B tech ee ii_ eee_ u-1_ dc circuit analysis_dipen patel

Unit I
DC Circuit Analysis
Course : B.Tech
Branch : EE
Semester : II
Subject : Elements of Electrical Engineering

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 Source
Single 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

Star Delta connection circuit
• The Y-Δ transform, also written wye-delta and also
known by many other names, 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 . This circuit
transformation theory was published by Arthur Edwin
Kennelly in 1899. It is widely used in analysis of three-
phase electric power circuits.
• The Y-Δ transform can be considered a special case of
the star-mesh transform for three resistors.

Cont..
Page No. 1.19 from Elements of Electrical Engineering ( J.N.Swamy)

Cont..
• The transformation is used to establish equivalence
for networks with three 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
complex as well as real impedances.
• Equations for the transformation from Δ-load to Y-
load 3-phase circuit
• The general idea is to compute the impedance at a
terminal node of the Y circuit with impedances , to
adjacent node in the Δ circuit by

Cont..
• where are all impedances in the Δ circuit. This yields the
specific formulae

Cont..
• Equations for the transformation from Y-load to
Δ-load 3-phase circuit.
• The general idea is to compute an impedance in
the Δ circuit by
• where is the sum of the products of all pairs of
impedances in the Y circuit and is the impedance
of the node in the Y circuit which is opposite the
edge with . The formula for the individual edges
are thus

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
loop

Equivalent Circuits:-
Source Transformation
Vs
+
-
Rs
Is Rs
sss IRV 
s
s
s
R
V
I 

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 Analysis
1. 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.
self making from Circuits and Networks (U.A.Patel)

1. Reference Node
The reference node is called the ground node
where V = 0
+
–
V 500W
500W
1kW
500W
500W
I1 I2

2. Node Voltages
V1, V2, and V3 are unknowns for which we solve
using KCL
500W
500W
1kW
500W
500W
I1 I2
1 2 3
V1 V2 V3

Steps of Nodal Analysis
1. 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.

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.

• 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
V
V
i
i
RRR
RRR
2132211 ,, iiIiIiI 

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

• 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.

• 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




























0
0
12
211
2195
6511
3
2
1
i
i
i

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.”
Superposition
_USI1
I1
I1
I1
I1 I1 I1 I1
 

R1
R1
R1 R1
IS 
R3=80W
R2=0.4W
+
_
VS=14V E2=12V
R1=0.5W
I2
I
 
2
313221
31
323121
3
2
22
E
RRRRRR
RR
V
RRRRRR
R
I
I
S
I
    







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 3
1 3
1 3
1 2 2 3 1 3
1 1 3
1 3
2 1 3
2
1 2 2 3 1 3
/ /
2
/ /
R R
R R
R R
R R R R R R
R R R
R R
E R R
I
R R R R R R


 
 


  
 
 
 
 
2 3
2 3
2 3
1 2 2 3 1 3
1 1 3
2 3
2 3
1 2 2 3 1 3
2 33
2
2 3 1 2 2 3 1 3
/ /
/ /
s
s
R R
R R
R R
R R R R R R
R R R
R R
V R R
I
R R R R R R
V R RR
I I
R R R R R R R R


 
 



 

   
  

Superposition
2kW1kW
2kW
12V
+-
I0
2mA
4mA

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 2
A
3 3
I I I
I I m
  W   W 
  
 0 2 1
2
2
3
4
A
3
I I I
m
         
 
 
I1 I2
Mesh 2

Superposition
P2.7
2kW1kW
2kW
I’’0
4mA
I1
I2
KVL for mesh 2:
 2 2 1 21k 0 2k 0I I I I W     W 
2 0
0o
I
I

 
Mesh 2
0 2I I  

Superposition
P2.7
2kW1kW
2kW
12V
+-
I’’’0
I2
Mesh 2
2oI I 
KVL for mesh 2:
2 21k 12V 2k 0I I W    W 
2
12
4 A
1k 2k
I m 
W  W
4 AoI m

Superposition
I0 = I’0 +I’’0+ I’’’0 = -16/3 mA
2kW1kW
2kW
12V
+-
I0
2mA
4mA
2kW1kW
2kW
12V
+-
I0
2mA
4mA

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

No Independent Sources
Circuit without independent sources
RTh
Thevenin equivalent circuit
Thevenin’s theorem

REFRENCES-IMAGES
• http://www.school-for-
champions.com/science/images/electricity_o
hms_law_dc_circuit.gif
• http://upload.wikimedia.org/wikipedia/comm
ons/7/72/Ohm%27s_Law_with_Voltage_sourc
e.svg
• http://www.engineersblogsite.com/wp-
content/uploads/2013/03/active-elements.jpg

REFERENCE- BOOK
• B.L.Theraja, “Electrical Technology Vol.1”, S.Chand Publication.
• D.P.Kothari, “Basic Electrical Engineering”, Tata McGraw-Hill
publication.
• U.A.Patel “Circuits and Networks”.
WEB REFRENCE
WWW.SCRIBD.COM
WWW.AUTHORSTREAM.COM

B tech ee ii_ eee_ u-1_ dc circuit analysis_dipen patel

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