Topic: DC Circuit Analysis Subject: Elements of Electrical Engineering

Prepared by: Prof. Dipen Patel

• Concept of DC circuit. • Current and voltage sources.• Kirchhoff's current law• Kirchhoff's voltage law• Mesh and nodal analysis• Superposition theorem• Thevenin’s Theorem

Topics

What is DC circuit?• Direct current (DC) circuits basically consist

of a loop of conducting wire (like copper) through which an electric current flows. An electric current consists of a flow of electric charges, analogous to the flow of water (water molecules) in a river. In addition to the copper wire in a circuit there usually are components such as resistors which restrict the flow of electric charge, similar to the way rocks and debris in a river restrict the flow of the river water.

Continue..

• Common DC circuit diagram is shown in figure containing resistors and battery.

Fig 1

Voltage source• A voltage source is a two terminal device

which can maintain a fixed voltage. An ideal voltage source can maintain the fixed voltage independent of the load resistance or the output current. However, a real-world voltage source cannot supply unlimited current. A voltage source is the dual of a current source. Real-world sources of electrical energy, such as batteries, generators, and power systems, can be modeled for analysis purposes as a combination of an ideal voltage source and additional combinations of impedance elements.

Cont..

A schematic diagram of a real voltage source, V, driving a resistor, R, and creating a current I

Fig 2

Ideal voltage source• An ideal voltage source is a two-terminal device

that maintains a fixed voltage drop across its terminals. It is often used as a mathematical abstraction that simplifies the analysis of real electric circuits. If the voltage across an ideal voltage source can be specified independently of any other variable in a circuit, it is called an independent voltage source. Conversely, if the voltage across an ideal voltage source is determined by some other voltage or current in a circuit, it is called a dependent or controlled voltage source.

Cont..

• A mathematical model of an amplifier will include dependent voltage sources whose magnitude is governed by some fixed relation to an input signal, for example. In the analysis of faults on electrical power systems, the whole network of interconnected sources and transmission lines can be usefully replaced by an ideal (AC) voltage source and a single equivalent impedance

Cont..

Ideal Voltage Source

Controlled Voltage SourceSingle cell

Battery of cells

Fig 3

Current sources• A current source is an electronic circuit that delivers or

absorbs an electric current which is independent of the voltage across it.

• A current source is the dual of a voltage source. The term constant-current 'sink' is sometimes used for sources fed from a negative voltage supply. Figure 1 shows the schematic symbol for an ideal current source, driving a resistor load. There are two types – an independent current source (or sink) delivers a constant current. A dependent current source delivers a current which is proportional to some other voltage or current in the circuit.

Cont..

Ideal Current Source

Controlled Current Source

Fig 3

Dependent and independent source

• Dependent sources:-

• In the theory of electrical networks, a dependent source is a voltage source or a current source whose value depends on a voltage or current somewhere else in the network.

• Dependent sources are useful, for example, in modeling the behavior of amplifiers. A bipolar junction transistor can be modeled as a dependent current source whose magnitude depends on the magnitude of the current fed into its controlling base terminal.

Cont..

• An operational amplifier can be described as a voltage source dependent on the differential input voltage between its input terminals. Practical circuit elements have properties such as finite power capacity, voltage, current, or frequency limits that mean an ideal source is only an approximate model. Accurate modelling of practical devices requires using several idealized elements in combination.

Classification

Dependent sources can be classified as follows: a)Voltage-controlled voltage source: The source delivers

the voltage as per the voltage of the dependent element. b)Voltage-controlled current source: The source delivers the

current as per the voltage of the dependent element. c)Current-controlled current source: The source delivers the

current as per the current of the dependent element. d)Current-controlled voltage source: The source delivers the

voltage as per the current of the dependent element.

Circuits

Voltage-controlled voltage source

Voltage controlled current source

Current controlled current source

Current controlled voltage source

Fig 4

Independent sources

• An independent voltage source maintains a voltage (fixed or varying with time) which is not affected by any other quantity. Similarly an independent current source maintains a current (fixed or time-varying) which is unaffected by any other quantity. The usual symbols are shown in figure

Symbols

• Symbols for dependent sources

Kirchhoff's laws• Kirchoff’s current law:-• This law is also called Kirchhoff's first law, Kirchhoff's

point rule, or Kirchhoff's junction rule (or nodal rule).• The principle of conservation of electric charge implies

that:• At any node (junction) in an electrical circuit, the sum

of currents flowing into that node is equal to the sum of currents flowing out of that node, or:The algebraic sum of currents in a network of conductors meeting at a point is zero.Recalling that current is a signed (positive or negative) quantity reflecting direction towards or away from a node, this principle can be stated as:

Cont..

• n is the total number of branches with currents flowing towards or away from the node.

• The law is based on the conservation of charge whereby the charge (measured in coulombs) is the product of the current (in amperes) and the time (in seconds).

Cont..

The current entering any junction is equal to the current leaving that junction. i2 + i3 = i1 + i4

Page No: 1.16 from Elements of Electrical Engineering ( J.N.Swamy)

Kirchoff’s voltage law• 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:

Cont..

The sum of all the voltages around the loop is equal to zero. v1+ v2 + v3 - v4 = 0

Page No: 1.17 from Elements of Electrical Engineering ( J.N.Swamy)

Cont..• 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 equal the amount of energy lost per unit charge, as energy and charge are both conserved.

Nodal Analysis • Circuit Nodes and Loops:-

• Node:- A node is a point where two or more circuit elements are connected.

• Loop:- A loop is formed by tracing a closed path in a circuit through selected basic circuit elements without passing through any intermediate node more than once

Example: Find the Nodes

+

-Vs

node

Page No: 2.35 self making from Circuits and Networks (U.A.Patel)

Example: Find the loops

loopPage No: 2.35 self making from Circuits and Networks (U.A.Patel)

Equivalent Circuits:-

Source Transformation

Vs

+

-

Rs

Is Rs

sss IRV s

ss R

VI

Page No: 2.61 self making from Circuits and Networks (U.A.Patel)

Methods of Analysis• Introduction• Nodal analysis• Nodal analysis with voltage source• Mesh analysis• Mesh analysis with current source• Nodal and mesh analyses by inspection• Nodal versus mesh analysis

Steps of Nodal Analysis1. Choose a reference (ground) node.2. Assign node voltages to the other nodes.3. Apply KCL to each node other than the reference

node; express currents in terms of node voltages.4. Solve the resulting system of linear equations for

the nodal voltages.

Common symbols for indicating a reference node, (a) common ground, (b) ground, (c) chassis.

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1. Reference Node

The reference node is called the ground node where V = 0

+

–

V 500W

500W

1kW

500W

500WI1 I2

Page No: 2.53 self making from Circuits and Networks (U.A.Patel)

2. Node Voltages

V1, V2, and V3 are unknowns for which we solve using KCL

500W

500W

1kW

500W

500WI1 I2

1 2 3

V1 V2 V3

Page No: 2.37 self making from Circuits and Networks (U.A.Patel)

3. Mesh Analysis

• Mesh analysis: another procedure for analyzing circuits, applicable to planar circuit.

• A Mesh is a loop which does not contain any other loops within it

(a) A Planar circuit with crossing branches,(b) The same circuit redrawn with no crossing branches.

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• Steps to Determine Mesh Currents:1. Assign mesh currents i1, i2, .., in to the n meshes.

2. Apply KVL to each of the n meshes. Use Ohm’s law to express the voltages in terms of the mesh currents.

3. Solve the resulting n simultaneous equations to get the mesh currents.

Figure:A circuit with two meshes.

Page No: 1.53 from Circuits and Networks (U.A.Patel)

• Apply KVL to each mesh. For mesh 1,

• For mesh 2,123131

213111

)(

0)(

ViRiRR

iiRiRV

223213

123222

)(

0)(

ViRRiR

iiRViR

• Solve for the mesh currents.

• Use i for a mesh current and I for a branch current. It’s evident from Fig. 3.17 that

2

1

2

1

323

331

VV

ii

RRRRRR

2132211 , , iiIiIiI

• Find the branch current I1, I2, and I3 using mesh analysis.

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• For mesh 1,

• For mesh 2,

• We can find i1 and i2 by substitution method or Cramer’s rule. Then,

123

010)(10515

21

211

ii

iii

12

010)(1046

21

1222

ii

iiii

2132211 , , iiIiIiI

• Use mesh analysis to find the current I0 in the circuit.

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• Apply KVL to each mesh. For mesh 1,

• For mesh 2,126511

0)(12)(1024

321

3121

iii

iiii

02195

0)(10)(424

321

12322

iii

iiiii

• For mesh 3,

• In matrix from become

we can calculus i1, i2 and i3 by Cramer’s rule, and find I0.

02

0)(4)(12)(4

, A, nodeAt

0)(4)(124

321

231321

210

23130

iii

iiiiii

iII

iiiiI

00

12

21121956511

3

2

1

iii

Mesh Analysis with Current Sources

A circuit with a current source.

Page no. 2.36 self making from Circuits and Networks (U.A.Patel)

• Case 1– Current source exist only in one mesh

– One mesh variable is reduced• Case 2– Current source exists between two meshes, a

super-mesh is obtained.

A21 i

• Superposition is a direct consequence of linearity• It states that “in any linear circuit containing multiple

independent sources, the current or voltage at any point in the circuit may be calculated as the algebraic sum of the individual contributions of each source acting alone.”

2

313221

31

323121

32

22

ERRRRRR

RRV

RRRRRR

RI

I

S

I

Superposition R3=80

R2=0.4

+

_ VS=14V E2=12V

R1=0.5

I2

I

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Superposition Theorem:-

How to Apply Superposition?

• To find the contribution due to an individual independent source, zero out the other independent sources in the circuit.– Voltage source short circuit.– Current source open circuit.

• Solve the resulting circuit using your favorite techniques.– Nodal analysis– Loop analysis

Superposition

For the above case:

Zero out Vs, we have : Zero out E2, we have :

R1 R3

E2

R2

I2’’

R1 R3

E2

R2

I2’I

+_Vs

1 31 3

1 3

1 2 2 3 1 31 1 3

1 3

2 1 32

1 2 2 3 1 3

/ /2

/ /

R RR R

R R

R R R R R RR R R

R R

E R RI

R R R R R R

2 32 3

2 3

1 2 2 3 1 31 1 3

2 3

2 3

1 2 2 3 1 3

2 332

2 3 1 2 2 3 1 3

/ /

/ /

s

s

R RR R

R R

R R R R R RR R R

R R

V R RI

R R R R R R

V R RRI I

R R R R R R R R

Superposition

2kW1kW

2kW12V

+-

I0

2mA

4mA

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Superposition

2kW1kW

2kW

I’o

2mA

0 2 1

1 2 A

I I I

I m

KVL for mesh 2: 2 1 2

2 1

1k 2k 0

1 2A

3 3

I I I

I I m

0 2 1

22

3

4A

3

I I I

m

I1 I2

Mesh 2

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Superposition

P2.7

2kW1kW

2kW

I’’0

4mAI1

I2

KVL for mesh 2:

2 2 1 21k 0 2k 0I I I I

2 0

0o

I

I

Mesh 2

0 2I I

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Superposition

P2.7

2kW1kW

2kW12V

+-

I’’’0

I2

Mesh 2

2oI I

KVL for mesh 2:

2 21k 12V 2k 0I I

2

124 A

1k 2kI m

4 AoI m

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Superposition

I0 = I’0 +I’’0+ I’’’0 = -16/3 mA

2kW1kW

2kW12V

+-

I0

2mA

4mA

2kW1kW

2kW12V

+-

I0

2mA

4mA

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Thevenin's theorem

• Any circuit with sources (dependent and/or independent) and resistors can be replaced by an equivalent circuit containing a single voltage source and a single resistor

• Thevenin’s theorem implies that we can replace arbitrarily complicated networks with simple networks for purposes of analysis

Thevenin’s theorem

Circuit with independent sources

RTh

Voc

+

-

Thevenin equivalent circuit

Independent Sources

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No Independent Sources

Circuit without independent sources

RTh

Thevenin equivalent circuit

Thevenin’s theorem

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