E. Gluskin, J Electr Eng Electron Technol 2013, 2:2
http://dx.doi.org/10.4172/2325-9833.1000108
Journal of Electrical
Engineering & Electronic
Technology
Research Article
An Extended Frame for
Applications of the HelmholtzThevenin-Norton Theorem
E. Gluskin *
1
Abstract
The Helmholtz-Thevenin-Norton theorem is widely used in circuit
calculations. A recent focusing on a specific 1-port topology that is
usually ignored, but can well complete the frame of application of
the classical theorem, led us to extension of this frame. The circuit
argument on which the extension is based is very clear, and, in
our opinion, the extended scheme (Figure 3) is important for basic
circuit theory.
Keywords
Circuit theory; Circuit simplification; Ideal source
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any control) source for the circuit. Notice that “independent source”
necessarily means “ideal source”, but not conversely, i.e. ideal source
can be dependent, and then one has to observe and remember from
where the controls are coming.
Based on some consultations with engineers, I can suppose that
the reason for the theoretical miss is that one intuitively supposes that
even if an ideal source is dependent (i.e. electronically controlled) then
it will keep its (voltage or current) function defined by the control(s)
also when the load is involved. In fact, the issue is more complicated,
but the central argument that allows inclusion of the cases of such
an output-source into the extended frame is remarkably simple, and
one’s intuitive understanding of the equivalent circuits, is corrected
in a natural way. In simple words, for the dependent ideal source (at
the output) the point is just whether or not the load can influence
the source’s controls. If the load cannot influence the controls, then
the source function remains unchanged for any otherwise legitimized
(i.e. not causing a too strong current that can overheat the device,
etc.) load.
For certainty, we shall focus on the case when this source is a
voltage source, and speak about Thevenin’s (originally, Helmholtz’s)
version of the equivalent 1-port. Norton’s version can be then
obtained (Appendix 2) by duality.
Introduction
General
As is well known, simplifications of linear circuits obtained by
using Thevenin’s theorem (the series vTh-RTh circuit), or Norton’s
equivalent (the parallel iN||RN circuit) are very useful in many
problems [1-4]. As far as we speak about linear circuits, the following
results can be seen as a rather simple extension of (completion to) the
Thevenin’s and Norton’s theorems. While for the actual circuits that
are relevant to the classical theorems the particular case when vTh = 0
(or iN = 0) is often obtained [2], in our situation we obtain a very rare
“opposite” case of RTh = 0 (or RN = ∞). The usual passing, in the linear
case, from the Thevenin circuit to the Norton circuit and back can
not be done here on the basis of the equivalent generator theorem in
which the equality iN = vTh/RTh is used, and each of the circuits arising
here has only one equivalent, either of Thevenin, or of Norton, i.e. the
sources shown in Figures 2 and 6 have to be derived independently,
or using the duality argument.
The circuit with the terminating dependent source
Figure 1 schematically shows the 1-port of [5] which interests us.
Figure 2 shows the equivalent for the circuit of Figure 1, an ideal
independent voltage source, relevant/valid, however, only when some
limitations on the internal functional dependencies in the original
circuit are provided. These limitations are very important, of course.
Considering that an ideal voltage source can conduct any current,
i.e. that the current of the source depends on the load, we have for the
circuit of Figure 1 two (and only two) possibilities:
A. If there is no feedback-influence of the current of the
terminating voltage source on its controls (i.e. when the
voltage on this source cannot be influenced by load’s
current), then with respect to the load or any other externally
connected circuitry, the whole one-port appears to be an ideal
independent voltage source.
B. If there is such an unwanted internal current feedback, which
In [5], the case, missed in [1-4,6] and many other sources, is
considered where at the output of a one-port there is a dependent
source. As in the usual Thevenin case, the controls of this dependent
source are inside the one-port.
Using the concept of ideal source, we recall that that a source is
ideal means that when its function is made zero by the controls, it
becomes a short circuit if it is a voltage source, and a disconnection if
it is a current source. In any case, it is not an “independent” (without
*Corresponding author: E. Gluskin, Kinneret College in the Jordan Valley (on
the Sea of Galilee), 15132, Israel, E-mail: gluskin@ee.bgu.ac.il
Received: April 17, 2013 Accepted: August 29, 2013 Published: September
02, 2013
International Publisher of Science,
Technology and Medicine
a
+
+
v
N
-
b
Figure 1: The 1-port (including, in particular, also some independent sources)
to the left of a-b should have, when detached, a unique solution. The controls
of the dependent voltage source terminating the 1-port are solely internal,
belonging to the 1-port. We consider the voltage v = vab on this source (the
“port’s voltage”).
All articles published in Journal of Electrical Engineering & Electronic Technology are the property of SciTechnol, and is
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Citation: Gluskin E (2013) An Extended Frame for Applications of the Helmholtz-Thevenin-Norton Theorem. J Electr Eng Electron Technol 2 :2.
doi:http://dx.doi.org/10.4172/2325-9833.1000108
we shall name “load-influences-controls”, shortly LICfeedback, which lets the load’s current internally influence the
1-port (the voltage controls are our concern, of course), then
the 1-port does not appear as an independent voltage source,
i.e. v = vab depends on the parameters of the load or any other
externally connected circuitry. The equivalent scheme will be
vTh - RTh.
The meanings of the arguments A and B are very lucid, and case
B is even obvious. The formal proof for the case “A” can be based on
the fact that vab can be determined without the nodal equation for
the node a (Figure 1), and thus no independent Kirchhoff’s equation
including the load’s current arises. The proof assumes only existence
of some functional relations, not their linearity. This proof, given in
Appendix 1, requires the possibility to resolve the circuit equations
for vab, i.e. the circuit should be solvable.
That the original circuit need not be linear is clear from the fact
that the final equivalent scheme of the independent ideal source
(Figure 2) does not include any linear element. This caused [5] to
suggest (suppose) that an ideal source of a chaotic process can thus
be obtained.
a
+
N
voc
b
Figure 2: The equivalent for the circuit in Fig.1, obtained with some limitations
(case A in the main text). For the initially given linear circuit, this equivalent
can be seen as “just” a case of Thevenin’s theorem for RTh = 0, but since the
equivalent does not include any linear element, i.e. there is no evidence of
circuit linearity, this equivalent can also represent a 1-port having nonlinear
internal dependencies.
Active 1-ports
Terminating-source
topology;
No LIC-feedback
Terminating-source
topology;
LIC-feedback acts
Regular Topology
(*)
Equivalent ideal source
(Rth = 0)
Regular Thevenin's
equivalent
(*)The usual external test-source must here be of the proper type, this is an
important specialty of the circuit with a terminating source. For instance, if we
deal with terminating current source, then the test must be done by a voltage
source.
Figure 3: The logical scheme of the correct complete classification for
Thevenin theorem, which we suggest to become mandatory in circuit theory.
Volume 2 • Issue 2 • 1000108
As far as only linear circuits are concerned, the results are also
relevant for “impedance” circuits formulated in the ω- and s- (Laplace
variable) domains, just as it is for the circuits to which the classical
Thevenin’s theorem is applied.
2).
All the said is easily “translated” to the current source (Appendix
The Completed Classification, for Applications, of
Helmholtz-Thevenin-Norton Theorem
Figure 3 shows the extended (completed) frame for HelmholtzThevenin-Norton theorem, including our results.
We think that this frame is important for a designer of electronic
sources and for basic teaching of circuit theory.
We are also in the opinion that when considering some important
nonlinear (affine) features of the common 1-ports [7], one should
take into account the completed classification.
Conclusions
1. Apart from the often obtained case when Thevenin (or
Norton) equivalent circuit is just a passive resistor, the
opposite case when vTh, or iN is nonzero, but the resistor is
of zero (or infinite) value, can be observed, and is worth
considering. This case should be useful in creating of the
ideal controllable voltage or current sources.
2. The equivalent-generator theorem is irrelevant, i.e. the
Thevenin and Norton cases are “split” here, but the duality
argument (Appendix 2) can be applied.
3. The absence of the linear resistors RTh or RN in such an
equivalent circuit makes the proved theorem(s) relevant to
nonlinear internal structure of the source.
4. For the output voltage source, current feedback from the
terminating source is prohibited, and for the dual version
of the output current source, voltage feedback is prohibited.
These simple and easily justified (almost directly following
from the very definition of the ideal source) instructions for
the designer are the only limitations on application of the
obtained results. The clarity of these limitations should make
the results simple in use, and we find this clarity to be the
aesthetic point of the circuit situation.
5. The classical Thevenin/Norton theorem still includes
interesting research aspects to which the attention of the
electronics and circuits specialists should be drawn. The
research scheme obviously includes many possibilities for
computer (and PSpice, etc.) simulations, and a teacher can
find here a good material for students’ projects. Especially
interesting would be cascade connection of such 1-ports (for
instance, consider the source vs in Figure 4 in Appendix 1 as
the previous such 1-port), and also thus creating of chaotic
function (process). In our opinion, this material should be
included in the text-books on basic circuit theory.
6. The circuits with terminating dependent sources are often
met in the theory of amplifiers. We think, however, that
such examples should be treated also at the much earlier
educational stage when the basic Thevenin theorem is
introduced, and the expected circuitry is much more general.
• Page 2 of 4 •
Citation: Gluskin E (2013) An Extended Frame for Applications of the Helmholtz-Thevenin-Norton Theorem. J Electr Eng Electron Technol 2 :2.
doi:http://dx.doi.org/10.4172/2325-9833.1000108
Appendix 1: The formal proof of equivalence of the
circuits of Figs 1 and 2 in the case when the load does
not influence the controls of the output source
Continuing to be focused at the output voltage source (Figure
1), we shall use nodal voltages analysis, writing KCL equations for
branch currents. Case A in Section 1.1 is relevant here.
For node a (Figure 1) we can separately write
va = f ( x1 , x2 ,..., x p )
(A1)
where x1 , x2 ,..., x p are the p variables that control the output
source (below “the controls”). It can be that among the controls there
are some of the nodal voltages, our main unknowns.
It is realistic to require that for all p controls zero,
f (0, 0,..., 0) = 0 .
(A2)
We assume, furthermore, that it is possible to present
x1, x2 ,..., x p as some known functions ξ1, ξ 2 ,..., ξ p
nodal voltages v1, v2 ,..., vn , i.e.
of the
x1 = ξ1 ( v1 , v2 ,..., vn )
x2 = ξ 2 ( v1 , v2 ,..., vn )
..............................
(A3)
va = F ( v1 , v2 , ..., vn −1 , va )
va = ψ ( v1 , v2 , ..., vn −1 )
The following simple examples illustrate the said, and the reader
will easily find (construct) more examples.
Consider the simple circuit (Figure 4) for which the output
voltage is defined only by the internal elements of the 1-port.
- v1 +
(A4)
+ v
s
R2
i1
i
+
(A4a)
A1.1. Some structural detailing
For some applications (e.g. for equivalent schemes of amplifiers)
it is reasonable to consider two possible cases of the circuit topology.
The first case is when there are no nodes in the 1-port directly
connected to node a, i.e. apart of its high-impedance control
inputs, the source just has a common ground with the rest of the
circuit. (Node a is connected only to the output source.) Then, we
have an independent problem for all the “internal” nodes, i.e. can
independently find v1 , v2 , ..., vn−1 , using n-1 KCL equations, and
then (A4a) simply gives va .
The other possibility is that some of the internal nodes are
connected to a via some simple branches, say via some resistors.
a
N
kv1
va ,
with a known function ψ of only n-1 variables. Since va is the
voltage that the load “feels”, -- if the load will not influence (in any
way), v1, v2 ,..., vn −1 , then the voltage generated by the output
source cannot be influenced by the load. According to (A1), this
good situation will be ensured also if it is directly known that the load
does not influence any of the controls x1 , x2 ,..., x p . The crucial
question for design is thus whether or not there is any “feedback” that
allows the current of the voltage source (which certainly is influenced
by the load) to influence the controls of the output voltage.
Volume 2 • Issue 2 • 1000108
In the practical examples, it is sufficient to derive, in any way, the
formula for the output voltage and observe whether this voltage does
not (as we want) depend on the output current, or depends on it. Of
course, direct inclusion of the parameters of the load, e.g. Rload, in a
final expression is sufficient for the conclusion of non-ideality.
R1
= va . Substituting (A3) into (A1), we obtain
with a known ‘F’, and after resolving (A4) for
In either case, we have a complete circuit solution, and there is
no place for the external current to influence anything in the 1-port,
including va , i.e. the 1-port is as a whole is an ideal voltage source
independent of the outer conditions. It is just required it to be
possible to come to (A4a).
A1.2. Obtaining an ideal (independent) output source
x p = ξ p ( v1 , v2 ,..., vn )
where vn
Then voltage vn = va influences the currents of these branches,
i.e. appears in the KCL equations for the nodes that are connected
to a. Since this circumstance does not influence the possibility to
obtain (A4a), we can substitute va = ψ ( v1 , v2 , ..., vn −1 ) into the
latter KCL equations, thus obtaining, in total, n-1 equations including
only the unknowns v1 , v2 , ..., vn−1. Thus, v1 , v2 , ..., vn−1 can be
found, and then va = ψ ( v1 , v2 , ..., vn −1 ) becomes known too.
b
Figure 4: An example. Voltage vab is independent of i and the
circuit N. The whole circuit to the left from a-b appears to be an ideal
independent source.
KVL over the left mesh gives:
v + (1 / k ) vab − vab
vab = kv1 = − kR1i1 = − kR1 s
R2
(A5)
(we used that v1 = (1/k)vab), from which
vab =
kR1
vs
( k − 1) R1 − R2
(A6)
This expression is independent of i and of the structure of the external
circuit. This is, of course, also the open-circuit voltage voc (i.e. the
special case of i = 0) of the 1-port to the left of a-b. The load cannot
change vab and the whole circuit to the left from a-b is an ideal
source for any external circuitry.
A1.3. A non-ideal source
In the example of Figure 5 the current of the voltage source, i1,
(that depends on the load’s current) influences (because v3 depends
on i1) the internal voltage v2 that controls the output source. Thus, the
load is allowed to influence vab via the current feedback.
• Page 3 of 4 •
Citation: Gluskin E (2013) An Extended Frame for Applications of the Helmholtz-Thevenin-Norton Theorem. J Electr Eng Electron Technol 2 :2.
doi:http://dx.doi.org/10.4172/2325-9833.1000108
-
v2
v3= k3i1
+
i1 +
+
- v = kv
1
2
+
vs
a
-+
R
-
equivalent of the circuit of Fig. 6 is the circuit of Figure 7, i.e. the
1-port is an ideal independent current source.
i2
(vTh)
-
+
a
vL
RLoad
b
Figure 5: The circuit with an internal dependent source, influenced
(controlled) by the current of the output dependent voltage source.
In this case, vTh ≠ v L , which shows that v L = vab = v1 depends
on the load. Observe, for (A8) and (A9), that the physical dimension
[k3] = Ω.
Indeed, simple calculations show that for this circuit, for
vTh = voc , we have
vs
,
vTh =
k3 1
− +1
kR k
(A8)
but when the load is connected, then for v Load = vab we have
v Load =
k3 (
1
kR
vs
+
1
RLoad
)−
1
k
.
(A9)
+1
We see that vL depends on RLoad and vTh ≠ v L . The circuit to
the left from a-b is a non-ideal voltage source. Such a case is reduced
to that of the usual Thevenin theorem, as is shown in the scheme of
figure 3.
Appendix 2: The dual circuit with current output source
Consider now the circuit (that also need not be linear) shown in
Fig.6, with a dependent output current source.
Figure 7: The equivalent of the circuit in Figure 6.
The physical reason for this conclusion is the ability of an ideal
current source withstand any voltage drop, and it is just prohibited
the designer to use this voltage drop as a control for any internal
dependent source. Since the equivalent-generator transformation
is not possible in these circuits, we cannot obtain this conclusion
automatically from Appendix 1, but can use concepts that are dual
with respect to those of the previous proof. That is, we have to use
the mesh currents instead of nodal voltages, and the associated KVL
equations instead of the KCL equations. Then, the steps of the proof
for the circuit of Appendix 1 are essentially repeated. In particular,
we use now mesh-current analysis, expressing the nodal voltages via
the mesh currents as the unknowns. Then, at the output loop, we
have that the mesh current, -- which is the current of the dependent
source, -- is already expressed via the internal state-variables, so that
no KVL equation is needed for this mesh, and thus vtest (or vload)
cannot be equationally included.
We finally conclude that the output current is only due to the
internal independent sources, i.e. iout = isc (short-circuit current) and
the circuit of figure 7 indeed is equivalent to the circuit of Fig. 6.
According to the fact that figure 7 does not include any linear
element, nonlinear circuit versions can be relevant here too.
Acknowledgments
I am grateful to Professor Michael Werner for some practical considerations
and to Professors Wolfgang Schwarz and Albrecht Reibiger for some comments
on [7].
a
N
b
Figure 6: The circuit to be simplified to an ideal current source.
The output current of the 1-port is of special interest. We can connect
a voltage-test source, vtest, thus defining vab, but can neither leave the
circuit open, nor connect any series source of itest. These are the only
limitations to the external circuit N. The point is to show that if there
is no LIC-feedback, vab does not influence the output current of the
1-port.
Note that the source-circuit cannot be detached from the external
circuit (a duality condition!), but we can replace the external circuit
by some vtest, in particular, short-circuit it.
With the limitations on the circuit structure, or its internal
functional connections, dual to those in Appendix 1 (see also Section
1.2), meaning that vab may not influence any control source, the
Volume 2 • Issue 2 • 1000108
N
isc
-
References
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2. Chua LO, Desoer ChA, Kuh ES (1987) Linear and Nonlinear Circuits, McGraw
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3. Irwin JD, Nelms RM (2008) Basic Engineering Circuit Analysis, Wiley, New
York.
4. Hayt WH, Kemmerly JE (1993) Engineering circuit analysis, McGraw-Hill,
New York.
5. Gluskin E, Patlakh A (2010) An ideal source as an equivalent one-port, East
Journal of Electronics and Communication 5: 79-89.
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Proceedings of the IEEE, 91: 636-640.
7. Gluskin E (2012) On the affine nonlinearity in circuit theory, Proceedings of
NDES, Wolfenbuttel, Germany.
Author Affiliation
1
Top
Kinneret College in the Jordan Valley (on the Sea of Galilee), 15132, Israel
• Page 4 of 4 •