1E6 Electrical Engineering
DC Circuit Analysis
Lecture 3: Fundamental Electric Circuit Laws
3.1 Introduction
Up to now electricity has been considered largely in its own right in terms
of atomic charge, potential, electric field and current as the flow of charge.
However, electricity does not appear in everyday life in an abstract form but is
harnessed and used to provide a source of energy, usually with the aim of
converting this energy into another form such as mechanical energy. In this
context it is used to do work as when used in domestic or industrial equipment
or machinery. Alternatively, on a lower scale of energy, it can be used in
electronic engineering to provide the source of power needed to control
semiconductor devices and integrated circuits used in so much of today’s
instrumentation, communication and computing applications.
In this regard, electricity is used to provide the electromotive force
required to enable current to flow through electrical loads of various forms. This
in turn requires an electric or electronic circuit to connect the source of emf and
the load together in a loop so that current can circulate between the two. This
essentially constitutes the formation of an electric circuit. Electricity can be
exploited in two forms in electric circuits. The first of these is direct current,
commonly known as dc, where the source emf or voltage is constant in polarity
and magnitude, as is the current flowing in the circuit under steady-state
conditions. The second is alternating current, commonly known as ac, where the
direction of the source emf and the resulting current which flows continually
reverses direction and varies instantaneously in magnitude. The easiest of these
to understand is dc and the fundamental circuit laws are more easily assimilated
in the context of dc circuits. Therefore dc circuits will be considered and
analysed first and later attention will be turned to ac circuits.
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3.2 Ohm’s Law
Fig. 1 shows the simplest form of dc electric circuit where a dc battery,
which will be examined in more detail later, is used as the source of emf or
voltage, E, to drive current, I, through the load in the form of the resistor, R,
developing the potential difference across it measured as the voltage, V.
It was the German physicist Georg Ohm (1789 - 1854) who first formally
described the relationship between the voltage developed across a load and the
current which flows through the load. The law which bears his name is the most
fundamental of the laws governing electric circuits and was first formally
published in 1827, almost two hundred years ago.
Ohm’s Law states that: ‘the potential difference which is developed across a
conductor at any point in an electric circuit is proportional to the current flowing
through the conductor at that point and the resistance of the conductor is the
constant of proportionality’
current I (Amps)
+
dc battery as
voltage source
E (Volts)
+
potential difference
developed across load
V (Volts)
_
load resistance
R (Ohms)
_
Fig. 1 A dc Battery Connected to a Resistive Load
Ohm’s Law is formally written as:
V
V
or
R
R
I
with V in Volts I in Amps and R in Ohms
V  IR
or
I
Note that the battery voltage, E, is the source of emf which provides the
electric force to drive current around the circuit. The voltage, V, is the potential
difference, often referred to as the voltage drop, across the resistor as a result of
its resistance to current flow. The emf, E, exists in its own right as a source of
energy, for example as the terminal voltage of a charged battery. However, the
potential drop, V, exists only as a result of the current flowing in the circuit
which in turn is caused by the emf, E. If there were no emf present no current
would flow in the circuit and there would be no potential drop across the
resistor, R. In this case, because there is only a single resistor as the load in the
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circuit, the full emf appears across this resistor as a potential drop and E = V. In
a more complex circuit this would not necessarily be the case.
Note also that term ‘point’ used in the statement of the law above would
more accurately be term ‘branch’ or ‘arm’ in a circuit as it does not refer to a
sinle node but rather a conducting element connected at a location within a
circuit.
Fig. 2 shows Ohm’s law as a linear relationship for different values of
resistance, R, where R1 > R2 > R3 > R4. Note that the higher the value of the
resistance, the lower the value of current which flows through it for the same
potential drop across it.
current
I (A)
R4
R3
R2
R1
voltage V (V)
Fig. 2 Ohm’s Law Plotted for a Range of Values of Resistance
3.3 Case Study 1
(i)
Determine the potential drop across a resistance of 15Ω when a current of
2.5A flows through it.
V  IR  2.5  15  37.5 V
(ii) Determine the current that flows through a resistance of 200Ω when an
emf of 12V is applied across it. The emf when applied across the single resistance
becomes the potential drop across the resistance so that:
I
V 12

 0.06 A  60 mA
R 200
(iii) Determine the value of the resistor needed to limit the current drawn from
a voltage source of 7.5 V to 1.5mA.
R
V
7.5
7.5
3



10
 5k
3
I 1.5  10
1.5
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3.4 Kirchhoff’s Circuit Laws
Next to Ohm’s Law in the fundamental rules which govern the behaviour
of electric circuits are Kirchhoff’s Circuit Laws. These were formulated by
Gustav Kirchhoff (1824 – 1887), another German physicist and published in
1845. They are referred to as his circuit laws as he contributed laws in other
related fields also, such as Electrochemistry and Radiation. He formulated two
circuit laws, one which essentially establishes the conservation of charge and the
other which establishes the conservation of potential.
3.5 Kirchhoff’s Current Law
Kirchhoff’s first law, known as Kirchhoff’s Current Law, KCL, or
sometimes as Kirchhoff’s Junction Rule, essentially expresses the conservation
of charge, which can be thought of as the conservation of matter if the charge is
considered as a quantity of charged particles. This implies that charge cannot
appear from nothing at any point in a circuit, neither can it disappear into
oblivion at any point.
Kirchhoff’s Current Law states that….‘the sum of the currents flowing at a node
in an electric circuit is zero’.
N
I
n 1
n
0
In modern terms this is restated as:
Kirchhoff’s Current Law states that….‘the sum of the currents flowing into a node
in an electric circuit is equal to the sum of the currents flowing out of that node’.
Ni
I
ni 1
No
ni
  I no
no 1
In using this law in circuit analysis it is essential to adopt a consistent sign
convention with regard to the polarity of the currents. The normal and most
reliable convention is that currents flowing into the node are considered positive
while currents flowing out of the node are considered negative. The diagram in
Fig. 3 shows several currents at a single node in a circuit. The dot represents the
node while the arrows are used both to represent wires connected to the node
and the direction of the current carried by each wire into or out of the node.
Each current is individually labelled, Ii , which is intended to indicate the
magnitude of the current while the arrow head specifies the direction of flow.
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I2
I1
I3
I6
I4
I5
Fig. 3 A Circuit Node With Several Associated Currents
Kirchhoff’s Current Law gives:
N
I
n 1
n
0
I1  I 2  I 3  I 4  I 5  I 6  0
or
or in its alternative form gives:
Ni
I
ni 1
No
ni
  I no
or
no 1
I1  I 2  I 5  I 3  I 4  I 6
Note that both forms are completely mathematically consistent.
Wires are used in a circuit to connect points of the same potential together
and are considered to be perfect conductors having no resistivity and therefore
no potential drop along them when carrying current. Therefore wires can be
used to enlarge the location of a node for visual enhancement in a circuit
diagram but without any effect on the unique potential at the node. The
schematic layout in Fig. 4 shows this where all of the points in this diagram are
at the same potential as the single nodal point shown in Fig. 3.
I3
I1
I2
I6
I4
I5
Fig. 4 An Equivalent Representation of the Node of Fig. 3
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3.6 Case Study 2
In the circuit of Fig. 4 the magnitudes of the currents are as follows:
I1 = 2.5 A, I2 = 4 A, I4 = 7.5 A, I6 = 6 A and I3 = 2I5.
Determine the values of I2 and I5.
Using Kirchhoff’s Current Law gives:
I1  I 2  I 3  I 4  I 5  I 6  0
2.5  4  I 3  7.25  I 5  6  0
 I 5  I 3  6.75  0
I 5  2 I 5  6.75 A
 I 5  6.75 A
I 5  6.75 A
I 3  2 I 5  13.5 A
Substituting the values obtained back in again to check gives:
I1  I 2  I 3  I 4  I 5  I 6  0
2.5  4  13.5  7.25  6.75  6  0
20  20  0
Note that in analysing a circuit the labels and directions of currents are
often assigned arbitrarily. However, Kirchhoff’s Law must be applied to the
analysis consistently with the assignment. Then any value of current calculated
which works out to be negative simply indicates that in practice the current is
actually flowing in a direction opposite to that assigned in the schematic
diagram of the circuit.
For the example in question this means that the current I3 evaluated as
equal to -13.5A is actually a current of magnitude 13.5A flowing out of the node
of Fig. 4 rather than into it as assigned in the schematic diagram.
Similarly, the current I5 evaluated as equal to -6.75A is actually a current
of magnitude 6.75A flowing into the node of Fig. 4 rather than out of it as
assigned in the schematic diagram.
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3.7 Kirchhoff’s Voltage Law
Kirchhoff’s second circuit law, known as Kirchhoff’s Voltage Law, KVL,
or sometimes as Kirchhoff’s Loop Rule, essentially formulates the conservation
of energy in the form of electric potential around a circuit in which current is
flowing. This means that no net voltage can be created or destroyed around the
loop of a closed circuit.
Kirchhoff’s Voltage Law states that…. ‘the sum of the potentials around a closed
electric circuit is zero’
N
V
n 1
n
0
In modern terms this can be restated as:
Kirchhoff’s Voltage Law states that…. ‘the sum of the emfs around a closed
electric circuit is equal to the sum of the potential drops around the same circuit’
N EMF
E
nEMF 1
nEMF

N PD
V
nPD 1
nPD
This essentially means that the total emf in a closed circuit, which may be
the sum of a number of emfs at different locations and of different magnitudes
and polarities, must equal the sum of the potential differences generated across
all of the conducting elements due to the current circulating around the circuit.
Fig. 5 shows the application of Kirchhoff’s Voltage Law to a simple
circuit. As with the first law, it is essential to have agreed conventions with
regard to the directions and polarities of potentials and to apply these
conventions consistently. The easiest way to accomplish this is to assign a
direction to the net current considered to be flowing around the circuit loop.
This could be for example either clockwise or anticlockwise, with the choice
being arbitrary. It is also often convenient to nominate some point in the circuit
as a reference ground or 0V point and to consider the loop as beginning and
ending at this reference point.
Following this, the potentials, both emfs and potential drops across
conducting elements, must also be assigned around the loop according to the
convention. The polarities of batteries serving as emfs are intrinsically
determined by the orientation of their terminals in their connection into the
circuit. The long-terminal is the positive side and the short terminal is the
negative side of the battery.
Emfs, shown in red, are summed around the loop of the circuit as being
positive when they support the direction of the assigned current flow (i.e. tend to
generate a current in this direction) and as negative when they oppose the
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direction of assigned current flow (i.e. when they tend to generate a current
flowing in the opposite direction).
Potential drops can be considered as losses of the overall emf in the loop
by way of its distribution among the conducting elements. The convention is to
assign all potential drops, shown in blue, across conducting elements in a
polarity consistent with the assigned direction of the current flowing through
them. In this case they will all be treated as negative potentials around the
circuit loop since their polarity appears as opposite to that of the direction of
current flow. However, if the second form of Kirchhoff’s Voltage Law is used
they will appear as positive potentials on the right hand side of the summation
equation, so that both forms are mathematically consistent. The labels En and Vn
are intended to represent the magnitude of the associated potentials or voltages.
E2
R1
- +
V1 =
+
+
E1
assigned
current
direction
=
=+
V3
+ +
= R3
R2
V2
=
=
+
=
E3
E4
Fig. 5 Kirchhoff’s Voltage Law applied to a Closed Circuit
Kirchhoff’s Voltage Law gives:
N
V
n 1
n
 0 or
E1  V1  E2  V2  E3  E4  V3  0
or in its alternative form gives:
N EMF
E
nEMF 1
nEMF

N PD
V
nPD 1
nPD
or
E1  E2  E3  E4  V1  V2  V3
3.8 Case Study 3
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In the circuit of Fig. 5 the magnitudes of the potentials are as follows:
E1 = 12V, E2 = 2V, E3 = 5V, E4 = 4.5V, V1 = 2V and V2 = 2V3.
Determine the values of the potentials V2 and V3.
Using Kirchhoff’s Voltage law gives:
E1  V1  E2  V2  E3  E4  V3  0
12  2  2  V2  5  4.5  V3  0
12  2  2  5  4.5  V2  V3
V2  V3  7.5V
3V3  7.5V
V3  2.5V
V2  2V3  5V
Using the Voltage Law again to check the values obtained:
E1  V1  E2  V2  E3  E4  V3  0
12  2  2  5  5  4.5  2.5  0
16.5  16.5  0
Note that, just as with Kirchhoff’s Current Law, consistency must be
exercised when applying Kirchhoff’s Voltage Law to the closed circuit.
However, in more complicated circuits a particular conducting element may
appear as branch of more than one closed loop and in this case the potential
drop may appear in one loop in an inconsistent direction with that of the
assigned current flow. Nonetheless, only one direction can be assigned to a given
potential and this principle must be adhered to. When this is the case, this
potential drop may be evaluated as a negative value which simply means that its
actual polarity in practice is opposite to that assigned for the purpose of
analysis.
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