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Introduction
The interconnection of various electric elements in a prescribed manner comprises as an
electric circuit in order to perform a desired function. The electric elements include controlled and
uncontrolled source of energy, resistors, capacitors, inductors, etc. Analysis of electric circuits refers to
computations required to determine the unknown quantities such as voltage, current and power
associated with one or more elements in the circuit. To contribute to the solution of engineering
problems one must acquire the basic knowledge of electric circuit analysis and laws. Many other
systems, like mechanical, hydraulic, thermal, magnetic and power system are easy to analyze and
model by a circuit. To learn how to analyze the models of these systems, first one needs to learn the
techniques of circuit analysis. We shall discuss briefly some of the basic circuit elements and the laws
that will help us to develop the background of subject.
Charge:
In an object comprised of many atoms, the net charge is equal to the arithmetic sum, taking
polarity into account, of the charges of all the atoms taken together. In a massive sample, this can
amount to a considerable quantity of elementary charges. The unit of electrical charge in
the International System of Units is the coulomb (symbolized C), where 1 C is equal to approximately
6.24 x 1018 elementary charges. It is not unusual for real-world objects to hold charges of many
coulombs.
An electric field, also called an electrical field or an electrostatic field, surrounds any object
that has charge. The electric field strength at any given distance from an object is directly proportional
to the amount of charge on the object. Near any object having a fixed electric charge, the electric field
strength diminishes in proportion to the square of the distance from the object (that is, it obeys the
inverse square law).
When two objects having electric charge are brought into each other's vicinity, an electrostatic
force is manifested between them. (This force is not to be confused with electromotive force, also
known as voltage.) If the electric charges are of the same polarity, the electrostatic force is repulsive. If
the electric charges are of opposite polarity, the electrostatic force is attractive. In free space (a
vacuum), if the charges on the two nearby objects in coulombs are q1 and q2 and the centers of the
objects are separated by a distance r in meters, the net force F between the objects, in newtons, is given
by the following formula:
F = (q1q2) / (4
o
r2)
Where o is the permittivity of free space, a physical constant, and is the ratio of a circle's
circumference to its diameter, a dimensionless mathematical constant. A positive net force is repulsive,
and a negative net force is attractive. This relation is known as Coulomb's law.
Electric Current:
An electric current is a flow of electric charge. In electric circuits this charge is often carried by
moving electrons in a wire. It can also be carried by ions in an electrolyte, or by both ions and
electrons such as in a plasma.
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The SI unit for measuring an electric current is the ampere, which is the flow of electric charge across
a surface at the rate of one coulomb per second. Electric current is measured using a device called
an ammeter.[2]
Electric currents can have many effects, notably heating, but they also create magnetic fields, which
are used in motors, inductors and generators.
Electric power:
Electric power, like mechanical power, is the rate of doing work, measured in watts, and
represented by the letter P. The term wattage is used colloquially to mean "electric power in watts."
The electric power in watts produced by an electric current I consisting of a charge of Q coulombs
every t seconds passing through an electric potential (voltage) difference of V is
Where
Q is electric charge in coulombs
t is time in seconds
I is electric current in amperes
V is electric potential or voltage in volts
Electrical elements are conceptual abstractions representing idealized electrical components,
such as resistors, capacitors, and inductors, used in the analysis of electrical networks. Any electrical
network can be analyzed as multiple, interconnected electrical elements in a schematic
diagram or circuit diagram, each of which affects the voltage in the network or current through the
network. These ideal electrical elements represent real, physical electrical or electronic components but
they do not exist physically and they are assumed to have ideal properties according to a lumped
element model, while components are objects with less than ideal properties, a degree of uncertainty in
their values and some degree of nonlinearity, each of which may require a combination of multiple
electrical elements in order to approximate its function.
Fig 1.1: Different types of circuit element
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Three passive elements:



Resistance , measured in ohms – produces a voltage proportional to the current flowing
through the element. Relates voltage and current according to the relation
.
Capacitance , measured in farads – produces a current proportional to the rate of change of
voltage across the element. Relates charge and voltage according to the relation
.
Inductance , measured in henries – produces the magnetic flux proportional to the rate of
change of current through the element. Relates flux and current according to the relation
.
Basic Elements & Introductory Concepts
Electrical Network: A combination of various electric elements (Resistor, Inductor, Capacitor,
Voltage source, Current source) connected in any manner what so ever is called an electrical network.
We may classify circuit elements in two categories, passive and active elements.
Passive Element: The element which receives energy (or absorbs energy) and then either converts it
into heat (R) or stored it in an electric (C) or magnetic (L ) field is called passive element.
Active Element: The elements that supply energy to the circuit is called active element. Examples of
active elements include voltage and current sources, generators, and electronic devices that require
power supplies. A transistor is an active circuit element, meaning that it can amplify power of a signal.
On the other hand, transformer is not an active element because it does not amplify the power level and
power remains same both in primary and secondary sides. Transformer is an example of passive
element.
Bilateral Element: Conduction of current in both directions in an element (example: Resistance;
Inductance; Capacitance) with same magnitude is termed as bilateral element.
Fig 1.2: Sign convention of current in the resistive circuit
Unilateral Element: Conduction of current in one direction is termed as unilateral (example: Diode,
Transistor) element.
Fig 1.3: Biasing of a diode
Meaning of Response: An application of input signal to the system will produce an output signal, the
behavior of output signal with time is known as the response of the system.
Linear and Nonlinear Circuits
Linear Circuit: Roughly speaking, a linear circuit is one whose parameters do not change with
voltage or current. More specifically, a linear system is one that satisfies (i) homogeneity property
[response of ()utα equals α times the response of , ()ut(()Sut α = (())Sutα for all α; and ] (ii) additive
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property [that is the response of system due to an input (()ut1122()()ututαα+) equals the sum of the
response of input 11()utα and the response of input22()utα, 1122(()(Sutut αα+ = 1122(())(())SutSutαα+.]
When an input is applied to a system “”, the corresponding output response of the system is observed
as respectively. Fig. 3.1 explains the meaning of homogeneity and additive properties of a system.
Fig 1.4: Block diagram of linear circuit
Non-Linear Circuit: Roughly speaking, a non-linear system is that whose parameters change with
voltage or current. More specifically, non-linear circuit does not obey the homogeneity and additive
properties. Volt-ampere characteristics of linear and non-linear elements are shown in figs. 3.2 - 3.3. In
fact, a circuit is linear if and only if its input and output can be related by a straight line passing
through the origin as shown in fig.3.2. Otherwise, it is a nonlinear system.
Fig 1.5: V-I characteristics of linear circuit
Fig 1.6: V-I characteristics of non-linear circuit
Potential Energy Difference: The voltage or potential energy difference between two points in an
electric circuit is the amount of energy required to move a unit charge between the two points.
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Open Circuit and Short Circuit
Open Circuit:
An electric circuit that has been broken, so that there is no complete path for current flow. A
condition in an electric circuit in which there is no path for current between two points; examples are a
broken wire and a switch in the open, or off, position.
Open-circuit voltage is the potential difference between two points in a circuit when a branch
(current path) between the points is open-circuited. Open-circuit voltage is measured by a voltmeter
which
has
a
very
high
resistance
(theoretically
infinite).
Short Circuit :
A low-resistance connection established by accident or intention between two points in an
electric circuit. The current tends to flow through the area of low resistance, bypassing the rest of the
circuit.
Common usage of the term implies an undesirable condition arising from failure of electrical
insulation, from natural causes (lightning, wind, and so forth), or from human causes (accidents,
intrusion, and so forth).
In circuit theory the short-circuit condition represents a basic condition that is used analytically
to derive important information concerning the network behavior and operating capability. Thus, along
with the open-circuit voltage, the short-circuit current provides important basic information about the
network at a given point.
The short-circuit condition is also used in network theory to describe a general condition of
zero voltage.
Resistivity and Conductivity:
Resistivity is the electrical resistance offered by a homogeneous unit cube of material to the
flow of a direct current of uniform density between opposite faces of the cube. Also called specific
resistance, it is an intrinsic, bulk (not thin-film) property of a material. Resistivity is usually
determined by calculation from the measurement of electrical resistance of samples having a known
length and uniform cross section according to the following equation where ? is the resistivity, R the
measured resistance, A the cross-sectional area, and l the length.
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In the mks system (SI), the unit of resistivity is the ohm-meter. Therefore, in the equation
below,
resistance
is
expressed
in
ohms,
and
the
sample
dimensions
in
meters.
Resistivity is also temperature dependent. Resistivity increases by about 0.4%/K at room temperature
and is nearly proportional to the absolute temperature over wide temperature ranges. The roomtemperature resistivity of pure metals extends from approximately 1.5 × 10^-8 ohm-meter for silver,
the best conductor, to 135 × 10^-8 ohm-meter for manganese, the poorest pure metallic conductor.
Most metallic alloys also fall within the same range. Insulators have resistivity within the approximate
range of 10^8 to 10^16 ohm-meters. The resistivity of semiconductor materials, such as silicon and
germanium, depends not only on the basic material but to a considerable extent on the type and amount
of impurities in the base material. Large variations result from small changes in composition,
particularly at very low concentrations of impurities. Values typically range from 10^-4 to 10^5 ohmmeters.
Conductivity:
Electrical conductivity is a measure of a material's ability to conduct an electric current. When
an electric potential difference is placed across a conductor, its movable charges flow, giving rise to an
electric current. The conductivity σ is defined as the ratio of the current density J to the electric field
strength E:
Conductivity is the reciprocal (inverse) of electrical resistivity and has the SI units of Siemens
per metre (S·m-1) i.e. if the electrical conductance between opposite faces of a 1-metre cube of
material is 1 Siemens then the material's electrical conductivity is 1 Siemens per metre. Electrical
conductivity is commonly represented by the Greek letter σ, but κ or γ are also occasionally used.
Classification of materials by conductivity:

A conductor such as a metal has high conductivity.

An insulator like glass or a vacuum has low conductivity.

The conductivity of a semiconductor is generally intermediate, but varies widely under
different conditions, such as exposure of the material to electric fields or specific frequencies of light,
and, most important, with temperature.
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Temperature Dependence:
Electrical conductivity is more or less strongly dependent on temperature. In metals, electrical
conductivity decreases with increasing temperature, whereas in semiconductors, electrical conductivity
increases with increasing temperature. Over a limited temperature range, the electrical conductivity can
be approximated as being directly proportional to temperature. In order to compare electrical
conductivity measurements at different temperatures, they need to be standardized to a common
temperature. This dependence is often expressed as a slope in the conductivity-vs-temperature graph,
and can be used:
Where
σT′ is the electrical conductivity at a common temperature, T′
σT is the electrical conductivity at a measured temperature, T
α is the temperature compensation slope of the material,
T is the measured temperature,
T′ is the common temperature.
The temperature compensation slope for most naturally occurring waters is about 2 %/°C,
however it can range between (1 to 3) %/°C. This slope is influenced by the geochemistry, and can be
easily determined in a laboratory.
Series and Parallel Circuits
Series and parallel electrical circuits are two basic ways of wiring components. The names
describe the method of attaching components, which is one after the other or next to each other. It is
said that two circuit elements are connected in parallel if the ends of one circuit element are connected
directly to the corresponding ends of the other. If the circuit elements are connected end to end, it is
said that they are connected in series. A series circuit is one that has a single path for current flow
through all of its elements. A parallel circuit is one that requires more than one path for current flow in
order to reach all of the circuit elements.
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With each of the two basic circuit (series and parallel) configurations, we have specific sets of
rules describing voltage, current, and resistance relationships.
Series circuits:

Voltage drops add to equal total voltage.

All components share the same (equal) current.

Resistances add to equal total resistance.
Parallel circuits:

All components share the same (equal) voltage.

Branch currents add to equal total current.

Resistances diminish to equal total resistance.
Resistances in Series:
To find the total resistance of all the components, add the individual resistances of each
component:
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for components in series with resistances R1, R2, etc. To find the current I use Ohm's law:
To find the voltage across a component with resistance Ri, use Ohm's law again:
Vi = I×Ri
Where I is the current, as calculated above. The components divide the voltage according to
their resistances, so, in the case of two resistors, V1/V2=R1/R2.
Resistances in Parallel:
Voltages across components in parallel with each other are the same in magnitude and they also
have identical polarities. Hence, the same voltage variable is used for all circuits’ elements in such a
circuit. The total current I is the sum of the currents through the individual loops, found by Ohm's Law.
Factoring
out
the
voltage
gives
Notation
The parallel property can be represented in equations by two vertical lines (as in geometry) to
simplify equations. For two resistors,
Resistors
To find the total resistance of all components, add the reciprocals of the resistances R i of each
component and take the reciprocal of the sum:
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To find the current in a component with resistance Ri, use Ohm's law again:
The components divide the current according to their reciprocal resistances, so, in the case of two
resistors, I1/I2=R2/R1
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Characteristics of circuit element
Electrical Resistance
The electrical resistance of an electrical conductor is the opposition to the passage of
an electric current through that conductor. The inverse quantity is electrical conductance, the ease
with which an electric current passes. Electrical resistance shares some conceptual parallels with the
notion of mechanical friction. The SI unit of electrical resistance is the ohm (Ω), while electrical
conductance is measured in Siemens (S).
An object of uniform cross section has a resistance proportional to its resistivity and length and
inversely proportional to its cross-sectional area. All materials show some resistance, except
for superconductors, which have a resistance of zero.
The resistance (R) of an object is defined as the ratio of voltage across it (V) to current through
it (I), while the conductance (G) is the inverse:
For a wide variety of materials and conditions, V and I are directly proportional to each other, and
therefore R and G are constant (although they can depend on other factors like temperature or strain).
This proportionality is called Ohm's law, and materials that satisfy it are called "Ohmic" materials.
In other cases, such as a diode or battery, V and I are not directly proportional, or in other words
the I–V curve is not a straight line through the origin, and Ohm's law does not hold. In this case,
resistance and conductance are less useful concepts, and more difficult to define. The ratio V/I is
sometimes still useful, and is referred to as a "chordal resistance" or "static resistance", as it
corresponds to the inverse slope of a chord between the origin and an I–V curve. In other situations,
the derivative
may be most useful; this is called the "differential resistance".
Conductors and Resistor
Substances in which electricity can flow are called conductors. A piece of conducting material of a
particular resistance meant for use in a circuit is called a resistor. Conductors are made of highconductivity materials such as metals, in particular copper and aluminum. Resistors, on the other hand,
are made of a wide variety of materials depending on factors such as the desired resistance, amount of
energy that it needs to dissipate, precision, and costs.
Relation to Resistivity and Conductivity
The resistance of a given object depends primarily on two factors: What material it is made of,
and its shape. For a given material, the resistance is inversely proportional to the cross-sectional area;
for example, a thick copper wire has lower resistance than an otherwise-identical thin copper wire.
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Also, for a given material, the resistance is proportional to the length; for example, a long copper wire
has higher resistance than an otherwise-identical short copper wire. The resistance Rand
conductance G of a conductor of uniform cross section, therefore, can be computed as
where is the length of the conductor, measured in metres [m], A is the cross-sectional area of
the conductor measured in square [m²], σ (sigma) is the electrical conductivity measured
in siemens per meter (S·m−1), and ρ (rho) is the electrical (also called specific electrical resistance) of
the material, measured in ohm-meters (Ω·m). The resistivity and conductivity are proportionality
constants, and therefore depend only on the material the wire is made of, not the geometry of the wire.
Resistivity and conductivity are reciprocals:
ability to oppose electric current.
. Resistivity is a measure of the material's
This formula is not exact, as it assumes the current density is totally uniform in the conductor,
which is not always true in practical situations. However, this formula still provides a good
approximation for long thin conductors such as wires.
Another situation for which this formula is not exact is with alternating current (AC), because
the skin effect inhibits current flow near the center of the conductor. For this reason,
the geometrical cross-section is different from the effective cross-section in which current actually
flows, so resistance is higher than expected. Similarly, if two conductors near each other carry AC
current, their resistances increase due to the proximity effect. At commercial power frequency, these
effects are significant for large conductors carrying large currents, such as busbars in an electrical
substation,[3] or large power cables carrying more than a few hundred amperes.
What determines resistivity?
The resistivity of different materials varies by an enormous amount: For example, the
conductivity of teflon is about 1030times lower than the conductivity of copper. Why is there such a
difference? Loosely speaking, a metal has large numbers of "delocalized" electrons that are not stuck
in any one place, but free to move across large distances, whereas in an insulator (like teflon), each
electron is tightly bound to a single molecule, and a great force is required to pull it away.
Semiconductors lie between these two extremes. More details can be found in the article: Electrical
resistivity and conductivity. For the case of electrolyte solutions, see the article: Conductivity
(electrolytic).
Resistivity varies with temperature. In semiconductors, resistivity also changes when exposed
to light. See.
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Measuring Resistance
An instrument for measuring resistance is called an ohmmeter. Simple ohmmeters cannot
measure low resistances accurately because the resistance of their measuring leads causes a voltage
drop that interferes with the measurement, so more accurate devices use four-terminal sensing.
Inductor
An inductor,
also
called
a coil or reactor,
is
a passive two-terminal electrical
component which resists changes in electric current passing through it. It consists of a conductor such
as a wire, usually wound into a coil. When a current flows through it, energy is stored temporarily in
a magnetic field in the coil. When the current flowing through an inductor changes, the time-varying
magnetic field induces a voltage in the conductor, according to Faraday’s law of electromagnetic
induction, which opposes the change in current that created it.
Fig 2.1: Different types of inductor
Capacitor
A capacitor (originally known as a condenser) is a passive two-terminal electrical
component used to store energy electrostatically in an electric field. The forms of practical capacitors
vary widely, but all contain at least two electrical conductors (plates) separated by a
dielectric (i.e. insulator). The conductors can be thin films, foils or sintered beads of metal or
conductive electrolyte, etc. The "nonconducting" dielectric acts to increase the capacitor's charge
capacity. A dielectric can be glass, ceramic, plastic film, air, vacuums, paper, mica, oxide layer etc.
Capacitors are widely used as parts of electrical circuits in many common electrical devices. Unlike
a resistor, an ideal capacitor does not dissipate energy. Instead, a capacitor stores energy in the form of
an electrostatic field between its plates.
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When there is a potential difference across the conductors (e.g., when a capacitor is attached
across a battery), an electric field develops across the dielectric, causing positive charge +Q to collect
on one plate and negative charge −Q to collect on the other plate. If a battery has been attached to a
capacitor for a sufficient amount of time, no current can flow through the capacitor. However, if a
time-varying voltage is applied across the leads of the capacitor, a displacement current can flow.
An ideal capacitor is characterized by a single constant value for its capacitance. Capacitance is
expressed as the ratio of the electric charge Q on each conductor to the potential difference V between
them. The SI unit of capacitance is the farad (F), which is equal to one coulomb per volt (1 C/V).
Typical capacitance values range from about 1 pF (10−12 F) to about 1 mF (10−3 F).
The capacitance is greater when there is a narrower separation between conductors and when the
conductors have a larger surface area. In practice, the dielectric between the plates passes a small
amount of leakage current and also has an electric field strength limit, known as the breakdown
voltage. The conductors and leads introduce an undesired inductance and resistance.
Capacitors are widely used in electronic circuits for blocking direct current while
allowing alternating current to pass. In analog filter networks, they smooth the output of power
supplies. In resonant circuits they tune radios to particular frequencies. In electric power
transmission systems, they stabilize voltage and power flow.
Ohm's law
Ohm's law states that the current through a conductor between two points is directly
proportional to the potential difference across the two points. Introducing the constant of
proportionality, the resistance, one arrives at the usual mathematical equation that describes this
relationship:
Where I is the current through the conductor in units of amperes, V is the potential difference
measured across the conductor in units of volts, and R is the resistance of the conductor in units
of ohms. More specifically, Ohm's law states that the R in this relation is constant, independent of the
current.
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Sign Convention
In electrical engineering, the passive sign convention (PSC) is a sign convention or arbitrary
standard rule adopted universally by the electrical engineering community for defining the sign
of electric power in an electric circuit. The convention defines electric power flowing out of the
circuit into an electrical component as positive, and power flowing into the circuit out of a component
as negative. So a passive component which consumes power, such as an appliance or light bulb, will
have positive power dissipation, while an active component, a source of power such as an electric
generator or battery, will have negative power dissipation. This is the standard definition of power in
electric circuits.
To comply with the convention, the direction of the voltage and current variables used to
calculate power and resistance in the component must have a certain relationship: the current variable
must be defined so positive current enters the positive voltage terminal of the device. These directions
may be different from the directions of the actual current flow and voltage.
Reference direction
The power flow p and resistance r of an electrical component are related to the voltage v and
current i variables by the defining equation for power and Ohm's law:
Like power, voltage and current are signed quantities. The current flow in a wire has two
possible directions, so when defining a current variable i the direction which represents positive
current flow must be indicated, usually by an arrow on the circuit diagram. This is called the reference
direction for current i. If the actual current is in the opposite direction, the variable i will have a
negative value. Similarly in defining a voltage variable v, the terminal which represents the positive
side must be specified, usually with an arrow or plus sign. This is called the reference direction for
voltagev.
To understand the PSC, it is important to distinguish the reference directions of the
variables, v and i, which can be assigned at will, from the direction of the actual voltage and current,
which is determined by the circuit. The idea of the PSC is that by assigning the reference direction of
variables v and i in a component with the right relationship, the power flow in passive components
calculated from Eq. (1) will come out positive, while the power flow in active components will come
out negative. It is not necessary to know whether a component produces or consumes power when
analyzing the circuit; reference directions can be assigned arbitrarily, directions to currents and
polarities to voltages, then the PSC is used to calculate the power in components. If the power comes
out positive, the component is a load, converting electric power to some other kind of power. If the
power comes out negative, the component is a source, converting some other form of power to electric
power.
Sign conventions
The above discussion shows that choosing the relative direction of the voltage and current
variables in a component determines the direction of power flow that is considered positive. The
reference directions of the individual variables are not important, only their relation to each other.
There are two choices:
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





Passive sign convention: Defining the current variable as entering the positive terminal means that
if the voltage and current variables have positive values, current flows from the positive to the
negative terminal, doing work on the component, as occurs in a passive component. So power
flowing into the component from the line is defined as positive; the power variable represents
power dissipation in the component. Therefore
Active components (power sources) will have negative resistance and negative power flow
Passive components (loads) will have positive resistance and positive power flow
This is the convention normally used.
Active sign convention: Defining the current variable as entering the negative terminal means that
if the voltage and current variables have positive values, current flows from the negative to the
positive terminal, so work is being done on the current, and power flows out of the component. So
power flowing out of the component is defined as positive; the power variable represents
power produced. Therefore:
Active components will have positive resistance and positive power flow
Passive components will have negative resistance and negative power flow
This convention is rarely used, except for special cases in power engineering.
In practice it is not necessary to assign the voltage and current variables in a circuit to comply with
the PSC. Components in which the variables have a "backward" relationship, in which the current
variable enters the negative terminal, can still be made to comply with the PSC by changing the sign of
the constitutive relations (1) and (2) used with them. A current entering the negative terminal is
equivalent to a negative current entering the positive terminal, so in such a component
, and
Kirchhoff’s Laws
Kirchhoff’s laws are basic analytical tools in order to obtain the solutions of currents and
voltages for any electric circuit; whether it is supplied from a direct-current system or an alternating
current system. But with complex circuits the equations connecting the currents and voltages may
become so numerous that much tedious algebraic work is involve in their solutions. Elements that
generally encounter in an electric circuit can be interconnected in various possible ways. Before
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discussing the basic analytical tools that determine the currents and voltages at different parts of the
circuit, some basic definition of the following terms are considered.
Fig 3.1: A simple resistive circuit




Node- A node in an electric circuit is a point where two or more components are connected
together. This point is usually marked with dark circle or dot. The circuit in fig. 3.4 has nodes a, b,
c, and g. Generally, a point, or a node in a circuit specifies a certain voltage level with respect to a
reference point or node.
Branch- A branch is a conducting path between two nodes in a circuit containing the electric
elements. These elements could be sources, resistances, or other elements. Fig.3.4 shows that the
circuit has six branches: three resistive branches (a-c, b-c, and b-g) and three branches containing
voltage and current sources (a-, a-, and c-g).
Loop- A loop is any closed path in an electric circuit i.e., a closed path or loop in a circuit is a
contiguous sequence of branches which starting and end points for tracing the path are, in effect,
the same node and touches no other node more than once. Fig. 3.4 shows three loops or closed
paths namely, a-b-g-a; b-c-g-b; and a-c-b-a. Further, it may be noted that the outside closed paths
a-c-g-a and a-b-c-g-a are also form two loops.
Mesh- a mesh is a special case of loop that does not have any other loops within it or in its interior.
Fig. 3.4 indicates that the first three loops (a-b-g-a; b-c-g-b; and a-c-b-a) just identified are also
‘meshes’ but other two loops (a-c-g-a and a-b-c-g- a) are not.
With the introduction of the Kirchhoff’s laws, a various types of electric circuits can be analyzed.
Kirchhoff’s Voltage Law or Kirchhoff’s loop rule
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Fig 3.2: Circuit diagram for the KVL
This law is also called Kirchhoff’s loop rule. It is a consequence of the principle of
conservation of energy. It states that “The algebraic sum of the potential differences around a circuit
must be zero".
Considering that electric potential is defined as line integral of electric field, Kirchhoff's voltage law
can be expressed equivalently with equation
Kirchhoff’s Current Law (KCL)
Fig 3.3: Circuit diagram for the KCL
This law is also called Kirchhoff's junction rule or Kirchhoff's point rule. This law states
that “At any point in an electrical circuit where charge density is not changing in time, the sum of
currents flowing towards that point is equal to the sum of currents flowing away from that point.
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A charge density changing in time would mean the accumulation of a net positive or negative charge,
which typically cannot happen to any significant degree because of the strength of electrostatic forces:
the charge buildup would cause repulsive forces to disperse the charges.
KCL states that at any node (junction) in a circuit the algebraic sum of currents entering and
leaving a node at any instant of time must be equal to zero. Here currents entering (+ve sign) and
currents leaving (-ve sign) the node must be assigned opposite algebraic signs.
Fig 3.2: Illustration of Kirchhoff’s laws
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This law is also called Kirchhoff's second law, Kirchhoff's loop (or mesh) rule, and Kirchhoff's
second rule.
 The principle of conservation of energy implies that
 The directed sum of the electrical potential differences (voltage) around any closed network is zero,
or:
 More simply, the sum of the emfs in any closed loop is equivalent to the sum of the potential drops in
that loop, or:
 The algebraic sum of the products of the resistances of the conductors and the currents in them in a
closed loop is equal to the total emf available in that loop.
Similarly to KCL, it can be stated as:
Here, n is the total number of voltages measured. The voltages may also be complex:
This law is based on the conservation of energy whereby voltage is defined as the energy per
unit charge. The total amount of energy gained per unit charge must be equal to the amount of energy
lost per unit charge, as energy and charge are both conserved.
Example:
Assume an electric network consisting of two voltage sources and three resistors.
According to the first law we have
The second law applied to the closed circuit s1 gives
The second law applied to the closed circuit s2 gives
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Thus we get a linear system of equations in
:
Assuming
the solution is
has a negative sign, which means that the direction of
direction defined in the picture).
is opposite to the assumed direction (the
1. Solve the circuit by nodal analysis and find Va .
Solution
a) Choose a reference node, label node voltages:
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b) Apply KCL to each node:
Node 1:
−Is2+ (V1−V2 ) /R3+(V1−V3) /R1 = 0 → 6V1−V2−5V =5
Node 2:
Is3+V2−V1R3+V2R2=0→−4V1+9V2=−40
(1)
(2)
Node 3:
−Is1−Is3+(V3−V1 ) /R1=0→V3−V1=4
(1), (2) and (3) imply that V1=37V,V2=12V and V3=41V .
(3)
c) Find the required quantities:
If we apply KVL in the loop shown above:
−V1−Va+V2=0→Va=−25V.
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Practical Voltage and Current Sources
Ideal and Practical Voltage Sources
 An ideal voltage source, which is represented by a model, is a device that produces a constant
voltage across its terminals (V=E) no matter what current is drawn from it (terminal voltage is
independent of load (resistance) connected across the terminals).
Fig 5.1: Ideal DC voltage source
For the circuit shown in fig, the upper terminal of load is marked plus (+) and its lower terminal is
marked minus (-). This indicates that electrical potential of upper terminal is volts higher than that of
lower terminal. The current flowing through the load RL is given by the expression
VS =VL
=ILRL and we can represent the terminal V−I characteristic of an ideal dc voltage as a straight line
parallel to the x-axis. This means that the terminal voltage VL remains constant and equal to the source
voltage VS irrespective of load current is small or large. The V−I characteristic of ideal voltage source
is presented in Figure.
 However, real or practical dc voltage sources do not exhibit such characteristics in practice. We
observed that as the load resistance RL connected across the source is decreased, the corresponding load
current IL increases while the terminal voltage across the source decreases (see eq.3.1). We can realize
such voltage drop across the terminals with increase in load current provided a resistance element (Rs)
present inside the voltage source. Figshows the model of practical or real voltage source of value Vs.
Fig 5.2: V-I characteristics of an Ideal voltage source
The terminal V− I characteristics of the practical voltage source can be described by an equation
VL = Vs - ILRs
and
this equation is represented graphically as shown in fig.3.16. In practice, when a load resistance RL
more than 100 times larger than the source resistance Rs, the source can be considered approximately
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ideal voltage source. In other words, the internal resistance of the source can be omitted. This
statement can be verified using the relation RL = 100 Rs in equation. The practical voltage source is
characterized by two parameters namely known as (i) Open circuit voltage (Vs) (ii) Internal resistance
in the source’s circuit model. In many practical situations, it is quite important to determine the source
parameters experimentally. We shall discuss briefly a method in order to obtain source parameters.
Fig 5.3: Practical dc voltage source
Method-: Connect a variable load resistance across the source terminals (see fig.). A voltmeter is
connected across the load and an ammeter is connected in series with the load resistance. Voltmeter
and Ammeter readings for several choices of load resistances are presented on the graph paper (see fig.
5.4). The slope of the line is −Rs, while the curve intercepts with voltage axis (at IL = 0) is the value of
Vs.
Fig 5.4: V−I characteristic Practical dc voltage source
The V−I characteristic of the source is also called the source’s “regulation curve” or “load line”. The
open-circuit voltage is also called the “no-load” voltage, Voc. The maximum allowable load current
(rated current) is known as full-load current IFl and the corresponding source or load terminal voltage
is known as “full-load” voltage. We know that the source terminal voltage varies as the load is varied
and this is due to internal voltage drop inside the source. The percentage change in source terminal
voltage from no-load to full-load current is termed the “voltage regulation” of the source. It is defined
as
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For ideal voltage source, there should be no change in terminal voltage from no-load to full-load and
this corresponds to “zero voltage regulation”. For best possible performance, the voltage source should
have the lowest possible regulation and this indicates a smallest possible internal voltage drop and the
smallest possible internal resistance.
Example: - A practical voltage source whose short-circuit current is 1.0A and open-circuit voltage is
24 Volts. What is the voltage across, and the value of power dissipated in the load resistance when this
source is delivering current 0.25A?
Solution:
From fig.
(Short-circuit test)
(Open-circuit test)
Therefore, the value of internal source resistance is obtained as
Let us assume that the source is delivering current IL = 0.25A when the load resistance RL is connected
across the source terminals. Mathematically, we can write the following expression to obtain the load
resistance RL.
.
Now, the voltage across the load RL = 0.25 × 72 =18 volts.
And the power consumed by the load is given by PL = IL× RL =0.625 × 72 = 4.5 watts
Example- A certain voltage source has a terminal voltage of 50 V when I= 400 mA; when I rises to its
full-load current value 800 mA the output voltage is recorded as 40 V. Calculate (i) Internal resistance
of the voltage source (Rs). (ii) No-load voltage (open circuit voltage Vs). (iii) The voltage Regulation.
Solution- From equation (VL = Vs - ILRs) one can write the following expressions under different
loading conditions.
50 = Vs − 0.4Rs & 40 = Vs − 0.8Rs → solving these equations we get Vs = 60V & Rs =25 Ω
=
60  40
 100  33.33%
60
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Ideal and Practical Current Sources
There are several voltage sources as well as current sources encountered in our daily
life. Batteries , DC generator or alternator all are very common examples of voltage source. There are
also some current sources encountered in our everyday life, such as photo electric cells,
metadynegenerator etc.
The sources can be categorized into two different types – independent source and dependent source.
• Another two-terminal element of common use in circuit modeling is current source` as depicted in
fig.3.17. An ideal current source, which is represented by a model in fig., is a device that delivers a
constant current to any load resistance connected across it, no matter what the terminal voltage is
developed across the load (i.e., independent of the voltage across its terminals across the terminals).
Fig 5.5: Ideal current source with variable load
It can be noted from model of the current source that the current flowing from the source to the
load is always constant for any load resistance (see fig.) i.e. whether R L is small (VL is small) or RL is
large (VL is large). The vertical dashed line in fig. 5.6 represents the V-I characteristic of ideal current
source.
Fig 5.6: V-I characteristic of ideal current source
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Fig 5.7: Practical current source with variable load
In practice when a load is connected across a practical current source, one can observe that the
current flowing in load resistance is reduced as the voltage across the current source’s terminal is
increased, by increasing the load resistance RL. Since the distribution of source current in two parallel
paths entirely depends on the value of external resistance that connected across the source (current
source) terminals. This fact can be realized by introducing a parallel resistance R s in parallel with the
practical current source Is, as shown in fig. 5.7. The dark lines in fig.5.6 show the V-I characteristic
(load-line) of practical current source. The slope of the curve represents the internal resistance of the
source. One can apply KCL at the top terminal of the current source in fig. 5.7 to obtain the following
expression.
The open circuit voltage and the short-circuit current of the practical current source are given by
VOC = ISRS and Ishort = Is respectively. It can be noted from the fig.5.6 that source 1 has a larger
internal resistance than source 2 and the slope the curve indicates the internal resistance R s of the
current source. Thus, source 1 is closer to the ideal source. More specifically, if the source internal
resistance Rs ≥100 RL then source acts nearly as an ideal current source.
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Independent Voltage Source
Output of an independent source does not depend upon the voltage or current of any other part
of the network. When terminal voltage of a voltage source is not affected by the current or voltage of
any other part of the network, then the source is said to be an independent voltage source . This type
of sources may be referred as constant source or time variant source. When terminal voltage of an
independent source remains constant throughout its operation, it is referred as time–invariant or
constant independent voltage source .
Again independent voltage source can be time–variant type, where the output terminal voltage of the
source changes with time. Here, the terminal voltage of the source does not vary with change
of voltage or current of any other part of the network but it varies with time.
Independent Current Source
Similarly,
output current of
independent current
source does
not
depend
upon
the voltage or current of any other part of the network. It is also categorized as independent timeinvariant and time-variant current source .
Symbolic representations of independent time-invariant and time-variant voltage and current source s
are shown below.
Fig 6.1: Different types of independent sources
Now we will discuss about dependent voltage or current source. Dependent voltage source is
one that’s output voltage is the function of voltage or current of any other part of the circuit. Similarly,
dependent current source is one that’s output current is the function of current or voltage of any other
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parts of the circuit. The amplifier is an ideal example of dependent source where the output signal
depends upon the signal given to the input circuit of the amplifier.
Dependent Voltage Source & Dependent Current Source
There are four possible dependent sources as are represented below,
1. Voltage dependent voltage source .
2. Current dependent voltage source .
3. Voltage dependent current source .
4. Current dependent current source .
Dependent voltage source s and dependent current sources can also be time variant or time invariant.
That means, when the output voltage or current of a dependent source is varied with time, referred as
time invariant dependent current or voltage source and if not varied with time, it is referred as time
variant.
a. VCVS
b. ICVS
c. VCIS
d. ICIS
Fig 6.2: different types of ideal dependent sources
Ideal Voltage Source
In every practical voltage source , there is some electrical resistance inside it. This resistance is
called internal resistance of the source. When the terminal of the source is open circuited, there is
no current flowing through it; hence there is no voltage drop inside the source but when load is
connected with the source, current starts flowing through the load as well as the source itself. Due to
the resistance inside the voltage source , there will be some voltage drop across the source. Now if any
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one measures the terminal voltage of the source, he or she will get the voltage between its terminals
which is reduced by the amount of internal voltage drop of the source. So there will be always a
difference between no-load (when source terminals are open) and load voltages of a practical voltage
source . But in ideal voltage source this difference is considered as zero that means there would not be
any voltage drop in it when current flows through it and this implies that the internal resistance of an
ideal source must be zero. This can be concluded that, voltage across the source remains constant for
all values of load current.
The V-I characteristics of an ideal voltage source is shown below.
Fig 6.3: VI Characteristics of Ideal Voltage Source
There is no as such example of ideal voltage source but a lead acid battery or a dry cell can be
considered an example when the current drawn is below a certain limit.
Ideal Current Source
Ideal current sources are those sources that supply constant current to the load irrespective of their
impedance. That means, whatever may be the load impedance; ideal current source always
gives same current through it. Even if the load has infinite impedance or load, is open circuited to the
ideal current source that gives the same current through it. So naturally from definition, it is clear that
this type of current source is not practically possible.
Current Source to Voltage Source Conversion
All sources of electrical energy give both current as well as voltage. This is not practically
possible to distinguish between voltage source and current source . Any electrical source can be
represented as voltage source as well as current source . It merely depends upon the operating
condition. If the load impedance is much higher than internal impedance of the source, then it is
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preferable to consider the source as a voltage source on the other hand if the load impedance is much
lower than internal impedance of the source; it is preferable to consider the source as a current source .
Current source to voltage source conversion or voltage source to current source conversion is always
possible. Now we will discuss how to convert a current source into voltage source and vice-versa.
Let us consider a voltage source which has no load terminal voltage or source voltage V and
internal resistance r. Now we have to convert this to an equivalent current source . For that, first we
have to calculate the current which might be flowing through the source if the terminal A and B of the
voltage source were short circuited. That would be nothing but I = V / r. This current will be supplied
by the equivalent current source and that source will have the same resistance connected across it.
Fig 6.4: Voltage to Current source conversion
Similarly a current source of output current I in parallel with resistance r can be converted into an
equivalent voltage source of voltage V = Ir and resistance r connected in series with it.
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Nodal Voltage Analysis
In this method, we set up and solve a system of equations in which the unknowns are the voltages at
the principal nodes of the circuit. From these nodal voltages the currents in the various branches of the
circuit are easily determined.
Steps in the nodal analysis method:
1.
Count the number of principal nodes or junctions in the circuit. Call this number n. (A principal
node or junction is a point where 3 or more branches join. We will indicate them in a circuit diagram
with a red dot. Note that if a branch contains no voltage sources or loads then that entire branch can be
considered to be one node.)
2. Number the nodes N1, N2, . . . , Nn and draw them on the circuit diagram. Call the voltages at these
nodes V1, V2, . . . , Vn, respectively.
3. Choose one of the nodes to be the reference node or ground and assign it a voltage of zero.
4. For each node except the reference node write down Kirchhoff’s Current Law in the form "the
algebraic sum of the currents flowing out of a node equals zero". (By algebraic sum we mean that a
current flowing into a node is to be considered a negative current flowing out of the node.)
Express the current in each branch in terms of the nodal voltages at each end of the branch using
Ohm's Law (I = V / R). Here are some examples:
Fig 7.1: Nodal voltage analysis for given branches
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The result, after simplification, is a system of m linear equations in the m unknown nodal voltages
(where m is one less than the number of nodes; m = n - 1). The equations are of this form:
where G11, G12, . . . , Gmm and I1, I2, . . . , Im are constants.
Example :
Fig 7.2: Circuit diagram for analysis of the theorem in the example
Number of nodes is 4.
We will number the nodes as shown above.
We will choose node 2 as the reference node and assign it a voltage of zero.
Write down Kirchhoff’s Current Law for each node. Call V1 the voltage at node 1, V3 the voltage at
node 3, V4 the voltage at node 4, and remember that V2 = 0. The result is the following system of
equations:
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The first equation results from KCL applied at node 1, the second equation results from KCL
applied
at
node
3
and
the
third
equation
results
from
KCL
applied
at
node
4.
Solving the above system of equations using Gaussian elimination or some other method gives the
following voltages:
V1 = -35.88 volts,
V3 = 63.74 volts and
V4 = 0.19 volts
Mesh or Loop Analysis for Electric Circuits
This is also called Loop Analysis. This method uses simultaneous equations, Kirchhoff's
Voltage Law, and Ohm's Law to determine unknown currents in a network.
Mesh Analysis only works for planar circuits: circuits that can be drawn on a plane (like on a paper)
without any elements or wires crossing each other. In some cases a circuit that looks non-planar can be
made in to a planar circuit by moving some of the connecting wires.
The first step in the mesh Current method is to identify "loops" within the circuit encompassing
all components. Represent all the loops with different loop currents in one direction. The choice of
each loop current's direction is entirely arbitrary.
The next step is to label all voltage drop polarities across resistors according to the assumed
directions of the mesh currents.
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Next write the KVL equations for each mesh and solve all the equations for mesh (loop)
currents.
Example:
Fig 7.3: Circuit diagram for analysis of the theorem in the example
KVL equation for Loop1: -28 + 2(I1+I2) + 4×I1 =0.
KVL equation for Loop2: -2(I1+I2) + 7 - 1I2=0.
Solving these 2 equations we get I1=5A I2=-1A.
The solution of -1 amp for I2 means that our initially assumed direction of current was
incorrect. In actuality, I2 is flowing in a counter-clockwise direction at a value of (positive) 1 amp.
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Superposition Theorem
The superposition theorem for electric circuits states that the total current in any branch of a
bilateral linear circuit equals the algebraic sum of the currents produced by each source acting
separately throughout the circuit.
To ascertain the contribution of each individual source, all of the other sources first must be
"killed" (set to zero) by:

replacing all other voltage sources with a short circuit (thereby eliminating difference of
potential. i.e. V=0)

replacing all other current sources with an open circuit (thereby eliminating flow of
current. i.e. I=0)
This procedure is followed for each source in turn, and then the resultant currents are added to
determine the true operation of the circuit. The resultant circuit operation is the superposition of the
various voltage and current sources.
Example:
Fig 8.1: Circuit Diagram for analysis of the superposition theorem
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Fig 8.2: Circuit Diagram for application of the superposition theorem
Fig 8.3: Circuit Diagram for analyzing of the superposition theorem for only 7V
Fig 8.4: Circuit Diagram for analyzing of the superposition theorem for only 28V
When superimposing these values of voltage and current, we have to be very careful to
consider polarity (voltage drop) and direction (electron flow), as the values have to be added
algebraically.
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Fig 8.5: Circuit after superimposing voltage
Fig 8.6: Circuit after superimposing voltage
Thus the total voltage or current can be obtained in the circuit by just recombining the circuits
with different sources.
Another point that should be considered is that superposition only works for voltage and
current but not power. In other words the sum of the powers of each source with the other sources
turned off is not the real consumed power. To calculate power we should first use superposition to find
both current and voltage of each linear element and then calculate the sum of the multiplied voltages
and currents.
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Thevenin's Theorem:
This theorem was first discovered by German scientist Hermann von Helmholtz in 1853, but
was then rediscovered in 1883 by French telegraph engineer Léon Charles Thévenin (1857-1926).
Fig 9.1: Circuit diagram for conversion of any given circuit to a Thevenin’s Equivalent circuit
Any black box containing only voltage source, current source or resisters can be converted to
Thevenin’s Equivalent circuit.
This theorem states that any linear bilateral circuit with combination of voltage sources ,
current sources and resistors with two terminals is electrically equivalent to a single voltage source V Th
(Thevenin’s Voltage) and a single series resistor RTh (Thevenin’s Resistance).This equivalent is called
Thevenin’s Equivalent.
Calculating the Thevenin’s equivalent:
To calculate the equivalent circuit, one needs a resistance and a voltage - two unknowns. And
so, one needs two equations. These two equations are usually obtained by using the following steps,
but any conditions one places on the terminals of the circuit should also work:
1.
Calculate the output voltage, VAB, when in open circuit condition (no load resistor -
meaning infinite resistance). This is VTh.
2.
Calculate the output current, IAB, when those leads are short circuited (load resistance
is 0) RTh equals VTh divided by this IAB.
Case 2 could also be thought of like this:
2a. Now replace voltage sources with short circuits and current sources with open circuits.
2b. Replace the load circuit with an imaginary ohm meter and measure the total resistance, R,
"looking back" into the circuit. This is RTh.
The Thevenin’s-equivalent voltage is the voltage at the output terminals of the original circuit.
When calculating a Thevenin’s-equivalent voltage, the voltage divider principle is often useful, by
declaring one terminal to be Vout and the other terminal to be at the ground point.
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The Thevenin’s-equivalent resistance is the resistance measured across points A and B "looking back"
into the circuit. It is important to first replace all voltage- and current-sources with their internal
resistances. For an ideal voltage source, this means replace the voltage source with a short circuit. For
an ideal current source, this means replace the current source with an open circuit. Resistance can then
be calculated across the terminals using the formulae for series and parallel circuits.
A Norton equivalent circuit is related to the Thevenin’s equivalent by the following equations:
RTh = RNo , VTh = INo RNo
Example of a Thevenin’s equivalent circuit
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In the example, calculating equivalent voltage and resistance:
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Norton's Theorem :
Norton's theorem is an extension of Thevenin's theorem and was introduced in 1926 separately
by two people: Hause-Siemens researcher Hans Ferdinand Mayer (1895-1980) and Bell Labs engineer
Edward Lawry Norton (1898-1983). Mayer was the only one of the two who actually published on this
topic, but Norton made known his finding through an internal technical report at Bell Labs.
Fig 10.1: Circuit diagram for conversion of any given circuit to a Norton's Equivalent circuit
Any black box containing only voltage source, current source or resisters can be converted to
Norton's Equivalent circuit.
This theorem states that any linear bilateral circuit with combination of voltage sources, current
sources and resistors with two terminals is electrically equivalent to an ideal current soure, INo, in
parallel
with
a
single
resistor,
RNo.
This
equivalent
is
called
Norton
Equivalent.
Calculation of Norton Equivalent
To calculate the equivalent circuit:
1.
Calculate the output current, IAB, when a short circuit is the load (meaning 0 resistance
between A and B). This is INo.
2.
Calculate the output voltage, VAB, when in open circuit condition (no load resistor -
meaning infinite resistance). RNo equals this VAB divided by INo.
The equivalent circuit is a current source with current INo, in parallel with a resistance RNo.
Case 2 can also be thought of like this:
2a. Now replace independent voltage sources with short circuits and independent current
sources with open circuits.
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2b. For circuits without dependent sources RNo is the total resistance with the independent
sources removed.
* Note: A more general method for determining the Norton Impedance is to connect a current
source at the output terminals of the circuit with a value of 1 Ampere and calculate the voltage at its
terminals; this voltage is equal to the impedance of the circuit. This method must be used if the circuit
contains
dependent
sources.
This
method
is
not
shown
below
in
the
diagrams.
To convert to a Thevenin’s equivalent circuit, one can use the following equations:
RTh = RNo
VTh = INo RNo
Example of Norton Equivalent circuit
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Maximum Power Transfer Theorem
This theorem was first discovered by Moritz von Jacobi which is referred to as "Jacobi’s law".
The theorem states that in a circuit maximum power is transferred from source to load when the
resistance of the load is same as that of the source.
The theorem applies only when the source resistance is fixed. If the source resistance were
variable (but the load resistance fixed), maximum power would be transferred to the load simply by
setting the source resistance to zero. Raising the source impedance to match the load would, in this
case, reduce power transfer. This is the case when driving a load such as a loudspeaker with a modern
amplifier. In this case, the load presented by the loudspeaker is fixed (typically, 8 ohms for home
audio) and maximum power occurs with an impedance bridging connection. This type of connection
also serves to maximize control of the speaker cone (due to high damping factor), which serves to
lower distortion.
It is important to note that maximum efficiency is not the same as maximum power transfer. To
achieve maximum efficiency, the resistance of the source (whether a battery or a dynamo) could be
made close to zero.
Fig 11.1: circuit for analysis of Maximum power Transfer
The condition of maximum power transfer does not result in maximum efficiency. If we define
the efficiency η as the ratio of power dissipated by the load to power developed by the source, then it is
straightforward to calculate from the circuit diagram that
Consider three particular cases :
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
If RLoad = RSource then η = 0.5

If RLoad = infinity then η = 1

If RLoad = 0 then η = 0
The efficiency is only 50% when maximum power transfer is achieved, but approaches 100% as the
load resistance approaches infinity (though the total power level tends towards zero). When the load
resistance is zero, all the power is consumed inside the source (the power dissipated in a short circuit is
zero) so the efficiency is zero.
Voltage Source :
When a load resistance RL is connected to a voltage source VS with series resistance RS,
maximum
power
transfer
to
the
load
occurs
when
RL
is
equal
to
RS.
Under maximum power transfer conditions, the load resistance R L, load voltage VL, load current IL and
load power PL are:
RL = RS
VL =
IL =
VS
2
VS
VL
=
2  RS
RL
PL = V L ×
VS
VL
= VS ×
4  RS
RL
Current Source:
Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering
46
Basic Electrical Engineering Class notes
When a load conductance GL is connected to a current source IS with shunt conductance GS,
maximum
power
transfer
to
the
load
occurs
when
GL
is
equal
to
GS.
Under maximum power transfer conditions, the load conductance GL, load current IL, load voltage V L
and load power PL are:
GL = GS
IL =
VL =
IS
2
IS
IL
=
2  GS
GL
PL = IL×
IS
IL
= IS ×
4  GS
GL
Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering
47
Basic Electrical Engineering Class notes
Y-Δ transformation & Δ-Y transformation
Delta-Star Transformation
The Y-Δ transform, also written Y-delta, Wye-delta, Kennelly’s delta-star transformation, starmesh transformation, T-Π or T-pi transform, is a mathematical technique to simplify the analysis of an
electrical network. The name derives from the shapes of the circuit diagrams, which look respectively
like the letter Y and the Greek capital letter Δ. In the United Kingdom, the wye diagram is known as a
star.
The transformation is used to establish equivalence for networks with 3 terminals. Where three
elements terminate at a common node and none are sources, the node is eliminated by transforming the
impedances. For equivalence, the impedance between any pair of terminals must be the same for both
networks. The equations given here are valid for real as well as complex impedances.
Equations
for
the
transformation
from
Δ-load
to
Y-load
3-phase
circuit
the general idea is to compute the impedance Ry at a terminal node of the Y circuit with
impedances
where
RΔ
R',
are
all
R''
to
impedances
adjacent
in
the
Δ
nodes
in
the
circuit.
This
yields
Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering
Δ
the
circuit
specific
by
formulae
48
Basic Electrical Engineering Class notes
Equations for the transformation from Y-load to Δ-load 3-phase circuit
The
general
idea
is
to
compute
an
impedance
RΔ
in
the
Δ
circuit
by
Where Rp = R1R2 + R2R3 + R3R1 is the sum of the products of all pairs of impedances in the Y
circuit and Ropposite is the impedance of the node in the Y circuit which is opposite the edge with RΔ.
The
formulae
for
the
individual
Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering
edges
are
thus
49