UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
1.0
Kirchoff’s Law
Kirchoff’s Current Law (KCL) states at any junction in an electric circuit the total current flowing
towards that junction is equal to the total current flowing away from the junction, i.e. Σ I = 0
Thus , referring to figure 1:
∑ current towards = ∑ current flowing away
I1 + I2+ I3 = I4 + I5
I1 + I2 + (- I3 ) + (- I4 ) + (-I5 ) = 0
∑I= 0
Figure 1
Kirchoff’s Voltage Law (KVL) states in any closed loop in a network, the algebraic sum Figure 2
of the voltage drops (i.e. products of current and resistance) taken around the loop is equal to
the resultant e.m.f. acting in that loop.
E = IR1 + IR2
E = I(R1 + R2 )
E + (- IR1 ) + (- IR2) = 0
Figure 2
1.1
Mesh analysis
Analysis using KVL to solve for the currents around each closed loop of the network and
hence determine the currents through and voltages across each elements of the
network.
Mesh analysis procedure:
1. Assign a distinct current to each closed loop of the network.
2. Apply KVL around each closed loop of the network.
3. Solve the resulting simultaneous linear equation for the loop currents.
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UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
Example 1
Find the current flow through each resistor using mesh analysis for the circuit below.
Figure 3
Solution:
Step 1: Assign a distinct current to each closed loop of the network.
Figure 4
Step 2: Apply KVL around each closed loop of the network.
Loop 1:
Loop 2:
------------ equation 1
--------------- equation 2
Step 3: Solve the resulting simultaneous linear equation for the loop currents.
Solve equation 1 and 2 using matrix
Matrix form:
From KCL :
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UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
Example 2
Find the current flow through each resistor using mesh analysis for the circuit below.
Figure 5
Solution:
Step 1: Assign a distinct current to each closed loop of the network.
Figure 6
Step 2: Apply KVL around each closed loop of the network.
Loop 1:
Loop 2:
------------ equation 1
--------------- equation 2
Step 3: Solve the resulting simultaneous linear equation for the loop currents.
Solve equation 1 and 2 using matrix
Matrix form:
From KCL :
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UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
1.2
Nodes analysis
Analysis using KCL to solve for voltages at each common node of the network and hence
determines the currents through and voltages across each elements of the network.
Nodal analysis procedure:
1. Determine the number of common nodes and reference node within the network.
2. Assign current and its direction to each distinct branch of the nodes in the network.
3. Apply KCL at each of the common nodes in the network
4. Solve the resulting simultaneous linear equation for the nodal voltages.
5. Determine the currents through and voltages across each the elements in the
network.
Example 3
Find the current flow through each resistor using mesh analysis for the circuit below.
Figure 7
Solution:
Step 1: Determine the number of common nodes and reference node within the network (Figure 8).
1 common node (Va) , reference node C
Step 2: Assign current and its direction to each distinct branch of the nodes in the network (Figure 8).
Figure 8
Step 3: Apply KCL at each of the common nodes in the network
KCL:
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UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
Step 4: Solve the resulting simultaneous linear equation for the nodal voltages.
! "
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! "
∴ Step 5: Determine the currents through each elements
Example 4
Find the current flow through each resistor using mesh analysis for the circuit below.
Figure 9
Solution:
Step 1: Determine the number of common nodes and reference node within the network (Figure 10).
1 common node (Va) , reference node C
Step 2: Assign current and its direction to each distinct branch of the nodes in the network (Figure 10).
Figure 10
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UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
Step 3: Apply KCL at each of the common nodes in the network
KCL:
Step 4: Solve the resulting simultaneous linear equation for the nodal voltages.
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$ %& $ %
∴ ##
Step 5: Determine the currents through each elements
##
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UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
TUTORIAL 1
Find the current through each resistor for the networking below using Mesh Analysis and Nodal
Analysis.
a)
d)
b)
e)
c)
R1
4kΩ
R3
3kΩ
V1
R2
2kΩ
30V
V2
25V
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UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
2.0
Thevenin’ s Theorem
Thevenins Theorem states:
"Any linear circuit containing several energy source and resistances can be replaced by just a
Single Voltage in series with a Single Resistor".
Thevenins equivalent circuit.
Figure 11
Thevenin’s theorem procedure:
1. Open circuit RL and find Thevenin’s voltage (VTH).
2. Find Thevenin’s resistance (RTH) when voltage source is short circuit or current source is
open circuit and RL is open circuit.
3. Draw the Thevenin’s equivalent circuit such as in figure 11 with the value of VTH and RTH.
Find the IL which current flow through the RL.
Example 5
Find the current flow through RL equal to 30Ω for the circuit in Figure 12.
Figure 12
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UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
Solution:
Step 1:
Open circuit RL and find Thevenin’s voltage (VTH).
'( )*Ω
Using VDR find VTH
'( $ Figure 13
Step 2: Find Thevenin’s resistance (RTH) when voltage source is short circuit
'( ++ $ '( '( Ω
Figure 14
Step 3: Draw the Thevenin’s equivalent circuit with the value of VTH and RTH
'(
'( ,
, , , Figure 15
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UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
Example 6
Find current flow through R4.
R2
60Ω
R1
30Ω
Is
300mA
R3
90Ω
R4
25Ω
Figure 16
Solution :
Step 1 : Open circuit RL and find Thevenin’s voltage (VTH).
'( -*Ω $ Using CDR, find I2
$ .
$ Figure 17
'( $ Step 2: Find Thevenin’s resistance (RTH) when current source,IS is open circuit.
'( ++ '( ++ '( ++ $ '( '( Ω
Figure 18
Step 3: Draw the Thevenin’s equivalent circuit with the value of VTH and RTH
RTH
45Ω
VTH
IL
RL
25Ω
4.5V
Figure 19
MARLIANA/JKE/POLISAS/ET101-UNIT4
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TUTORIAL 2
1. Refer to figure 1, find the current flow
through resistor 12Ω using Thevenin’s
Theorem.
Figure 1
2. Find the current flow through resistor 15Ω
for the circuit in figure 2 using Thevenin’s
Theorem.
5. Calculate the current flow in 30Ω resistor
for the circuit in figure 5 using Thevenin’s
Theorem.
Figure 5
6. Refer to figure 6, find the current flow
through 50Ω using Thevenin’s Theorem.
Figure 6
Figure 2
3. Count value stream IL by using Thevenin’s
Theorem.
7. Use Thevenin’s Theorem, find the current
flow through resistor R=10Ω.
Figure 7
Figure 3
4. Use Thevenin’s Theorem to find the current
flowing in 5Ω resistor shown in figure 4.
Figure 4
8. Use Thevenin’s Theorem, find the current
flow through resistor R=10Ω.
Figure 8
3.0
Norton’s Theorem
Nortons Theorem states:
"Any linear circuit containing several energy sources and resistances can be replaced by a single
Constant Current generator in parallel with a Single Resistor".
Figure 20
Norton’s theorem procedure:
1. Remove RL from the circuit. Find IN by shorting links output terminal.
2. Find RN by short-circuit voltage source or open-circuit current source.
3. Draw the Norton’s equivalent circuit such as in figure 20 with the value of IN and RN. Find
the IL which current flow through the RL.
Example 7
Find the current flow through RL equal to 30Ω for the circuit in Figure 21.
Step 1:
Figure 21
Remove RL from the circuit. Find IN by shorting links output terminal.
Figure 22
UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
' ++ $ ' ' /
Step 2:
' ' 0 $ Find RN by short-circuit voltage source.
0 11
0 $ 0 /
Figure 23
Step3:
Draw the Norton’s equivalent circuit with the value of IN and RN. Find the IL which current
flow through the RL.
Using CDR, find IL
, 0
$ 0
0 ,
, $ Figure 24
Example 6
Find current flow through R4.
Figure 25
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UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
Solution:
Step 1: Remove RL from the circuit. Find IN by shorting links output terminal.
Current flow at 90Ω is 0A, so 23 2&*4.
R2
60Ω
Is
R1
30Ω
R3
90Ω
IN
R4
25Ω
0 $ 300mA
Step 2:
Figure 26
Find RN by open-circuit current source.
R2
60Ω
R1
30Ω
R3
90Ω
RN
0 11
0 11
$ 0 0 /
Figure 27
Step3:
Draw the Norton’s equivalent circuit with the value of IN and RN. Find the IL which current
flow through the RL.
Using CDR, find IL
Figure 28
MARLIANA/JKE/POLISAS/ET101-UNIT4
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UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
TUTORIAL 3
1. Refer to figure 1, find the current flow
through resistor 12Ω using Norton’s
Theorem.
Figure 1
2. Find the current flow through resistor 15Ω
for the circuit in figure 2 using Norton
Theorem.
5. Calculate the current flow in 30Ω resistor
for the circuit in figure 5 using Norton
Theorem.
Figure 5
6. Refer to figure 6, find the current flow
through 50Ω using Norton Theorem.
Figure 6
Figure 2
3. Count value stream IL by using Norton
Theorem.
7. Use Norton Theorem, find the current flow
through resistor R=10Ω.
Figure 7
Figure 3
4. Use Norton Theorem to find the current
flowing in 5Ω resistor shown in figure 4.
Figure 4
MARLIANA/JKE/POLISAS/ET101-UNIT4
8. Use Norton Theorem, find the current flow
through resistor R=10Ω.
Figure 8
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4.0
Maximum Power Transfer theorem
The maximum power transfer theorem states:
‘A load will receive maximum power from a linear bilateral dc network when its total resistive
value equal to the Thevenin’s or Norton resistance of the network as seen by the load.’
Figure 29
For the Thevenin equivalent circuit above, maximum power will be delivered to the load when:
56 578
For the Norton equivalent circuit above, maximum power will be delivered to the load when:
56 53
There are four conditions occur when maximum power transfer took place in a circuit:
1.
2.
3.
4.
Value of RL equal to RTH (RL=RTH).
Value of current is half of the current when RL is short circuited.
Value of load voltage is half the Thevenin’s voltage (VL = ½VTH).
Percentage of efficiency,η% = 50%.
Where:
,
,
$ 9 $ 9
9η '( ,
'(
Example 7
Refer to figure 30, determine the load power for each of the following value of the variable load
resistance and sketch the graph load power versus load resistance.
a) 25Ω
b) 50Ω
c) 75Ω
d) 100Ω
e) 125Ω
Figure 30
UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
Solution:
, '(
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a)
b)
c)
, /
# :, ;
, /
#
# :, # ;
d)
, /
# :, ;
, /
# :, ;
e)
, #/
#
# #
:, # # ;
RTH
75Ω
75Ω
75Ω
75Ω
75Ω
75Ω
RL
0
25Ω
50Ω
75Ω
100Ω
125Ω
I
0.133A
0.1A
0.08A
0.067A
0.057A
0.05A
VTH
10V
10V
10V
10V
10V
10V
VL=IRL
0V
2.5V
4V
5.0V
5.7V
6.5V
%η
η
0%
25%
40%
50%
57%
65%
PL
0W
0.25W
0.32W
0.336W
0.325W
0.312W
Load Power (W)
Load Power (PL) vs Load Resistance(RL)
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
50, 0.32
75, 0.336
100, 0.325 125, 0.312
25, 0.25
0, 0
0
20
40
60
80
100
120
140
Load Resistance (Ω)
Figure 31
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UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
Example 8
For the network in figure 32, determine the value of R for maximum power transfer to R and
hence calculate the maximum power using Thevenin’s equivalent circuit.
Figure 32
Solution:
Open circuit R and find Thevenin’s voltage (VTH).
'( Ω
Using VDR find VTH
'( $ Figure 33
Find Thevenin’s resistance (RTH) when voltage source is short circuit
'( ++ $
'( '( Ω
Figure 34
Draw the Thevenin’s equivalent circuit with the value of VTH and RTH
Maximum power transfer occur when R=RTH.
So, the value of R is 10Ω.
, , Figure 35
MARLIANA/JKE/POLISAS/ET101-UNIT4
Maximum power transfer,
:,< = , ;
18
5.0
Superposition Theorem
The superposition theorem states:
‘In any network made up of linear resistances and containing more than one source of e.m.f, the
resultant current flowing in any branch is the algebraic sum of the currents that would flow in
that branch if each source was considered separately, all other sources being replaced at that
time by their respective internal resistances.’
Removing the effect of voltage and current source
Voltage source
Current source
Example 9
Determine the current through resistor R2=5Ω for the network in figure 36 using superposition
theorem.
Figure 36
Solution:
Step 1: V active , I inactive. So current source is open circuit.
Figure 37
UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
Step 2: V inactive, I active. So voltage source is short circuit.
Using CDR
> $
$ Figure 38
Step 3: Total current through R2=5Ω.
Ia
1A
Ib 6A
? > #
Example 10
Find the current flow through each resistor for the network in figure 39.
Figure 39
Solution:
Step 1: V1 active, V2 inactive
' 11
' /
@ ' @ $ @ Figure 40
MARLIANA/JKE/POLISAS/ET101-UNIT4
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Step 2: V1 inactive, V2 active
' 11
' /
@@ #
' $ # #
@@ @@ Figure 41
Step 3: Total current flow through each resistor
IR1 ⇒ I1’=0.429A
I1’’=0.571A
So
@@ @ # I2’’=0.714A
IR2 ⇒ I2’=0.286A
@@
@
So
? # IR3 ⇒ I3’=0.143A
So
I3’’=0.143A
? @@ @ $ # UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
TUTORIAL 4
Find the current through each resistor for the networking below using Superposition Theorem.
b)
d)
b)
e)
c)
R1
4kΩ
R3
3kΩ
V1
R2
2kΩ
30V
V2
25V
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