DC Circuits and Network Theorems

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1EEX 17215 : Elements of ElectronicsFY Diploma E&TC DC Circuits and Network Theorem



1. Introduction :

Network is an electrical configuration in which various components such as resistors, capacitors,

inductors etc. and voltage sources or current sources are electrically connected to each other. A

network can also be called as a circuit.

The networks can be DC or AC depending on the type of source involved. For the DC circuits,the voltage and current sources are DC whereas AC sources are connected in the AC circuits.

Example of a DC source is a battery and that of an AC source is the AC mains supply.

2EEX 17215 : Elements of ElectronicsFY Diploma E&TC DC Circuits and Network Theorem



2. Fundamentals of DC Circuit :

1. Review of Ohms Law :

Statement :

The Ohm’s law gives the relation between the voltage, current and resistance for good

conductors.

It states that the electric current (I) passing through a capacitor is directly proportional to the

potential difference (V) applied across the conductor. i.e. I ⍺ V.

Mathematically it is given by

I = V/R

Where R = constant of proportionality known as resistance of the conductor in ohm.

EEX 17215 : Elements of ElectronicsFY Diploma E&TC DC Circuits and Network Theorem 3



.

Explanation :

The equation I = V/R may also be expressed as in the following ways

V =I.R

The meaning of Eq. is that whenever there is a current through the conductor, the potential

difference (or voltage drop) across its two ends is equal to the product of the current (I) passing

through it and the resistance (R) of the conductor.

Similarly, the Eq. I = V/R says that the ratio of potential difference (V) across the two ends of a

conductor to the current (I) passing through the conductor is constant and is equal to the

resistance (R) of the conductor.




.

2. Electrical Circuits :

Introduction : In actual practice, a charge of many billions of electrons or protons is necessary for common

applications of electricity.

An electron charge has a potential to do some work by moving another charge either by

attraction or repulsion.

An electric charge is a result of work done in separating electrons and holes.

If one electric charge is different from the other, then there is a difference of potential between

them.

It means no work can be done in moving electrons between the two identical electrical charges.

The potential difference is generally called voltage and its unit is volt (V).

The potential difference is also called as electro motive force (e.m.f.).

Voltage is used to denote the voltage drop across the passive components (i.e. resistor,

capacitor, inductor).




.

The e.m.f. is used to denote the potential difference of voltage source.

The unit of voltage or e.m.f. is volt (V).

The movement of the electric charge is called as an electric current.

The electric current is always produced due to the potential difference between the two charges.




.

Concept :

The combination of electrical devices, voltage source and load connected together to form a

connecting path for the flow of electric current is called as an electric circuit. It is also known as

an electrical network.

The electric circuits fulfill a desired function, such as to light bulb, heat a toaster etc.




.

Characteristics :

There must be a source of voltage. Without the applied voltage, the electric current cannot flow.

There must be a complete path for electric current to flow from one side of the applied voltage

source, through the external circuit and returning to the other side of the voltage source.

Usually, the current path has some value of resistance, whose purpose is either to generate the

current or limit the amount of current.






.

An open circuit gives rise to the following two effects ;

1. The resistance of the open circuit is infinitely high.

2. There is no flow of current in an open-circuit.

Fig. shows an open-circuit in a load resistance (i.e. load).

As a result the resistance between two points A and B is infinity (i.e. RAB =∞) and there is no

flow of electric current through the load resistance (i.e. IAB = 0).

If there is ‘open’ in a series d.c. circuit, then there is no voltage across the resistor.

The open in a parallel d.c. circuit may be of two types, namely an open in the main line and an

open in the branch line.

If there is an ‘open’ in any one of the main lines, then there is no flow of current in all the

branch lines but the voltage across that open line is equal to the applied source voltage.

If there is an ‘open’ in any one of the branch lines, then there is no flow of electric current in

that branch lines only.




.

Short Circuit :

An electrical circuit is said to be ‘short’ when the voltage source has a closed path across its

terminals.

The short circuit results in too much flow of electric current in the electric circuit.

Usually, the short circuit is a bypass across the load resistance.

A short may occur due to a component failure or wiring error in the electrical circuit.

A short may cause some problem due to excessively high current in the electric circuit, even

may result burning out the circuit components.






.

4.Kirchhoff’s Current and Voltage Law :

Kirchhoff’s Voltage Law :

Statement : It states that the algebraic sum of the products of current (I) and resistance (R) in

each of the conductors in any closed loop or mesh in a network plus the algebraic sum of the

emfs in that loop is zero.

Circuit Diagram :




.

Consider the above circuit E is a voltage source across which a resistor connected.

There will be a flow of current I through the resistor R.

There will be a voltage drop across the resistor R due to the flow of current I from high potential

(+) to low potential (-).

This is termed as voltage drop and it is considered to be negative (i.e. –IR).

The same current I flows through the voltage source from low potential (-) to high potential (+)

and this is termed as voltage rise.

The voltage rise is considered to be positive (i.e. +E).

∑ Voltage rise = ∑ Voltage drop

E = IR

E –IR = 0

Thus in any closed loop circuit, the algebraic sum of voltage rise is equal to the algebraic sum of

voltage drop.




.

Kirchhoff’s Current Law :

Statement : It states that the algebraic sum of currents at any point (junction or node) in a

network is zero.

Kirchhoff’s current law is also called as point law.

It means that the total current leaving a junction (or node) is equal to the total current entering

it in a network because the junction (or node) cannot produce or consume any current.

∑ Incoming currents = ∑ Outgoing currents

I1+I2+I3-I4-I5 = 0

i.e. I1+I2+I3 = I4+I5




5. Maxwell’s Loop Current Method :

The Maxwell’s loop current method is best suited for the electrical circuit having voltage sources

rather than current sources.

Maxwell’s loop current theorem is an extension of Kirchhoff’s voltage law (KVL) but here loop

currents rather than the branch currents are considered in solving the problems.

Consider the network comprising of three loops 1,2 and 3, and two voltage sources V1 and V2

as shown in the fig.


.



.

If we have to solve the network by KVL, we also have named different branch currents as I1, I2

and I3 as shown in Fig. The currents in different branches are as follows

Current in branch AB = I1

Current in branch BE = I2

Current in branch BC = I1-I2

Current in branch EG = I3

Current in branch EF = I2-I3

According to Maxwell’s loop current theorem, we would name the currents in different loops as

i1, i2 and i3.

The currents in different loops are as follows.

current in loop ABCD = i1

current in loop BEFC = i2

current in loop EGHF = i3




.

From this, if we find the branch currents for BC and EF by inspection, it turns not to be (i1-i2)

and (i2-i3) respectively.

The KVL and Maxwell’s loop current methods for analysis of network are therefore similar and

may be used as required.

In this method, the loop voltage equation is to be written by using KVL.




3. Node Analysis :

1. Sources for Electronic Circuit :

Introduction :

Every electronic circuit requires some source of energy for its operation.

The electrical energy may be supplied either in the form of a voltage or current.

The electrical device which supplies the electrical energy required for the operation of an

electronic circuit is called as a source.




.

Types of Sources :

The sources required for the electronic circuits are of the following two types :

1. Voltage source.

2. Current source.

All the voltage and current sources may be classified into to categories as under :

1. Direct current(d.c.).

2. Alternating current( a.c.).




.

Internal Resistance of a Source :

All sources of electrical energy, either voltage or current, have some internal resistance. Thevalue of this resistance is few ohms.

The internal resistance of a source may be due to the following reasons :

1. The resistance of an electrolyte between the electrodes in the case of dry cell.

2. The resistance of an armature winding in the case of d.c. generator or an alternator.

3. The output resistance of an active device like transistor in case of capacitor, signal generator

or regulated d.c. supply.

The d.c resistance offered by a source is called as an internal resistance.

The internal resistance of a voltage source and a current source should be very low and very

high respectively.

The internal resistance of a source (voltage or current), may be due to following reasons :

1. the resistance of an electrolyte between the electrodes in the case of dry cell.

2. The resistance of armature winding in the case of d.c. generator or an alternator.




.

3. The output resistance of an active device like a transistor in case of oscillator, signal

generator or regulated d.c. supply.




.

2. Voltage Source :

Introduction :

The source, which supplies electrical energy in the form of a voltage, is known as a voltage

source.

All voltage sources in practice must have very low internal resistance.

All voltage sources may be broadly classified as d.c. voltage source, a.c. voltage source,

depending upon whether a source supplies d.c. or a.c. to the load.

Some of the commonly used d.c. sources are batteries, d.c. generators and regulated d.c.

power supplies.






.

The total resistance inn the circuit is given by

RT = Rs + RL

The current through the variable load is given by,

I = Vs/RT = Vs/(Rs + RL)

Therefore, voltage across the terminals (which is also equal to the voltage across the load) is

given by,


VAB = VL = I.RL

= Vs

+ .

=Vs

1+Rs

RL



.

It may be noted from the equation, that the ratio Rs/RL is small as compared to unity, then the

terminal voltage remains the same as that of the source voltage.

Under this condition, this source behaves as a good voltage source.

Thus even if the load resistance changes, the terminal voltage VAB of the voltage source

remains practically constant, provided the ratio Rs/RL is very much less than unity (i.e.

Rs/RL<<1).




.

Ideal Voltage Source :

A voltage source, which produces a constant voltage (or load voltage), which does not dependon the value of load resistance, is known as an ideal voltage source.

This may be possible only if the internal resistance of a voltage source is zero.

Fig.




.

Fig. a and b shows the circuit symbol and reference direction for currents of d.c. and a.c.

voltage sources.

It may be noted that the marking of positive and negative on the a.c. source does not mean the

same thing as that of a d.c. source.

In an a.c. source, it means that the upper terminal of the ideal voltage source is positive, with

respect to the lower terminal at that particular instant.

In the next half cycle of a.c. voltage, the marking will become opposite.

Fig. c shows the V-I characteristic of an ideal voltage source.

It may be noted that as the value of load resistance is reduced, the load current increases and

vice versa.

The open circuit voltage remains constant for the variation of either load resistance or current.




.

Practical Voltage Source :

It will be interesting to note that in actual practice, there is no voltage source, which is an ideal

one.

The ideal voltage source does not exist in actual practice, because it is not possible to have zero

value of the internal resistance for the voltage source.

A practical voltage source, known as real voltage source, consists of an voltage source in series

with an internal resistance of the voltage source.

Fig .




.

Fig. a and b show the circuit symbols and the reference direction for current of the practical d.c.

and a.c. voltage sources.

The practical (i.e. real) voltage source has very small value of internal resistance.

It may be noted that the open circuit voltage (Vs) decreases as the load current increases and

vice-versa. This is shown in fig. c by the dotted line.




.

3. Current Source :

Introduction : The source, which supplies electrical energy in the form of an electrical current, is known as a

current source.

All current sources, in practice, may be broadly classified as d.c. current source or a.c. current

source, depending upon whether a source supplies direct current (d.c.) or alternating

current(a.c.) to the load.

Some of the commonly used a.c. sources are alternators. oscillators or signal generators.




.

Concept :

Consider the d.c. current source connected as a variable load resistance as shown in fig.

Let Is = Short circuit (i.e. a current through the source terminals when they are shorted)

RL = Variable load resistance connected across a current source.

Rs = Internal resistance of the source.

The current Is divides it into parts, namely IL and I1.




..

The part IL flows through the load RL and I1 through the internal resistance Rs.

The source current is given byIs = I1 + IL

I1 = Is – IL

The voltage drop across the internal resistance Rs is given by

V = I1.Rs

The voltage drop across the load resistance is given by

VL = IL.RL

Since the resistance Rs and RL are in parallel, therefore, the voltage drop across them should be

equal.

so, Is . Rs = IL . RL(Is – IL).Rs = IL . RL

(Rs+RL).IL = Is . RS

Therefore, IL = Is/(1+[RL/Rs])




.

The above equation indicates that the load current remain almost equal to the short circuit

current Is, provided the ratio RL/Rs is small as compared to unity.

In other words, large the value of internal resistance RS, as compared to the load resistance RL,

smaller is the ratio RL/Rs and hence better is the current source.

The condition for a good current source is given by,

RL/Rs <<1.




.

Ideal Current Source :

A current source, which supplies a constant current to a load even if its resistance varies, isknown as a constant source. It is also known as an ideal current source.

Ideally, the current supplied by a current source should remain constant, irrespective of the load

resistance.

Fig.




.

This may be possible only if the internal resistance of a current source is infinite.

Fig. a shows the circuit symbol of an ideal current source. The arrow inside the circle indicates the direction in which the current flows in the circuit when a

load is connected across it.

Fig. b shows the V-I characteristics of an ideal current source.

The load current, supplied to the load by the current source, remains constant for all values of

the load resistance RL.

It may be noted that as the value of load resistance is increased, the terminal voltage VAB also

increases.

It is due to the fact, that the variation of terminal voltage is shown in the same direction as that

of the load resistance along the horizontal axis of the V –I characteristic.




.

Practical Current Source :

In actual practice, there is no current source, which is an ideal one, because it has very highvalue of n internal resistance.

The ideal current source does not exist in actual practice, because it is not possible to have an

infinite value of the internal resistance for the current source.

A practical current source, known as a real current source, consist of an ideal current source in

parallel with a resistance known as the internal resistance of the current source.

Fig.




.

Fig. a and b show the circuit symbols and reference direction for current of the practical d.c. and

a.c. current source.

The practical (i.e. real) current source has very high value of the internal resistance.

It may be noted that the load current Is decreases below the short circuit current, as the loas

resistance RL is increased. It is shown in Fig. b by dotted line.




.

4. Equivalence Between Sources :

We know that the source may either work as a voltage source or a current source, depending

upon the working conditions.

As a matter of fact, a voltage source is not different from a current source.

If the value of load resistance RL is very high as compared to the internal resistance Rs, the

source is treated as a voltage source.

If the value of the load resistance RL, is very low as compared to the internal resistance RL, the

source is treated as a current source.

In order to establishment between a voltage source and a current source, consider an a.c.

source connected to the load resistance RL as shown in Fig.

The equivalent circuit of a practical voltage source is as shown in fig. a and it consist of an ideal

voltage source Vs in series with a source resistance Rs.

The equivalent circuit of a practical current source is as shown in fig. b and it consists of an ideal

current source Is in parallel with a source resistance Rs.

Since both equivalent circuits are of the same source, they give the same results.




.

It may be noted that the source resistance in both the equivalent circuits are the same.

Rs1 = Rs2

Also indicates that whatever may be the value of load resistance RL, the current drawn by the

load resistance in either voltage source or current source, equivalent circuits will be the same.

IL1 = IL2

Thus the equivalence between a voltage source and current source is completely established.




.

5. Conversion of Sources :

Consider a practical voltage source Vs with an internal resistance Rs as shown in fig..Fig.




.

In order to convert it into its equivalent current source representation, let us short circuit the

terminals A and B and then determine the current through the short circuit. The short circuit current is given by

Is = Vs/Rs

We know that the source resistance in either voltage source representation or current

representation in the same.

Therefore, the current source has the same internal resistance at that a voltage source.

Thus the equivalent current source may be represented by a current source Is in parallel with a

source resistance Rs as shown in fig. b.






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7. Delta to Star and Star to Delta Conversions :

Introduction : Kirchhoff’s laws or Maxwell’s loop current theorem is very useful in solving network problems.

But if a network with considerable number of branches is to be analyzed, then there are a large

number of simultaneous equations and a great difficulty may have to be forced in finding the

solution.

Such complicated networks may be simplified by successfully replacing delta or mesh circuit

with equivalent star circuit and vice versa.




.

Delta to Star Conversion :

Delta to star conversion is useful for quick reduction of the given circuit. Consider the delta network as shown in fig. a below.




.

It is connected between points 1,2 and 3, and the resistance are denoted as R12, R22 and R31.

This delta circuit can be transformed into a star circuit as shown in fig. b with the resistance R1,R2 and R3.

In order that the two networks be exactly equivalent, the resistance between any two terminals

of the network must be equal to the resistance between the corresponding pair of terminals of

the other network.

For delta to star conversion, we can find the values for R12, R21, R31 in terms of R1, R2 and

R3.

R12 = R1+R2+(R1R2)/R3

R23 = R2+R3+(R2R3)/R1

R31 = R3+R1+(R3R1)/R2




.

Star to Delta Conversion :

Consider a star network as shown in the fig. b. It is connected between points 1, 2 and 3 and the resistances are denoted as R1, R2 and R3.

Fig.




.

Conversion from star to delta can be easily obtained by use of the resistance R12, R23, R31 in

terms of R1, R2, R3.R12 = (R1.R2+R2.R3+R3.R1) / R3

R23 = (R1.R2+R2.R3+R3.R1) / R1

R31 = (R1.R2+R2.R3+R3.R1) / R2

The star network can be transformed into a delta network as shown in fig. a with the resistance

R12, R21 and R23.




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8. Network Terminology :

Any interconnection of electrical components like resistors, inductors, capacitors,semiconductor devices, transformers and sources of e.m.f. is known as a network. A network

may comprise of one or more circuit elements.

Types of network are as follows :

1. One port network.

2. Two port network.

A network having one pair of terminal is called one port network.

e.g. :- a series, parallel or series-parallel combination of resistors, capacitors etc.

A network having two pairs of terminals (i.e. four terminals) is called two port network.

e.g. :- an amplifier, attenuator, filter, transformer etc.




.

8.1 Linear Network :

If the resistance, capacitance or inductance offered by the passive elements does not change

with a change in the applied e.m.f. or the circuit current , the element is called as a linear

element.

The linear element shows a linear relation between voltage and current.

The network comprising of one or more linear number of elements is known as a linear network.




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8.2 Non-linear Network :

A non-linear circuit element is one in which the current does not change linearly with a changein applied voltage at a given frequency.

The non-linear elements include components like VDR (voltage dependent resistor), varactor

(voltage dependent capacitor), LDR (light dependent resistor), thermistor (temperature

dependent resistor) etc.

A network comprising of one or more non-linear elements is known as a non-linear network.




.

8.3 Passive Network :

An element that is not capable of amplifying or processing an electrical signal is known aspassive element.

Resistor, capacitors, inductors, transformers are few examples of passive elements.

A network comprising of one or more number of passive elements is known as a passive

network.




.

8.4 Active Network:

An electronic element which itself is capable of amplifying (or processing) an electrical signal is

known as an active element.

The active element includes semiconductor devices such as diodes, BJTs, FETs etc.

A network comprising one or more number of active elements is known as an active network.




.

8.5 Unilateral Network :

If the magnitude of the current flowing through a circuit element is affected when the polarity of

the applied e.m.f is changed, then the element is known as a unilateral element.

An example of the unilateral element is semiconductor diode.

The unilateral elements offer different resistance to the flow of current in different direction.

A network comprising of one or more number of unilateral elements is known as a unilateral

network.




.

8.6 Bilateral Network :

If the magnitude of the current flowing through a circuit element is not affected when the

polarity of the applied voltage is changed, then the element is known as a bilateral element.

The bilateral elements offer the same impedance in both directions.

A resistor is a bilateral element.

A network comprising of one or more number of bilateral elements is known as a bilateral

network.




4. Network Theorem :

1. Superposition Theorem :

Statement :

The superposition theorem states that in any linear network containing two or more

sources, the response (current) in any element is equal to the algebraic sum of the response(current) caused by individual sources acting alone, while the other sources are inoperative.

Or

The voltage or current, present in a component, is equal to the sum of voltages currents,

which exist independently.




.

Diagram :




.

Explanation :

This theorem is useful where more than one voltage or current source is present. It may be

used to determine the current in a component, which exist as a result of each of the sources.

It is important to note that while calculating the results for each source, the other sources must

be set to zero and replaced by their internal resistances. In other words voltage sources are

replaced by a short circuit and current sources by an open circuit.

If there are any dependent sources, then leave them undisturbed in the circuit.

For example, let us consider the circuit as shown in fig. 1(b) . Now the first step is that we have

to draw the individual circuits for each source for calculation purposes.

Thus fig. 1(b) shows the circuit for source V1, fig. 2(a) for source V2 and fig. 2(b) for current

source I.

The next step is to calculate the effects of each source from the individual circuits.

Then they finally find the results caused by all the combined sources.

In using this theorem the polarity of each voltage and current is very important , because end

result is the algebraic sum of the individual results.




.

Applications :

The superposition theorem is useful in circuit analysis, when the circuit has large number of

independent sources.

It helps up to determine the voltage across a component or current through branch by

calculating the effect of each source individually and determining the combined effect of these

sources.

The superposition theorem ca be used for a.c. as well as d.c. networks.




.

Drawbacks :

The superposition theorem is not applicable to the circuits consisting of non-linear elements e.g.

diode, transistors.

The superposition theorem cannot be applied to circuits containing dependent sources only.

The superposition theorem is not useful for circuit analysis, for circuits containing less than two

independent sources.

Superposition theorem cannot be used for calculating power because power is proportional tosquare of voltage or current i.e. nonlinear.




.

2. Thevenin’s Theorem :

Statement :

Thevenin’s theorem states that any two terminal linear(complex) network containing

energy sources and impedance can be replaced by an equivalent circuit consisting of an ideal

voltage source VTH in series with a resistance RTH.




.

Explanation :

Consider a box containing a network of impedance and generators connected to its output

terminals A and B across which load resistance RL is connected as shown in fig.




.

Let V0 be the open circuit voltage between terminals A and B of the network and Z is the

impedance measured between these two terminals when all energy sources are eliminated.

In any circuit, consider the load resistor R1 is removed and with ends kept open, calculate the

open circuit voltage Vx. This equals the Thevenin’s resistance RTH.

Next remove voltage source leaving only its internal resistance, calculate equivalent resistance

across the terminals. This equals the Thevenin’s resistance RTH.

According to Thevenin’s theorem, the entire network connected to A and B terminals can be

replaced by a single voltage source VTH in series with a single resistance RTH across the same

terminals as shown in fig. b.




.

Applications :

The Thevenin’s theorem makes it possible to convert a complicated network to a simple

equivalent network. Using the simple network we can determine voltage, current, power

delivered to the load.

It can be used to justify the concept of input and output resistances of an amplifier.




.

Drawbacks :

The Thevenin’s theorem is not applicable to the networks containing unilateral elements e.g.

diode.

The Thevenin’s theorem is not applicable to circuits consisting of load with dependent or

controlled sources.

The Thevenin’s theorem is not applicable to networks containing non linear elements e.g. diode,

transistors.

The Thevenin’s theorem is not applicable to the networks containing magnetic coupling between

load and any other circuit element.




.

3. Norton’s Theorem :

Statement :Norton’s theorem states that any complex linear network can be replaced by a parallel

circuit consisting of an ideal current source and a resistance.




.

Explanation :

Consider a box containing a network of impedance and generators connected to its output

terminals A and B as shown in fig. a.

Then according to Norton’s theorem, the entire circuit connected between A and B can be

replaced by a single current source IN in parallel with a single resistance RN across the terminala

as shown in fig. b.




.

As a matter of fact, IN is the current, that actually flows between the terminals A and B, when

they are short circuited.

Similarly, RN is the equivalent resistance between the terminal A and B of the original network

with the load resistance removed and all the sources replaced by their internal resistances.




.

Applications :

The Norton’s theorem makes it possible to convert a complicated network to a simple equivalent

network. Using the network we can find voltage , current and power delivered to the load.




.

4. Maximum Power Transfer Theorem :

Statement :

The power transfer theorem states that for a given linear network represented by a

Thevenin’s equivalent circuit , the maximum power will be transferred by the network to the

resistive load, when the load resistance is equal to the Thevenin’s resistance.






.


∴ = ² 1

(+)²× 1 + [ −2+3]

∴ ² 1(+)² − 2+3 = 0

∴ 1

(+)² =2

+3

Or 2RL = RL + Ri

∴ RL = Ri

This is the condition for maximum power given by

P = ²

(2)²× =²4 ×

Pmax =²4



.

Applications :

In the power amplifiers, we use this theorem in order to match the loud speaker impedance to

the output impedance of amplifier in order to ensure maximum power transfer.

For impedance matching in any electrical circuit.




EEX 17215 El t f El t i

DC Circuits and Network Theorems

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