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PAPER PREPARATION GUIDELINES
UDC 621.3
COMPARISON OF METHODS FOR ELECTRIC CIRCUITS SIMULATION
J. Dziak, I. Tomčíková
Technical university in Košice
Park Komenského 3, 04200, Košice, Slovakia. E-mail: [email protected], [email protected]
This paper deals with few methods of electrical circuit analysis. It points the advantages and disadvantages of each
method and evaluates their use for electrical circuit simulation. In paper are outlined procedure for network analysis in
general and also the procedure for each method. It establishes criteria for comparison of methods. Paper compares
methods according to specified criteria.
Кey words: circuit simulation, computer simulation, network analysis.
ПОРІВНЯННЯ МЕТОДІВ МОДЕЛЮВАННЯ ЕЛЕКТРИЧНИХ КІЛ
Д. Дзіак, І. Томчікова
Технічний університет Кошице
вул. Парк Коменскего, 3, м. Кошице, 04200, Словаччина. Е-mail: [email protected], [email protected]
У статті розглянуто декілька методів аналізу електричних кіл. Було визначено переваги і недоліки кожного з
розглянутих методів та виконано оцінювання можливості їх застосування для моделювання електричних кіл. В
статті виокремлено загальну процедуру мережевого аналізу та процедуру аналізу для кожного з методів. Також
представлено критерії порівняння методів. У статті методи моделювання електричних кіл порівняно згідно
запропонованого критерію.
Ключові слова: моделювання електричних кіл, комп’ютерне моделювання, мережевий аналіз.
PROBLEM STATEMENT. Circuit simulation is a
technique for checking and verifying the design of electrical and electronic circuits and systems prior to manufacturing and deployment. It is used across a wide spectrum of applications, ranging from integrated circuits
and microelectronics to electrical power distribution
networks and power electronics. Circuit simulation
combines mathematical modeling of the circuit elements, or devices, formulation of the circuit equations
and techniques for solution of these equations. We will
focus mainly on the formulation and solution of the
network equations in this paper. We will focus mainly
on the formulation and solution of the network equations in this paper [1].
In a computer program the equations have to be
formulated automatically in a simple, comprehensive
manner. Once formulated, the system of equations has
to be solved. There are two main aspects to be considered when choosing algorithms for this purpose: accuracy and speed. In principle, accuracy depends on mathematical modeling of the circuit elements and speed
depends on method.
Consider a network having n nodes and m branches.
A network may be characterized of a set of network
equations. There are two types of equations: branch
equations and connection equations. Branch equations
are also called element equations. These equations describe a circuit element by means of a current - voltage
relationship. Element equations can by completely expressed as (1). Set of element equations consist of m
equations.
i
(1)
Z Y    s .
u 
Z – Matrix of branch (element) impedance
Y – Matrix of branch (element) admittance
i – Vector of branch currents
u – Vector of branch voltages
s – Vector of voltage and current sources
Table 1 – Examples of branch relations of basic elements in linear electric circuits with harmonic
sources in the complex form
Element Branch relation Matrix values
Y
1
resistor
U – R.I = 0
Z
–R
s
0
Y
1
inductor
U – jωL.I = 0
Z
– jωL
s
0
Y
jωC
capacitor
jωC.U – I = 0
Z
-1
s
0
Y
1
voltage
U = US
Z
0
source
s
Us
Y
0
current
I = IS
Z
1
source
s
IS
Connection equations that consist of Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL) from
the structure of the network, they don’t due to the properties of any network element. Formulation of the connection equations is done by applying KCL and KVL to
the network. It is possible to capture KCL and KVL in
two forms. The first form is in terms of branch currents
and branch voltages. Set of equations (2) consist of m
independent equations in 2m unknowns.
 A 0   i  0 
 0 B  u   0 .

   
A –Incidence matrix of nodes
B –Incidence matrix of meshes
(2)
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Matrix A expresses relationships between nonreference nodes and branches (elements). These relationships can take the forms as it shown in Figure 1
options a), b) and c). Matrix B expresses relationships
between meshes and branches. They can take the forms
as it shown in Figure 1 options d), e), and f). Matrix
values for all combinations of network relationships are
in Table 2 – Matrix values of network relationships.
Figure 1 – Relations between node and branch and between mesh and branch.
Table 2 – Matrix values of network relationships
Option
a)
b)
c)
1 
-1 
0 
A
Value
 A     A    








Option
Value
d)
1 
B

 
e)
-1 
B

  
f)
0 
B

  
The second form is more compact with more equations and variables which are branch currents, branch
voltages and node voltages. Set of equations (3) consists
of (m+n-1) independent equations with (2m+n-1) unknowns.
i 
0    0
A 0
 0 E  AT  u   0 .
(3)

 v   
 
E – Unit matrix
AT – Transpose reduced incidence matrix of nodes
v – Vector of node voltages
The connection equations written in the both forms
(2) and (3) are due to the network topology only, and
independent of any element’s property. The form (2) is
not preferred in practice for handling large networks,
because this set of equations is not sparse. The form (3)
is highly sparse and this is major advantage for solving
large circuits.
EXPERIMENTAL
PART
AND
RESULTS
OBTAINED. There are several methods for electrical
circuit analysis. Six methods were selected for compared: superposition theorem, kirchhoff’s law analysis,
nodal analysis, mesh analysis, modified nodal analysis,
sparse tableau analysis.
SUPERPOSITION THEOREM. Superposition theorem (ST) is used to simplify networks containing two or
more sources. In a network containing more than one
source, the current at any one point is equal to the alge-
braic sum of the currents produced by each source independently (4) [2].
k
i
i .
i
(4)
i 1
ii – Matrix of currents produced by each source
k – Number of sources in network
ST can be used only in linear circuits. If we use ST,
then is not possible to formulate circuit equations automatically. Automatic formulation of equations is a very
important for computer-based circuit simulation. Therefore we will not use this method for simulation.
KIRCHHOFF’S LAW ANALYSIS. In Kirchhoff's law
analysis (KLA) are used directly Kirchhoff's laws. Circuit equations consist of two sets of equations. Connection equations formed by using Kirchhoff's voltage law
(KVL) and Kirchhoff's current law (KCL) (2) and element equations described the voltage - current relationship of elements (1). Aim of KLA is do find all unknown’s branch currents (5) [2] [3].
 A.s I 
 A 
 B.Z i     B.s  .


 U
sI – Vector of current sources
sU – Vector of voltage sources
(5)
Circuit description consists of relations between the
branches (elements) and nodes corresponding matrix A,
and relations between branches and meshes corresponding matrix B. It complicates and prolongs the computation and thus adversely affects the speed of the algorithm.
NODAL ANALYSIS. The nodal analysis (NA) is
based on the following idea. Instead of finding for circuit variables, current and voltage of each element, we
seek node voltages in this case, which automatically
satisfy KVL. We do not need to write KVL equations
we need only to apply KCL equations to nonreference
nodes. The aim of nodal analysis is to determine the
voltage at each node relative to the reference node (6)
[2] [3].
Y v   A
s 
Y  I  .
 sU 
(6)
NA is a fairly comprehensive method of solving circuits. But it has complication for computer circuit simulation. In NA we have a problem with the voltage
sources. If all the voltage sources in network do not
have a common node, an algorithm for automatic computation is difficult. It extends time of computation and
negatively affects the speed of algorithm. Therefore,
modified nodal analysis has been developed in order to
remove problems of NA.
MESH ANALYSIS. The mesh analysis (MA) is analogy of the NA. We solve for a new set of variables,
mesh currents that automatically satisfy KCL. As such,
mesh analysis reduces circuit solution to writing KVL
PAPER PREPARATION GUIDELINES
equations. The aim of mesh analysis is to determine the
currents at each independent mesh (7) [2] [3].
B.Z.B i
T
M
   B
s 
B.Z  U  .
 sI 
(7)
iM – Vector of mesh currents
BT – Transpose incidence matrix of meshes
Lack of MA is that we can MA use just for planar
circuits. Element or connecting wire can't intersect other
element or connecting wire in plane. MA also has a
problem with circuit description. Circuit is described by
using relations between branches and meshes. Instead of
simply matrix A we need more complicated matrix B. It
adversely affects the speed of the algorithm.
MODIFIED NODAL ANALYSIS. NA is used for
formulating circuit equations. However, several limitations exist in this method including the inability to process voltage sources and current dependent circuit elements in a simple and efficient manner. A modified
nodal analysis (MNA) retains the simplicity and other
advantages of nodal analysis while removing its limitations [4].
To handle voltage sources, the key idea is to not insist on eliminating their currents, but to retain those
currents as additional variables. For of these new variables, we add a new equation, namely the branch equation for that (voltage source) element. The size of the
matrix equation will grow, by as many equations as we
have voltage sources, compared to nodal analysis. We
can retain more currents (other currents) than just the
voltage source currents [1]:
• All voltage source currents, be they independent or
controlled.
• Any current that is a control variable for current
control voltage source (CCVS) or current control
current source (CCCS)
• Any current that is a user-specified simulation output.
We have a system of equations for the currents according NA, a system of equations for the other currents. Merging both obtained MNA system (8).
 A.Y . A 1

T
  AO
AO   u   A.sU 
   
.
Z O  iO   s I 
(8)
iO – Vector of other branch currents
AO – Node incidence matrix of other currents
AOT – Transpose node incidence matrix of other currents
ZO – Matrix of branch impedance of other currents
Level of MNA is between STA, where no currents
were eliminated, and NA, where all were eliminated.
This creates a disadvantage of MNA. MNA doesn’t
contain information about all currents and voltages in
electric circuit [5].
SPARSE TABLEAU ANALYSIS. In this approach the
only matrix operation required is that of repeatedly
solving linear algebraic equations of fixed sparse structure. For non-linear circuits the partial derivatives and
numerical integration are done at the branch level leading to complete generality and maximum sparsity of the
characteristic coefficient matrix [6].
Sparse tableau analysis (STA) described in [6], involves the following steps: write KCL, write KVL in
form (2) and write the element equations (1). The combination of these three sets of algebraic equations leads
to the sparse tableau system (9) [1].
0   i  0
A 0
 0 E  AT  u   0 .

   
Z Y
0  v   s 
(9)
This formulation has some key features consisting in
fact that it can be applied to any circuit in a systematic
fashion. The equations can be assembled directly from
the input (circuit specification). The coefficients matrix
is very sparse with mostly zero elements, although it is
larger in dimension than the MNA matrix [1].
STA makes possible a simple yet truly general purpose computer program. By combining the concept of
variability type with sparse matrix techniques, this program can achieve practically optimum efficiency by
choosing the order of elimination so as to minimize the
total operations count required for simulation and/or
automated optimization [6].
COMPARISON OF METHODS. In this paper, have
been studied options of circuit analysis for computer
simulation. All investigated methods of analyses could
be divided into three groups. First is group of methods
that are unsuitable for computer simulation, the second
one is group of methods fewer suitable and the last one
is group of methods that are very suitable for computer
simulation.
Unsuitable for computer simulation is superposition
theorem. It is not possible to formulate circuit equations
using these methods automatically. This method (called
"ad hoc") is inappropriate to analysis of all circuits.
Instead, we need a systematic and automatic approach
for formulating and solving the circuit equations.
Into group of methods fewer suitable were included
Kirchhoff's laws analysis, nodal analysis and mesh
analysis. They can be used for computer simulation, but
their algorithm will be more complicated and / or must
be used for circuits with restrictions. MA has both disadvantages. Circuit must be planar and relationships
must be defined between branches and nodes and moreover between branches and meshes. KVL has the same
complications as MA with circuit description. NA isn't a
complex method because it has a problem with the voltage sources and the algorithm for solving all types of
networks is tedious and difficult.
Unlike previous methods, the methods in the group of
methods very suitable can be used for a general solution
to electrical circuits. It is a modified nodal analysis and
sparse tableau analysis. MNA is more compact than
STA, preparation and creation of analysis is faster than
STA. MNA doesn’t contain information about all cur-
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rents and voltages in electric circuit. Handicap of STA
is that preparation of STA analysis is slower than MNA,
but STA contains information about all voltages and
currents.
ACKNOWLEDGMENT. The paper has been prepared under support of Slovak grant projects KEGA
No. 005TUKE-4/2012.
REFERENCES
1. Najm F. N. – Circuit Simulation, John Wiley &
Sons, Inc., Hoboken, United States of America, 2010,
ISBN 978-0-470-53871-5.
2. Mayer D. – Úvod do teorie elektrických
obvodů, SNTL / ALFA, Prague, Czech Republic, 1978,
ISBN 04–536–78.
3. Šimko V., Kováč D.: Učebné texty z
teoretickej elektrotechniky I, elfa, Košice, Slovakia,
1998, ISBN 80-88786-79-7.
4. Ho Ch., Ruehli A. E., Brennan P.A. – The
modified Nodal Approach to Network Analysis. IEEE
Transactions on circuit and systems, 1975
5. Vansáč M., Vince T. – Sparse tableau analysis
of electrical circuits, XIV International PhD Workshop
OWD, Poland, 2012.
6. G.D, Brayton R.K., Gustavson F.G.: The
Sparse Tableau Approach to Network Analysis and
Design. IEEE Transactions on circuit theory, 1971.
СРАВНЕНИЕ МЕТОДОВ МОДЕЛИРОВАНИЯ ЭЛЕКТРИЧЕСКИХ ЦЕПЕЙ
Д. Дзиак, И. Томчикова
Технический университет Кошице
ул. Парк Коменскего, 3, м. Кошице, 04200, Словакия. Е-mail: [email protected], [email protected]
В статье рассмотрено несколько методов анализа электрических цепей. Определены достоинства и
недостатки каждого из рассмотренных методов и проведена оценка возможности их применения для задач
моделирования электрических цепей. В статье выделена процедура сетевого анализа и процедура анализа для
каждого метода. Также представлены критерии сравнения методов. Методы моделирования электрических
цепей, представленные в данной статье, были сравнены согласно предложенного критерия.
Ключевые слова: моделирование электрических цепей, компьютерное моделирование, сетевой анализ.