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CHAPTER 2
DC CIRCUITSAND ANALYSIS
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CONTENTS
Ohms Law
Kirchhoffs Law
Series resistors & voltage division
Parallel resistors & current division
Y -
transformation
Method of Analysis Nodal and Mesh
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INSPIRING CREATIVE AND INNOVATIVE MINDS
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DC Circuits and method of analysis
Simplify dc circuits by combining resistances inseries and parallel or star-delta transformation(where applicable) circuit reduction.
- Apply the voltage-division and current-
division principles to determine thevoltages or currents.
- Usually, the circuit contains only one voltagesource or one current source.
Solve and simplify a more complex circuits usingthe node-voltage analysis and mesh-currentanalysis
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Ohms Law
Property of a material to resist a flow of current known as resistance
A
lR - measured in ohms ()
- Resistivity of the material
l - length of the material
A - Cross section area of the material
Mathematically,
+ V
i
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Ohms Law
Ohmss Law: A voltage across a resistor is directly proportional tothe current flowing through a resistor
+ V
i
v i
Constant of proportionality between v and i is the resistance, R ()
v = i R
Must comply with passive sign convention
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Ohms Law
Fixed resistors
Wirewound type
carbontype type
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Ohms Law
Variable resistors
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Ohms Law
Two extreme values of resistance:
Short circuit
Open circuit
0i
0
i
vR
ov
ivR
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Ohms Law
Conductance: reciprocal of resistance
v
i
R
1G - measured in siemens (S)
Conductance: ability of an element to conduct current
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Ohms Law
Power in a Resistor
vip
+ V
i
Rii)iR(p 2R
v)
R
v(vp
2
Always absorbs power
Always positive
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Kirchoffs Laws
Introduced in 1847 by Gustav RobertKirchoff German physicist
Formally known as
i) Kirchoffs current law (KCL)
ii) Kirchoffs voltage law (KVL)
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Kirchhoffs Law
Network topology
A branch represents a single element such as a
voltage source or a resistor.
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Kirchhoffs Law
Network topology
A node is the point of connection between two
or more branches.
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Kirchhoffs Law
Network topology
A branch represents a single element such as a
voltage source or a resistor.
A node is the point of connection between two
or more branches.
A loop is any closed path in a circuit.
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Kirchhoffs Law
Network topology
Two or more elements are inseries if they exclusively
share a single node and consequently share the same
current
Two or more elements are inparallel if they areconnected to the same two nodes and consequently
have the same voltage across them
1 & 2 - parallel
10V & 4 - parallel
5 in series with (1 and 2 in parallel)
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Kirchhoffs Law
Kirchhoffs Current Law (KCL)
Kirchhoffs current law (KCL) states that the algebraic sum
of currents entering a node (or a closed boundary) is zero
01
N
n
niMathematically,
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Kirchoffs current law (KCL) KCL states that the algebraic sum of currents entering a
node is zero.
01
N
nni
0)()()(54321
iiiii
i1 i2
i3
i4
i5
node
outin ii
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a
b
IT
IT = I1 I2 + I3Equivalent circuit
IT
I1 I
2I3
a
b
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INSPIRING CREATIVE AND INNOVATIVE MINDS
Example
Determine the current I for the circuit shown in the figure below.
I + 4-(-3)-2 = 0
I = -5A
This indicates that the
actual current for I is
flowing in the opposite
direction.
We can consider the whole
enclosed area as one node.
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Kirchhoffs Law
Kirchhoffs Voltage Law (KVL)
Kirchhoffs voltage law (KVL) states that the algebraic sum
of all voltages around a closed path (or loop) is zero.
Mathematically,0
1
M
m
nv
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Kirchoffs voltage law (KVL)
KVL states that the algebraic sum of all voltagesaround a closed path (or loop) is zero.
01
M
mmv
0)()()(54321
vvvvv
V4V1
+ V2 - + V3 -
- V5 +
i
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INSPIRING CREATIVE AND INNOVATIVE MINDS
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Example 1 Find Iand Vab
DC DC
DC
+
Vab
-
30V
3
10V
5
8V
I
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SERIES RESISTOR
26
R1 R2R3 RN
V
+ V1 - + V2 - + V3 - + VN -
Current flow through each elementVoltage across each element (voltage
drop)
I
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DC+ v1 - + v2 - + v3 - + vN -
R1 R2 R3 RNi
v
11 iRv 22iRv
33 iRv NN iRv KVL:
Nvvvvv ........
321
NiRiRiRiRv ......321
N
RRRRi ......321
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NRRRRiv ......321
eqiR
Neq RRRRR ...321
N
nn
R1
eqR
vi
DC
+ v1 - + v2 - + v3 - + vN -
R1 R2 R3 RNi
v
DC
Req
v
i
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DC
+ v1 - + v2 - + v3 - + vN -
R1 R2 R3 RN
i
v
1
321
11
... R
RRRRviRv
N
vRRRR
Rv
N
...
321
1
1
vRRRR
Rv
N
...
321
2
2
v3 = ? ; vN= ?
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Series resistor
vRRRR
R
vN
n
n ...
321
Voltage division
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Example 2 Find
a)Voltage across 2 and 3
b)Power absorbedc)Power supplied DC 20V
2
3
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DC 20V
2
3
I
+ V2 - +
V3
-
KVL :
A4
2032020 32
I
IIVV
V1243
V842
3
2
V
V
Voltage division
V82032
22
V V1220
32
33
V
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Parallel resistors
DC
Current flow through each elementVoltageacross each element (voltage
drop)
21 vvv
21iii KCL
V R1R2
i1
i i2
+
v1
-
+
v2
-
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2211RiRiv
;22
2
2
11
1
1R
v
R
vi
R
v
R
vi
eqR
v
RRv
R
v
R
v
iii
2121
21
11
V R1R2
i1
i i2
+
v1
-
+
v2
-
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21
111
RRReq
21
21
RR
RRR
eq
21 GGGeq
DC v
+
vReq
-
Req
i
V R1R2
i1
i i2
+
v1
-
+
v2
-
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21
21
RR
RiRiRv
eq
21
2
1
1i
RR
R
R
vi
21
1
2
2i
RRR
Rvi
iGG
Gi
21
1
1
iGG
G
i21
2
2
Current divider
V R1R2
i1
i i2
+
v1
-
+
v2
-
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NRRRR
1111
R
1
321eq
NeqGGGGG ....
321
R1 RNR3R2V+
V1
-
+
V2
-
+
V3
-
+
VN
-
i
i3 i
Ni
2i
1
NVVVVV 321
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iGGGG
Gi
N
....
321
1
1
iGGGG
Gi
N
N
N
....
321
R1 RNR3R2V+
V1
-
+
V2
-
+
V3
-
+
VN
-
i
i3 i
Ni
2i
1
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Example 3 For each figure,
calculate:
- Currents- Power on each
elements.
10 212A
i1 i2
10 212A
i1 i2 i3
5
10 212A
i1 i2 i3
5
3
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10 212A
i1 i2 A212210
21
i
A1012210
10
2
i
Current Division
A212)21()101(
1011
i
A1012)101()21(
212
i
POWER
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Example 3 Find v1 and v2, P3k and P20k, and
Power supplied by the source
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i1
i2i3
mA510)20/1()5/1()41(
411
i
V15
)105)(103( 331
v
mA110)20/1()5/1()4/1(
20/13
i
V20
)101)(1020( 332
v
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Series-parallel circuit
a
b
R1
R2 R3Rab= Req
R1
R2//R3
32
32
1
RRRRRRR
eqab
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Example 4
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Example 5
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Find v1 and v2, i1 and i2 ,P12 and P40
Example 6
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Example 7
1
3
1.6
2
4
a
b
Find the resistance Rab
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Example 8
Calculate Vo andIo in the circuit below:
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I0I1
I2
I1 = I0 + I2
0 V
70 30
20 5 +V0
-
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Example 9
For the circuit in Figure below, obtain the
Equivalent resistance at terminal a-b.
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Example 10
Find Reqand io in the following circuit
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Example 11 Find i and V0 in the circuit below:
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Example 12 FindRab
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Example 13
FindRab
3k
1k
400
600
a b
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Example 14 Find the equivalent resistance at terminals a-b .
Y t f ti
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Y transformationStar delta transformation
How can we combine R1 to R7 ?
Y transformation
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)(1
cba
cb
RRR
RRR
)(2
cba
ac
RRR
RRR
)(3
cba
ba
RRR
RRR
1
133221
R
RRRRRRRa
2
133221
R
RRRRRRRb
3
133221
R
RRRRRRRc
Delta -> Star Star -> Delta
Y transformationStar delta transformation
Y transformation
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Y transformationStar delta transformation
example
Y transformation
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Y transformationStar delta transformation
example
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INSPIRING CREATIVE AND INNOVATIVE MINDS
Obtain the equivalent resistance Rab for the circuit below
Example 15
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INSPIRING CREATIVE AND INNOVATIVE MINDS
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INSPIRING CREATIVE AND INNOVATIVE MINDS
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Source transformation
Another circuit simplifying technique
Itis the process of replacing a voltage source vS in serieswith a resistor R by a current source iS in parallel with a
resistor R, or vice versa
+
R
vs
a
b
Terminal a-b sees:
Open circuit voltage: vs
Short circuit current: vs/R
For this circuit to be equivalent, it
must have the same terminal
charateristics
Ris
a
b
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Source transformation
Another circuit simplifying technique
Itis the process of replacing a voltage source vS in serieswith a resistor R by a current source iS in parallel with a
resistor R, or vice versa
+
R
vs
a
b
Ris
a
b
ix
iy
Note: current through R (hence power) for both circuits is not the same
i.e. ix iy
l
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Find vo in the circuit shown below using source transformation
Example 16
l 1
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Example 17
E l 18
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Find io in the circuit shown below using source transformation
Example 18
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