Modified no al analysis:

an essential addition to
electrical circuit theory and
anaIysis
by L. M. Wedepohl and L. Jackson
T k e known limitations of classical mesh and nodal methods ofanalysing linear electrical circuits are
described before considering an established modijcation ofthe nodal approach. The method, known as
‘tnodijed nodal analysis’, has none ofthe limitations ofthe basic nodal technique and is well suited both
to symbolic and numeric analysis of complex circuits using modern matrix-based soffware. T h e simplicity o f
incorporating into the matrix equations all types ofpasrive and active circuit elements is demonstrated and
examples are used to illustratefurther the e@cacy ofthe method. It is emphasised that the absence ofthis
circuit analysis techniquefrom many academic engineering courses is totally at variance with its widespread
application in modern circuit simulation packages.

capabilities, is a much more powerfid analytical tool.

S
ince the days of Maxwell the development of
methods for the solution of electrical networks The method, known as ‘modiGed nodal analysis’”, is
with large numbers of elements has proceeded applied quite widely in electrical circuit simulation
systematically. Maxwell’s cyclic current, so packages. However, many undergraduate courses and
named in his honour, was fundamental to the method texts do not acknowledge this key development so that
of mesh analysis, which was the earliest method of engineers entering the profession have been denied the
formulating equations for electrical circuits and is now opportunity of studying it.
more conunonly identiGed with Kirchhoffs voltage
law (KVL). Nodal analysis was subsequently Classical nodal analysis
introduced as the topological dual ofmesh analysis and
is identified with Kirchhoffs current law (KCL). The In classical nodal analysis currents are not explicitly
nodal method has become the mainstay of analysis of calculated so the number of unknowns, namely the
large electrical systems principally because it has two voltages and currents of branch elements, is halved. A
overriding advantages over mesh analysis. Firstly further reduction is made hy speclfylng nodal voltages
problems of crossovers in nonplanar networks are with respect to a reference node so that voltages
eliminated thus avoiding the need for tree-graph between nodes are expressed as the difference between
theory to formulate the equations. Secondly the the associated node voltages. Thus, there are as many
number of equations is smaller with the nodal unknowns as there are non-trivial nodes.
approach, reflecting the fact that the number of nodes A consistent set ofsiniultaneous equations is obtained
in a network is generally less than the number of by invokmg Kirchhoffs current law for each node in
branches. Notwithstanding these advantages nodal ~ r n The
. currents are expressed in terms of all the
analysis has certain limitations which complicate its branch currents leaving a node equated to the forcing
application in some types of circuit. currents entering the node. Voltage sources,
It is the purpose of this paper to consider briefly the transformers and dependent sources do not enter into
sahent features of the more favoured nodal analysis, nodal analysis in a natural way and have to be
w h c h has been in use for more than 100 years, and transformed using a variety of algorithm, such as
then to review a known method, derived from nodal Norton’s theorem, to get the equations into suitable
analysis, which, combined with modern computing form (although it must be emphasised that ideal voltage

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84
sources and transformers cannot be transformed).Such
transformations result invariably in loss of informahon
because topologically connected nodes are reduced in
number. Also, certain variables, such as source or
transformer currents, have to be obtained by post-
processing. It is these aspects ofnodd analysis, requiring
non-standard pre- and post-processing procedures to
complete a solution,which have been a constant source
of confusion and irritation and whch are mastered
only through long experience.
Before developing the theory of momfied nodal Fig. 1 Independent voltage source
analysis it is helpful first to review the properties of
nodal analysis. Thus, in classical nodal analysis in which
matrix methods are employed, each row in the matrix In MNA such nodes retain their identity So, having
represents a constraint equation, namely the Kirchhoff removed the non-natural elements, the resulting system
current equation for that node. On the left-hand side is described by the basic equation:
of the equation, the unknown nodal voltages are
multiplied by the adnutrance of the associated passive W = I (1)
components that are connected to the node arid on the
right-hand side is the known forcing current injected The process ofreintroducing the non-natural elements
into that node. Since there are as many nodes as is described in the following.
unknown uodal voltages a consistent set ofequations is
obtained and a solution can be effected. If the Voltage sources
numbering of the nodal voltages is consistent with the Introducing an ideal voltage source, E,, and series
order of entry of the equations, the nodal admittance impedance, Zs,between nodes p and q, as in Fig. 1,
matrix is symnetrical. This meam that the element at affects the original set ofequations in two ways. Firstly,
the intersection of mw m and column n is the same as it introduces a new constraint equation defining the
the element at the intersection of row nand column m. relationship between the two nodal voltages V,, and V+
It is important to remember, especially when Secondly, it permits an unknown current I, to flow
developing the modified nodal analysis algorithm, that Thus the new constraint equation increasesthe number
the rows arid columns have quite different meanings. of rows in the matrix by one and the equations for
Modified nodal analysis (MNA) was forniulated in nodes p and q are modified to take account of the
the nlid-1970s and developed subsequently for the additional currents in those two nodes. This maintains
analysis of analogue filters and the simulation of the consistency of the set of simultaneous equations.
electronic circuits; it is used in one form or ariother in Assuiuing that the positive end ofthe voltage source E,
n m y modern simulation patkages, such as SPICE. It is connected to node p and that the unknown current
is not well known in the analysis of power systems. I, is dnected toward the positive ternunal of the source
However, it has many advantages over classical nodal and hence flows away from node p and toward node q.
analysis, principal amongst which are the following: thrn the nodal equations forp and q are modified and
a new equation or row added, namely:
(a) All linear circuit elements, both passive and active,
ideal and non-ideal, are accomnodated by MNA
with virtually no pre- and post-processing.
(b) The passive (admittance) element of the matrix is
symmetrical, which greatly facilitates the checking
of input and output data.
(c) Diakoptics, or partitioning of large system, is very
easy to define and apply Note that for completeness the forcing currents I, and
I, entering nodes p and q, respectively, have been
Modified nodal analysis included, although they are zero when voltage sources
only are present in the circuit. Clearly, the consequence
In MNA, the first step is to remove completely all non- of introducing the voltage source is to augment the
natural elements in order to write a conventional set of o r i g i d matrix by one row and one column, the
nodal equations. These elements are voltage sources, column being the transpose of the row, and the internal
transformers and dependent sources (other than impedance, with negative sign, being at the inter-
voltagr-controllrd current sources). The resulting section ofthe two. A short-circuit may be simulated by
adnllttance matrix differs from that obtained by using setting Z,and E<equal to zero and can be used as an
classical nodal analysis in which, for exauple, the nodes ideal ammeter inserted between nodes p and q.
on either side ofa voltage source become topologically Using Gaussian elimination to remove the
identical and are short-circuited together. augmented row and column, by pivoting at -Zs,

ENGINEERING SCIENCE AND EDUCATION JOURNAL ]LINE 2002

85
 will be zero.
(b) Both ideal and non-ideal sources are accom-
modated by a single equation, i.e. Z, can be zero or
finite.
(c) It is possible to access the junction of the source
impedance and source voltage by entering the
impedance separately as an admittance in the nodal
nlatrix and then entering the (ideal) voltage source.

Voltapdependent voltage source

The most common example ofa voltage-dependent
voltage source is an operational amplifier. A schematic
representation is shown in Fig. 2 for which the
constraint equation is:

Fig. 2 Voltage-dependent voltage source V p - V q =G(K- + Z& or

-K+ ~+DVP-DVq-DZ&=O (4)
modifies eqn. 2 as follows: where 0 = 116 and E, = G(K - 4)

The equations are arranged to use the inverse of the

(3) gain G in order to fachtate the analysis of ideal
aniplifiersin which the gain is infinite. It is assumed that
the input impedance of the amplifier is infinite. If not,
where Y i i , Y,2 etc. are the elements of Y in eqn. 2. the impedance may be entered as a passive impedance
This is the classical nodal matrix that would have externally The action of introducing the amplifier
been obtaiued if the source voltage and source results in an unknown current ofmagnitude 4 entering
impedance had been subjected to a Norton trans- the amplifier. Thus the constraint equation augments
formation, which would not have been possible if the the system matrix by one row and the current
source impedance had been zero. In this case, the augments the matrix by one column, maintaining the
matrix could still have been reduced by first removing consistency of the system equations. The structure of
the last row by pivoting about the unit value in either the augmented row and column is:
of the columns and then removing the last column by
pivoting about the unit value in the corresponding
rows. In this case one of the nodes, p or q, would
disappear, because they are topologically identical
when joined by an ideal voltage source. This principle
of equivalence is demonstrated further with a simplc
example in the section on applications of MNA.
The following additional features of the voltage
source are worthy of note:
Note that eqn. 4 is for a non-inverting amplifier ifnode
(a) If one of the nodes, p or q, is taken as a reference p is chosen as the output node and an inverting
then the corresponding unit values in the matrix amplifier ifnode q is chosen.
If now the additional row and colunln are removed
by pivoting at -DZ, then the classical nodal matrix is
recovered and takes the form of eqn. 6:

In the right-hand matrix, E2K = CU,(K - V,) and

therefore the terms Vj and V, can be incorporated into
the left-hand matrix, resulting in an asymmetric
modification to the original adrmttance matrix. Such
modification can make it extremely difficult to
set up the admittance matrix by inspection, even for
Fig. 3 Voltage-dependent current source relatively siniplr circuits, so that one of the very

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86
useful features of the nodal method is lost

Voltqc-dependent current source

A voltage-dependent current source, shown sche-
matically in Fig. 3, may. in its simplest form, be a
transistor or may be a more complex circuit
arrangement. It should be noted however that this
dependent source can be acconmodated directly hy
classical nodal analysis.
The constraint equation is:

4 = G,,(V,- y ) or
V , - 5 RJ, = 0
- (7)
where K,, = 1/6,

which is defined in terms of R,,, rather than 6, in order Fig. 4 Current-dependent current source
to consider ideal sources in which the trans-
conductance is infinite. q would be the collector (output).
The structure of the matrix equation is:
Current-dependenr u o h p source
This is the one case in w h c h the element has to be
defined in ternx of two augmented rows and two new
unknowns. The reason is that the input current has to
be defined as an unknown by including a zero voltage
source as was done for the current-dependent current
source. However the output voltage source is also
accompanied by an unknown current which is not
related to the input current. Unlike the above examples
It should be noted that ifa non-inverting configuration of dependent sources the current-dependent voltage
is required, node p should be chosen as output node; source is not realised in any natural physical device.
node q should be chosen for an inverted output. However, if necessary such an element can be defined
by hvo augmenting rows and columns.
Ciirr~,rrr-depmdcrrtcwreiir sourcc
The most common, and simple, realisation ofsuch a Two-winding transformers
source is a transistor operated as 3 current-dependent The introduction ofa transformer connected between
element, but it can ofcourse he realised in many forms; four nodes, as in Fig. 5, creates one new constraint
Fig. 4 shows a general schematic representation. equation which is balanced by one additional unknown,
An essential constraint is that the voltage between namely the secondary current. The primary current
nodes i and j shall be zero. This allows an unknown will be determined by the ampere-turn balance equation.
current li, to flow into node i and out of node j . The The constraint equation is:
output current, being proportional to I,, is known.
However in order to permit an ideal amplifier with AT( V, - V, + NZJJ = Vp- Vq, or (10)
infinite current gain to be analysed, the output current -NK + NV, + Vp - V4- N2Z,I,= 0
I, will be designated as unknown. Thus, the input
current li, = DI,, where D is the reciprocal ofthe gain,
and the structure of the augmented row and column
becomes:

I Y
i D K

i l &
.... ._...... .... ___.
l - l O O i O I , ;-I
i - D y
~

....
4
-1 & = I p
L

I*
0

The device will be non-inverting if node p is chosen

as output. For a grounded emitter transistor, which
is inverting, the base would be node i, the emitter
(9)

would be nodes j and p (the same node) and node Fig. 5 Wo-winding voltage transformer

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87
 Here the two-winding transformer of ratio N is
connected between nodes 1 , 2 , 3 and 4 and Y, = 1/Z,.
This conversion is only possible if Z, is non-zero,
precluding the use ofthe EMTP model when studying
the hehaviour of ideal transformers. No such
restriction applies to eqn. 11, where Z, = 0 does not
create a problem.
Having established the modified matrix for a
simple transformer it is then possible to introduce
the concept of a phase-shifting transformer. Such a
device is encountered in symmetrical component
analysis whereby an ideal star-delta (or zig-zag)
transformer has the effect of introducing a phase
shift from one side to the other without any loss of
power. The accompanying matrix is modified to
the extent that the augmenting row in eqn. 11
becomes:

-NLQ, NLQ,1, -1, -N2Zt (13)

and the column becomes the complex conjugate of

the row:

-NL-Q, NL-g, 1-1, -N2Z, (14)

Fig. 6 Single (tapped) winding voltage transformer Auto-transformers
An auto-transformer is the same as a two-windmg
In addition, currents -NL, NIr,I, and -Il leave nodes i, transformer with one node 'shared' by primary and
j , p and q, respectively This means that the system secondary. This is easily taken into account hy
matrix must be augmented by addmg a row: sumnung the elements of the augmented row and
column to the appropriate nodes. A simple
-
arrangement is shown in Fig. 6 for which the constraint
equation is:

(1 1) ( K - Q = N ( y - V k ) , or (15)
K-(N+l)Y+NVb=O

Thus, the matrix is augmented as in eqn. 16, in which

the add~tional column is, again, the transpose of
It must also be augmented by a column to the voltage constraint row, as may be seen €-om
accommodate the variable '4' with the column being the magnitude and direction of current flow in
the transpose of the row and with the parameter -NZZt Fig. 6:
being at the intersection of the row and column. With
the normal consistent numbering between node
voltages and node numbers, this WIII ensure that the
curren6 will appear at the correct nodes with the (16)
correct polarity An ideal transformer may be simulated
by setting the impedance Z, to zero. 1 -(N+l) N , 12

If Z, is non-zero, the augmented row and column

may he removed using Gaussian elimination and Entering data to the augmented rows and columns may
choosing the parameter -N2Zfas pivot. Ifth~sis done, be made identical to that for a two-winding
and it is assumed for clarity that the elements of Y in transformer with discrete nodes. The augmented row
eqn. 11 are zero, it will automatically generate the and column are first set to zero. The entries are then
standard EMTP' nodal equation for a floating two- summed into the appropriatelocations. Ths eliminates
winding transformer, namely: the need for an auto-transformer to be defined as a
special case.

Grounded tap-changing auto-tranrfrmer

In this case there are only two nodes for each
transformer. The matrix equation is the following:

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88
 This may he recognised as a classical nodal equation
I
I-(N+l) N
.... ~ ~ ..
.... ....
0
incorporating a voltage source. The source has been
implicitly transformed by Norton$ theorem and
node 4 has disappeared because nodes 3 and 4 are
topologically the same by virtue of the voltage source
In the particular case that N = 0 the solution is trivial connecting them
and correct, namely Vb = 4. The following points may be noted

Multiwindirg ideal tran$urrners (a) Pre-processing ofboth the system nodal matrix and
A three-windmg transformer has two constraint the right-hand-side known vector are required
equations, one defining the voltage transformation before the matrix equation can be solved.
between primary and secondary and the other the (b) The nodal matrix has to be modified by adding
primary to tertiary transformation. The two new together rows 3 and 4 and then columns 3 and 4.
unknowns are the currents in the secondary and The Norton transformation of the voltage source
tertiary windings. The voltage in the primary is and the resultant additional driving currents
derived from unpere-turn balance in all the windings, in the right-hand-side vector presents problems.
so that the matrix is augmented finally by two rows There are, confusingly, many different methods
and two columns. These, in fact, will show the for explaining how this is to be done and it
arrangement to be cquivalent to two separate two- is quite common for polarity mistakes to be made.
winding transformers. Such equivalence can be (c) Both the voltage of node 4 and the current in the
extended to a transformer with W windings, for voltage source are suppressed, and have to be
which the matrix is extended by W-1 rows, to calculated by post-processing after the remaining
acconmiodate the voltage ratio constraints, and W-1 nodal voltages have been determined.
columns, to accommodate the W-1 unknown (d) MNA is totally straightfonvard, the matrices are
currents. As in the case of the auto-transformer this easy to compile and no knowledge of special
procedure eliminates the need to enter this theorems is required nor is any pre- or post-
arrangement as a special case. processing required.

Applications of modified nodal analysis There is another important property of MNA that has
significant ranlifications in the analysis of very large
Examplel: Relationship between MNA and clasikal systems. Before augmenting eqn. 18 with the
nodal analysis constraints for the voltage source, the pure nodal matrix
There is a fundamental relationship between MNA could have been reduced to order 2 by eliminatingrows
and nodal analysis which is best deiiionstrated by way and columns 1, 2, 5 and 6 by standard Gaussian
ofa simple example in which two separate three-node elimination (pivoting on the diagonal elements 1 , 2 , 5
subsets of a system are connected together by a single and 6). This is a very efficient process because the two
voltage source benveen nodes 3 and 4. The MNA halves of the system are decoupled. The reduced
matrix equation is: equation, after reintroducing the constraint equation
- for the voltage source, is:
XI -E?
-x3 0 0 0
-Ylz E2 -E3 0 0 0
-x3-Y23 Y33 0 0 0
0 0 0 E4 -&5 -&fi
0 0 0 -E5 E5 -E6
0 0 0 -fie -86 E6 V3, Vq and I may now be found and the remaining
- 0 0 1 -1 0 0 voltages found by back substitution in the usual way By
postponing the introduction of the voltage source,
Two successive reductions may be made. Firstly, I is ‘fill-ins’ have been substantially reduced. In large
eliminated using the fourth equation (pivoting about systems this dramatically improves the efficiencyof the
the term ‘-1’ at the intersection ofrow 4 and column 7) soluaon. I t also directs the ordering of the elimination
and, secondly, V4 is eliminated using the last equanon of nodes in a sparsity algorithm. Processing of the
(pivoting about the term ’-1’ at the intersection ofmw augmented rows and columns due to voltage sources,
7 and column 4). The result is: transformers and dependent sources should be
postponed until the nodal coniponent of the MNA
matrix has been reduced by Gaussian elimination to
those nodes associated with these elements. It should
be noted that this procedure is an example of
dakopticss although there is no need to store the
subcomponents of the nodal matrix separately unless
storage is limited.

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 a b

Fig. 7 (a) Active bandpass filter; (6)its equivalent circuit

Example 2 : Deperident sources involung a Norton equivalent of the input voltage

A Sallen and Key inverting bandpass filter is shown source and resistance RI in which the input current
in Fig. 7 a . The equivalent circuit in Fig. 7 b shows the into node 2 would he W R I .Also, incorporating the
operational amplifier as a voltage-dependent voltage dependent source (the amplifier) requires additional
source with an infinite input impedance, zero output processing, which destroys the symmetry of the
impedance and voltage gain 6.In fact the impedance admittance part of the matrix. Of course, for hand
parameters are easily acconmiodated hut are omitted analysis, there is obvious benefit in reducing the order
here for clarity. The resulting matrix equation of the matrix using whatever processing achieves that
follows from the two simple constraint equations effect. However, one of the principle virtues of MNA
which are: lies in its simplicity and the ease with which numerical
analysis can be undertaken using computer-based
VI = v, matrix methods. That is not to say that symbolic
Vj = DV4 analysis based on MNA has any less value. Indeed,
where D = 1/G a number of modern symholic circuit simulators
use MNA in one form or anothr? but further
(Note that if G were assumed infinite, then D = 0 and consideration of such methods is beyond the scope of
v3 = 0.) this paprr.
The matrix equation incorporates these constraints
to become: Example 3: Star-delta transformer
Multiphase transformers present students with
logistical problems in defiuing a consistent set of
equations for solution. The star-delta collfiguration
is particularly troublesome. To demonstrate the ease
-G '(G+! with which such a problem may be defined using
0 0 0 0 I, MNA, consider the circuit arrangement of Fig. 8
-1 -D 0 0][lA in which the transformation ratio from primary
to secondary is assumed to be 1:2. For the 9 designated
nodes and the associated admittances there will be
where a 9 x 9 admittance matrix, which is then supplemented
with 6 rows and associated columns for the voltage
G I = ~ / R IG, = l / R z , SI=jwC,, Sz=jWCz (23) sources and the 3-phase transformer. In this instance
the example can be simplified without loss of clarity
As emphasised earlier, the admittance part ofthe matrix by considering only one primary phase, say red, and
retains its symmetry about the diagonal and the the associatedsecondary (delta) components. Thus, the
augmented raws and columns are made possible by simple constraint equations are:
incorporating the currents in the two voltage sources as
additional unknowns. This hybrid nature ofthe matrix VI = E, (24)
equation makes it possible to define more easily, in t h i s v,-v6-2vz=O
example, the effective input and output impedances.
Classical nodal analysis could be applied only by and the associated matrix elements are:

ENGINEERING SCIENCE AND EDUCATION JOURNAL JUNE 2002

90
 k; -y o . . o . . 1 . . 0 . . 11= 4-16L-5, 12=2.37L-137-78,13 3,088114063
-k; y 0 . . 0 . . 0 . . - 2 . . 1" = 8.321-5, l b = 4751-137'78, IC= 6'18L140.63
0 0 Y l " 0 .' 0 " 1 "
(26)
.. .. .. .. .. .. .. .. .. ..
.. .. .. .. .. .. .. .. .. ..
The unbalance in the currents is due of course to the
fact that the delta secondary is grounded through
0 1, unequal resistors.
.. .. .. .. .. .. .. .. .. .. . .
0 -2 1 0 .. 0 .. 0 11
.. .. .. .. .. .. .. .. .. .. .. Implications for numerical analysis
..
.................... "
I, - It may appear f" the foregoing that MNA requires
more storage and computation than conventional
where, on the left-hand side of the equation, each nodal analysis. This is not so ifa good sparsity routine'&
colunm ofthe system matrix corresponds with each of is used for loading the system data and solving the
the components identified in the unknown colunui equations. This stenis fioni the fact that, although there
m?trix. Uy following the above procedure the are more nodes, the MNA equations are much more
remaining nvo phases can be entered into the sparse than their conventional nodal equivalents with
appropriate locations of the matrix. the result that the number of numerical operations for
The rase of setting up the system matrix is apparent the two methods is the same.
and, again, attention is drawn to the high degree of A particularly subtle benefit derives from the fact
sparsity which, given a sparse matrix routine, nuniniises that MNA can result in a dramatic reduction in
computation by bypassing trivial operations. A simple numerical processing if the equations are carefully
numerical example serves to illustrate the power ofthis ordered so that augmented rows and columns are dealt
approach. Thus, consider a balanced source in which with last, as indicated in example 3 . A further increase
the three phase voltages are each 100 V and the source in efficiency may be achieved if voltage sources that
resistors are each 1 R, and consider an unbalanced change from one shidy to the next are ordered so that
secondary in which the three resistors connecting the they are dealt with last. An example of this would be
delta to ground are l / Y l = 1 0 0, 1/Y2 = 20 R and where a large complex system comprises passive and
l/Y, = 40 R. The solution autonlatically delivers the active elements but the configuration and number of
primary and secondary currents as well as the such elements may vary due to switching operations
transformer primary voltages finm which the in, for example, an electrical power system. The
secondary voltages are then easily derived. For this switches can be represented simply as zero voltage
particular case the various cornponents are: sourccs. If the ordering is such that switches are
entered last, then the partitioned form of the system
Vr = 91-71LO.45, Vs = 95.49L-119.13. matrix takes the form:
Vg = 94.25L118.68
V3=69.331-19.56, Vi,= 120.64L-168.21,
Vi= 14436Ll00.02

Fig. 8 Three-phase, star-

delta, voltage transformer

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91
Here A is that part of the system matrix that does not behaviour of ideal transformers, w h c h do not have a
contain any switches, although it may contain, for classical nodal equivalent circuit, it being necessary to
example, voltage sources and transformers. B represents include series impedance, so that the device then
the element? acting on the unknown currents flowing becomes non-ideal. Also, once the matrix equation
in the switches, C represents the constraint equations has been established, the student has access to all the
for the switches and D contains the resistance values of required unknowns without resorting to further
the switches, which may or may not be zero. If processing.
triangularisation is applied from the top left, then the Where large system are being analysed, MNA
operations on A are independent of B, C and D and indicates the most efficient ordering for the Gaussian
may be computed once and then stored. Furthermore, triangularisation process, which conserves sparsity and
there are only two non-zero elements in each column minimises the number of arithmetic operations in
of B and each row of C. By entering the nodes to obtaining the solution.
which the switches are connected to be at the bottom It is hoped that this short paper will help stimulate
of B, then the nodes in the upper half of B and the left some development in the teachmg of circuit analysis
half of C will be zero. Consequently, in the with the aim of filling the gap that has developed
triangularisation process, no operations will be required between some undergraduate and graduate courses
on the elements in the upper halfofB. This process will and established engineering practice.
reduce W i n s and will also acheve a major saving in Finally, it is hoped also that more practising
computation and could well be considered one of the engineers who use traditional methods will consider
overriding advantages of MNA compared with other MNA as an alternative technique.
methods.
Acknowledgments
Conclusions
The authors wish to acknowledge past discussions with
Modified nodal analysis removes all of the limitations Dr. Brian Stott and Dr. AEed Brameller, formerly
of the classical nodal method and is most appropriate with UMIST, Dr. Hermann Dommel, formerly with
for the symbolic and numeric analysis of linear University of British Columbia, Messrs. Denis
electrical circuits. The only overhead in adopting Woodford and Garth Irwin of the Elektranix
MNA is in the increased order ofthe matrix equation, Corporation in Winnipeg, and Mr. Rum Wierckx of
hut this is countered by the fact that such equations are RTDS Technologiesin Winnipeg. Thanks are due also
generally highly sparse and amenable therefore to to Professor Edwin Powner of Sussex University.
efficient solution using modern matriu-based
computing packages. These make use of powerful References
sparsity routines whereby only nonzero elements
1 HO, C-W., RUEHLI, E., and BRENNAN, P A,: ‘The
are stored and trivial operations (such as multiplication
modified nodal approach to network analysis‘, IEEE Pons.
by, and addition of, zero) are avoided. The fact that Circuits Syif.. June 1975, CAS-25, pp.504-509
MNA is applied widely in commercial circuit 2 VLACH, J., and SINGHAL, K.: ‘Computer rnerhods for
simulators, and has been for some years, emphasises circuit analysis and design’ (Van Normand-Rrinhold,
the importance of this powerful analytical tool and NewYork, 1983)
3 VLACH, J.: ‘Basic network theory with computer
yet most engineering courses, and many associated applications’ (van Nostrand-Reinhold. New York, 1992)
academic texts, fail even to acknowledge the topic. 4 DOMMEL, H. W: ‘EMTP theory book‘ (Micronan Power
The reason for this is difficult to understand given System Analysis C o p , Vancouver, Canada, 1996,2nd edn.)
the inescapable advantages of the method, amongst 5 GIELEN, G., WAMBACQ, l?, and SANSEN, W:
w h c h is the overriding benefit ofspecifying all circuit ‘Symbolic analysis methods and applications for analog
circuits a tutorial overview’, Roc. IEEE, Febmary 1994, 82,
elements, without exception, by an irreducible (2), pp.287-301
number of parameters. For example, the necessary 6 TINNEY, W., and SATO, N.: ‘Techniques for exploiting
parameters for the ubiquitous voltage source are the the sparsity of the network admittance matrix’, IEEE
positive and negative terminations (nodes) and the Tam. Power Appar. Syii., December 1963, PAS-82,
source magnitude and phase. Sirmlarly, a two-winding pp.944-950
7 TINNEY, W., and WALKER, J.: ‘Direct solutions ofspane
transformer is defined simply by the start and finish network equations by optimally ordered trianplar factoriza-
nodes of the primary and secondary windings and the tion’, fioc. IEEE, November 1967, 55, pp.1801-1809
associated turns ratio. 8 BKAMELLER, A,, ALLAN, R. N., and HAMAN, Y, M.:
From an educational viewpoint MNA enables ‘Sparsity’ (Pimian Press, New York, 1976)
students to focus on the fundamentals of circuit
analysis rather than spend valuable time learning how OIEE:2002
to transform circuit elements and manipulating
equations. But perhaps the most important benefit for Dr. Wedepohi (E-mail: wedepohi@ihaw.ca), who i s now
retired, was fornirrly Dean of Applied Science, University of
the student is the fact that ideal circuits can he analysed Biitish Columbia. Dr. Jachon (E-mail:Ijm@jr’Jack.ndo.co.uk),
that have no representation in classical nodal analysis. now retired, was formerly with British Gas a i President of
This is particularly important in understanding the Pipeline lntegrity International.

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