PART I
ELECTRIC CIRCUIT CONCEPT
AND ANALYSIS
Circuits as modelling tools
Techniques for solving DC circuits
A.C. Linear electric circuits
Three-phase circuits
uC (t )
Chapter
2
Circuits as modelling tools
Table of Contents 2.1 Introduction ........................................................................................................................... 5 2.2 Definitions ............................................................................................................................. 7 2.3 The charge conservation and Kirchhoff’s current law ........................................................... 8 2.3.1 The charge conservation law ........................................................................................................ 8 2.3.2 Charge conservation and circuits .................................................................................................. 9 2.3.3 The electric current ..................................................................................................................... 11 2.3.4 Kirchoff’s Current Law formulations ........................................................................................... 12 2.4 The circuit potential and Kirchhoff’s voltage law ................................................................ 15 2.4.1 The electric field inside conductors ............................................................................................. 15 2.4.2 Kirchhoff’s Voltage Law formulations.......................................................................................... 18 2.5 Solution of a circuit .............................................................................................................. 19 2.5.1 Determination of linearly independent Kirchhoff’s equations (loop‐cuts method) ..................... 19 2.5.2 Constitutive equations ................................................................................................................ 22 2.5.3 Number of variables and equations ............................................................................................ 23 2.5.4 Example ...................................................................................................................................... 24 2.6 The substitution principle .................................................................................................... 25 2.7 Kirchhoff’s in comparison with electromagnetism laws ..................................................... 26 2.8 Power in circuits .................................................................................................................. 27 Example ...................................................................................................................................... 29 2.9 Historical notes .................................................................................................................... 30 2.9.1 Kirchhoff’s short biography ......................................................................................................... 30 2.9.2 Tellegen’s short biography .......................................................................................................... 30 2.10 Reference list ....................................................................................................................... 30 2.8.1 2.4
Chapter 2: Circuits as modelling tools
For the teacher
This chapter, in comparison with similar books’ approaches, contains the basic innovation
of making a clear distinction between physical systems and models.
Since no uniform terminology in the books exists, here the following was adopted:
- circuital systems was the name adopted for physical systems constituted by devices
connected by wires, that are good candidates to be modeled using lumped-component
models
- circuits was adopted for are actual (lumped-component) models.
This way of thinking has several advantages:
- it helps the student, i.e. the future engineer, to get accustomed to the importance of the
activity of modelling a physical system, and the corresponding need to evaluate the
approximations done in the process. This activity will accompany their whole working
life;
- it allows to clearly distinguish between the electromagnetism laws, i.e. Maxwell’s
equations written in more or less complicated forms, and Kirchhoff’s laws, that are
assumed to be valid, by definition, to all circuits.
The authors took the decision to make this shift with respect to what most frequently found
in books, after years of teaching and meeting every-day students’ needs and doubts.
They hope it will be appreciated by the teachers. If so, it will be useful and appreciated by
their students as well.
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
2.5
2.1 Introduction
Electrical engineers spend a large part of their time in working with Electric Circuits, or
simply circuits.
The word circuits is used to indicate graphical/mathematical tools that are used in description
of systems in which charges flow constantly (the so-called direct-current circuits), or sinusoidally
(the so-called alternating-current circuits), or in which they vary in a general way. They are able
to model adequately electronic boards, home and industrial electrical installations, the inner
behaviour of electrical machines, and so on.
The physical systems circuits are intended to model, whenever precision is necessary, will be
called circuital systems.
More in Depth
It is very common in written text (textbooks, articles, brochures) not to evidence the difference
between physical systems and their model. This is widely erroneous, because physical systems have
a given behaviour that can be modelled at different levels of precision.
For instance the wheel of a car can be just an ideal rigid rolling cylinder for some purposes, but if
one wants to evaluate the forces generated by collisions, elasticity must be taken into account; to
evaluate energy losses also rubber hysteresis must be considered, and if one wants to evaluate the
response to lateral forces, further characteristics must be taken into account.
Making a difference between physical systems and their models is very important because to draw
a model from a system a certain set of assumptions is to be made, and the results of the model’s
analysis can be applied to the given system only if these assumptions are met to a sufficient degree
of precision.
That is why in this book a lot of care has been put in distinguishing between physical systems and
their models, i.e. circuital systems and circuits.
Consider for instance the simple system shown in fig. 2.1.
1
G
2
Fig. 2.1. A simple circuital system.
It shows an electric sinusoidal generator feeding two lamps with the interposition of a couple
of wires, that are represented “thick”, because in a physical system they have not only a length
but also width and depth.
Obviously, the analysis of this system would be greatly simplified if, instead of having to
analyse simultaneously all points of space using the electromagnetics laws (the four laws quoted
in Appendix A, and that are often referred to as Maxwell’s equations) it would be possible to
write independent equations of the involved individual elements and linking them by some
congruence additional equations.
A qualitative analysis of this figure shows that the generator connects with the lamps with
long wires, while short connections are present at the two horizontal sides of the circuit.
It is intuitively understood that, if the effects of space around the wires is not significant for
what happens inside the system components, the system of fig. 2.1 can be studied as a system
constituted by connection of its main elements, as indicated in fig. 2.2. The elements are the
2.6
Chapter 2: Circuits as modelling tools
generator, the two lamps, and the two long wires, while the short wires used for connecting the
lamps to the longer wires are neglected. All elements have terminals, that are the points of them
used to connect them to the other circuits elements. In the figure terminals are indicated as small
white circles. Thin lines represent ideal wires, i.e. graphical symbols indicating that what happens
in the circuit is exactly the same as if the components at the ends of them were directly connected
to each other. Thin lines are therefore like ideal wires, that are able to transfer electric charge
from point to another of space in an ideal way, i.e. in such a way that charges present at one end
are immediately transferred to the other end.
The passage from the system in figure 2.1 to that of figure in 2.2 is dramatically important: it
implies that we have modelled a spatially distributed system (for which all the fields connected to
electromagnetics phenomena, such as electric field E, magnetic field H, current density field J,
et. have a value at each point of space) as a lumped elements system, whose behaviour is
determined by the behaviour of the boxes connected by ideal wires, that will be expressed in
terms of equations that will be called constitutive equations of the components (or elements), and
by the effect of the connections made by the wires.
More specifically, we have modelled the circuital system from which we started, that was a
distributed parameter one, through a circuit, that is a lumped parameter model, and whose
analysis is much simpler.
ideal
wire
G
circuit
element
lower
conductor
lamp 2
lamp 1
upper
conductor
terminal
node
Fig. 2.2. Circuital approximation of (=circuit modelling) the system of fig. 2.1
The ease of analysis of electromagnetical systems using circuits will become more and more
clearer as far as the study of the book proceeds. But what kind of systems can be studied as
circuits?
In a very simplified way, it can be answered that systems that can be studied as circuits are
those we called earlier as circuital systems: i.e. systems that are physically constituted by devices,
(such as the generator, the lamps and the long wires of figure 2.1) that perform complex actions,
others that could play a marginal role (such as the short wires), and some space around these
elements, whose effects on the electrical phenomena inside them can be neglected.
These qualitative considerations often are too simplified. It must be stressed that if a circuit
does not model correctly a physical system, the results of its analysis (the calculated currents and
voltages) will not be close to the actual values of the original circuit, and this could lead to big
mistakes.
Therefore in this chapter, along with the next two ones, discussion is reported on what the
basic hypotheses allowing a circuital system to be modelled as a circuit are.
The construction of a lumped parameter model of a physical system is not unique. For
instance, if the current between upper and lower conductors of the system reported in fig. 2.1
cannot be neglected, the model reported in fig. 2.3 can be considered, in which the transmission
line is a unique lumped component, that can take into account also phenomena occurring between
its upper and lower conductors.
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
2.7
Transmission
line
lamp 2
G
lamp 1
ideal
wire
circuit
element
terminal
node
Fig. 2.3. A circuit modelling the system of fig. 2.1 different from that of figure 2.2.
2.2 Definitions
To deal with circuits effectively, precise definitions of the terms used are needed.
Although there is some uniformity in textbooks about terms and definitions, some significant
differences exist. Therefore it is useful to summarise here all the major definitions related to
circuits used in this book.
The authors have carefully consulted the International Electrotechnical Vocabulary (IEV) [4] that is the most
authoritative source of terminology standardisation for all electrical engineers all over the world. Therefore the
definitions reported here normally comply with the ones reported in the IEV. Only the wording reported here is original,
since it has the purpose of best explaining the meanings given the book’s scope and audience. Where significant
differences exist from the IEV small noted are added explaining the reasons beyond the choice of partly deviating from
standards.
The following definitions are reported in alphabetical order for the reader’s convenience.
Branch
A branch is a circuit element having two terminals.
Branch-based circuit
A branch-based circuit is an electric circuit in which all elements, except nodes, have two
terminals
The IEV does not give any definition for circuits having only two terminal components, except nodes. his has required a
definition specific to this book, because of the importance of these circuit topology.
Circuit (electric circuit)
An electric circuit, or simply circuit, graphical-mathematical tool that constitutes a
lumped-component model of a circuital system, consisting of lumped components
connected to each other. These components are the circuit elements.
Since circuits are models of real systems, circuits of different levels of accuracy can be
produced for the same physical system.
NOTES:
1. See the note on the circuital systems definition.
2. The circuit behaviour is defined by the inner behaviour of circuit elements and by their interconnection:
no influence is possible between what happens outside and what happens inside circuit wires and
elements.
3 In this definition no constraints are reported on the inner structure or behaviour of the circuit elements.
However ideal transformers, very special circuit elements that will be discussed in chapter 4, play a
special role in circuits: they are elements of interconnection of circuits to form electric networks.
Circuit element
A circuit element is a component of a circuit that is connected to other components by
means of some connection points, called terminals.
2.8
Chapter 2: Circuits as modelling tools
Circuital system
A circuital system is a physical system containing elements connected to each other
through wires in such a way that one or more closed that loops are formed.
A circuital system is a spatially-distributed (three-dimension) system.
IEV does not provide special terminology to distinguish between physical systems and mathematical models. This
distinction is however very important in practice, and therefore the author felt obliged to introduce this new term to
refer to the physical systems, while circuits (and networks) are used for lumped-component mathematical models of
circuital systems.
Ideal wire
An ideal wire is a branch, the terminals of which have the same potential value, i.e. such
that the voltage across its terminals is zero.
Node
A node is a point in a circuit in which three or more circuit elements are connected to each
other.
Although this definition does not coincide with the one reported in the IEV, it is very commonly used in textbooks
Network (electric network)
An electric network, o simply network, is a set of circuits separated from each other by ideal
transformers. Two circuits separated by ideal transformers are also normally called
magnetically coupled circuits.
NOTES
1. in IEV electric circuits and electric networks are considered to be equivalent. The availability of two
names is exploited in this book to indicate two different concepts, thus easing the description of the
circuit theory and analysis techniques
2. networks and ideal transformers will be introduced in this book in chapter 4.
2.3 The charge conservation and Kirchhoff’s current law
2.3.1 The charge conservation law
The reader should be already acquainted with the idea of charge conservation.
The charge conservation principle states that “the charge in a given region of space remains
constant over time”. However, studies carried out mainly in the XIX century showed that the
charge intended as sum of individual charged elements (electrons, ions) does not actually
conserve, but can accumulate in elements of space. The issue was solved by Maxwell, who has
defined a new form of charge, called displacement charge, that enables us to state the following:
Law: total charge conservation
The total charge (sum of conduction and displacement charge) in any given region of space
remains perfectly constant over time
Conduction charge is the charge that moves through conductor wires (electrons) or
conductive solutions (ions), while the displacement current due to the variation of the
displacement field over time. This is detailed in the following more in depth box, as well as in the
next chapters.
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
2.9
More in Depth
this block is intended to those that have some previous knowledge of the electromagnetic behaviour
of capacitors. and some basic knowledge of the behaviour of a simple R-C circuit in which a
capacitor is charged
sw
R
Battery
i
L
C
armature
dielectric
armature
In the circuit shown aside, if the
armatures of the capacitor C are
initially without any charge, when
switch sw is closed, some current
flows in the circuit, and neat charge
enters the closed curve L.
Charge is thus accumulated inside the armature, and therefore the charge conservation principle
does not apply to it. If however we hypothesise that a special current, called displacement current
flows in the dielectric, i.e. the space between the armatures, and that this current is exactly equal to
the current i flowing in the conductors outside the capacitor, then the total charge conservation
applies, the total charge being due to both conduction and displacement currents. In this case the
armature total charge will not change since the dielectric will absorb a displacement current that is
perfectly equivalent to the conduction current entering the upper conducting wire.
2.3.2 Charge conservation and circuits
In the qualitative analysis shown in section 2.1, it was said that circuital systems may be
correctly modelled by means of circuits when “the effects of space around the wires is not
significant for what happens inside the system components”.
If the space around system components must not affect what occurs inside them, no charge
must flow in that space. Consider again the system reported in fig. 2.1, reported also in fig. 2.4.
G
1
2
Fig. 2.4. The system of fig. 2.1, with indication of some stray currents.
The reader should already know that, although materials are commonly classified in
conducting and insulating ones, a perfect conductive material does not exist1, as well as a
perfectly insulating one. Therefore same stray currents can flow also though the air surrounding
the wires, e.g. flowing from the upper conductor towards the lower one (as indicated in fig. 3.4),
and vice-versa.
If the system of fig. 2.1 is represented by the circuit of fig. 2.2, it should be possible to
neglect these stray currents. Indeed this is very often acceptable although in some rare cases stray
currents are considered and different models are used.
There is more than just considering the air surrounding the conductors to be perfectly
conductive in the passage from figure 2.1 to 2.2. It must be noted that the system of figure 2.1 is
subject to variable quantities, since the generator is a component, that we have not discussed yet
in depth, but that has to be imagined to be able to determine at its ends a variable voltage.
therefore all the quantities of the system of fig. 2.1 (currents, voltages, fields at all the points of
space) vary with time. When quantities vary with time displacement currents can flow between
upper and lower wires even though the air could be considered to be perfectly insulating. Indeed
1
the case of “superconductivity”, that occurs at very low temperatures is very special, and is not considered here.
2.10
Chapter 2: Circuits as modelling tools
upper and lower conductors can be considered as being as the two cylindrical armatures of a
capacitor whose dielectric is the air between (fig. 2.5).
A
upper
armature
1
G
2
lower
armature
dielectric
A-A view
A
Fig. 2.5. Stray currents through perfectly insulating means can be due to capacitive effects, or
displacement currents between upper and lower wires.
Again, the lumpisation process that leads to circuits requires that not only conductive but
also displacement currents must not flow through the air (or space) interspersed between the ideal
wires of any circuit.
Consider again one of the lumped-component models proposed, for the system of fig. 2.1, i.e.
the circuit reported in fig. 2.3 and in fig, 2.6 also. Here, for simplicity, the terminals are not
evidenced anymore (the reader should keep in mind that they always exist).
Transmission
line
lamp 1
G
lamp 2
1
3
2
Fig. 2.6: Possible closed curves around circuit parts.
In this figure three different closed curves crossing the circuit are considered: a curve
surrounding a circuit node (type 1), one surrounding a single lumped element (type 2), one
surrounding a group of elements and wires (type 3).
For them the charge conservation principle applies and therefore the global charge entering
the curves, of either conductive or displacement types must be equal to zero. But it was also said
that whenever a circuit is created, all displacement currents between wires must be neglected.
That is equivalent to saying that displacement currents can occur only inside circuit elements.
As a consequence of this it can be concluded that the total (conductive) charge entering
through some of the wires that traverse any of the considered curves must be exactly equal to the
sum of those that exit it.
As a consequence of this it can be concluded that, for any of the considered curves, the total
(conductive) charge entering through the wires that traverse it must be exactly equal to zero.
This is equivalent to saying the sum of the charge entering from some of the wires is
identically equivalent to the sum of the charge that exits from the others.
This is the rationale behind so called Kirchhoff’s law, that is the charge conservation law for
circuits. In the next sections it will be expressed in a more formal way (and more useful for
practical computations.)
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
2.11
2.3.3 The electric current
From their basic electromagnetics knowledge the readers should be already aware that
electric charges present inside conductor materials (namely the electrons) are not linked to atoms,
but can freely move. If therefore a wire is made of conducting material, these charges can freely
move from an end to another of it.
Also in other conducting means, such as electrolytic solutions there are charge free to move;
while in conductor media the charge carriers are just electrons, and are negatively charged, in an
electrolytic solution the charges are carried by ions, that can be either positive or negative.
Therefore, in general it may happen that in a conducting wire charge carriers move from left
to right or right to left, and they can be positively or negatively charged.
It is experimentally verified that a positive charge moving from left to right is totally
equivalent to a charge having the same size but opposite sign moving form right to left (fig.
3.7a)2, and that the combined effect of a positive charge Q+ and a negative charge Q- is equivalent
to a generic charge Q, , assumed to be positive, moving in the same direction of Q+ 3.7b).
Q

QQ+

-Q
is
equivalent
to:
is
equivalent
to:

Q
a)

b)
Q=Q+ -Q-
Fig. 2.7. Charge flows equivalence:
a): a positive Q from left is equivalent to the opposite charge -Q from right
b): a comprehensive charge Q=Q+-Q- from left is equivalent to Q+>0 from left and Q-<0 from right.
Therefore in the general case in which a conductor medium there are both positive charges
and they move in either direction, the analysis can be performed computing just an equivalent
positive charge defined as:
Q  Q1  Q2
in which
 Q1 is the algebraic sum of charges in the same direction as the direction assumed for Q
 Q2 is the algebraic sum of charges moving in direction opposite to the direction assumed
for Q
Consider this equivalent charge Q (flowing in the time t through a cross section  of a
conductor (fig. 2.7b), the current I through flowing in the conductor is defined as:
2
As usual in textbooks, for simplicity it is often said also in this book that “a charge moves” instead of “a charge
carrier moves.”
2.12
Chapter 2: Circuits as modelling tools
Q
for continuous flow at constant rate
t
In case the flow of charge varies over time, an infinitesimal time interval dt can be considered in
which the charge flowing through  is infinitesimal as well and can be indicated as dq
(remember, chapt.1, that by convention lowercase letters indicate quantities that vary with time).
In this case, the current i(t), by definition is:
dq (t )
i (t ) 
for any flow (constant or variable).
dt
So, the electric current is defined as the rate at which the charges flow. Since charges are
measured in coulomb (symbol: “C”) in the SI the electric current will therefore be measured in
coulomb per second, Because of its importance, to the electric current has a unit of measure of its
own, called ampere (symbol: A): one ampere is one coulomb per second.
I
Definition: the ampere (unit of measure of current)
The unit of measure of current is the ampere (symbol: “A”).
One ampere is the a charge flow of one coulomb per second.3 In formula: 1A=(1C)/(1s)
As a further clarification of the definition and its sign, it can be said that a current of 1A
moving from left to right can be equivalently due to a one positive coulomb charge crossing a
given surface in one second from left to right, or to one negative coulomb charge crossing a given
surface in one second from right to left.
2.3.4 Kirchoff’s Current Law formulations
Consider a generic closed curve drawn in a circuit, in such a way that it does not cross any
circuit lumped component, but only wires.
A general representation of this generic curve is reported in fig. 2.8 a, where just the curve
and the traversing wires are reported (the parts of the considered circuit inside and outside the
curve are omitted). Since we are considering a circuit, that was already defined as a lumped
component representation of a physical system in which no charge can flow outside conductors
and circuit lumped elements, only conductive charge can flow from the inner of the curve to the
outer and vice-versa, through wires.
Needless to say, even though, for graphical ease, only five wires traverse the closed curve displayed in the
figure, the reasoning we are going to carry is valid for any number of wires.
dq 4
i 4(t)
dq 4 i 4(t)
dq 3
i 5(t)
i 3(t)
dq 5
dq 3
i 5(t)
i 3(t)
dq 5
i1(t)
dq 1
dq 2
i2(t)
a)
i1 (t )  i2 (t )  i3 (t )  i4 (t )  i5 (t )  0
i1(t)
dq 1
dq 2
i2(t)
b)
 i1 (t )  i2 (t )  i3 (t )  i4 (t )  i5 (t )  0
Fig. 2.8. Some n-terminal element charge conservation law
(arrows may indicate “actual” or “reference” charge flow directions).
3
the student should know from the electromagnetics study that this is not the definition of ampere of the S.I.,
although, obviously non in contrast with it; here this is proposed because it is more effective for the student’s
understanding at this point of the book.
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
2.13
Imagine that charge (considered to be positive charge, as discussed earlier) flow from the
outside of the loop toward the inside according to the arrows reported in figure near the currents.
Let dqk, be the infinitesimal charge entering the curve in the infinitesimal time interval dt
from wire k.
The charge conservation for circuits (sect. 2.3.2) will therefore imply that:
dq1  dq2  dq3  dq4  dq5  0 (charge flows directed as per figure a)
dq1  dq2  dq3  dq4  dq5  0 (charge flows directed as per figure b)
and, dividing by the corresponding time interval dt:
i1 (t )  i2 (t )  i3 (t )  i4 (t)  i5 (t)  0 (currents directed as per figure a)
 i1 (t )  i2 (t )  i3 (t )  i4 (t )  i5 (t )  0 (currents directed as per figure b)
Note that in some cases the dependence on time t was explicitly shown with the writing i(t), in other cases
was implicit. In all cases, however, the equations indicating the charge conservation law, either expressed in
terms or charges or currents it valid at any time.
We want to express this concept in a general way, without having the need of knowing a
priori the actual flow directions. To do that, first the following convention is set:
Convention: Current sign.
The number indicating a current i is a real number whose module refers to the absolute
charge flow itself, and whose positive or negative sign indicates respectively that the flow
direction is in agreement or not with an arrow reported near the current name i. The arrow
is called reference direction of current i.
Using this sign convention, instead of considering the actual charge flow directions, we can
use the reference directions. Therefore the current equations of figs 2.8 are still valid considering
the arrows not being the actual current directions, but just reference (or, by some authors,
assumed) current directions. These equations correspond to other ones, related to differently
reference direction, for example to
i1 (t )  i2 (t )  i3 (t)  i4 (t)  i5 (t )  0 (flows using the flow variable sign convention and reference
directions all entering the node)
This is a very useful result: we can state the law describing the inability of the element to
accumulate charge simply as follows: the sum of currents of all terminals of a circuit element,
assumed entering, must always be identically zero. And, equivalently, the sum of currents of all
terminals of an element, assumed exiting, must always be identically zero.
In the following, whenever this does not cause ambiguity, the word or “assumed” is omitted and therefore for
a branch current indicated with a symbol the expression “current assumed entering ...” and “current assumed
exiting-- ” will be substituted with “current entering...” and “current exiting...”.
When analysing a circuit, it often occurs that current reference directions are not all entering
or exiting a curve. So, alternative formulations must be issued to be with generic assumed
directions. They can be as follows (subscript “in” indicating assumed sign entering the curve,
“out” indicating exiting currents):
n _ in
1)
i
k 1
k ,in
(t ) 
n _ out
i
k 1
k ,out
the sum of all currents entering a closed curve in a circuit
(t )  0 identically equals the sum of all currents exiting it.
if all currents are considered to be entering, the algebraic sum of
all currents entering a closed curve in a circuit identically equals
k 1
k 1
zero.
Note, that in formulation 2, we introduced the notion of algebraic sum and of current
considered to be entering: it obviously means that when the assumed direction enters the closed
n _ in
2)
i
k ,in
(t ) 
n _ out
i
k ,out
(t )  0
2.14
Chapter 2: Circuits as modelling tools
curve, the corresponding current will be taken with a positive sign, otherwise will be preceded by
a negative sign.
When charge conservation introduced is applied to a curve of type 1, 2 and 3 in figure 2.6,
specialised versions of it are immediately derived, stating that the sum of currents assumed to be
entering any node, any circuit element, any subcircuit is always identically equal to zero.
In the previous circuit examples, for instance those shown in figures 2.2 and 2.3 circuit
elements had two or more terminals, by which to exchange charge with their exterior. It will be
seen that, except for nodes, in the large majority of cases the considered circuit elements will
have always two terminals. Elements with two terminals are therefore of the biggest importance,
and will need a name of their own. They will be therefore called branches4.
When considering circuit branches, the charge conservation law is immediately taken into
consideration from the very beginning stage of analysis, simply using a unique value for the
currents at their two ends. Consider fig. 2.9. It is clear from it that the usage of a unique current
flowing in the branch A, e.g. the iA, shown in part b) of the figure, is very convenient instead of
using different names for the currents at the two sides of element A, as indicated in part a), and
complement this with the addition of equations stating their algebraic equality.
i’A2
i’A1
i’A1=-i’A2
A
i”A1
i”A2
A
a)
iA
i”A1=i”A2
A
B
b)
Fig. 2.9. Charge conservation for branches:
a) stated by equations; b) implicitly defined using a single branch current
Now we can state Kirchhoff’s current law in three forms that are used in practice in circuit
analysis, all equivalent.
Since in the large majority of cases in every-day work Kirchhoff’s law is applied to nodes, it
is expressed with reference to nodes, even though, as should be now clear, it is still valid
substituting the word “node” with “closed curve”:
4
The reader is suggested to refer to sect. 2.2 for definitions of branches and branch-based circuits.
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
2.15
Law: Kirchhoff’s Current Law (KCL)
In any circuit it happens that
form 1: the algebraic sum of all of the currents assumed entering a node in a circuit is
identically null;
form 2: the algebraic sum of all of the currents assumed exiting a node in a circuit is
identically null;
form 3: the sum of the currents assumed entering a node in a circuit identically equals the
sum of the currents assumed exiting it.
KCL applies to any circuit, that is a mathematical-graphical model of a physical system. Any circuit models
the behaviour of a physical system within some degree of approximation.
2.4 The circuit potential and Kirchhoff’s voltage law
2.4.1 The electric field inside conductors
Consider the very simple circuital system reported in figure 2.10, containing just a battery, a
conductor and an electric load in the form of a lamp. It is to be recalled that inside the conductor,
as everywhere in space, proportionality exists between current density J and the total electrical
field Et:
Et   J (Et=Eb+Ec)
(2.1)
where =1/ is the resistivity of the material existing where Et and J are evaluated.
This should be already known to readers, and is however rapidly recalled in sect. A.7 A.7 of Appendix A.
The Eb is due to the input of external power that the battery comes from chemical potential
energy while electric field Ec is due to charges distributed in the surfaces of conductor wires, and
is therefore conservative and its work along the circuit is null.
These surface charges might have the aspect of those shown in figure by plus (+) and minus (-) signs.
++
++
+ +
+ +
Ec
I
+
+
I
f b + +b
++++ ++
Eb
Ec
c
battery
- - - - - -
f a - -a
Ec
d
- - -
- - -
I
Ec
-
Fig. 2.10. A simplified version of the circuital system of figure 2.1:
here the generator generates a constant voltage
Integrating (2.1) around the loop constituting the circuit gives:

 E t  dl    J  dl    b

lb
l
l 
  c c   l l  I  ( Rb  Rc  Rl ) I
Sb
Sc
Sl 
(2.2)
In the previous equation the symbols Rb, Rc and Rl indicated physical characteristics of the
battery b, conductor c and of the lamp l, that will be further discussed in the next chapter and that
are called resistances.
Exploiting the conservativity of Ec: (2.2) gives:
2.16
Chapter 2: Circuits as modelling tools
b
a E b  dl  ( Rb  Rc  Rl ) I
The equation shows that the power generated by the battery is dissipated in the circuit
elements (including battery inner resistance) by effect of their resistivity.
The example given, however, has also the very important purpose of showing that in this
circuit an electric field Ec is generated all around the loop that is due to surface charges, and
therefore has the nature of an electrostatic field, is conservative and therefore has a potential
function V.
Between any two points of the circuit a difference of potential or voltage U can be defined.
For instance between a and b or c and d it is:
Uba=Vb-Va Ucd=Vc-Vd
A voltage expressed as potential difference is often called voltage across terminals. For
instance Uba is the voltage across terminals b and a.
A natural circuit representing the system of figure 2.10 is the one shown in figure 2.11.
b’
c’
upper
conductor
c
lamp
b
Battery
a
d
a’
lower
conductor
d’
Fig. 2.11: A circuit representation of the system of figure 2.10.
The ideal wires are needed to permit the four basic elements to be parted from each-other,
but obviously the potentials at the ends of ideal wires must be the same, e.g. Va=Va’, etc.
If the consecutive branch voltages are summed to each other it is:
(2.3)
Uba+Ucb+Udc+Uad=(Vb-Va )+ (Vc-Vb )+ (Vd-Vc )+ (Va-Vd )=0
where the equality of zero is due to the fact that all the potential values appear twice, with
opposite signs.
Eq (2.3) says that the consecutive voltages across terminals (i.e. potential differences) in the
loop constituted by the circuit has a sum that it is equal to zero. This is very similar to the well
known law of physics stating that the work performed by a conservative field in any closed loop
is zero: the considered sum of the voltages is indeed the work, per unit charge, performed by the
electric field Ec present in the conductor.
Eq. (2.3) is interesting but its usefulness would be much greater if it can be expanded to
larger families of circuital systems and the related circuits.
A question that arises immediately is: does this result depend on the fact that the electric
source is a battery, i.e. a constant voltage, or is it valid also in case of time-varying sources, such
as the sinusoidal voltage source considered in fig. 2.1? Indeed result (2.3) comes from the
physical model of the system in figure 2.10 that has made us consider the presence of the electric
field Ec inside the conductor only created by the charges appearing on the surface of the circuit
conductors. It was implicitly assumed that no other fields were induced by possible presence of
other electromagnetic phenomena outside the system shown in figure 2.10, for instance time
varying magnetic fields produced by other systems not shown, that, by Faraday’s law would have
induced additional contribution to the conductor inner electric field.
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
2.17
Analysis of this topic requires knowledge that will be developed in chapter 4. Here the result
that will be obtained there is just anticipated.
The result (2.3) will be valid also in cases in which the voltage source varies with time, given
that the self-induction phenomena have negligible effects, or: the consecutive voltages across
terminals (i.e. potential differences) in a one-loop circuital system have a sum that it is equal to
zero whenever induction and self-induction phenomena in the loop have negligible effects.
This is congruent with the initial rule that has been stated for creating circuits from circuital
systems, i.e. that any action on the space around the wires on what happens in the wires
themselves should be negligible: in this chase the induced electric field by the presence of timevarying magnetic field around the circuit, and in the space occupied by the circuit loop.
A second very important question that arises from the analysis made on the simple circuit
reported in figure 2.10 is: Can the result (2.3) be extended also to more complex circuits, that
contain more than a single loop, such as the system reported in figure 2.1 ?
Consider again the system of figure 2.1, reported again in fig. 2.12, slightly modified. At a
given instant the situation inside the conductors can be imagined as reported in the figure: the
field inside the generator EG plays the same role of Eb in the previous example, and at any given
point of the conductors field Ec is present. Note that the value of Ec varies with the point of the
conductor considered (it is an electric field having a value at any point of space) and both EG and
Ec vary with time. As in the previous example Ec is due to the surface charges present in the
conductors including the surfaces of the conductive parts fa and fb that interface G with the wires.
Ec
b
EG
a
fb
Ec
L1
fa
Ec
c
L2
d
Ec
e
f
L3
Fig. 2.12: A different view of the circuital system of figure 2.1.
Ec has therefore the nature of electrostatic field, even though it varies with time, and
therefore will admit a potential v(p), function of the point p of the circuit considered, variable
with time and therefore reported using a lowercase symbol).
Since v the function of the point, the potential differences in any closed loop that might be
considered sum up to zero in a way that is totally similar to the process that brought to eq. (2.3).
For instance, for the loops L1, L2, L3 that might be considered in the system of figure 2.12, it will
be:
L1: uba+ucb+udc+uad=(vb-va )+ (vc-vb )+ (vd-vc )+ (va-vd )=0
L2: ucd+uec+ufe+udf=(vc-vd )+ (ve-vc )+ (vf-ve )+ (vd-vf )=0
L3: uba+ueb+ufe+uaf=(vb-va )+ (ve-vb )+ (vf-ve )+ (va-vf )=0
(2.4)
where the equality of zero is due to the fact that all the potential values appear twice, with
opposite signs.
The voltages (or potential differences) can be reported in the circuit corresponding to the
circuital system, as shown for the generator, the lamps and the lower conductor in fig. 2.13.
2.18
Chapter 2: Circuits as modelling tools
Whenever a voltage is reported in a circuit the reference polarities must be reported: a plus sign
or a couple plus-minus signs. They indicate the references towards which to measure the
corresponding voltages: the number indicating a voltage is equal to the difference of potentials
between the point of circuit marked “+” and the one marked “-”.
For instance, in the figure it is uG(t)=vb(t)-va(t). Since the negative sign must always be at the
opposite side of the positive one, it can be omitted, as done in the figure for the conductor
voltages uuc and ulc. A time varying voltage u(t) will typically assume, as time passes, positive
and negative values; for what said, the part of circuit marked with “+” will actually have a higher
potential the one marked with “-” when u(t) has a positive value.
For its importance the voltage sign convention is reported in the following box:
Convention: Voltage sign
The number indicating a circuit voltage u is equal to the difference of potentials existing at
the circuit wire carrying the mark “+” and the one carrying the mark “-”.
+
uuc
uG G
-
a
+
ulc
lower
conductor
+
lamp 2
b
lamp 1
+
upper
conductor
uL1=uL2
L3
Fig. 2.13. The circuit of figure 2.2, with indication of some voltages,
along with the corresponding polarities.
2.4.2 Kirchhoff’s Voltage Law formulations
It should be now clear that the results obtained in analysing figures 2.10 and 2.12 can be
extended to circuital systems and related circuits of any complexity and number of loops.
These results are summarised by equations (2.3) and (2.4), telling that in all circuits a
function V of terminals exists, called potential, such as the voltages across terminals are
expressed as the difference of the corresponding potential values.
The very existence of V implies that the sum of consecutive branch voltages around any loop
is identically equal to zero.
In this section these results are expressed in a more formal and practical way. Before doing
this, the concepts of voltage rises and voltage drops must first be introduced. Consider again
figure 2.12. Before writing eqs (2.4) possible loops in the circuit were indicated, and reference
directions in which to follow these loops were assumed (indicated by the arrows reported on the
loop-characterising lines). This reference direction is arbitrary. Then, the loop is fully followed,
starting from a point of it and returning to the same point, moving according the reference
direction assumed for that loop. When the loop is followed, branches are traversed. It may happen
that the negatively marked terminal of a traversed branch is met before the positively marked one,
or after. In the first case the branch voltage will be considered to be a voltage rise, in the second
one a voltage drop.
For instance, considering loop L3, in 2.13, uG and ulc are a voltage rises, while uuc and uL2 are
voltage drops. The third of eq. (2.4) could be written, using the symbols reported in fig. 2.13, as
follows:
L3: uG+-uuc+-uL2+ulc=0
or, equivalently:
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
2.19
L3: uG+ulc =uuc+uL2
Now we can express the rationale behind loop equations as those in eq. (2.4) in a general and
formal way. This constitutes a fundamental law of the circuits, that is called Kirchhoff’ Voltage
Law (KVL). Similarly to KCL, can be expressed in three possible forms, all equivalent:
Law: Kirchhoff’s Voltage Law (KCL).
In any circuit, if any loop is traversed in an arbitrary direction (either clockwise or
counterclockwise), it happens that
form 1: the algebraic sum of voltages across terminals, all considered as voltage rises, is
identically null;
form 2: the algebraic sum of all voltages across terminals, all considered as voltage drops,
is identically null;
form 3: the sum of all voltage rises across terminals identically equals that of all voltage
drops across terminals.
KVL applies to any circuit, that is a mathematical-graphical model of a physical system. Any circuit models
the behaviour of a physical system within some degree of approximation.
In chapter 4 the circuit concept will be expanded, and the possibility of having sectioned
circuits, i.e. circuits containing more sections will be introduced. It will be seen there that KVL
applies individually to circuit sections, but not to the whole circuit: i.e. each section will have a
potential function of its own, and the difference between potential of terminals belonging to
different sections is meaningless.
2.5 Solution of a circuit
Solving a circuit normally means to find a value for all node potentials and wire currents. For
branch-based circuits this means just to find the values of all branch voltages and currents.
Some of these equations will be Kirchhoff’s equations (KCL and KVL equations), others
will give the description of the branches inner behaviour and will be referred to as constitutive
equations.
In this section information on how to write linearly independent Kirchhoff’s equations and
constitutive equations will be provided, as well some discussion on whether the equations written
are in number sufficient to determine all currents and voltages.
2.5.1 Determination of linearly independent Kirchhoff’s equations (loop-cuts
method)
Let us consider a branch-based circuit having b branches and n nodes. to fix ideas, consider
the circuit shown in fig. 2.14 with names for nodes and some possible loops. Note that the
displayed loops are not all the possible ones for the circuit (for instance a loop traversing
sequentially B, C, F, G, D could also be considered).
2.20
Chapter 2: Circuits as modelling tools
L4
iA
+
uA A
-
+
N2
+
uB B
L1 iB
uC
N3
C
iC
N1 +
iFG
+
F uF
+
+
E uE
L2
iE
iD
D
uD
L3
G uG
-
- N4
Fig. 2.14. A circuit with nodes and loops evidenced.
Consider the KCL for all the circuit nodes (written using form 1):
N1:  iA  iB  iD  0
N2:
N3:
N4:
iA  iB  iC  0
(2.5)
i D  i E  i FG  0
i D  i E  i FG  0
It can immediately verified that these equations are not linearly independent, since it is:
N4=N1+N2-N3.
It is also easy to verify that (using the linear algebra usual techniques to solve linear systems,
e.g. computing the determinant of the system 2.5) that only three of the equations in (2.5) are
linearly independent.
The four KVL equations related to the loops shown in fig. 2.14 can also be written, using
again form 1, as follows:
L1: uA uB  0
L2: uB  uC  uE  uD  0
(2.6)
L3: uE uF  0
L4:  u C  u F  u D  u A  0
Again, it is easy to find that these equations are not linearly independent, since it is:
L4=L1+L2+L3
and it is easily verified that only four of the equations (2.6) are linearly independent.
Let us now introduce a technique allowing to write only linearly independent Kirchhoff’s
equations.
The simplest case of circuits without nodes has obviously no KCL equations and a single
KVL equation.
In the other cases the following simple procedure, that in this book will be called method of
loop-cuts, can be used:
 The KCL can be applied to any set of nodes amounting to the total number of nodes n
diminished by one
 The KVL can be determined in the following recursive manner:
a an equation is be written considering an arbitrary circuit loop; once it is written, a cross
can be written (or imagined to be written) to a branch of that circuit
b if more loops are present in the circuit, return to step a, but loops must not contain
branches already crossed
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
2.21
c in case of branches in parallel, if a unique voltage name is considered across the branches,
a cross must be put on all branches in parallel but one, before starting to write loop
equations.
It is apparent that the procedure proposed allows determination of linearly independent equations. If for
instance the KVL is considered, crossing a branch ensures that the equation later chosen does not contain the
voltage across the crossed branch, and therefore are linearly independent with the previous one. Since this
reasoning can be repeated recursively, it can be concluded that the set obtained is constituted by linearly
independent equations. With similar considerations it can also demonstrated that the number of equations that
can this way be obtained is equal to b-n+1.
Rule: Determination of independent Kirchhoff’s equations (loop-cuts method).
1) independent KCL equations can be obtained applying the law to all the circuit nodes
except one (arbitrarily chosen)
2) independent KVL equations can be obtained by crossing, whenever a loop equation is
considered, an arbitrary branch of the loop and choosing the next loop equation in such a
way that no crossed branch is traversed.
As an example, fig. 2.15 shows two possible ways to determine KVL sets, that are linearly
independent. After L1 is determined, cross C1 excludes branch A from the subsequent loops, and
after L2 is determined crossing C2 excludes branches B1 C and D from the subsequent, final loop
L3, that excludes previously crossed or unconnected branches.
C1
C2 +
uC
-
C
+
+
uB
uA A
-
+
+
F uF
E uE
B
L2
-
-
-
L3
L1
D
+
+
uD
+
L1
uA A
-
-
uC
-
N3
C
+
uB
C2
+
+
F uF
E uE
B
-
-
-
C1
L3
L2
D
+
uD
-
Fig. 2.15. Graphical procedure to determine linearly-independent KVL equations.
The corresponding sets of linearly independent equations are:
L1 : uA  uB1  0

Set1: L2 : uB  uC  uE  uD  0
L : u  u  0
F
 3 E
L1 : uA  uB  0

Set 2 : L2 : uA  uC  uE  uD  0
L : u  u  0
 3 E F
The fact that the two sets define the same space of functions is confirmed by the fact that set
2 can be obtained from set 1 by substituting L2 with L1+L2.
2.22
Chapter 2: Circuits as modelling tools
2.5.2 Constitutive equations
Clearly, Kirchhoff’s equations themselves cannot be sufficient to determine the full
behaviour of a circuit, i.e. all the currents and voltages, since information must be given also
about how the circuit elements, (i.e., for branch-based circuits, the circuit branches) operate
internally.
It may be said that by Kirchhoff’s laws only the topology of the circuit is defined; no
information is given about the inner behaviour of the circuit branches.
This behaviour is introduced by means of equations, called constitutive equations; they will
be often referred to with their acronym.
In the next two chapters constitutive equations for the most important components will be
introduced and discussed. Here, it can just be said that a constitutive equation is a relation
between the branch current of voltage5. Examples of constitutive equations are therefore the ones
reported in table 2.I.
The reader might recognise the equations of typical circuit components they have already
met in their previous studies; for constitutive equations in which a current and a voltage appear,
for the coefficients to be positive (i.e. R, L, C) branch voltage and current must be associated
according in such a way that the assumed direction of current enters the branch in the positively
marked terminal. This combination of reference signs will be later referred to as the load sign
convention:
Convention: load and generator sign convention.
References for current and voltage in a branch follow the load sign convention when the
assumed current direction enters the branch from the positively marked terminal.
If the current exits the branch from the positively marked terminal, the references follow
the generator sign convention
Table 2.I. The constitutive equation types used in this book
(in equations containing both current and voltage, load sign convention is used).
ELEMENT
EQUATION
resistive element
ub=Rb ib
voltage source
ub=us
current source
ib=is
self inductive element
ub=L dib/dt
capacitive element
ib=C dub/dt
DESCRIPTION
branch voltage and current proportional (the “+” sign must be used
if the current is assumed entering the branch from the positively
marked terminal)
voltage is equal to us (subscript stands for “source”) regardless of
any other circuit quantity.
current is equal to is (subscript stands for “source”) regardless of
any other circuit quantity
branch voltage proportional to time derivative of branch current
(the “+” sign must be used if the current is considered entering the
branch from the positively marked terminal)
branch current proportional to time derivative of branch voltage
(the “+” sign must be used if the current is assumed entering the
branch from the positively marked terminal)
Not all constitutive equations are possible in all positions in a circuit.
It is unpractical to treat this issue in a general way. As the more significant examples,
however, consider that:
 since two branches in series share the same current they cannot be characterised by
current source constitutive equations since the two currents might differ even slightly and
5
Note that for any circuit it is assumed that the effects on the behaviour of any branch of terminal potentials at its
terminals p1 and p2 are always and only through the potential difference u=vp1-vp2). This is coherent with the
assumption, valid for any potential, that single potential values do not have a meaning of their own, only potential
differences have measurable meaning.
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
2.23
this would contradict the topologic constraint imposing perfect equivalence of the two
currents.
 since two branches in parallel share the same voltage they cannot be characterised by
voltage source constitutive equations since the two voltages might differ even slightly and
this would contradict the topologic constraint imposing perfect equivalence of the two
voltages.
In case some quantity is constant over time, it is indicated using uppercase symbols. For
instance a constant voltage source can be indicated to be equal to Us.
The list of constitutive equations in table 2.I does not consider branches whose current or
voltage is a function of the current or voltage present in another branch. Different kinds of them
exist, for instance current-controlled voltage sources or voltage-controlled current sources, that
have theoretical and practical significance.
However their usage is outside the scope of this book. With one important exception: the
ideal transformer component. Consideration of this component requires an expansion of the very
definition of circuit given in this chapter, and therefore it was not included in 2.I. This
component, along with these new, expanded circuits, i.e. the magnetically coupled circuits will be
discussed in chapter 4.
2.5.3 Number of variables and equations
It was stated that solving a circuit means to find a value for all node potentials and wire
currents, that for branch-based circuits means just to find the values of all branch voltages and
currents.
A branch-based circuit containing b branches has, in general, b voltages and b currents as
independent variables, for a total of 2b variables. Solving this kind of circuits therefore requires
to find a set of 2b independent equations creating mathematical connection among all the branch
voltages and currents.
It could be demonstrated that the problem of finding solution of a circuit is well posed since
b Kirchhoff’s equations and b constitutive equations can be written.
However, such a large number of equations is typically written only by algorithms for
computerised solution of the circuits; for manual solution this number is typically dramatically
reduced by visual analysis of the circuit.
Take for instance the circuit shown in 2.16 a). It contains 5 branches, and therefore in general
a system of 10 equations in 10 unknowns can be written.
If the circuit has to be manually solved, however, the person in charge of solving it will
immediately understand that branches A, B and C share the same voltage, and therefore a unique
variable can be defined for it, and that branches D and E share the same current, and therefore a
unique variable can be defined for it, giving rise to the circuit shown in fig. 2.16 b) whose
variables (currents and voltages) are now seven instead of ten.
Obviously enough, the reduction of the system variables implies a corresponding reduction
of Kirchhoff’s equations, so that the number of equations and variables will remain balanced, in
this case 7x7.
Further simplifications can be done on the circuit, and will be treated directly in more
concrete cases in the next two chapters. In a case like the system of figure 2.16 it would be easy
to perform simplifications so that the system is reduced to a system of four equations in four
variables.
2.24
Chapter 2: Circuits as modelling tools
+
+
uA A
iD D u D
iA
iC C u C
iB
-
-
u ABC A
-
+
iE
iA
+
+
+
uB B
-
+
D uD
+
u ABC B
CiC
iB
iDE
+
E uE
E uE
-
a)
b)
Fig. 2.16. An example showing circuit simplification and reduction of variables.
2.5.4 Example
Consider the circuit shown in fig. 2.17a. In it, for some of the currents the numerical values
are reported: they are known; the others, namely i1, i2, i3 are unknowns. The circuit contains four
nodes, that lead to 4-1=3 independent LCL equations, e.g.:
N1 (form 1): i3+2+3=0 (that implies i3=-5A)
N3 (form 1): i1+1+1=0 (that implies i1=-2A)
N2 (form 2): 3+i2+i1=0 (that implies i2=- i1-3=-1 A)
So, it is verified that the n-1 equations (n=4 being the number of circuit nodes) were linearly
independent and allowed to determine the three unknown currents; moreover the assumed
directions for the currents, were useful for writing the equations, but no way indicated the actual
currents. In the example of fig. 2.17, the actual flow directions are all opposite to the assumed
ones.
3A
N1
i1
N2
C
N3
E
i2
B
A
F
2A
a)
G
1A
H
I
i3 N 0 1 A
2V +
C
+
1V A
+
-
L1
-
L2
5V D
-
B
H
+
+
L3
3V +
E
+
+
F
-
G 4V
L4
-
I
b)
+
Fig. 2.17. A basic circuit example to show the usage of KCL and KVL.
In figure 2.17b, the four voltages uB, uF, uH, uI are unknowns. Using somewhat wise choice
of loop equations, four linearly independent of them can be determined, e.g.:
L2 (form 1): uB +2-5=0 (that implies uB=3 V; the cut c2 is made)
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
2.25
L1 (form 1): -uH+1- uB =0 (that implies uH= 1- uB =-2 V; the cut c1 is made)
L3 (form 2): -uF+3 +5=0 (that implies uF=8 V; the cut c3 is made)
L4 (form 3): uI+4= uF (that implies uI= uF -4= 4 V)
More in Depth: Number of Kirchhoff’s equations
It could be demonstrated that for any circuit the number of Kirchhoff’s equations are equal to the
number of branches, computing branches that share the same current only once.
In the example of figure 2.17 A and H, and G and I share the same current, therefore the total
independent equations should be equal to the total number of branches reduced by two, i.e. seven.
In fact, we wrote three KCL equations and four KVL ones.
In the example some currents and voltages were considered to be known, and exactly seven
remained unknown.
In more frequent cases, instead of some currents or voltages, the constitutive equations of seven
equivalent branches would have been known, considering a unique equivalent branch for the couple
A-H and G-I, thus leading again to a balanced system of equations.
An equivalent branch is easily found, summing up the constitutive equations expressed in terms of
voltages a function of currents. If, for instance all these four branches are resistive components, the
four equations would be
UA=RAIA UB=RBIB UG=RGIIGI UI=RII
that allow the determination of the following constitutive equations of the equivalent branches:
UAB=(RA+RB)IAB UGH=(RG+RH)IGH.
2.6 The substitution principle
Consider a branch-based circuit.
Without loss of generality attention can be given to the one shown in fig. 2.16, reported
again, slightly simplified, in fig. 2.18.
Any given closed curve such as the curve C shown in figure that selects a part of the circuit
that is connected with the remaining part by only two wires has the structure of a 2-terminal
element, i.e., a branch.
Is it possible to consider it an actual branch, so that a single box Eq can be substituted to the
whole circuit section inside C, without changing the solution of the remaining part of the circuit?
iA
A
N
+
uB B
iB
C
uC + i
C
+
u
E
F
uC + i
C
A
+
uB B
-
+
u
Eq
D
Fig. 2.18. A sample circuit for showing the so-called substitution principle.
Consider the full solution of the left part of the circuit. It will allow to determine, in
particular, voltage u and current i.
2.26
Chapter 2: Circuits as modelling tools
It is apparent that KCL equations can be written of the part of the circuit outside C, and of
the part inside it using the value of i instead of the missing part. For instance equation of node N
will be in both cases iA+iB+i=0
It is also apparent that KVL equations can be written of the part of the circuit outside C, and
of the part inside it using the value of u instead of the missing part. For instance the KVL
equation of the loop involving B, C, u will be in both cases uB+uC-u=0
Is should be easy to convince that this conclusion is valid in general for any circuit.
It can be concluded that in any circuit, a circuit portion having two terminals can be always
substituted by an equivalent branch, that is a branch having as constitutive equation the same
relation between its voltage and current as in the subcircuit it substitutes.
This result is often called the “substitution principle” of circuits.
2.7 Kirchhoff’s in comparison with electromagnetism laws
The reader should already know the electromagnetism laws are basically four (also called the
four Maxwell’s equations):
 Gauss’s law
 Ampère-Maxwell’s law
 Faraday’s law
 Solenoidality of magnetic field.
Those who wish to recall the consents of these laws are suggested to revert to Appendix A.
Kirchhoff’s laws are not included in the list.
How Kirchhoff’s equations compare to electromagnetism? This is a question that very few
people are able to answer immediately. But the reader of this book should be able to answer
straightforward, since they will now know that:
Kirchhoff’s laws apply to circuits, i.e. to mathematical models of physical systems, while the
above-recalled electromagnetism laws (and therefore Maxwell’s equations) apply to actual
physical systems.
Therefore, in this view, the applicability of Kirchhoff’s laws to circuits is guaranteed by
definition, while their applicability to physical systems is undefined.
I.e., the question whether Kirchhoff’s equations are valid for a given circuit or a circuital
system is not well posed: the actual problem that is to be posed is whether and when a given
system can be modelled using a given circuit.
It was seen in this chapter that in the so called DC circuits, where all quantities are constant
with time, circuits can be used as models of physical systems whenever the space between wires
can be considered perfectly insulating, and this is very often the case.
On the other hand, in the so called AC circuits, where all quantities are vary with time,
circuits can be used as models of physical systems, whenever conductive and displacement
currents between wires have negligible effects, and voltages induced in the circuit loops, by effect
of mutual induction or self induction, can be neglected.
It will be seen in chapter four that this is not a big limitation and allows circuits to correctly
model a large quantity of systems, but not all. For instance in long transmission lines, in which
conductors remain parallel to each other for kilometres, self inductance cannot be neglected. This
special case can still be solved with some special extension of the circuit concept, as will be
discussed in chapter 4 and appendix B.
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
2.27
2.8 Power in circuits
It has been said several times that physical systems that are modelled though circuits, will
have to be independent from other systems, in the sense that no mutual induction with external
systems must occur.
In this case, any physical system modelled through circuit will be energetically autonomous,
in the sense that the energy generated by some parts of the considered systems will be absorbed
to others.
As usual, consider again the basic example proposed throughout this chapter, i.e. the system
reported in figure 2.1. It is apparent that the generator will introduce in the system some power,
that is drawn from an external source (e-g. a rotating shaft), and that power will be equal to the
power dissipated in wires and lamps, i.e. converted into heat and luminous energy.
This power neutrality of system is a very important characteristic, and will have to be
retained in any well made model of them. It will be soon demonstrate that this power neutrality
exists in circuit, i.e. the energy conservation principle applies to them.
Before doing that, let us first to consider how to mathematically evaluate the power flowing
in a section of a circuital system.
Consider the representation of the system of figure 2.1 already presented in figure 2.12, and
reproduced, with some modifications in figure 2.19.
i
B
b
EG
a
fb
+
Ec
fa
Ec
u BA
Ec
c
d
A
Ec
e
f
p
Fig. 2.19. A physical circuit used to introduce transferred power p(t).
The work that the electric field inside the conductor makes onto the conductor charges, per
unit charge is the integral of Ec  dl by an arbitrary path connecting A to B.
Note that if only Ec is to be considered in the following equation, the path should be chosen in the right part
of the circuit, e.g. B-c-d-A or B-r-f-A.
dL
B
  A E c  dl  v B  v A  u BA
dq
where V is the potential of electric field Ec.
The work per unit of time, i.e. the power that traverses the circuit and goes from the left part
of the couple of points AB to the right part of them is therefore:
P
dE d L dq

 u BA i
dt
dq dt
(3.7)
Eq. (3.7) says that the power flowing through two wires in a circuital system is equal to the
product voltage times current. Attention is to be given to the signs, since the voltage uba=vb-va is
the opposite of uab: the power is measured by p=ui is towards the right part of the circuit, when i
2.28
Chapter 2: Circuits as modelling tools
represent a current that enters in the right part of the circuit though the positively marked terminal
that gives the reference polarity for u.
When a circuit models a circuital system, it shares with the point of interconnection of
elements voltages and currents with the system to be modelled.
Let us now demonstrate that circuits are power-neutral, i.e. the power delivered by branches
perfectly equals the power absorbed by the others, i.e. the conservation principle applies.
Since this occurs to any circuit topology, with components of any type, this must be a
consequence of Kirchhoff’s equations.
Indeed it is exactly like this. The demonstration that we are providing is so called Tellegen’s
theorem.
Consider a generic circuit containing n nodes, named by consecutive natural numbers, taking
as reference, just to fix ideas, the five-node circuit reported in fig. 2.20. Without loss of generality
let the circuit have only a single branch (possibly equivalent of several branches6) between each
couple of nodes.
vj 2
iij
vi
3
1
0
4
Fig. 2.20. A generic circuit used for demonstration of Tellegen’s theorem.
Consider the current between node i and j to have the assumed direction indicated in figure,
i.e. from i to j.
Although we are talking of the abstract circuit, that is not explicitly linked to the actual behaviour of any
physical circuit, it is apparent that this name was chosen in such a way that, when abstract circuits are used as
models of physical circuits, the quantities pik will be actual powers.
Tellegen’s theorem states that:
if
 {v0, v1, ...vn} is a set of node voltages satisfying KVL for the considered circuit
 {i01, i02, ...i0n, ... i12, i13, ...i1n, ... i1n-1,n, i1n} is a set of currents potentially flowing in the circuit
branches (from the first node to the second one) satisfying KCL for the considered circuit
then
 u ij iij  0
i  0, n; j  0, n
(2.8)
i j
Tellegen’s theorem can be very easily demonstrated writing the branch voltage uij as the
difference of potential vi and vj (this can be done because the KVL applies) as follows.
6
Equivalent branches will be discussed in the next chapters. This sentence will thus be more clear at a second
reading of this chapter.
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering

pij
i  0, n ; j  0, n
i j


uij iij
i  0, n ; j  0, n
i j


(vi  v j )iij
i  0, n ; j  0, n
i j

2.29


vi iij
i  0, n; j  0, n
i j

v j iij
i  0, n; j  0, n
i j

  vi iij    v j iij   vi  iij   v j  iij  0
i  0, n j  0, n
j i
j  0, n i  0, n
i j
i  0, n
j  0, n
j i
j  0, n
i  0, n
i j
In the latest passage the KCL been used, respectively in the form 1 and 2:
 node j ,  iij  0 ,  node i,  iij  0 .
i  0, n
j  0, n
There is a special application case of Tellegen’s theorem, that occurs when uij and iij not only
are compatible with Kirchhoff’s laws but also are present simultaneously in a circuit. In this case,
for hat has been said at the beginning of this section, the quantity
pij=uijiij=(vi-vj)iij
(2.9)
is the power absorbed by the branch existing between note i and j.
Therefore eq. (2.8) can be written as:
pij  0

i  0,n;
(2.10)
j 0,n
Equation (2.8), that is an immediate consequence of Tellegen’s theorem, states exactly what
anticipated:
Law: energy conservation in circuits
In any circuit, the sum of energy absorbed by all branches is identically null.
This is equivalent to saying that the power delivered by the branches that act as generators,
identically equals the power absorbed by those that act as loads.
2.8.1 Example
2V +
C
-
+
+
1V A
-
-2 A
+
3V
B
2A
-
3A
H +
-2 V
-
3V +
E
-1 A
5V D
-
-
+
8V +
F
G 4V
-
1A
-5 A
1A
-
I
4V
+
Fig. 2.21: Currents and voltages computed in the example sect. 2.5.4.
As an example consider again the circuit used in fig. 2.17, reported again in fig. 2.21,
containing the current reference signs and voltage polarities used in the same figure, and the
corresponding numerical values of voltages and currents as already computed in sect. 2.5.4. For
each branch, the product ui will be a power produced or absorbed by the branch, depending on
the combination of references in the circuit: they are absorbed if the reference sign for current
enters the positively marked terminal.
Voltages, currents (measured as entering into the positively marked terminals) and absorbed
powers are reported in table 2.II, that shows that the power conservation applies to the circuit: the
algebraic sum of the powers absorbed by all branches is equal to zero.
2.30
Chapter 2: Circuits as modelling tools
Table 2.II. Power balance of the circuit reported in fig. 2.21.
pabs /W
u/V
i/A
5
1
-5
A
-6
3
-2
B
6
2
3
C
-5
5
-1
D
6
3
-2
E
-8
8
1
F
-4
4
-5
G
10
-2
1
H
-4
4
1
I
TOTAL
0
2.9 Historical notes
2.9.1 Kirchhoff’s short biography
Gustav Robert Kirchhoff7 (1824-87), was a German physicist and inventor of the two laws
that carry his name.
Teacher of physics at the University of Berlin, then of mathematical physics at the University
of Berlin.
He worked mainly in the fields of spectroscopy, electric circuit theory and thermodynamics.
Along with Bunsen, in 1860 he found the luminous spectra signature of chemical elements.
Thus he founded the spectroscopy analysis and made it possible to discover new chemical
elements.
In circuit analysis he enunciated the two circuit laws that carry his name: Kirchhoff’s Current
Law and Kirchhoff’s Voltage Law, while updating the results previously obtained by Ohm
(whose short biography is in chapter 3) and eliminating their contrast with the already generally
accepted electromagnetics theory.
In thermodynamics he studied the radiation emitted by the so-called black body.
2.9.2 Tellegen’s short biography
Bernard D. H. Tellegen (Winschoten, the Netherlands, 1900 - Eindhoven, the Netherlands,
1990) was an electrical engineer and inventor of the penthode and the gyrator. In circuit theory he
has produced the very important theorem: Tellegen's theorem.
He obtained his master’s degree in electrical engineering in 1923, and then joined the Philips
Research Laboratories in Eindhoven.
In the period 1946-1966 Tellegen was professor of circuit theory at the University of Delft.
He invented the penthode vacuum tube (adding a fifth electrode to the tetrode) in 1926, and
the gyrator around 1948. Since its discovery, Tellegen’s theorem has gained increasing
consideration in circuit theory, and now is one of the main pillars of this theory.
2.10 Reference list
[1] C. A. Desoer, E. S. Kuh; “Basic Circuit Theory”, Mc Graw hill, 1984, (1st edition: 1969)
[2] M. E. Van Valkenburg; “Network Analysis”, Prentice Hall, 2007 (1st edition: 1974).
[3] P. Feldmann, R. A. Roher: “Proof of the Number of Independent Kirchhoff Equations in an Electrical
Circuit. IEEE Transactions on circuits and Systems, Vol. 38, N. 7, July 1991.
7
Pronounce (from Oxford English Dictionary): /’kɪɘtʃɒf/ or /’kɪrxhɔf/
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
2.31
[4] The International Electrotechnical Vocabulary (IEV), accessible at the Electropedia Internet site,
http://www.electropedia.org.