Graphical technique for analyzing series-parallel networks by rectangular diagrams in solving power distribution problem

Ding, J. L.

Erdey, M. R. A.

English

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The Schwaigerian Driver Transfer Technique 
and The Thevenin's and The Norton's
 
Theorem
 
M. R. A. Erdey, Professor
 
and
 
J. L. Ding, Research Assistant
 
Electrical Engineering Department
 
School of Engineering
 
Tuskegee Institute
 
Alabama 36088
 
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https://ntrs.nasa.gov/search.jsp?R=19710006468 2020-03-11T23:54:49+00:00Z
S fWMARY 
Schwaiger 1 2 introduced a graphical technique in order to 
networks in the form of rectangular diagramsanaiise series-p3rallel 
' with the aid of these in a forward process he was able to obtain 
equ talent driver's while in the reverse process he obtained the solution 
e network0 In this article it is shO if t-at ;h.e equivalent driverz 
that Sch., "-2!r obtained by a L.cp-by-step graphical transfer of drves 
is nothing else .. ,,u-- ,-jlTh 0 subsequently the rectangular 
diagram proofs of t ,e Thevenin's and the Norton s theorem's are deduced, 
IN TRODU611IO 1 NOT REPRODUc[BL
 
Rectaijtular diEgrams were first introduced by Schwaiger 
His method will be exposeq in this article, an indirect form of re­
ctangular diagrams to solVw power distribution systems0 In his in­
direct form, the diagrams ar- used like a slide rule to aid the solution0 
In the direct reading form, that we will also use in this article, the
 
element values, voltagesq cuirents, and even the consumed and supplied
 
powers, can be directly read tom the diagrams. These type of diagrams
 
were used by Magyari 1 0 to solve linear resistive and reactive networks, 
by Xovattana9 , Cherry1 and Erd;-5 to solve non-linear resistive networks. 
Linear and non-linear net ,k.jc properties and theorems in rectangular dia­
gram form were exposed by Cherry1, fdey2 3 4 7 Magyari10 and Mayne 1K
 
2 
On Pig, 1-a a linear resistive network is given. Its re­
ctangular diagram is given on Fig. 1-b. On this diagram each element 
is represented with a component rectangle, the vertical side of which 
represents the voltage across, and the horizontal side represents the 
current through, while its area gives the power through the element. 
In cases of linear resistors as in the given example the slope of the, 
diagonal gives the element value while the non-linear elements are re­
presented with their general v-i characteristic curves, Studying re­
ctangular diagrams with dual non-linear elements with the aid of re­
ctangular diagrams led Cherry to his "Classes of 4-Pole IMetworks having 
Non-Linear Transfer Characteristics but Linear Iterative Impedances". 
In cases of linear resistive networks there is a straight 
forward construction technique coined by one of the authors as Ilocus 
method" since the loci of the vertices of the individual rectangles move 
on straight lines, as the size of the rectangle is proportionally in­
creased or decreased0 Pig. 2 shows the case when the size of a rectangle 
is proportionally enlarged into the rectangle drawn with the broken sides. 
The proportionality ensures that the diagonals will have the same slopes, 
which ensures that the element value of the resistor is not changed by 
this procpss0 The loci of the four vertices always intersect each other 
in one common point (point P on Pig. 2), The physical meaning of this is 
that in the case when no current is flowing through a resistor the voltage 
across it will also become zero (rectangle shrinks into a point)0 This 
method is a routine operation in case of planar networks but becomes a 
little more cumbersoue in case of non-planar networks8 
3
 
The Schwaigerian Method of Step-byStep Transfer of Drivers 
Let's take the two series connected R and elements of
2 
Fig 3-a in which only R1 is excited by the ideal current driver ia 
On Fig 3-b an equivalent current driver i is given which produces the 
e 
same voltage across nodes A and C that i produces across the terminalsg 
on Fig, 3-a, The Schwaigerian graphical solution of the equivalent current 
is given on Fig. 3-c. On this diagram the two series connected elements 
are treated as if the same i current is flowing through both of their, 
in order to define the R12 resultant resistance. The opera'ing point of 
the R resistance of Fig. 3-a is at point P1 Since the R2 element in 
this network is idling the voltage across the terminals A and C is the 
same as that of the B and C. In order to obtain the same voltage across 
the A and C nodes on Fig. 3-b. we Lave to move the P, point to the left 
until it intersect the R12 resultant resistance line at P2 ' The per­
pendicular from P2 to the base cuts the length of the ie equivalent 
current, and which we All shos that is equal to the Nortonian equivalent 
current, which is obtained by short - circuiting the pair of terminals, 
By short - circuiting terminals A and C we obtain Fig. 3-d, where the 
short - circuit current is equal to the current flowing through element 
R2' We have to show that this current is equal to the one Ln Fig. 3-b. 
This is obtained graphically on Fig.. 3-e. 
4 
On Pig. 3-e the excited resistor of ig. 3-a is given by 
PIP2P3 p4 /7 while Pig. 3-b is represented by the PP2P? __o3 9 

Since both the rectangles have the same height (1P2P3) -the f-eieiffnal' 
voltages are the same Th both the cases as required, The parallel
 
connected R1 and R2 esistors of Fig, 3d are given by PIP 2 SP7
 
/ 7o It can be noticed that the current through R2 element (re­
presented by P8P- remained the same in -igs. 3-b and d. and thus we
 
have shown that the Sahwaigerian equivalent driver and the Nortonian
 
are the same for this particular case.
 
The Schwaigerian graphical driver transfer technique was de­
monstrated for a series connected elements on Pig. 3. It is very east 
to demonstrate that the same technique can be applied to any series 
parallel connected network. 
5
 
The Thevenin's and The Norton's Theorem.
 
It was just pointed out with a particular example the 
Schwaigerian equivalent current driver , which is obtained by a step­
by-step .transfer of drivers to a preditermined pair of terminals is 
nothing else but current driver in the Morton's equivalent. Since the 
Thevenin's and The Morton's equivalents of a two terminal active networks 
are closely interrelated to each other, we will expose these theorems in 
their graphical forms in order to obtain a visual image and understanding 
of these basic equivalencies. 
Let's take the two-terminal linear active network N of Pig, 
4 which a Ic46d L (which may be a second network) is connected across its 
terminals, The active network N can be represented with a voltage driver. 
in series wi.:th an impedance (Thevenin representation) or with a current 
driver sunted by the same impedance (Norton's representation). When 
the load is disconnected the terminal voltage of N is equal to the ter­
minal vuttage of the ideal voltage driver in the Thevenin's equivalent. 
When th, load is replaced with a short - circuit4 the current through 
the sirt - circuit is equal to the terminal current of the ideal current 
drive: in the Norton's equivalent. The series impedance in the Thevenin 
repreentation and the shunt impedance of the Norton's representation are 
equa7I to each other and equal to the input impedance of network N
. 
looking 
into it at its terminals, when all internal drivers have zero value, 
6
 
It can be shown that simisoidally excited RLC networks can 
be represented with rectangular diagrams too since their vector diagrams 
can be broken into inphase and quadrature components. Since the com­
ponents themselves are real quantities naturally the component re­
ctangular diagrams have the same characteristics as the rectangular 
diagrams of resistive networks, Tnherefore, without any loss from 
generality we can confine ourselves to the rectangular diagrams of 
resistive networks when we prove The Theveni's and The Norton's theorem. 
The simplest case of the %hevenintsand The Norton's equivalent 
is usually referred by driver or source transformation in the literature. 
Its simple graphical proof is given on Pig. 5 as follows- Let us consider 
a practical voltage driver. As indicated on Pig 5-a it can be decomposed 
into an ideal driver series with a resistor (Thevenin equivalent). This 
decomposition can be justified by measurements0 If we measure the open 
circuit voltage of the practical driver, we can assign this vO value
 
for the ideal driver on Pig 5-a. When a load is connected across its
 
terminal (Pig 5-b) then from the load resistance R2 and the vL voltage 
across the load according to the rectangular diagram of Fig . S-eO the 
internal resistance can be defined0 When the terminals of this
 
practical driver are short - circuited its corresponding diagram is 
given in Fig0 S-do Further, when this driver is loaded by another re­
sistor RL Pig 5-e the diagram for the complete network can be constructed 
as shown in Pig, 5-f. In this figure; the broken lines complete the 
diagram into that given in Fig. 5-d and one can see that given il Fig, 
5-d 	and one can see that the terminal conditions for R. will not change 
if R. were connected in parallel to RF rather than series to it and also 
if the voltag3 driver- vO is replaced by a current driv;er i as is shown 
in Fig. 5-h. Sis diagram corresponds to the network in Fig. 5-g and 
its left sides is called the Norton's equivalent of the practical driver 
which is lodel by RLO
 
J.e All prove the general ease of the Thevenin's a d The 
Nfaton Is he~rem with the aid of superposition theorem; that- is we are 
-taking inco account the effects of each driver separately We can as 
sume that all drivers are current drivers, If this would not be the 
case with the given driver transformation technique, that we have just 
seen or Fi;. 5, In case if we take into account each driver- separately 
we havia tio po5,sibilities: 
a, 	The equivalent driver has a common node with 
the original driver 
b. 	The equivalent driver has no common node with 
the 	original driver.
 
SCLUTION OF THE TWO SUBOASES
 
If we have the first case at hand then we can eliminate
 
s transformation* ex­
cept the 3 vertices that are involved in the problem , After this process 
the network will have the form of Fig. S-a. The rectangular diagram 
solution to this problem is given or. Pig0 6-c. When nodes B and C are 
all vertices with the aid of of or Rosent t 

shorted the network has the form of Fig. 6-b., the solution of which is 
gien in Pig0 6-c by P3 P4 P5 P6 /:: , In Fig, 6-d an )euivalent driver 
is connected across terminals B and C. The rectangu diagram solution 
of the network ?dith the equivalent driver is given in Fig, 6-c with 
P2P7Ps%9 Q= the width of which is the same as the width of the 
R rectangle in the short - circuited case0 Thus3 we have proved 
that the short - circuit current Is equal to the Norton's equivalent
 
current. 
In the case of the Thevenin's equivalent we have to find the
 
open circuit voltage at the required pail- of terminals. In Fig. 7-c
 
with the P1P2P3P4 =-, the conditions of the origiani loaded network
 
of Fig. 6-a are repeated. When the terminals are open circuited (R. 
removed) Pig, 7-a is obtained. The form of the solution in this case 
is given by PIP14P15P4 j . In order to adjust-the diagram to the 
applied current I the diagram has to be proportially enlarged into
 
PiP2 P! 2 Pl3 7, In this enlarged diagram the open circuit voltage 
(voltage across the R3 element) is given by Pj0 o Let's show that
 
the voltage VO thus obtained is as an ideal voltage driver connected
 
in series to the internal resistance of the network of -9iig 7-e 
The graphical form of the & and of its generalization9 the
 
Rosenvs transformation, was presentedby the authors in another paper 7 
9
 
and applied to thie RL, load (as given on Fig. 7-b) then the solution for 
R remains the same as the one we had in the orginal network. Let's 
add in series R2 of the P8PsP3 Pll =7 to RI of the PllP3P6P7 = 
Where the resultant R12 line intersects the R3 line we obtai the 
point Qlo It can be shown that Q is at the same level ofE Thus
. 
the rectangular diagram solution of the network in Fig, 7-b is given 
by P2PoP 10 2f- Since the original rectangle of RL is an intact 
subrectangle of the obtained solution we have just verified the validity 
of the Norton's equivalent for the case a.
 
In the case of b after the elimination of the vertices to
 
which neither the given driver nor the load is connected we obtain 
the form in Fig. 8-a. Note that in the given form any of the five
 
01 to G5 conductances can be zero. This will mean that the pertinent
 
rectangle will have a zero width, but otherwise the procedure will be
 
the same.
 
i ig° the P 2 P 72 P31  is the solution of the 
network of Fig. 8-a, When nodes B and C are short circuited in order
, 

to obtain the Nortonts equivalent eurent, the Fig. 8-b is obtained.
 
In this case in the rectangular diagram the intersection point of the
 
resistance lines of the parallel R1 and R2 lines is at P21 and
 
similarly the R3 and R4 lines intersect each other in point P2' The 
horizontal distance between these two pointsis equal to P1 P2 and is 
equal to the short - circuit current. On Pig 8-e this current is used 
10
 
in the Wortonts equivalent across the B and C terminals parallel
 
to the internal resistance of the de-energized network of Fig. 8-a 
looking into it at terminals B and 0. This equivalent network is 
driving the % load. The rectangular diagram solution except relative 
positions of the rectangles of the network on Pig. 13-e is given by the 
P6P7PsP5 Q on Fig. 8-co -justfor the sake of ocarity the solution 
is redrawn on Fig. 8-d once again. Since the size of the R1, rectangle 
on this diagram remained the same as before, thus it is proved that 
as far as the load is concerned the two networks (Figs U.-a and e) are 
equivalent 
It is sufficient to obtain the Norton's equivalent to ny 
configuration , since the driver transformation shoWn on Fic,, 5 can 
convert it into the Thevenin's form. Just to show the reade, how the 
Thevenins equivalent is obtained the additional construction flnes n 
Pig. 8-c are related to the finding of the Norton's equivalent Nn-ltage 
is found in the p.rticular case when G. = 0 and R, is the resultatmt 
of the previous parallel RL and R.. In this particular case, the ox.en 
circuiting procedure makes R1 and and similarly R2 and I 4 ,!e .ents 
series to each other. The intersection of the pertinent vesistence 
lines is at P22 and P23. In order to eliminate the gaps betweeL the 
left and right sections te resultant R, and R2 4 rectangles are 
proportionally enlarged till they intersect each other at Po
 
After the enlargement r22 moved to P4 and P23 to P30 
 hus the 
open circuit voltage for this particular case is given by P3P. 
The reader should notice that the Norton's equivalent in this parti­
cular case is the same as before. Thus the Pl7P8l9P2o0 
represents the Is VO rectangle the diagonal which should represent 
the input resistance across terminals B and C. This can be seen
 
from the fact that this rectangle breaks down into a series of
 
subrectangles of elements R3 and R4 and parallel to the" are the series
 
rectangles of elements R1 and R2o
 
The General Case 
In the above procedures the drivers were taken into 
account individually. This is possible since we were dealing with 
linear networks and thus superposition theorem is applicable. 
Since
 
the equivalents each time are taken across the same pair of terminals 
the input impedance of the network remains the same and the short 
circuit current will be the same as the short - circuit currents with 
the individual drivers.
 
REFERMCES 
1. 	 cherry, E. C. "Classes of 4- Pdle Networks Having Non-Linear
 
Transfer Characteristics but-Linear Iterative Impedances% Pro­
ceedings I.E.E. vol. 107., part B, pp. 26-30 (1960).
 
2. 	 ERdey., M. R.A0 - "Rectangular Diagrams and Their Applica-tlorlS to
 
Network Theorems", Proceedings of the Sixth Midwest Syaposium
 
on Circuit Theory, pp. F-1 to F-17, Madison, Wisconsin (1963). 
3. 	Lrdey., MR.A. "The Misrepresented Duality and the Rectangular

Diagram", Proceedings of the IEEE RegionIII Convention, pp.
 
9.51-9.5.10, New Orleans, Louisiana (1968).
 
4. 	 Erdey, M.R.A: "Rectangular Diagrams in Graph Theory", Proceedings
of the Eleventh Midwest Symposium on Circuit Theorypp 441-456,
Notre Dame, Indiana (1968). 
5. 	 _Exdeyq M.R.A.: "Iterative Solution of Same Non-Linear Resistive
Networks with the Aid of the Rectangular Diagrams'", Proceedings

of the Eleventh Midwest Symposium on Circuit Theory, pp. 371-3889
 
Notre Dame, Indiana (1968). 
6. 	Erdey, N.R. and lurty, B. V.G.Co: "Graphical Methods in the
 
Design of Resistor - Transistor - Logic Circuits", Proceedings

of the 12-th Midwest Symposium on Circuit Theoryq pp. YIV 4.-1 
- YaV. 410, Austin, Texas (1969). 
7. 	 Erdey M.RoA. and Ding, J. Lo: "Graphical Form of Rosents Trans­
formation", Proceedings of the 12-th Midwest Symposium on Circuit 
Theory, pp. XIV.3.1-XIV 3.14, Austin,-Texas (1969). 
8. Erdeyq M. P A. and Ding, J. L.: "Graphical Solution of Non-Linear
 
Networks7, First Southeste,. Symposium on System - Theory pp. 
F4-1 	to F4-19 (1969). 
9. 	 Kovattana, T. and Barker J. R. : "Solution of Ladder Networks 
Containing Non-Linear Resistance", British Journal of Applies
PWsics, Vol. ll pp. 437-4392 (September 1960). 
10. 	Magyari, E. "Grafijus Szamitasok a Hiradas Teebnikaban"
 
Muszaki Konyvkiado, Budapest, Hungary (Vol. 1. in 1958, Volo12 
in 1966)0 
11. 	 Mayne, D.OQ.: "An Analogy Between Non-Linear Resistive and Linear 
A. C. Necworks". Proceedings IoEoEo, Vol. 107; part B, pp. 410­
411 (1960). 
32 	 Schwaiger, A. : "Elektrische Leitungeng "Leibnitz-Verlag, Munchen 
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The Schwaigerian driver transfer technique and the Thevenin's and the Norton's theorem Final report

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