Unit II Electrical Circuits

An electric circuit is a path in which electrons from a voltage or current source flow. The point
where those electrons enter an electrical circuit is called the "source" of electrons. In other
words a circuit is a path between two or more points along which an electrical current can be
carried. This means that a circuit is an unbroken loop of conductive material that allows
electrons to flow through continuously without beginning or end. If a circuit is “broken,” that
means its conductive elements no longer form a complete path, and continuous electron flow
cannot occur in it.

Basic Circuit element and their combinations:

The interconnection of various Electrical Circuit Elements in a prescribed manner to form a

closed path is called an electric circuit. The system in which electric current can flow from
source to load through one path and after delivering energy at load the current can return to the
other terminal of source through another path is referred as electric circuit. The main parts of
an ideal electric circuit are

1. Electrical Sources (for delivering electricity to the circuit and these are mainly
electric generators and batteries)
2. Controlling Devices (for controlling electricity and these are mainly switches, circuit
breakers, MCBs, and Potentiometer like devices etc.)
3. Protection Devices (for protecting the circuit from abnormal conditions and these
are mainly electric fuses, MCBs, Switchgear systems).
4. Conducting Path (to carry current one point to other in the circuit and these are
mainly wires or conductors)
5. Load

Thus Voltage and Current are the two basic features of an Electric Element. Various
techniques by which voltage and current across any element in any electric circuit are
determined is called Electric Circuit Analysis.

In this figure shows a simple Electric Circuit Containing

• A battery of 30 V
• A carbon resistor of 5kΩ

Due to this a current I flows in circuit and a potential drop of V volts across resistor.
Basic Properties of Electric Circuits

• A circuit is always a closed path.

• A circuit always contains an energy source which acts as source of electrons.
• The electric elements include uncontrolled and controlled source of energy,
resistors, capacitors, inductors, etc.
• In an electric circuit flow of electrons takes place from negative terminal to positive
terminal.
• Direction of flow of conventional current is from positive to negative terminal.
• Flow of current leads to potential drop across the various elements.

Types of Electric Circuit

The electric circuit can be categorized in three different ways

1. Open circuit
2. Closed circuit
3. Short circuit

Open Circuit

If due to disconnection of any part of an electric circuit if there is no flow of current the circuit
is said to be open circuited.

Closed Circuit

If there is no discontinuity in the circuit and current can flow from one part to another part of
the circuit then the circuit is said to be closed circuit.
Short Circuit

If two or more phases, one or more phases and earth or neutral of AC system or positive and
negative wires or positive or negative wires and earth of DC system touch together directly or
connected together by a zero impedance path then the circuit is said to be short circuited.

Electric circuits can further be categorised according to their structural features.

1. Series Circuit
2. Parallel Circuit
3. Series Parallel Circuit.

Series Circuit

When all elements of a circuit are connected one after another in tail to head fashion and due
to which there will be only one path of flowing current then the circuit is called series circuit.
The circuit elements are said to be series connected. In series electrical circuit same current
flows through all element connected in series.

Parallel Circuit

If components are connected in such a way that the voltage drop across each component is
same is known as parallel circuit. In parallel circuit the voltage drop across each component is
same but the current flowing through each component is different. The total current is the sum
of currents flowing through each element. An example of a parallel circuit is the wiring system
of a house. If one of the lights burns out, current can still flow through the rest of the lights and
appliances. In a parallel circuit the voltage is the same for all elements. When the resistors are
connected in parallel:- To find the total resistance of all components, add the reciprocals of the
resistances of each component and take the reciprocal of the sum. When the inductors are
connected in parallel:- Total inductance of non-coupled inductors in parallel is equal to the
reciprocal of the sum of the reciprocals of their individual inductances. When the capacitors
are connected in parallel:- The capacitors connected in parallel acts as series combination. The
total capacitance of capacitors in parallel is equal to the sum of their individual capacitances.

Types of Electric Circuit Elements

• Passive Element

The element which receives energy (or absorbs energy) and the either converts it into heat or
stored it in an electric or magnetic field called Passive Element. Example: Resistor, Inductor,
Capacitor etc.

• Active Element

The elements which supply energy to the circuit is called Active Element. Examples: Include
Voltage and Current sources, Generators etc. Note: A transistor is an active circuit element,
meaning that it can amplify the power of a signal. On the other hand, Transformer is not an
active element because it does not amplify the power and power remains same both in primary
sides and secondary sides. Transformer is an example of passive element.

In summary of:

Active: Those devices or components which produce energy in the form of Voltage or Current
are called as Active Components

Passive: Those devices or components which store or maintain Energy in the form of Voltage
or Current are known as Passive Components

What is the difference between active and passive components?

1. Active devices inject power to the circuit, whereas passive devices are incapable of
supplying any energy

2. Active devices are capable of providing power gain, and passive devices are incapable
of providing power gain.
 3. Active devices can control the current (energy) flow within the circuit, whereas passive
devices cannot control it.

• Bilateral Element

Conduction of current in both directions in an element with same magnitude is termed as

Bilateral Element. Example: Resistor, Inductor, Capacitor etc.

This
figure shows a bilateral element can conduct from both sides and offers same resistance for
current from either side

• Unilateral Element

Conduction of current in one direction is termed as Unilateral Element. Example: Diode,

Transistor etc.

This figure shows the unilateral element. When diode is forward biased it offers very small
resistance and conducts. While when reverse biased it offers very high resistance and don't
conduct.

• Lumped Elements

When the voltage across and current through the element doesn't vary with dimension of
element, it is called Lumped Elements. Example: Resistor connected in any electrical circuit.

• Distributed Element

When the voltage across and current through the element changes with dimension of element,
it is called Distributed Element. Example: Resistance of the transmission line.

Types of system available in electric circuit:

There are two types of system available in electric circuit, single phase and three phase
system. In single phase circuit, there will be only one phase, i.e the current will flow through
only one wire and there will be one return path called neutral line to complete the circuit. So in
single phase minimum amount of power can be transported. Here the generating station and
load station will also be single phase. This is an old system using from previous time.In 1882,
new invention has been done on polyphase system, that more than one phase can be used for
generating, transmitting and for load system. Three phase circuit is the polyphase system
where three phases are send together from the generator to the load. Each phase are having a
phase difference of 120°, i.e 120° angle electrically. So from the total of 360°, three phases are
equally divided into 120° each. The power in three phase system is continuous as all the three
phases are involved in generating the total power. The sinusoidal waves for 3 phase system is
shown below-
The three phases can be used as single phase each. So if the load is single phase, then one phase
can be taken from the three phase circuit and the neutral can be used as ground to complete
the circuit.

Why Three Phase is preferred Over Single Phase?

There are various reasons for this question because there are numbers of advantages over single
phase circuit. The three phase system can be used as three single phase line so it can act as
three single phase system. The three phases generation and single phase generation is same in
the generator except the arrangement of coil in the generator to get 120° phase difference. The
conductor needed in three phase circuit is 75% that of conductor needed in single phase circuit.
And also the instantaneous power in single phase system falls down to zero as in single phase
we can see from the sinusoidal curve but in three phase system the net power from all the
phases gives a continuous power to the load.
Till now we can say that there are three voltage source connected together to form a three phase
circuit and actually it is inside the generator. The generator is having three voltage sources
which are acting together in 120° phase difference. If we can arrange three single phase circuit
with 120° phase difference, then it will become a three phase circuit. So 120° phase difference
is must otherwise the circuit will not work, the three phase load will not be able to get active
and it may also cause damage to the system.
The size or metal quantity of three phase devices is not having much difference. Now if we
consider the transformer, it will be almost same size for both single phase and three phase
because transformer will make only the linkage of flux. So the three phase system will have
higher efficiency compared to single phase because for the same or little difference in mass of
transformer, three phase line will be out whereas in single phase it will be only one. And losses
will be minimum in three phase circuit. So overall in conclusion the three phase system will
have better and higher efficiency compared to the single phase system.
In three phase circuit, connections can be given in two types:

1. Star connection

2. Delta connection

Star Connection

In star connection, there is four wire, three wires are phase wire and fourth is neutral which is
taken from the star point. Star connection is preferred for long distance power transmission
because it is having the neutral point. In this we need to come to the concept of balanced and
unbalanced current in power system.
When equal current will flow through all the three phases, then it is called as balanced current.
And when the current will not be equal in any of the phase, then it is unbalanced current. In
this case, during balanced condition there will be no current flowing through the neutral line
and hence there is no use of the neutral terminal. But when there will be unbalanced current
flowing in the three phase circuit, neutral is having a vital role. It will take the unbalanced
current through to the ground and protect the transformer. Unbalanced current affects
transformer and it may also cause damage to the transformer and for this star connection is
preferred for long distance transmission.
The star connection is shown below-
In star connection, the line voltage is √3 times of phase voltage. Line voltage is the voltage
between two phases in three phase circuit and phase voltage is the voltage between one phases
to the neutral line. And the current is same for both line and phase. It is shown as expression
below

Delta Connection

In delta connection, there is three wires alone and no neutral terminal is taken. Normally delta
connection is preferred for short distance due to the problem of unbalanced current in the
circuit. The figure is shown below for delta connection. In the load station, ground can be used
as neutral path if required.

In delta connection, the line voltage is same with that of phase voltage. And the line current is
√3 times of phase current. It is shown as expression below,

In three phase circuit, star and delta connection can be arranged in four different ways-
 1. Star-Star connection

2. Star-Delta connection

3. Delta-Star connection

4. Delta-Delta connection

But the power is independent of the circuit arrangement of the three phase system. The net
power in the circuit will be same in both star and delta connection. The power in three phase
circuit can be calculated from the equation below,

Since, there is three phases, so the multiple of 3 is made in the normal power equation and the
PF is power factor. Power factor is a very important factor in three phase system and sometimes
due to certain error, it is corrected by using capacitors.
Analysis of Series/parallel resistance circuits:

When all the devices in a circuit are connected by series connections, then the circuit is
referred to as a series circuit. When all the devices in a circuit
are connected by parallel connections, then the circuit is
referred to as a parallel circuit. A third type of circuit involves
the dual use of series and parallel connections in a circuit;
such circuits are referred to as compound circuits or
combination circuits. The circuit depicted at the right is an
example of the use of both series and parallel connections
within the same circuit. In this case, light bulbs A and B are connected by parallel
connections and light bulbs C and D are connected by series connections. This is an example
of a combination circuit.

When analyzing combination circuits, it is critically important to have a solid understanding

of the concepts that pertain to both series circuits and parallel circuits. Since both types of
connections are used in combination circuits, the concepts associated with both types of
circuits apply to the respective parts of the circuit. The main concepts associated with series
and parallel circuits are organized in the table below.

Series Circuits Parallel Circuits

• The current is the same in every resistor; • The voltage drop is the same across
this current is equal to that in the battery. each parallel branch.
• The sum of the voltage drops across the • The sum of the current in each
individual resistors is equal to the voltage individual branch is equal to the current
rating of the battery. outside the branches.
• The overall resistance of the collection of • The equivalent or overall resistance of
resistors is equal to the sum of the the collection of resistors is given by
individual resistance values, the equation
Rtot = R1 + R2 + R3 + ... 1/Req = 1/R1 + 1/R2 + 1/R3 ...

Each of the above concepts has a mathematical expression. Combining the mathematical
expressions of the above concepts with the Ohm's law equation (ΔV = I • R) allows one to
conduct a complete analysis of a combination circuit.

Analysis of Combination Circuits

The basic strategy for the analysis of combination circuits involves using the meaning of
equivalent resistance for parallel branches to transform the combination circuit into a series
circuit. Once transformed into a series circuit, the analysis can be conducted in the usual
manner. Previously in Lesson 4, the method for determining the equivalent resistance of
parallel are equal, then the total or equivalent resistance of those branches is equal to the
resistance of one branch divided by the number of branches.

This method is consistent with the formula

1 / Req = 1 / R1 + 1 / R2 + 1 / R3 + ...

where R1, R2, and R3 are the resistance values of the individual resistors that are connected in
parallel. If the two or more resistors found in the parallel branches do not have equal
resistance, then the above formula must be used. An example of this method was presented in
a previous section of Lesson 4.

By applying one's understanding of the equivalent resistance of parallel branches to a

combination circuit, the combination circuit can be transformed into a series circuit. Then an
understanding of the equivalent resistance of a series circuit can be used to determine the
total resistance of the circuit. Consider the following diagrams below. Diagram A represents
a combination circuit with resistors R2 and R3 placed in parallel branches. Two 4-Ω resistors
in parallel is equivalent to a resistance of 2 Ω. Thus, the two branches can be replaced by a
single resistor with a resistance of 2 Ω. This is shown in Diagram B. Now that all resistors are
in series, the formula for the total resistance of series resistors can be used to determine the
total resistance of this circuit: The formula for series resistance is

Rtot = R1 + R2 + R3 + ...

So in Diagram B, the total resistance of the circuit is 10 Ω.

Once the total resistance of the circuit is determined, the analysis continues using Ohm's law
and voltage and resistance values to determine current values at various locations. The entire
method is illustrated below with two examples.

Example 1:

The first example is the easiest case - the resistors placed in parallel have the same resistance.
The goal of the analysis is to determine the current in and the voltage drop across each
resistor.

As discussed above, the first step is to simplify the circuit by replacing the two parallel
resistors with a single resistor that has an equivalent resistance. Two 8 Ω resistors in series is
equivalent to a single 4 Ω resistor. Thus, the two branch resistors (R2 and R3) can be replaced
by a single resistor with a resistance of 4 Ω. This 4 Ω resistor is in series with R1 and R4.
Thus, the total resistance is
 Rtot = R1 + 4 Ω + R4 = 5 Ω + 4 Ω + 6 Ω

Rtot = 15 Ω

Now the Ohm's law equation (ΔV = I • R) can be used to determine the total current in the
circuit. In doing so, the total resistance and the total voltage (or battery voltage) will have to
be used.

Itot = ΔVtot / Rtot = (60 V) / (15 Ω)

Itot = 4 Amp

The 4 Amp current calculation represents the current at the battery location. Yet, resistors R 1
and R4 are in series and the current in series-connected resistors is everywhere the same.
Thus,

Itot = I1 = I4 = 4 Amp

For parallel branches, the sum of the current in each individual branch is equal to the current
outside the branches. Thus, I2 + I3 must equal 4 Amp. There are an infinite number of
possible values of I2 and I3 that satisfy this equation. Since the resistance values are equal, the
current values in these two resistors are also equal. Therefore, the current in resistors 2 and 3
are both equal to 2 Amp.

I2 = I3 = 2 Amp

Now that the current at each individual resistor location is known, the Ohm's law equation
(ΔV = I • R) can be used to determine the voltage drop across each resistor. These
calculations are shown below.

ΔV1 = I1 • R1 = (4 Amp) • (5 Ω)

ΔV1 = 20 V

ΔV2 = I2 • R2 = (2 Amp) • (8 Ω)

ΔV2 = 16 V

ΔV3 = I3 • R3 = (2 Amp) • (8 Ω)

ΔV3 = 16 V

ΔV4 = I4 • R4 = (4 Amp) • (6 Ω)

ΔV4 = 24 V

The analysis is now complete and the results are summarized in the diagram below.
Example 2:

The second example is the more difficult case - the resistors placed in parallel have a
different resistance value. The goal of the analysis is the same - to determine the current in
and the voltage drop across each resistor.

As discussed above, the first step is to simplify the circuit by replacing the two parallel
resistors with a single resistor with an equivalent resistance. The equivalent resistance of a 4-
Ω and 12-Ω resistor placed in parallel can be determined using the usual formula for
equivalent resistance of parallel branches:

1 / Req = 1 / R1 + 1 / R2 + 1 / R3 ...

1 / Req = 1 / (4 Ω) + 1 / (12 Ω)

1 / Req = 0.333 Ω-1

Req = 1 / (0.333 Ω-1)

Req = 3.00 Ω

Based on this calculation, it can be said that the two branch resistors (R 2 and R3) can be
replaced by a single resistor with a resistance of 3 Ω. This 3 Ω resistor is in series with R1 and
R4. Thus, the total resistance is

Rtot = R1 + 3 Ω + R4 = 5 Ω + 3 Ω + 8 Ω

Rtot = 16 Ω
Now the Ohm's law equation (ΔV = I • R) can be used to determine the total current in the
circuit. In doing so, the total resistance and the total voltage (or battery voltage) will have to
be used.

Itot = ΔVtot / Rtot = (24 V) / (16 Ω)

Itot = 1.5 Amp

The 1.5 Amp current calculation represents the current at the battery location. Yet, resistors
R1 and R4 are in series and the current in series-connected resistors is everywhere the same.
Thus,

Itot = I1 = I4 = 1.5 Amp

For parallel branches, the sum of the current in each individual branch is equal to the current
outside the branches. Thus, I2 + I3 must equal 1.5 Amp. There are an infinite possibilities of I2
and I3 values that satisfy this equation. In the previous example, the two resistors in parallel
had the identical resistance; thus the current was distributed equally among the two branches.
In this example, the unequal current in the two resistors complicates the analysis. The branch
with the least resistance will have the greatest current. Determining the amount of current
will demand that we use the Ohm's law equation. But to use it, the voltage drop across the
branches must first be known. So the direction that the solution takes in this example will be
slightly different than that of the simpler case illustrated in the previous example.

To determine the voltage drop across the parallel branches, the voltage drop across the two
series-connected resistors (R1 and R4) must first be determined. The Ohm's law equation (ΔV
= I • R) can be used to determine the voltage drop across each resistor. These calculations are
shown below.

ΔV1 = I1 • R1 = (1.5 Amp) • (5 Ω)

ΔV1 = 7.5 V

ΔV4 = I4 • R4 = (1.5 Amp) • (8 Ω)

ΔV4 = 12 V

This circuit is powered by a 24-volt source. Thus, the cumulative voltage drop of a charge
traversing a loop about the circuit is 24 volts. There will be a 19.5 V drop (7.5 V + 12 V)
resulting from passage through the two series-connected resistors (R1 and R4). The voltage
drop across the branches must be 4.5 volts to make up the difference between the 24 volt total
and the 19.5-volt drop across R1 and R4. Thus,

ΔV2 = V3 = 4.5 V

Knowing the voltage drop across the parallel-connected resistors (R1 and R4) allows one to
use the Ohm's law equation (ΔV = I • R) to determine the current in the two branches.
I2 = ΔV2 / R2 = (4.5 V) / (4 Ω)

I2 = 1.125 A

I3 = ΔV3 / R3 = (4.5 V) / (12 Ω)

I3 = 0.375 A

The analysis is now complete and the results are summarized in the diagram below.

Kirchhoffs Circuit Law

We saw that a single equivalent resistance, ( RT ) can be found when two or more resistors
are connected together in either series, parallel or combinations of both, and that these
circuits obey Ohm’s Law. However, sometimes in complex circuits such as bridge or T
networks, we can not simply use Ohm’s Law alone to find the voltages or currents circulating
within the circuit. For these types of calculations we need certain rules which allow us to
obtain the circuit equations and for this we can use Kirchhoffs Circuit Law.

In 1845, a German physicist, Gustav Kirchhoff developed a pair or set of rules or laws
which deal with the conservation of current and energy within electrical circuits. These two
rules are commonly known as: Kirchhoffs Circuit Laws with one of Kirchhoffs laws dealing
with the current flowing around a closed circuit, Kirchhoffs Current Law, (KCL) while the
other law deals with the voltage sources present in a closed circuit, Kirchhoffs Voltage Law,
(KVL).

Kirchhoffs First Law – The Current Law, (KCL)

Kirchhoffs Current Law or KCL, states that the “total current or charge entering a junction
or node is exactly equal to the charge leaving the node as it has no other place to go except
to leave, as no charge is lost within the node“. In other words the algebraic sum of ALL the
currents entering and leaving a node must be equal to zero, I(exiting) + I(entering) = 0. This idea by
Kirchhoff is commonly known as the Conservation of Charge.

Kirchhoffs Current Law

Here, the 3 currents entering the node, I1, I2, I3 are all positive in value and the 2 currents
leaving the node, I4 and I5 are negative in value. Then this means we can also rewrite the
equation as;

I1 + I2 + I3 – I4 – I5 = 0

The term Node in an electrical circuit generally refers to a connection or junction of two or
more current carrying paths or elements such as cables and components. Also for current to
flow either in or out of a node a closed circuit path must exist. We can use Kirchhoff’s
current law when analysing parallel circuits.

Kirchhoffs Second Law – The Voltage Law, (KVL)

Kirchhoffs Voltage Law or KVL, states that “in any closed loop network, the total voltage
around the loop is equal to the sum of all the voltage drops within the same loop” which is
also equal to zero. In other words the algebraic sum of all voltages within the loop must be
equal to zero. This idea by Kirchhoff is known as the Conservation of Energy.

Kirchhoffs Voltage Law

Starting at any point in the loop continue in the same direction noting the direction of all the
voltage drops, either positive or negative, and returning back to the same starting point. It is
important to maintain the same direction either clockwise or anti-clockwise or the final
voltage sum will not be equal to zero. We can use Kirchhoff’s voltage law when analysing
series circuits.
When analysing either DC circuits or AC circuits using Kirchhoffs Circuit Laws a number
of definitions and terminologies are used to describe the parts of the circuit being analysed
such as: node, paths, branches, loops and meshes. These terms are used frequently in circuit
analysis so it is important to understand them.

Common DC Circuit Theory Terms:

• • Circuit – a circuit is a closed loop conducting path in which an electrical current

flows.
• • Path – a single line of connecting elements or sources.
• • Node – a node is a junction, connection or terminal within a circuit were two or
more circuit elements are connected or joined together giving a connection point
between two or more branches. A node is indicated by a dot.
• • Branch – a branch is a single or group of components such as resistors or a source
which are connected between two nodes.
• • Loop – a loop is a simple closed path in a circuit in which no circuit element or node
is encountered more than once.
• • Mesh – a mesh is a single open loop that does not have a closed path. There are no
components inside a mesh.

Note that:

Components are said to be connected together in Series if the same current value flows
through all the components.

Components are said to be connected together in Parallel if they have the same voltage
applied across them.

A Typical DC Circuit

Kirchhoffs Circuit Law Example No1

Find the current flowing in the 40Ω Resistor, R3

The circuit has 3 branches, 2 nodes (A and B) and 2 independent loops.

Using Kirchhoffs Current Law, KCL the equations are given as;

At node A : I1 + I2 = I3

At node B : I3 = I1 + I2

Using Kirchhoffs Voltage Law, KVL the equations are given as;

Loop 1 is given as : 10 = R1 I1 + R3 I3 = 10I1 + 40I3

Loop 2 is given as : 20 = R2 I2 + R3 I3 = 20I2 + 40I3

Loop 3 is given as : 10 – 20 = 10I1 – 20I2

As I3 is the sum of I1 + I2 we can rewrite the equations as;

Eq. No 1 : 10 = 10I1 + 40(I1 + I2) = 50I1 + 40I2

Eq. No 2 : 20 = 20I2 + 40(I1 + I2) = 40I1 + 60I2

We now have two “Simultaneous Equations” that can be reduced to give us the values of I1
and I2

Substitution of I1 in terms of I2 gives us the value of I1 as -0.143 Amps

Substitution of I2 in terms of I1 gives us the value of I2 as +0.429 Amps

As : I3 = I1 + I2

The current flowing in resistor R3 is given as : -0.143 + 0.429 = 0.286 Amps

and the voltage across the resistor R3 is given as : 0.286 x 40 = 11.44 volts
The negative sign for I1 means that the direction of current flow initially chosen was wrong,
but never the less still valid. In fact, the 20v battery is charging the 10v battery.

Application of Kirchhoffs Circuit Laws

These two laws enable the Currents and Voltages in a circuit to be found, ie, the circuit is said
to be “Analysed”, and the basic procedure for using Kirchhoff’s Circuit Laws is as follows:

• 1. Assume all voltages and resistances are given. ( If not label them V1, V2,… R1,
R2, etc. )
• 2. Label each branch with a branch current. ( I1, I2, I3 etc. )
• 3. Find Kirchhoff’s first law equations for each node.
• 4. Find Kirchhoff’s second law equations for each of the independent loops of the
circuit.
• 5. Use Linear simultaneous equations as required to find the unknown currents.

As well as using Kirchhoffs Circuit Law to calculate the various voltages and currents
circulating around a linear circuit, we can also use loop analysis to calculate the currents in
each independent loop which helps to reduce the amount of mathematics required by using
just Kirchhoff's laws. In the next tutorial about DC circuits, we will look at Mesh Current
Analysis to do just that.

Current and voltage drop across the DC circuit elements:

Ohm's Law can be used to verify voltage drop. In a DC circuit, voltage equals current
multiplied by resistance. V=IR. Also, Kirchhoff's circuit laws state that in any DC circuit, the
sum of the voltage drops across each component of the circuit is equal to the supply voltage.

Single-Phase vs Three-Phase Power:

Single-phase power is:

▪ Used in most homes in the several countries.
▪ Able to supply ample power for most smaller customers, including homes and small, non-industrial
businesses
▪ Adequate for running motors up to about 5 horsepower; a single-phase motor draws significantly
more current than the equivalent 3-phase motor, making 3-phase power a more efficient choice for
industrial applications
With the wave form of single-phase power, when the wave passes through zero, the power supplied at
that moment is zero. In the U.S., the wave cycles 60 times per second.

3-phase power is:

▪ Common in large businesses, as well as industry and manufacturing
▪ Increasingly popular in power-hungry, high-density data centers
▪ Expensive to convert from an existing single-phase installation, but 3-phase allows for smaller, less
expensive wiring and lower voltages, making it safer and less expensive to run
▪ Highly efficient for equipment designed to run on 3-phase

3-phase power has 3 distinct wave cycles that overlap. Each phase reaches its peak 120 degrees apart
from the others so the level of power supplied remains consistent

Power Factor
For a DC circuit the power is P=VI, and this relationship also holds for the instantaneous
power in an AC circuit. However, the average power in an AC circuit expressed in terms of
the rms voltage and current is

where is the phase angle between the voltage and current. The additional term is called the
power factor

From the phasor diagram for AC impedance, it can be seen that the power factor is R/Z. For a
purely resistive AC circuit, R=Z and the power factor = 1.

Power Factor in AC transmition:

In AC circuits, the power factor is the ratio of the real power that is used to do work and the apparent
power that is supplied to the circuit. The power factor can get values in the range from 0 to 1. When
all the power is reactive power with no real power (usually inductive load) - the power factor is 0.

Importance of Power Factor

A power factor of one or "unity power factor" is the goal of any electric utility company since
if the power factor is less than one, they have to supply more current to the user for a given
amount of power use. In so doing, they incur more line losses. They also must have larger
capacity equipment in place than would be otherwise necessary. As a result, an industrial
facility will be charged a penalty if its power factor is much different from 1.

Industrial facilities tend to have a "lagging power factor", where the current lags the voltage
(like an inductor). This is primarily the result of having a lot of electric induction motors - the
windings of motors act as inductors as seen by the power supply. Capacitors have the opposite
effect and can compensate for the inductive motor windings. Some industrial sites will have
large banks of capacitors strictly for the purpose of correcting the power factor back toward
one to save on utility company charges.

Improving the power factor: saving energy and Money

Most AC electrical machines draw from the supply apparent power in terms of kilovolt amperes
(kVA) which is in excess of the useful power, measured in kilowatts (kW), required by the machine.
The ratio of these quantities is known as the power factor of the load, and is dependent upon the
type of machine in use. Assuming a constant supply voltage, this implies that more current is drawn
from the electricity authority than is actually required.

Power factor = (true power) / (apparent power) = kW / kVA

A large proportion of the electrical machinery used in industry has an inherently low power factor,
which means that the supply authorities have to generate much more current than is theoretically
required. This excess current flows through generators, cables, and transformers in the same
manner as the useful current. The motive power requirements are generally greater than the
resistive loads such as lighting and heating. If steps are not taken to improve the power factor of
the load, all the equipment from the power station to the factory sub-circuit wiring has to be larger
than necessary. This results in increased capital expenditure and higher transmission and
distribution losses throughout the whole supply network.

To overcome this problem, and at the same time to ensure that generators and cables are not
overloaded with wattless current (as this excess current is termed), the supply authorities often offer
reduced terms to consumers whose power factor is high, or impose penalties on those with low
power factor. Most supply authorities insist that a power factor of at least 0.90 is achieved.
Improving the power factor helps to reduce the overall consumption of electricity.