known interdependencies
[23]). It is clear that an
-variable that emerges from these sce-
narios may have a finite scaling factor.
1
This is true under the assumption that the circuit described is nondegenerate,
as will be the case for all physically realizable circuits.
2
We exclude discussion here of the case where both denominator coefficients
are zero; this will be discussed in Section IX.
When we allow for a finite scaling factor associated with an
-variable, it is necessary to ensure that the constraints imposed
by the complete set of
-variables are consistent. This require-
ment can be met by introducing, into the set of
-variables de-
scribing the nullor in (13), a row scaling factor
and a column
scaling factor
(16)
For this matrix, the constraint imposed by each row is iden-
tical and given by
and the constraint imposed by
each column is identical and given by
,. The set
of elements in (16) is a very general one that can represent the
class of 2-port circuits which do not, in the conventional sense,
possess an admittance matrix; this class includes the nullor and
short-circuit, and all the elements whose stamps we will derive
in Sections V–VII.
Following the procedure adopted in Section III-A for inter-
preting the nullor description in (7), let us divide row
and
column
of the matrix equation set corresponding to (16) by
to create row and column for use in performing row
and column operations
(17)
All other elements in the row
and column become zero,
including dependent currents on the LHS of the matrix equation
in the case where the source row corresponds to a port node.
If we had derived (17) starting from the second row or second
column of (16) instead of the first row and column, we would
have obtained an identical result.
Thus, the presence of
-variables in an admittance matrix
permits the carrying out of special row and column operations
that preserve port equivalence; these are in addition to the
general ones for finite NAMs described in Section II-B that are
equally applicable to matrices containing
-variables. The spe-
cial operations differ from the ordinary ones in that the source
row and column are not restricted to correspond to internal
nodes and in that it is only coefficients of
-variables that are
scaled and added to other rows and columns of the matrix. Thus,
the admittance matrix of a circuit containing nullors where all
of its nodes are accessible is not unique, as it is in the case of a
network of passive elements. Before developing theorems from
the special row and column operations, we consider the case
where two or more sets of
-variables share a row or column
of an admittance matrix.
D. Case Where Different
-Variables Co-Exist in a Single
Row or Column of [Y]
We amend the first row of the admittance matrix shown in (16)
by introducing a second set of
-variables with column
scaling factor
(18)
HAIGH et al.: SYMBOLIC FRAMEWORK FOR LINEAR ACTIVE CIRCUITS 2015
The nodal equation for this row may be written as
(19)
where
is the nodal current ( if is an internal node)
and the finite terms may exist in any columns. Let us divide (19)
by
(20)
Since
and are independent variables approaching the
limit of infinity, this equation must be true for all finite values
of
. Hence, in the limit, the solution to (20) yields two
separate equations
(21)
Thus, each set of
-variables yields a separate row, corre-
sponding to one of the expressions in (21), that may be used as
a source row for row operations
(22)
The same outcome arises where two sets of
-variables share a
column of the matrix and applies irrespective of the number of
sets of
-variables that share a row or column. Thus, each set
of
-variables that shares a row or column with other sets of
-variables generates its own row and column that may be used
for row and column operations.
IV. T
HEOREM FOR
MATRICES WITH
-VARIABLES
A. Arbitrary Element Theorem
From the general set of
-variables in (16), we may derive
an extra row
and column as in (17), scale the extra row
and column by arbitrary factors and then add them to any row
or column, respectively
.
.
.
(23)
Variables
and are arbitrary expressions and and repre-
sent any row or column including the source rows and columns,
and . We call this equivalence the arbitrary element the-
orem.If
and were to contain -variables, there would be
an inconsistency in the derivation; hence the arbitrary elements
and in (23) are restricted to be finite. We now consider a
particularly useful corollary of the arbitrary element theorem.
B. Element Shift Theorem
We apply the arbitrary element theorem in (23) in the special
case where there already exist matrix elements
and and
we let
and
.
.
.
.
.
.
(24)
The effect is to eliminate
and from their original posi-
tions and shift them as shown while scaling them by the appro-
priate row or column scaling factor
or , respectively. Where
and consist of a sum of admittance terms, then the shift
theorem may be applied to any sub-set of these terms. If
or are -variables, then (24)-LH already implies some con-
straints on
and ;
3
since these constraints are unchanged in
(24)-RH, it follows that, in this corollary of the theorem,
and are permitted to be -variables.
C. Arbitrary Element and Element Shift Theorems for the
Nullor
The description for the nullor corresponds to setting
in (16), in which case the arbitrary element theorem in (23)
takes the form
.
.
.
(25)
Under the same conditions, the element shift theorem of (24)
takes the following forms for finite and for
-variable elements:
.
.
.
.
.
.
(26)
.
.
.
.
.
.
(27)
Davies and others [24]–[26] have suggested a method of
analysis for circuits containing nullors in which the rows of the
matrix corresponding to the nodes of each norator are com-
bined into a single row and the columns corresponding to the
nodes of each nullator are combined into a single column. The
theorems in (25), (26), and (27) are consistent with this method
of analysis, because introduction of elements or movement of
3
LH and RH denote left-hand and right-hand matrices, respectively.
![Fig. 10. Sub-circuit that leads to zero pivot in [Y].](/figures/fig-10-sub-circuit-that-leads-to-zero-pivot-in-y-3diz87bo.webp)
