Chapter
3
Techniques for solving DC circuits
Table of Contents 3.1 Introduction ........................................................................................................................... 3 3.2 Modelling circuital systems with constant quantities as circuits .......................................... 3 3.2.1 The basic rule ................................................................................................................................ 3 3.2.2 Resistors: Ohm’s law ..................................................................................................................... 6 3.2.3 Ideal and “real” voltage and current sources ................................................................................ 8 3.3 Solving techniques ................................................................................................................. 9 3.3.1 Combined Kirchhoff’s‐constitutive equations basic usage .......................................................... 10 3.3.2 Nodal analysis ............................................................................................................................. 12 3.3.3 Mesh Analysis ............................................................................................................................. 12 3.3.4 Series, parallel and star/delta conversion of resistors ................................................................. 13 3.3.5 Voltage and current division ....................................................................................................... 16 3.3.6 Linearity and Superimposition .................................................................................................... 18 3.3.7 Thévenin’s theorem .................................................................................................................... 22 3.4 Power and energy, and Joule’s law ..................................................................................... 25 3.5 More examples .................................................................................................................... 27 3.6 Resistive circuits operating with variable quantities ........................................................... 33 3.7 Historical notes .................................................................................................................... 33 3.7.1 Ohm’s short biography ............................................................................................................... 33 3.7.2 Thévenin’s short biography ......................................................................................................... 33 3.7.3 Joule’s short biography ............................................................................................................... 34 3.8 Reference list ....................................................................................................................... 34 3.9 Proposed exercises .............................................................................................................. 35 For the teacher
This chapter, in comparison with similar books’ approaches, has the peculiarity that
distinction between Kirchhoff’s laws and constitutive equations, clearly stated in chapter 2,
is maintained for a while, to keep the students in touch with the fact that Kirchhoff’s
equations are valid to all circuits, not only DC ones.
But very soon the constitutive equations typical of DC circuits are shown integrated and
combined within Kirchhoff’s.
Because of the expected usage and audience of this book, a selection of the most effective
solving techniques, among all that are available, is to be made.
Here basic usage of combined Kirchhoff’s-constitutive equations, and nodal analysis are
selected: the first because it naturally completes the knowledge gained in chapter two, the
second because it allows a dramatic reduction in the number of the equations that need to
be simultaneously solved.
The mesh analysis, that for a circuit expert is also very important, is just outlined here as a
technique for which the existence is to be known. It is not intended to be used for problem
solving.
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
3.3
3.1 Introduction
In the previous chapter “circuits” were introduced as a mathematical-graphical tool useful to
analyse physical systems constituted by electric elements connected to each other by means of
wires.
It was also shown that, under given assumptions, circuits can effectively model physical
systems that work with either constant and variable quantities.
In this chapter the analysis of circuits working with all constant quantities (voltages, currents,
powers, etc.) is developed, so that the user is rendered able to solve these kinds of circuits.
Circuits operating with all constant quantities are by tradition called Direct Current circuits,
or DC circuits.
3.2 Modelling circuital systems with constant quantities as circuits
In the discussion presented in the previous chapter of circuital systems and circuits, the
introduction of KCL and KVL was made taking as reference examples in which all quantities in
the system were constant: currents, voltages, fields, etc. According to the tradition we call these
systems Direct-Current systems or DC systems, even though the name is evidently incorrect since
the world “direct” is not as appropriate as “constant” and making reference only to currents
instead off all quantities is limiting.
Since KCL and KVL were discussed using DC systems, it the applicability of the circuit
concept to circuital systems operating with constant quantities is rather natural.
However it must always be remembered that circuits are models of physical systems and the
activity of modelling always implies the issuing of some hypotheses, that are satisfied in actual
systems only up to a certain degree of precision.
This is discussed in the next section.
3.2.1 The basic rule
To evaluate what hypotheses must be issued to allow a circuit system to be modelled as a
circuit, consider again the example discussed in the previous chapter, and in particular the version
with constant quantities reported in fig. 2.12 and in fig. 3.1-a) as well.
3.4
Chapter 3: Techniques for solving DC circuits
N1
Ec
fb
b
EG
Ec
fa
a
c1
cu
B
L1
c3
cl
c2
c
e
d
f
a)
L2
c4
N2
three-termi na l
element
Upper co nd uctors (cu+c1+c2)
c
B
L1
e
a
d
two-termi nal
element
(bra nch)
L2
b
f
Lower co nd ucto rs (cl+c3+c4)
N1
B
L1
a
L1
cl
f
c4
c3
N1
c2
e
d
cl
B
c
L2
c1
cu
c)
L2
cu
b
b)
N2
d)
N2
Fig. 3.1. A sample circuital system (a) and some possible models of it (circuits: b to d).
The presence of the electric field Ec in the conductor, that is due only to the surface charges
on the conductors, allows KVL to be satisfied, as already discussed in chapter 2. Moreover, in
case the charge flows through wires or wires and elements can be neglected, KCL also applies.
Note that this is equivalent to saying that the air surrounding the wires has infinity as resistivity or zero as
conductivity.
Three different models for the same physical system are proposed in fig. 3.1 b to d.
In the model reported in fig. b, the terminals a, b, c, d, e, f are put into evidence, and the
subsystems having these terminals as borders towards other circuit elements are singled-out. Note
that two of the elements have three terminals. The reader might remember that circuits can have
multi-terminal elements, although more frequently just circuits having only two terminal
elements are considered, that are called in this book branch-based circuits.
The model of figure c) is equivalent to the model of figure b), but, putting into evidences the
two points N1 and N2 of the physical system to be modelled, the two nodes N1 and N2 are created,
and the circuit is now a branch-based circuit. Finally, the model of figure d) is a simplified
version of the model of figure c), therefore less approximated to the starting system, but easier to
deal with.
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
3.5
It must be remembered that in circuits KCL is postulated to be exactly valid. In the
transformation of system of fig. 3.1 a) in the circuits of fig 3.1 b to d, therefore any charge flow
through the air surrounding the wires and though the elements boundaries is neglected.
For DC circuits, therefore the rule for modelling a given system as a circuit is very simple
and is the following one:
Rule Modelling of DC systems through circuits
A circuital system composed by components that can be modelled as circuit elements
connected to each other by conducting wires can be modelled by a DC circuit if all charge
flow through the air between the wires and around the elements can be neglected.
Earlier, it was stated that DC circuits are circuits in which all the quantities are constant.
How come that they are constant? It only depends on the constitutive equations.
Consider a circuit whose branches have constitutive equations only of the types reported in
Table 3.I, that are a subsystem of those already reported in table 2.1.
Table 3.I: The constitutive equation types used in this book
(in equations containing both current and voltage, load sign convention is used).
ELEMENT
EQUATION
DESCRIPTION
voltage is equal to Us (subscript stands for “source”) regardless of
voltage source
ub=Us
any other circuit quantity.
current is equal to Is (subscript stands for “source”) regardless of
current source
ib=Is
any other circuit quantity
resistive element
branch voltage and current proportional (*)
ub=Rb ib
(*) It will be soon seen that resistive elements have a positive value of resistance Rb, when ub and ib are measured
assuming that the current enters the positively marked terminal)
In such a circuit some of the branches impose constant voltages at their terminals, other ones
constant currents (voltage and current sources) others proportionality between current and voltage
(resistive elements).
In this case it is intuitive that all voltages and currents of the circuit will be constant, and
might be formally demonstrated.
More in Depth: Equation-only formulation of DC circuit models
The reader will learn in this chapter the voltages and currents could be determined as the solution of a
linear system of equations of the type:
Ax  b
in which
x is the vector of unknowns (voltages across all branches that are not of the voltage source type and
currents through all the branches that are not of the constant current type)
b is the vector of known terms (voltages across all voltage source branches and currents through
constant current branches)
A is a matrix of coefficients that will contain constant terms, function of the resistances of the resistive
elements
Since A and b are both constant, the solution x will be constituted by a vector of constants
Therefore circuits containing elements having only the constitutive equations reported in
table 3.I will be DC circuits.
In the following sections, the three basic types of constitutive equations reported in table 3.I
will be discussed more in detail.
3.6
Chapter 3: Techniques for solving DC circuits
3.2.2 Resistors: Ohm’s law
Consider the load reported in fig. 3.2, and suppose that if the applied voltage U changes, the
current flowing into the load changes in proportion, i.e. whenever the voltage doubles the current
doubles, or in general, multiplying the voltage for a given factor, the current varies by the same
factor.
This implies direct proportionality between voltage and current:
U  kI
(3.1)
This example is far from rare in nature; indeed this law is valid to a high degree of
approximation to any load constituted by some conducting material of any shape, such as wire to
distribute energy along an electric installation (long cylinder of copper), a spiral (such as in
domestic heaters, ovens and bulb lamps).
The first one that formulated a law that resembles to (3.1) was Georg Simon Ohm in 1827
(cf. 3.7.1) and therefore today it is called Ohm’s law. The proportionality factor is a quality of the
conductor: the higher this factor the lower the current at a given voltage, or the higher the voltage
is needed to have a given current flow. Therefore it somehow measure the quality of a conductor
of resisting to charge transfer. It is therefore called “resistance” of the load, and it is consequently
normally given the symbol “R” . Ohm’s law is therefore written commonly as:
U  RI
(3.2)
Eq. (3.2) does not only relate the amplitudes of voltage and current of a given conductor, but
also the signs. U and I can therefore be positive or negative numbers; however, if the load sign
convention is used for the branch for which Ohm’s law applies, the resistance R is always a
positive number (cf. figure 3.2).
The resistance R has the dimensions of a voltage times a current; because of its importance
this quantity has a name of its own in the SI, that is the “ohm” (note the lowercase initial as for all
units of measure of the SI). Therefore a ohm is the resistance of a resistor in which, when
subjected to a voltage of 1 volt, a current of 1 ampere flows.
Definition: The ohm (the unit of measure of resistance)
The unit of measure of resistance is the ohm (symbol: “”).
One ohm is the resistance of a conductor that under a voltage of 1 V absorbs a current of 1
A. In formula: 1=(1V)/(1A)
feeder
feeder
+
U
I
branch
U=kI
to be
-
modelled
U
+
branch
I
to be
U=RI
U=-RI
load
sign convention
for branch
generator
sign convention
for branch
modelled
Fig. 3.2: Proportionality between voltage and current, and Ohm’s law for branch.
A conductor shows a resistance that is function of the characteristics of the material used,
and of the conductor’s geometry. For the most important case of a cylindrical conductor having
the length l, and the cross section S, the following relation applies:
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
R
3.7
l
l

S  S
where  and =1/ constitute a characteristic of the material with which the conductor is made,
and are called resistivity or conductivity of the material respectively.
The resistivity (and therefore also the conductivity), is rather dependent on temperature. Its
dependence is normally considered linear around the value at a base temperature, according to the
law:
   0 [1   (   0 )]
where 0 is the reference temperature,  is a coefficient characteristic of the material used that
can be defined as
 
(  0 ) / 0


 0 
 0
For the best conductors, the numerical values are reported in of tab. 3.II; among them, widely
used in electrical engineering applications are aluminium and copper.
Tab. 3.II: Some resistance parameters (reference temperature: 20 °C)
Material
silver
copper
gold
aluminium
resistivity
(n.m)
16.4
17.5
24
29
temperature coefficient
(°C-1)
0.004
0.004
0.0035
0.004
It is rather obvious that Ohms law can be written in a slightly different form than (3.2):
1
(3.3)
I  U  GU
R
Where G, by definition equal to the reciprocal of R, is called conductance of the element
having R as resistance, and has a unit of measure of its own, called siemens and whose symbol is
“S”:
Definition: The siemens (the unit of measure of conductance)
The unit of measure of resistance is the siemens (symbol: “S”).
One siemens is the conductance of a conductor that under a voltage of 1 V absorbs a
current of 1 A. In formula: 1S=(1A)/(1V)
Example 1
A voltage U=10 V is applied to a copper conductor, with a length l=200 m and a cross
section S=50 mm2.
Calculate the resistance R and the current I at 60°C.
From tab. 3.II, ρ20°C=17.5 nm.
ρ  l 17.5×10-9  200
R20C  20C =
= 0.07 Ω
S
50×10-6
R60C  R20C (1+   ( - 0 )) = 0.07  (1+ 0.004  (60 - 20)) = 0.0812 Ω
I=
V
10
=
= 123.1 A
R 0.0812
3.8
Chapter 3: Techniques for solving DC circuits
3.2.3 Ideal and “real” voltage and current sources
In previous chapter the “battery” was introduced, making reference to the physical intuition
that everyone has of a battery: a source of direct current voltage (and current and power). A reallife battery shows (cf. fig. 3.0) the presence of some voltage, as measured by the voltmeter Vm
displayed in figure, even when it is unconnected from any load. When it is connected to the load,
it begins delivering current, as measured by the ammeter Am displayed in figure, and the higher
the current delivered, the lower the voltage.
Ri
I
+
S
+ E
S closed
U
Am
Vm
+
A
I
N
+
E/Ri
Ri
U
-
Fig. 3.3: From a battery to a E-R idealised feeder.
The voltage-current relationship of a battery is very complex, and depends on several things
such as the type of battery considered, the ambient temperature, how charged the battery is, etc.
However, it is often sufficient to resort to simplified modelling. The simplest way of
modelling this complex behaviour is resorting to a linear constitutive equation, of the type:
(3.4)
U  E  Ri I
It is interesting to note that this simple constitutive equation allows it to be “translated” into
two very basic circuital elements: an ideal, constant voltage source element in series with an ideal
resistor having as resistance Ri: indeed, it is very easy to see that eq. (3.4) is immediately found
applying the KVL to the rightmost upper circuit of figure 3.3.
Another equivalent circuital interpretation of eq. (3.4) is in the rightmost lower circuit of
figure 3.3:
U  R i  I ( R i )  Ri ( E / R i  I )
where I(Ri) indicates the current flowing through Ri.
The characteristics of a real and ideal generators can be seen also in a U-I plane, as shown in
fig. 3.4. Consider the leftmost diagram. It is just a graphical representation of equation (3.4),
taking into account the fact that Ri must be a real positive quantity. However, in case real
batteries are considered, the good practice requires not to load the battery with too high currents;
therefore, normally, the operating point P and the voltage U near the no-load voltage E; therefore
the useful part of the left plot of figure (3.4) will correspond to the one shown in the central
diagram of that figure. For this part, it is more intuitive to use for the generator shown in the right
top part of figure 3.3, since it evidences the ideal generator E, plus the deviation from ideal
behaviour induced by Ri, that causes the actual voltage to be reduced with respect to the ideal
one.
In case the terminals of a real voltage source (i.e. composed by the ideal generator E plus the inner resistance
Ri) are directly connected to each other, the voltage between them becomes zero and the current flowing
E/Ri. This current will typically be very high and must be interrupted fast to avoid source damage and or
excessive generation of heat in the conductors with possible ignition of fire. These very high currents are
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
3.9
normally called short circuit currents; possible interrupting devices are called breaker and fuses. Some
information about them will be supplied in the Part IV of this book.
There are some cases, however (not frequent indeed) in which feeders are used in the vicinity
of point Q, i.e. when the voltage is near zero, In these cases, the useful zone of the leftmost
diagram of figure 3.3 is the one shown in the rightmost diagram of that figure. For this part, it is
more intuitive to use for the generator shown in the right-bottom part of figure 3.0, since it
evidences the ideal current generator E/Ri, plus the deviation from ideal behaviour induced by Ri,
that causes the actual current to be reduced with respect to the ideal one.
If the element Ri were omitted from the circuital representations, the actual characteristics of
the circuital equivalents reported in figure 3.3 would be the dashed ones in the middle and right
diagrams of figure 3.4.
U
P
E
U
E
Ri=tan

Ri=1/tan
I
I


Ri=tan
Q
E/Ri
U
E/Ri
I
Fig. 3.4: Ideal and real generators.
3.3 Solving techniques
In chapter 2 it was stated that solving a circuit means finding 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.
It was also stated that the equations to be written are Kirchhoff’s and constitutive equations,
and these are exactly in the number needed to solve the circuit.
However, having restricted, for DC circuits, the possible constitutive equations to the very
limited number reported in table 3.I makes it possible to make a lot of simplifications to this
general approach, that allow relatively easy manual computation in case of circuits that are not
very large.
While computer programs solve circuits just assembling Kirchhoff’s and constitutive
equations in a “blind” way, electrical engineers (and non-electrical engineers with some
knowledge of electrical engineering basics) use for solution of DC circuits specialised solving
techniques. In the following sections the easiest and more common of them are reported and
discussed. In particular:
 combined Kirchhoff’s-constitutive equations basic usage allow to limit the rank of the
system to be solved to the number of Kirchhoff’ equations themselves, since constitutive
equations are directly substituted into Kirchhoff’s
 nodal analysis allows further reduction of equations, giving rise to a system whose rank is
equal to the number of circuit nodes minus one
 mesh analysis is another method that allows reduction of the number of equations up to
the number of KVL equations
Because of the introductory nature of this book, mesh analysis is just quoted as a possible
technique and is not dealt with in detail, since in the majority of cases gives rise to a number of
equations that is larger than the number obtained with nodal analysis.
3.10
Chapter 3: Techniques for solving DC circuits
3.3.1 Combined Kirchhoff’s-constitutive equations basic usage
Consider as an example the system reported in fig. 3.5. It is a circuit in which instead of
generic branches specialised symbols of the types considered in table 3.I, i.e. constant voltage,
constant current, resistors, are used.
These symbols are commonly used in textbooks. Other symbols are often used, but those
adopted in this book are the ones recommended by international standardisation bodies (see
Chapter 1 for details). Note that the resistor symbol is similar to the generic branch symbol, but it
has a different aspect ratio: 3:1 instead of 2:1.
In the circuit all voltage polarities and current flow references are reported. The choice made
here, is to always consider positive currents when entering the positively-marked terminals. It is
hardly the case of stressing that any choice of voltage polarities and current references would
have been acceptable.
RF
RA
I
+
+
N3
Us
U0
IC
+
I1
RC
N1
IE
IB
+
-
+
A
+
+
RB
L1
N0
RE
L2
ID +
L3
+
Is
U1
-
N2
RD
Fig. 3.5: A sample circuit used for writing solving equations.
If one would have wanted to write a full set o Kirchhoff’s equations and constitutive
equations (CE), the following system would have been produced:
N1 : I C  I1  I E  0
N2 : I E  I D  I0  0
N3 :  I A  IC  I B  0
L1 : U 0  U A  U B  0
L2 : U B  U C  U E  U D  0
c.e. : U 0  U s ; U A  RA I A ; U B  RB I B ; U C  RC IC ; U D  RD I D ; U E  RE I E ; I1  I s
If blindly analysed, it is of 12 equations in 12 unknowns1:, that in this form appears lengthy
to solve. This system, however, can be immediately simplified by substituting the constitutive
equations into Kirchhoff’s , giving rise to the following system, much leaner:
N1 : IC  I s  I E  0
N2 : I E  I D  Is  0
N3 :  I A  IC  I B  0
(3.5)
L1 : U s  RA I A  RB I B  0
L2 : RB I B  RC I C  RE I E  RD I D  0
It is apparent that instead of first writing the full set of Kirchhoff’s equations and of
constitutive equations, and then making the substitution, it is much faster to directly write KVL
1
In principle the unknowns are all branch voltages and currents, i.e they would be 14 since branches are 7; however,
in the writing of the system it was already considered that Us is traversed by the same current of RA, and U1 shares its
current with RF.
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
3.11
equations in a way that voltages are already expressed, whenever resistors are involved, as
function of the corresponding currents:
Rule Combined Kirchhoff’s-constitutive equations
When dealing with DC circuits, combined Kirchhoff’s-constitutive equations are normally
written, in which resistor voltages are reported as products of currents and resistances (with
appropriate sign).
This way the rank of the system to solve is just the number of Kirchhoff’s equations.
The reader might have noticed that in the writing of Kirchhoff’s equations for the system of
figure 3.5 the loop-cuts rule introduced in chapter 2 would have caused an additional loop
equation to be written:
L3 : RE I E  RF I F  U1  0
(3.6)
But this equation, once added to the system (3.5) will raise its rank by one: this additional
equation and the new variable U1, that does not appear anywhere else in the system.
It is therefore wise to leave this equation out of system (3.5), solving it, and, in case U1 is
needed, finding it immediately from eq. (3.6).
This is a general situation whenever current source are present in the circuit. Therefore, in
DC circuits, the cut-loops rule is applied without considering when writing loops branches
containing current sources, to reduce the system rank:
Rule Kirchhoff’s-constitutive equations when current sources are present
When determining combined Kirchhoff’s-constitutive equations the loop-cuts method
described in chapter 2, any branch with current source can be considered already cut.
In case the current source voltage is needed, the loop containing it can be considered and evaluated
after solving the main system.
The system (3.5) is a linear system having the five unknowns IA, IB, IC, ID, IE. Once all of
them are determined by solving it, all the branch voltages are easily determined by trivial
application of the constitutive equations.
That system, as in general happens for any circuit, can also be represented in the standard
form of linear systems:
Ax  b
(3.7)
in which
x is the vector of unknowns
b is the vector of known terms
A is the matrix of coefficients
When circuit equations are considered, vector b contains the voltages across all voltage
source branches and currents through constant current branches, A is a function of the resistances
of the resistive elements, x contains the unknown currents.
For instance (3.5) can be written as follows:
 IC  I E  I s

I D  I E  Is

 I A  I B  IC  0
 R I  R I  U
B B
s
 A A
 RB I B  RC I C  RD I D  RE I E  0
3.12
Chapter 3: Techniques for solving DC circuits
that is:
I A 
0
I 
0
 B

Ax  b with: x   I C  A  1
 

IE 
 RA
I 
0

 F
0
0
1
0
0
1
1 1
 RB 0
0
0
RB  RC RD
1 
 Is 



1 
 I s 
0  b 0 



0 
 U s 
 0 
 RE 


3.3.2 Nodal analysis
In the basic usage of combined Kirchhoff’s and constitutive equations the CE’s are directly
inserted into Kirchhoff’s, giving rise to a system that has as rank the number of Kirchhoff’s
equations themselves.
In reality, this number can be further reduced, often in a dramatic way, using the solution
method called the nodal potentials method. This is somewhat an advanced usage of combined
Kirchhoff’s and CE equations.
The method consists of writing KCL equations in such a way that KVL are automatically
satisfied. The order of the obtained system of equations is therefore just n-1, where n is the
number of circuit nodes. For instance, the order of the circuit uses as example in section 3.3.1 can
be reduced from five (after all simplifications there described are made) to three.
Indeed the KVL was stated in chapter 2 to be equivalent to saying that a potential exists, i.e.,
that it is possible to define a potential function V that has a value at each circuit terminal, and
such that the constitutive equations are function of potential differences only.
From this it follows immediately that if the circuit is analysed in terms of terminal potentials
instead of branch voltages KVL is automatically satisfied.
Consider again the circuit of figure 3.5, in which the potential of N1 is assumed to be V1, of
N2 V2, etc.. Since constitutive equations are function only of branch voltages, i.e. potential
differences, the voltage of a terminal, usually one of those belonging to the circuit bottom line,
can take an arbitrary value, and is given the conventional value of zero for simplicity. It can for
instance assumed V0=0.
The equations of the circuit are now just the following ones:
V V V V
N1: 3 1  1 2  Is  0
RC
RE
N2: V0  V2  V1  V2  Is  0
RD
RE
N3: V3 Us  V3 V0  V3 V1  0
RA
RB
RC
Nodal analysis is therefore extremely powerful and tends to be the preferred method for
manual solution of circuits because of the reduced number of equations. For its importance, its
usage is summarised in the following
Rule: Number of nodal analysis equations
Nodal analysis allows DC circuits to be solved using a system of only n-1 equations, while
n is the number of circuit nodes.
3.3.3 Mesh Analysis
In the previous section the nodal analysis was presented and shown to be able to produce a of
simultaneous equations to be solved in a number as low as n-1, where n is the number of circuit
nodes.
Therefore this is the preferred method for manual solution of circuits.
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
3.13
In addition to this a lot of other techniques can be used, and indeed are used from circuit
experts; however, to avoid confusion in the reader of this book, for which circuit analysis is just a
tool towards electrical engineering knowledge, they will not be reported here.
It is only mentioned here that the so-called mesh analysis exists, which allows to write, for a
given circuit, a number equations to be simultaneously solved equal to the number of KVL
equations, that is typically larger than n-1.
The interested reader can find details of this technique in any of [2-5].
3.3.4 Series, parallel and star/delta conversion of resistors
Consider the left part of fig. 3.6. Branch currents of resistors R1, R2, ...Rn share the same
value, and can therefore be represented by a single variable name, branches sharing the same
current are called branches in series. The voltage across each resistor is proportional to that
current.
By simple mathematical computations it is seen that also the total voltage is proportional to
that current, and therefore the whole set of resistors in series behaves as an equivalent resistor
Reqs:
U sea  U 1  U 2  ...  U n  R1 I  R 2 I  ...  R n I  ( R1  R 2  ...  R n ) I  R eqs I
In any circuit containing branches in series, the whole series can be substituted by a single
branch having as resistance:
R eqs  R1  R 2  ...  R n
In the case, particular simple, of series of resistors having all equal resistance R it obviously
is:
R eqs  nR
Example 2
Calculate the equivalent resistance Reqs of the branch constituted by the series of the
following resistances: R1=20 , R2=10 , R3=5 .
Reqs= R1+R2+R3=20+10+5=35 .
Consider the right part of fig. 3.6. Branch voltages of resistors R1, R2, ...Rn share the same
value, and can therefore be represented by a single variable name, Branches sharing the same
voltage are called branches in parallel. The current entering each resistor is proportional to that
voltage.
By simple mathematical computations it is seen that also the total current is proportional to
that voltage, and therefore the whole set of resistors in parallel behaves as an equivalent resistor
Reqp. the computation is more easily made considering instead of the resistances in parallel the
corresponding conductances (equal to the reciprocal of the resistances):
U sea  U 1  U 2  ...  U n  G1 I  G 2 I  ...  G n I  ( G1  G 2  ...  G n ) I  G eqp I
In any circuit containing branches in parallel, the whole parallel can be substituted by a
single branch having as conductance:
G eqp  G1  G 2  ...  G n
In every-day work resistances are more commonly used than conductances; it is therefore
useful to report, although trivial to draw, the formula of the equivalent resistance of a parallel of
resistors:
3.14
Chapter 3: Techniques for solving DC circuits
1
1
1
1


 ... ;
Reqp R1 R2
Rn
Reqp
 1
1
1
 

 ...
Rn
 R1 R2



1
In the case particular simple of parallel of resistors having all equal resistance R it obviously
is:
R eqp  R / n
Moreover, when only two resistors in parallel are involved, the equivalence formula can be
written also as follows:
RR
Reqp  1 2
R1  R2
Ipeq
+
Rn
Un
I
-
R1
In
I2
I1
I
U
Rn
R2
+
+
+
Rpeq
U
-
-
+
R2
R1
Rseq
U2
+
Useq
Ipeq
-
In
I2
I1
U1
G1
-
G2
Gn
+
+
U
Gpeq
-
U
-
Fig. 3.6: Series (left) and parallel (right) composition of circuit branch.
Example 3
Calculate the equivalent resistance Reqp of the branch constituted by the parallel of the
following resistances: R1=4 , R2=2 , R3=1 .
1
1

 0.571 
1
1
1
1 1 1


 
R1 R2 R3 4 2 1
1
1
1
1
where Reqp 

 1.75  1
and Geqp  
Geqp
R1 R2 R3
Reqp=
Please note that Reqp < min(Ri).
Following the same approach, other equivalents can be found or particular circuit sections.
One important case is constituted by the so-called delta-star conversion (also named -Y or Wye conversion).
In circuital terms a star is a group of three resistors having one terminal in common and the
other free, while a delta is a connection of resistors one after the other, in such a way to form a
closed shape, resembling a triangle, or a Greek uppercase delta (fig. 3.7).
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
T1
T1
1
3.15
1
R1
R13
R2
R3
T2
T3
2
3
3
R1 
R12
R23
2
R12 R13
Rs
R12 
R1 R2
Rp
Rs  R12  R13  R23
T2
T3
 1
1
1 

R p   

R
R
R
2
3
 1
1
Fig. 3.7: Delta and star circuital 3-terminal components, and conversion formulas
The equivalence formulas must be found in such a way that their behaviour, as seen from the
outside terminals (shown in figure 2.7) is indistinguishable.
The conversion formulas are as follows (demonstration is omitted):
RR
R R
R12  1 2
R1  12 13
Rp
Rs
in which :
R s  R12  R13  R 23
 1
1
1 

R p   

R
R
R
2
3
 1
1
Since the numbering of terminals is arbitrary, formulas for the other resistors can be obtained rotating the
indexes: for instance adding one to each index of the first formula one obtains the formula reported below; in
the same scheme an intuitive sketch showing how to “add one to an index” is reported, showing that adding
one to three makes the count restart:
R R
R2  23 21
Rs
2
1
3
The reader can verify that, in case of three equivalent resistances, Rdelta=3Rstar.
Example 4
Considering fig.3.7, calculate R1, R2 and R3, if R12=10 , R23=8  and R31=5 .
Rs= R1+R2+R3=23 .
R R
10  5
 2.174 
R1= 12 31 
RS
23
R R
8 10
R2= 23 12 
 3.478 
RS
23
R R
58
 1.739 
R3= 31 23 
RS
23
These results can be verified recalculating the delta resistances using R1, R2 and R3:
Rp=(R1-1+R2-1+R3-1)-1=0.7561 
RR
R12= 1 2  10 
Rp
3.16
Chapter 3: Techniques for solving DC circuits
R23=
R2 R3
8
Rp
R31=
R3 R1
5
Rp
Example 5
Calculate the equivalent resistance “seen” from terminals A and B:
R5 and R6 are series connected, then R56=R5+R6=10 .
R3 and R56 are connected in parallel, thus R356=R3||R56= 3.333 :
R356 and R4 are series connected: RS=R356+R4=13.33 .
Since R2 and RS are in parallel, RAB=R1+R2||RS=3+(20-1+13.33-1)-1=11 .
3.3.5 Voltage and current division
When resistors are in series or in parallel, special relations occur between voltages and
current that are useful to investigate for their practical implications.
Consider several resistors in series having as resistances R1, R2, ...Rn (left part of fig. 3.6).
The ratio of the voltage applied to any of the individual branches to the total voltage applied
to the series is:
Uk
R I
R
 k  k
U eqs
Reqs I
Reqs
(3.8)
As trivial as it might appear eq. (3.8), for its usefulness has a name of its own: it is the
voltage division rule of resistors in series.
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
3.17
Example 6
Considering the following figure, calculate U1, U2 and U3, if U=100 V, R1=2 , R2=4 
and R3=14 .
+
Rn
I
Un
I
-
+
+
R2
R1
U2
+
Rseq
Useq
-
U1
-
U1= I  R1  U 
R1
2
 100 
 10 V
2  4  14
Reqs
U2= I  R2  U 
R2
4
 100 
 20 V
Reqs
2  4  14
U3= I  R3  U 
R3
14
 100 
 70 V
2  4  14
Reqs
Consider now several branches in parallel having as resistances R1, R2, ...Rn , or,
equivalently, as conductances G1, G2, ...Gn (right part of fig. 3.6).
The ratio of the current flowing into any of the individual branches to the total current
flowing into the parallel equivalent is:
Ik
I eqp

GkU
Gk

G eqpU G eqp
(3.9)
As trivial as it might appear, eq. (3.9), for its usefulness, has a name of its own: it is the
current division rule of resistors (or conductors) in parallel.
Since, as mentioned, resistances are more commonly used than conductances, the version of
(3.9) for the simpler case of just two branches in parallel can also be written in terms of
resistances:
I1
I eqp

Gk
1 / R1
R2
;


1
1
G1  G 2
R
1  R2

R1 R2
I2
R1

I eqp
R1  R2
I.e.: the current flowing into one of two branches in parallel is equal to the total current
entering the parallel multiplied by the resistance of the other branch and divided by the sum of
the resistances of the two branches in parallel.
Example 7
Considering the following figure, calculate I1, I2 and I3, if I=10 A, R1=2 , R2=4  and
R3=3 .
3.18
Chapter 3: Techniques for solving DC circuits
Ipeq
I1
R1
I2
In
R2
U
Rn
+
+
Rpeq
U
-
-
Ipeq
I1
G1
I2
In
G2
Gn
+
+
U
-
Gpeq
U
-
I1= U  G1  I 
G1
21
 10  1 1 1  4.615 A
2 4 3
Geqp
I2= U  G2  I 
G2
41
 10  1 1 1  2.307 A
2 4 3
Geqp
I3= U  G3  I 
G3
31
 10  1 1 1  3.077 A
2 4 3
Geqp
3.3.6 Linearity and Superimposition
It was in sect. 3.3.1. that set of Kirchhoff’s equations and constitutive equations of DC
circuits can be expressed in the standard form of linear systems:
Ax  b
(3.7)
The vector of known quantities b in circuits contains the numerical values of voltage source
voltages and current source currents, while the matrix A contains information on the circuit
topology carried by Kirchhoff’s equations and on the constitutive equations of the non-source
branches, i.e. the resistors.
To solve the system (3.7) any of the existing algorithms to solve linear systems might be
used. This can be visualised as per fig. 3.8.
inp ut: vector b
Algorithm to so lve
Ax=b
outp ut: vecto r x
Fig. 3.8: Circuit solution by means of numerical solution of a linear system.
In case b is constituted by a sum of n terms:
b  b1  b2 ,...bn
it will be:
Ax  b , with
x  x1  x2 ,...xn where Ax1  b1, ... Axn  bn
Therefore the solution of the given circuit can be thought to be the sum of partial solutions
x1, x2, xn obtained by individual application of a partial sets of sources b1, b2, bn:
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
3.19
Result: Superimposition principle
In a linear system, such as DC circuit, in case the input vector is the sum of several partial
input vectors, the output vector is equal to the sum of the partial output vectors.
The terminology of this result is explained in fig. 3.8.
It has been said several times that circuits are a mathematical/graphical tool to study circuital
systems. Their graphical nature makes it easier to perform several actions, one of which is
application of the superimposition circuit.
Instead of working with the matrix description (3.7) of the circuit, it appears much more
natural to work directly in the graphical representation of the circuit.
In circuit language, decomposition of input vector b into several addends is made applying
the sources one at a time, or in groups.
When a source is not applied it is deactivated, i.e. its value is equal to zero; in circuit
language for voltage sources this means that when it is deactivated it must be substituted by a
wire that directly connects its two terminals (so that the voltage across them is zero), for current
sources that it must be substituted by disconnection of any wire between the two terminal ends,
so that the current through them is zero (fig. 3.9).
In circuit terminology this is typically said that when a voltage source is to be deactivated, its
terminals must be short-circuited, while when a current source is deactivated they must be open
circuited. Two terminals connected to each other are called a short circuit, two terminals without
any connections constitute an open circuit. This allows to summarise superimposition in circuits
as follows:
Rule: Superimposition principle in circuits
Linear circuits can be solved applying voltage and current sources one at a time, and
summing partial results. It is possible to use them in groups.
When a voltage or current source is not applied (or it is deactivated) the corresponding
branch is to be short-or open-circuited, respectively
Us +
Is
A DC circuit
of any
complexity
Us not applied
implies
short-circuiting
of its terminals:
A DC circuit
of any
complexity
A DC circuit
of any
complexity
Is not applied
implies
open circuiting
of its terminals:
A DC circuit
of any
complexity
Fig. 3.9- Superimposition principle application to circuits.
This way of using the substitution principle can be effectively illustrated considering again
the circuit of figure 3.5, treated as reported in fig. 3.10. The inputs are Us and Is.
3.20
Chapter 3: Techniques for solving DC circuits
UA
IA
+
RA
+
+
Us +
UB
IB
U’A
I’A
RC
+
U D  U ' D U " D U E  U ' E U " E
+
RE
RB
ID
U A  U ' A U " A U B  U ' B U " B U C  U 'C U "C
UE
Is
I A  I ' A I"A
UD
U”
+
RA
+
U’B
I’B
I’C
U’C
+
U’D
U’E
IE
I C  I ' C  I "C
I E  I ' E  I "E
+
RA
+
+
+
RE
RB
I’D
I”A
RC
I B  I ' B  I "B
I D  I ' D  I "D
IE
RD
+
Us +
IC
UC
U”B
I”B
I”C
U”C
RC
RE
RB
I”D
+
+
U”D
U”E
Is
I”E
RD
RD
Fig. 3.10: An example showing circuital application of superimposition principle.
Figure 3.11 and fig. 3.12 show possible ways to apply of the superimposition principle to a
more complex circuit. Since this circuit has three sources, either can each of them considered
individually, and the result constituted by the three contributions or, they are grouped in a
different way, first the two voltage source, then the current source.
Any other grouping would have been acceptable, so the user can choose what it deems more
useful or practical.
Note that in figures 3.11 and fig. 3.12 the numerical value of the voltage of voltage sources is
indicated using the letter E. This is very common in circuit representation.
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
UA +
UC
IC +
RA
RC
+
E1 +
UB
IA
+
IB
UE
U’A +
RE
RA
E2 +
RB
Is
IE2
ID
3.21
E1 +
I’A
RA
U”B
I”A
+
RC
+
I’B
U”E
U”’A +
RE
RA
E2 +
RB
I’E2
I’D + U’D
I”D
+
I”’C+
I”’A
U”’C
U”’E
RE
Is
RB
I”’E2
I”’B
U”D
+
RC
+
U”’B
I”E2
I”B
RE
RD
U”C
I”C+
U’E
+
RB
RD
U”A +
U’C
RC
+
U’B
UD
+
I’C +
I”’D+
RD
U”’D
RD
U A  U ' A U " A U " ' A U B  U ' B U " B U " ' B U C  U 'C U "C U " 'C
U D  U ' D U " D U " ' D U E  U ' E U " E U " ' E
I A  I ' A I"A I"' A
I B  I ' B  I "B  I " ' B
I D  I ' D  I "D  I " ' D
I E  I ' E  I "E  I " ' E
I C  I 'C  I "C  I " 'C
Fig. 3.11: A first superimposition example involving three known quantities (sources).
UA +
RA
E1 +
+
UB
IA
IC +
UC
+ UE
RC
RE
E2 +
RB
IB
U A  U ' A U " A U B  U ' B U " B U C  U 'C U "C
U D  U ' D U " D U E  U ' E U " E
Is
I A  I ' A I"A
IE2
ID
+
I B  I ' B  I "B
I D  I ' D  I "D
UD
I C  I ' C  I "C
I E  I ' E  I "E
RD
U’A +
RA
E1 +
I’A
I’C +
+
U’B
I’B
U’C
+ U’E
RC
RE
U”A +
RA
E2 +
RB
I’D + U’D
RD
I”A
+ U”E
RC
RE
+
U”B
I’E2
I”C + U”C
Is
RB
I”E2
I”B
I”D +
U”’D
RD
Fig. 3.12: A second way to treat using superimposition the circuit analysed in fig. 3.11.
3.22
Chapter 3: Techniques for solving DC circuits
From the examples proposed it should be clear to the reader that usage of the superimposition principle to solve a circuit is normally not advantageous, since it requires the solution
of several circuits although simplified, instead of a single one.
It is however very important and useful in understanding how circuits work, and can give rise
to simplified computations in several practical situations.
Moreover, it is the basis for demonstrating the very important Thévenin theorem, that, this
one, is very useful and widely used in normal circuit-solving practice.
3.3.7 Thévenin’s theorem
Consider a circuit. If two wires are put in evidence, the circuit will be composed by two
subcircuits interfaced to each other by means of the two wires (fig. 3.13 a). Any of the two
subcircuits can contain an arbitrary number of voltage and current sources, and resistors,
connected to each other in an arbitrary manner; it is however mandatory to suppose one of the
two parts (let this be the left one) linear.
Let the current flowing in the upper wire in the right direction be I, and the voltages between
the two wires U12. By applying the substitution principle (sect. 2.5 ) the electric equilibrium of
the left part of the circuit can be obtained simply by substituting the right part with the current
source I (fig. 3.13, b).
Now the superimposition principle can be applied; since the grouping of sources can be
arbitrary, choice is made so that in the first circuit all the sources belonging to the left part of the
circuit are activated, and I is considered non-acting (therefore the corresponding branch is
eliminated from the circuit); then I is supposed acting but all other sources (i.e. all the sources
present in the left-part of the circuit) are deactivated.
Therefore it is (fig. 3.13, c):
U12=U’12+U”12
(3.10)
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
I
+
3.23
w1
w1
+
+
+
Any
subcircuit
U12
U12
w2
w2
a)
b)
w1
+
+
w1
w1
+
I
+
+
=
U12
I
I
U’12
+
w2
w2
+
U”12
w2
c)
RTh
+ UTh
Thevenin
equivalent
RTh  
U '12
I
U Th  U "12
d)
Fig. 3.13: Thévenin’s theorem.
The first term of the sum, U’12, is the voltage appearing at terminals at the left circuit when I
is applied and internal sources are de-activated; in these conditions the left circuit is a network of
resistances, for which the voltage to current ratio is a constant, function of the resistor network.
this constant, that has the dimensions of a resistance, is the so called equivalent resistance of the
left part of the circuit:
Therefore we can describe eq. (3.10) as follows:
Law: Thévenin’s theorem
In a circuit composed by two parts connected to each other by means of two wires one part,
supposed linear, can be substituted by the series of a constant-voltage branch and a resistor.
The constant voltage is the voltage appearing at this part then it is disconnected from the
right part, and the resistor has a s resistance, the ratio of voltage and current appearing at
its terminals when all the internal sources are deactivated.
The Thévenin equivalent of a subcircuit is therefore composed by the series of two simple
components (fig. 3.13, d). It is apparent that this theorem can induce great simplification to
circuit analysis: if knowledge of voltages and current internal to the left part are not of interest,
the whole part, that might be composed by an arbitrary number of branches, loops and inner
nodes, is substituted by a very simple equivalent, that allows complete analysis of the right part of
the circuit.
3.24
Chapter 3: Techniques for solving DC circuits
Example 8
Considering the following circuit, calculate I3:
a) using Kirchhoff’s laws
b) using nodal analysis
c) applying Thévenin’s theorem at nodes A and B.
a) The following set of equations can be written:
 I1  I 2  I 3
KCL at node A


 E1  R1 I1  R2 I 2
KVL at left mesh
-E  R I -R I
KVL at right mesh

3 3
2 2
 2
 I1  I 2  I 3

10  5 I1  25 I 2
-8  4 I -25 I
3
2


 I1  0.367 A

 I 2  0.326 A
 I  0.041 A
 3
b) The following equation can be written:
E1  U AB 0  U AB E2  U AB


 0 , deriving from KCL at node A.
R1
R2
R3
Substituting values:
10  U AB 0  U AB 8  U AB


 0  U AB  8.163 V . Finally:
5
25
8

E1  U AB 10  8.163

 0.367 A
 I1 
5
R
1


U AB 8.163

 0.326 A
I2 
25
R2


U  E2 8.163  8

 0.041 A
 I 3  AB
4
R3

M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
3.25
c) The Thévenin’s voltage UTh at nodes A and B can be easily calculated disconnecting
the right part of the circuit:
E1
10
 R2 
 25  8.333 V
5  25
R1  R2
The Thévenin’s equivalent resistance RTh is the resistance “seen” from nodes A and B,
when all generators are deactivated (in our case, only E1 is present):
RR
5  25
 4.166 
RTh=R1||R2= 1 2 
R1  R2 5  25
The left side of the circuit can now be substituted by its Thévenin’s equivalent, in order
to calculate I3:
Uth≡UAB0= (voltage division)=
This single-mesh circuit can be easily solved using KVL:
UTh-E2=(RTh+R3)I3
I3=(UTh-E2)/(RTh+R3)=0.041 A
The reader should note that UAB0≠UAB:
UAB=UTh-RThI3=8.333-4.1660.041=8.163, as calculated in b).
Comparing the three methods, we can conclude that the Thévenin’s theorem is very
powerful, in particular when a single current is requested.
3.4 Power and energy, and Joule’s law
Power and energy in circuits was already discussed in chapter 2. It was shown there that the
power that in a circuit flows from one part to another is
p=ui
In DC circuit everything is constant, uppercase letters are advantageously used to underline
this
P=UI
3.26
Chapter 3: Techniques for solving DC circuits
Attention must be given to signs, since P is directed towards the part of the circuit into which
the assumed current enters from the positively marked terminal (fig. 3.14).
I
+
subcircuit
1
subcircuit
2
U
P=UI
Fig. 3.14. Power between to circuit halves and its reference sign.
As usual, we must remember that the arrow for I and the “+” for U are only assumed
polarities: actual values of I and U will be positive or negative depending whether the actual
charge flow and voltage polarity2 are in agreement with the assumed signs or opposite to them.
This transfers to the value of power as well: In figure 3.14 P will be in accordance to the
assumed direction, visually represented by the shown arrow, if (and only if) its numerical value is
positive.
What happens in case the subcircuit 2 is constituted by just a single resistor? Ohm’s equation
U=RI applies, with U and I are again measured accord to the load convention, i.e. current
entering the positive terminal. Therefore:
P  UI  ( RI )  I  RI 2
(3.11)
So the resistor always absorbs power, whatever the sign of the applied current and voltage.
This power will be converted into heat in the resistor, and the thermal power generated is given
by (3.11). The law (3.11) is very important since it establishes a link between the worlds of
electricity and thermal phenomena and, since it was first stated by James Prescott Joule, it is
called Joule’s law.
2
Furthermore, remember that the current is considered to be composed by a flow of positive charges; electrons, i.e.
the negative charges inside conductors, in reality flow opposite the assumed sign for currents, when I is positive.
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
3.27
Law: Joule’s law
The power absorbed by any resistor is always positive, and equal to RI2=U2/R.
This power is converted into electromagnetic radiation, usually at a frequency
corresponding to heat.
Example 9
Considering the circuit of Example 8, calculate the power flowing across section AB,
from left to right:
PAB=UABI3=8.1630.041=0.335 W
PAB can be also calculated considering the power delivered by generators (PE=EI) and
the power consumed by resistors (PR=RI2), then imposing the power balance:
a) to the left side of the circuit:
PAB=PE1-PR1-PR2=E1I1-R1I12-R2I22=100.367-50.3672-250.3262=0.335 W
b) to the right side of the circuit:
PAB=-(PE2-PR3)=-(E3(-I3)-R3I32)=-(8(-0.041)-40.0412) =0.335 W
c) to the left side of the final circuit used in Thévenin’s analysis:
PAB=UThI3-RThI32=8.3330.041-4.1660.0412 =0.335 W
3.5 More examples
Consider again the example 2 discussed in section 2.4.4. , whose representation is reported in
fig. as well, for clarity:
C
+
+
E
+
+
A
+
D
B
H
+
+
+
F
I
G
+
Fig. 3.15. An example circuit (currents always considered entering branches from “+” signs).
It referred to a circuit in which the inner branch behaviour was not known, and therefore to
find all the quantities just making usage of Kirchhoff’s equations some of them needed to be set
as known in advance.
In case the constitutive equations are known, all voltages and currents can be obtained
integrating the latter in Kirchhoff’s equations, and possible reducing the set of equations even
more making use of the note voltage technique.
According to what already done in sect. 3.3.1, once the branch constitutive equations are
known, the circuit is written directly containing specific symbols for the branches. For instance
the circuit reported in fig. 3.16 is just a specific version of the one reported in fig. 3.15, where A
is a voltage source, B a current source, C a resistor, etc.
3.28
Chapter 3: Techniques for solving DC circuits
RC
N1
EA
+
+
IB
RE
N2
+
+
L1
+
N3
IF
+
+
RD
EG
L2
+
RH
N0
RI
+
Fig. 3.16. A specialised version (including constitutive equations)
of the circuit reported in fig. 2.15.
This circuit can be solved using one of the techniques introduced in this chapter:
 direct usage of constitutive equations in KCL and KVL (sect. 3.3.1);
 nodal analysis (sect. 3.3.2).
Other techniques such as mesh analysis can be used, but these were not presented in this
chapter.
The circuit can also be first transformed in a series of simpler circuits using the principle of
superimposition and then solving these simpler circuits with one of the circuit-solving techniques.
It was already noted, in section (0) that the usage of principle of superimposition is normally
a lengthy process, and is not recommended, except in special cases.
Therefore here the basic usage of combined Kirchhoff’s-constitutive equations as well as the
more advanced nodal analysis are used, both to show their practical application and show, once
again, that the node voltage analysis is much more effective, for manual computations.
Basic combined Kirchhoff’s-constitutive equations usage.
It brings to equations the structure of which depends on the choices of notes tor KCL and
loops for KVL equations. Choosing for instance nodes N1, N2, N3 for KCL and L1 and L2 for
KVL, the following equations are obtained (remember that currents are considered entering
branches from “+” signs):
N1: -IA+ IB + IC =0
N2: IC + ID – IE =0
N3: IE - IF + IG =0
L1: RHIA + EA +RCIC -RDID=0
L2: RIII + EG – REIE-RDID=0
(3.12)
The (3.12) is a system of five linearly independent equations in the five unknowns IA, IC, ID,
IE, IG. once solved, in case voltages across the current sources are needed, they can be found by
loop equations involving them, that allow immediate determination. For instance, considering
auxiliary loops La1 and La2 reported in fig. 3.17, it will be:
UF=RDID+ REIE
UB=RHIH+ EA
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
RC +
N1
EA
IB
+
RE
N2
+
RD
3.29
+
N3
IF
+
La2
EG
La1
+ R
H
V0
RI
+
Fig. 3.17: Additional loops to determine all quantities
from those obtained by direct application of Kirchhoff’s laws or the nodal analysis.
Nodal analysis
Nodal analysis will produce a set of n-1 equations, n being the number of circuit nodes.
Therefore in this case a set of only three equations in three unknowns is expected. These are the
following ones (V1=Voltage at note N1, etc):
V  EA
V V
N1 :  1
 IB  1 2
RH
RC
N2 :
V1  V2 V2 V2  V3


RC
RD
RE
N3 :
V2  V3
V  EG
 IF  3
0
RE
RI
(3.13)
To verify that the two systems (3.12) and (3.13) give the same results, some numerical
examples are reported in the table 3.III.
To obtain these results, if (3.12) is used, all currents are determined from it, while if (3.13) is
first used the node voltages are determined. The remaining quantities can then be obtained using
trivial equations.
For instance the following trivial equations can be written from Ohm’s law and/or from
auxiliary KVL equations, that allow the determination of the node voltages from the currents,
once they are computed using (3.12):
V2=RDID, (Ohm’s law on RD)
V1=RHIA+EA (KVL in loop La1 in fig.3.17)
(3.14)
V3=V2+REIE (KVL in loop La2 in fig. 3.17)
In a similar way, the following trivial equations can be written from Ohm’s law and/or KCL
equations, that allow the determination of the node voltages from the currents, once they are
computed using (3.13):
ID=V2/RD (Ohm’s law on RD)
IC=(V2-V1)/RC (Ohm’s law on RC)
IE=(V3-V2)/RE (Ohm’s law on RE)
IA=IC+IB (node N1 in fig.3.17)
IG=-IE+IF (node N3 in fig.3.17)
(3.15)
The full solution of the system is reported in table 3.III, with two different sets of values for
resistances, and voltage and current sources.
3.30
Chapter 3: Techniques for solving DC circuits
Tab. 3.III: Numerical data and results for the proposed examples.
Ex. num
1
2
Known values
RC=RD=RE=RH=RI=1 
IB=IF= 1A
EA=EG= 1 V.
RC=RD=RE=RH=RI=1 
IB=IF= 2A
EA=EG= 1 V
IA/A
IC/A
ID/A
IE/A
IG/A
V1/V
V2/V
V3/V
1/2
-1/2
1
1/2
1/2
3/2
1
3/2
5/4
-3/4
3/2
3/4
5/4
9/4
3/2
9/4
From the values of table 3. III the power absorbed by all branches can be easily computed, and
the energy conservation verified. The numerical data are reported in Table 3.IV.
In all cases resistors absorb power, as it must be.
In the example #1 the current generators deliver globally 3 W, while the voltage generators and the resistors
absorb 1 W and 2 W respectively. In the example #2 the current generators deliver globally 9 W, while the
voltage generators and the resistors absorb 2.5 W and 6.5 W respectively.
In both cases, obviously, the power delivered by the current generators perfectly equates that absorbed by the
other resistors and the voltage generators.
Table 3.IV: Power absorbed by all circuit components for the proposed example
Currents always entering the positively-marked terminals).
U/V
I/A
P/W
U/V
I/A
P/W
A
1
1/2
1/2
1
5/4
5/4
B
3/2
-1
-3/2
9/4
-2
-9/2
C
-1/2
-1/2
1/4
-3/4
-3/4
9/16
D
1
1
1
3/2
3/2
9/4
E
-1/2
-1/2
1/4
3/4
3/4
9/16
F
3/2
-1
-3/2
9/4
-2
-9/2
G
1
1/2
1/2
1
5/4
5/4
H
1/2
1/2
1/4
5/4
5/4
25/16
I
1/2
1/2
1/4
5/4
5/4
25/16
TOT
0
0
As an application of Thévenin’s theorem the current ID can be computed using it. Consider
figure 3.18.
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
RC +
N1
EA
RE
RD
+
RH
RC +
N1
IB
+
+
N3
IF
+
ID
ID
+
IB
+
EA
N2
3.31
V0
N2
+
ET h
Thé ve ni n
Equiva le nt
RD
+
RI
RE
EG
+
N3
IF
+
EG
+
+
RH
V0
ID
+
RI
RT h
N1
RC +
N2
RE
N3
+
+ ET h
RD
+
ZT h
+
+ R
H
V0
RI
+
Fig. 3.18: An example of application of Thevenin’s theorem.
Instead of solving the full circuit, the simple circuit containing Eth, RTh, RD can be solved.
The value of Eth is easily computed considering that this quantity is the open-circuit voltage,
i.e. the voltage that would appear at the RD terminals in case RD is disconnected from the circuit
(see the central circuit of fig. 3.18). Rth, in turn, is the resistance that is seen from the RD terminals
when the circuit to which RD is connected is made passive, by deactivation of all the inner
sources. This is shown in the bottom drawing of figure 3.18, in which deactivation of current
sources is made substituting them with open circuits, that forces to zero the current and
deactivation of electromotive force is made substituting then with ideal wires, that forces the
corresponding branch voltage to zero.
The central circuit of figure 3.18, to find Eth is much simpler to solve than the starting one
(on top of figure), and can be solved using any of circuit solving the techniques, i.e., for the
reader of this book, the direct application of Kirchhoff’s laws, or the nodal analysis.
However, the cases considered in the previous examples, reported in table 3.III, involved
special numerical values, for which:
EA=EG=E
IB=IF=I
RC=RE=RH=RI=R
These conditions create a special symmetry that can be exploited, with the help of the
superimposition principle (sect.0), to find the final solution with virtual no computations. Indeed
the circuit can be decomposed in two parts to be superimposed, as shown in figure 3.19.
3.32
Chapter 3: Techniques for solving DC circuits
In the upper circuit of figure 3.19 it is apparent that no current circulates, and therefore
ETh=E. In the down part, because of symmetry, the currents in the right part of the circuit must be
the mirrored version of the ones of the left part. Therefore the currents flowing in the right and
left wires connected top node N2, measured towards the right part of the circuit must be one the
opposite of the other. But their sum must also be equal to each other, because of the charge
conservation in node N2, and therefore both must be null. Consequently the currents of the current
generators flow in the bottom resistances, that implies E”Th to be equal to RI.
R
E
R
+
+
E’Th
+
IF
-
R
+
+
E
E’Th=E
R
+
+
R
+
R
N2
+
E”Th
I
+
I
-
R
E’Th =E’Th+E”Th=
=E+RI
E”Th=RI
R
+
+
Fig. 3.19: Calculation of Eth using the superimposition principle
As RTh is regarded it is nearly immediate (fig. 3.20) that is RTh=R.
R
R
IF
RTh
R
RTh
2R
R
2R
RTh
R
Fig. 3.20: Calculation of RTh for the example of fig. 3.18
in the case of resistances all equal to each other.
Finally, from the Thevenin equivalent circuit (right-most circuit in fig. 3.18 is immediate
that:
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
ID 
ETh
E  RI

RTh  RD
2R
3.33
I D ,ex1  1.0 A I D ,ex 2  1.5 A
that is the same numbers already computed and reported in table 3.III.
3.6 Resistive circuits operating with variable quantities
All the developments made in this chapter derive from the conclusions of chapter 2 valid for
any circuits, and for the list of constitutive equations reported in Table 3.I.
It could be very easy to demonstrate that all the results obtained here, e.g. the usage of
combined Kirchhoff’s-constitutive equations, the nodal analysis, superimposition and Thevenin
theorem, are still valid in case the constitutive equations are as reported in Table 3.V, i.e. when
current and voltage sources are not constant, but all non-source branches are resistive.
Table 3.V: The constitutive equation types for resistive circuit with variable quantities.
ELEMENT
(variable) voltage source
(variable) current source
resistive element
EQUATION
ub=us(t)
ib= is(t)
ub=Rb ib
All the branch voltages and currents, will be functions of time, and could be determined
using this chapter’s techniques.
3.7 Historical notes
3.7.1 Ohm’s short biography
George Simon Ohm3 (Germany) was a physicist that is mainly famous for the law that relates
voltage and potential in a conductor that carries his name.
He took his degree at the University of Erlangen in 1811, and then taught for several years at
primary and secondary schools. In 1826 he became physics professor first at the Military
Academy of Berlin, then at the universities of Nuremberg and Munich.
His main studies are related to current flow in conductors, that gave birth (after further
contributions of other scientists) to the law today called Ohm’s law.
He also made studies on light interference and on the ear perception of complex sounds.
3.7.2 Thévenin’s short biography
Léon Charles Thévenin (1857-1926) was a French engineer that is mainly famous for his
theorem for circuits. Born in Meaux, near Paris, he graduated from the Ecole polytéchnique in
1876. Two years later he joined France’s national electrical communication company (Posted et
Télégraphes) that he never left.
As a result of studying the already stated circuit laws (by Kirchhoff and Ohm), he developed
what is now called Thévenin’s theorem, which allowed people to reduce complex circuits into
simpler ones, containing the so-called Thévenin's equivalent.
Biographies of science have shown that the same theorem had already been stated thirty years earlier than
Thévenin, by Hermann von Helmholtz. Credit is given to Thévenin that he developed his own version of the
same theorem unaware of Helmholtz’s work.
3
Pronounce (from Oxford English Dictionary): /ɘʊm/ (Brit.), /oʊm/ (USA).
3.34
Chapter 3: Techniques for solving DC circuits
When Thévenin issued his theorem, controversy arose as to whether it was correct or not.
Shortly before his death, however, his theorem became accepted all over the world.
3.7.3 Joule’s short biography
James Prescott Joule4 (1818-1889) was born in Salford, Manchester. He spent his life in
managing his brewing factory in Salford and making studies on physics. Even though he did not
receive formal education, he had so important scientific achievements that received a degree
honoris causa from the University of Leida.
A very important achievement of his was a mathematical relation that links the heat produced
by a current-carrying conductor, its resistance and the current itself. This relation is today
universally referred to as Joule’s law. Also very important were his studies on the “mechanical
Equivalent of Heat” (as himself called in a paper of his) gave a fundamental impulse towards the
definition of the First Law of Thermodynamics. He also worked with W. Thompson (lord Kelvin)
to develop the absolute scale of temperature.
The S.I. unit of energy (and work and amount of heat), the joule, is named after him.
3.8 Reference list
[1] R. W. Chabay, B. A. Sherwood: Matter & Interactions II – Electric & Magnetic Interactions”, John
Wiley & Sons, Inc, ISBN-13 978-0-470-10831-4.
[2] Mulukutla S. Sarma: “Introduction to Electrical Engineering”, Oxford University Press, 2001 ISBN
0-19-513604-7
[3] Fawwaz Ulaby: Fundamentals of Applied Electromagnetics Prentice Hall 2007, ISBN 0132413264
[4] Giorgio Rizzoni: “Principles and applications of electrical engineering”, McGraw-Hill Higher
education, 2004 ISBN 0-07-121771-1 (ISE)
[5] C. Alexander, M. Sadiku: Fundamentals of Electric Circuits, McGraw-Hill Higher Education, third
edition, 2007, ISBN 13-978-0-07-197718-9
4
As the pronunciation is concerned the Oxford English Dictionary refers that although some people with this name
call themselves /dʒaʊl/ or /dʒɘʊl/ it is almost certain that J.P Joule used /dʒu:l/, that is therefore to be assumed as
the correct pronunciation.
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
3.35
3.9 Proposed exercises
3.1. A voltage U=12 V is applied to an aluminium
conductor, with a length of 300 m and a cross
section of 150 mm2.
Calculate the resistance R, the current I and the
absorbed power P, at 20°C and 55°. Use tab. 3.II.
3.2. Calculate the equivalent resistance “seen”
from nodes A and B:
3.3. Using the rule of current divider, calculate I1
and I2, I being 10 A. Verify the solution,
calculating UAB=Reqp*I and observing that
R1*I1=R2*I2=UAB.
3.4. Determine I and UAB.
If E1 and E2 represent two ideal batteries, which
one charge the other?
E1=120 V, E2=90 V, R1=R2=10 , R3=40 .
3.5. Calculate the resistance RE seen by the
generator and I1. Then, using the voltage division
rule, calculate I2 and I3.
Check the conservation of power, comparing what
is delivered by the generator and what is absorbed
by resistors.
E=12 V, R1=R2=2 , R3=8 , R4=6 .
3.36
Chapter 3: Techniques for solving DC circuits
a) using the superposition rule;
b) using Kirchhoff’s laws;
c) using the nodal voltage technique to calculate
UAB;
d) using the Thévenin’s theorem to find an
equivalent, left side or right side section AB.
E1=12 V, R1=0.5 , R2=5 , E2=9 V.
3.6. Solve again Ex.3.5:
a) using Kirchhoff’s laws;
b) using the nodal voltage technique in order to
calculate UAB.
3.7. Applying Thévenin’s theorem between A and
B, calculate the equivalent voltage UTh and the
equivalent resistance RTh:
3.10. Solve the following circuit:
a) using the superposition rule;
b) using Kirchhoff’s laws;
c) using the nodal voltage technique to calculate
UAB (write KCL at node A);
d) using the Thévenin’s theorem to find an
equivalent, left side section AB.
E=100 V, R1=20 , R2=30 , I=3 A.
3.11. In the previous exercise, calculate the
voltage UJ across the current generator. Then
verify the energy balance, comparing the power
delivered by the generators and the one absorbed
by resistors.
3.12. Find at least three ways to calculate I2 and I3.
Is R1 really required to determine currents? And to
calculate the power delivered by J1?
J1=5 A, J2=1 A, J3=8 A, R1=5, R2=1, R3=4 .
3.8. Considering Ex.3.5, determine the voltage
across R4, using Thévenin’s theorem.
3.9. Solve the following circuit:
M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering
3.13. The circuit shown in figure supplies a lamp
connected between nodes A and B and having the
following rated values: U=12 V, P=10 W, 100
lumen.
Calculate:
a) the light flow  delivered by the lamp,
assuming  to be proportional to I2
b) the energy W consumed by the lamp over 100h.
E=15 V, R1=2 , R2=10 , I=0.3 A.
Suggestion: apply Thévenin’s equivalent to nodes
A and B.
3.37
3.16. Calculate the Thévenin equivalent of the
following circuit:
E1=6 V, E2=2 V, R1=4 , R2=6 , R3=2 ,
R4=1 , J=1 A.
3.17. The previous circuit supplies a 10 
resistance. Calculate the power absorbed by the
load.
3.14. Calculate the current I in the following
circuit:
E1=6 V, E2=2 V, R1=1 , R2=R3=2 ,
J=3 A.
3.15. Calculate the Thévenin equivalent of the
following circuit:
E=2 V, J=2 A, R1=4 , R2=R3=1 .
3.18. Express the active power P absorbed by the
load, as a function of R, E and Rline. Demonstrate
that, E and Rline being fixed, P is maximum when
R=Rline (Theorem of maximum power transfer).
3.19. Find the load resistance R to be supplied by
the following circuit, in order to maximize the
power transfer to the load. Calculate such a power.
E=12 V, R1=1 , R2=15 , R3= 2 , I=2 A.
3.38
3.20. Calculate IA and UAB and determine the
power flowing through the section AB.
Verify the result using the principle of power
conservation.
E1=10 V, E2=4 V, R1=1 , R2=2 ,
RJ=100 , J=1A.
3.21. Find the load resistance R to be supplied by
the following circuit, in order to maximize the
power transfer to the load. Calculate such a power.
(before solving, read the text of ex. 3.18)
Chapter 3: Techniques for solving DC circuits