UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS · 2012-03-19 · 1.1 Mesh analysis ... Find...

UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS

MARLIANA/JKE/POLISAS/ET101-UNIT4 1

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:

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.

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.

∑ current towards = ∑ current flowing away I1 + I2+ I3 = I4 + I5

I1 + I2 + (- I3 ) + (- I4 ) + (-I5 ) = 0 ∑ I = 0

E = IR1 + IR2 E = I(R1 + R2 ) E + (- IR1 ) + (- IR2) = 0



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: ------------ equation 1

Loop 2: --------------- 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 :




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: ------------ equation 1

Loop 2: --------------- 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 :



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.


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:



Step 4: Solve the resulting simultaneous linear equation for the nodal voltages.

!

"

! #"

∴∴∴∴

Step 5: Determine the currents through each elements


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



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.

!

"

# $ %& $ % ∴∴∴∴ ##

Step 5: Determine the currents through each elements

##

##

##



TUTORIAL 1 Find the current through each resistor for the networking below using Mesh Analysis and Nodal Analysis. a)

b)

c)

R1

R2

R3

V1

V2

4kΩ

2kΩ

3kΩ

30V

25V

d)

e)



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



Solution: Step 1: Open circuit RL and find Thevenin’s voltage (VTH).

Figure 13

'( )*Ω

Using VDR find VTH

'( $

Step 2: Find Thevenin’s resistance (RTH) when voltage source is short circuit

Figure 14

'( ++

'( $

'( Ω

Step 3: Draw the Thevenin’s equivalent circuit with the value of VTH and RTH

Figure 15

, '('( ,

,

,



Example 6 Find current flow through R4.

60Ω

30Ω 90Ω 25Ω300mA

IsR1

R2

R3 R4

Figure 16

Solution : Step 1 : Open circuit RL and find Thevenin’s voltage (VTH).

Figure 17

'( -*Ω $

Using CDR, find I2

$ .

$

'( $

Step 2: Find Thevenin’s resistance (RTH) when current source,IS is open circuit.

Figure 18

'( ++ '( ++ '( ++

'( $

'( Ω

Step 3: Draw the Thevenin’s equivalent circuit with the value of VTH and RTH

RTH

RLVTH

4.5V

45Ω

25Ω

IL

Figure 19

, '('( ,

,

,

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.

Figure 2

3. Count value stream IL by using Thevenin’s Theorem.

Figure 3 4. Use Thevenin’s Theorem to find the current

flowing in 5Ω resistor shown in figure 4.

Figure 4

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 7. Use Thevenin’s Theorem, find the current

flow through resistor R=10Ω.

Figure 7 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.

Figure 21

Step 1: Remove RL from the circuit. Find IN by shorting links output terminal.

Figure 22



' ++ ' $

' /

' '

0 $

Step 2: Find RN by short-circuit voltage source.

Figure 23

0 11

0 $

0 /

Step3: Draw the Norton’s equivalent circuit with the value of IN and RN. Find the IL which current

flow through the RL.

Figure 24

Using CDR, find IL

, 00 ,

$ 0

, $

Example 6 Find current flow through R4.

Figure 25



Solution: Step 1: Remove RL from the circuit. Find IN by shorting links output terminal.

60Ω

30Ω 90Ω 25Ω300mA

IsR1

R2

R3 R4IN

Figure 26

Current flow at 90Ω is 0A, so 23 2&*4.

0 $

Step 2: Find RN by open-circuit current source.

60Ω

30Ω 90Ω

R1

R2

R3 RN

Figure 27

0 11 0 11

0 $

0 /

Step3: Draw the Norton’s equivalent circuit with the value of IN and RN. Find the IL which current flow through the RL.

Figure 28

Using CDR, find IL

, 00 ,

$ 0

, $



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.

Figure 2

3. Count value stream IL by using Norton Theorem.

Figure 3 4. Use Norton Theorem to find the current

flowing in 5Ω resistor shown in figure 4.

Figure 4

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 7. Use Norton Theorem, find the current flow

through resistor R=10Ω.

Figure 7 8. Use Norton Theorem, find the current flow

through resistor R=10Ω.

Figure 8

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. Value of RL equal to RTH (RL=RTH). 2. Value of current is half of the current when RL is short circuited. 3. Value of load voltage is half the Thevenin’s voltage (VL = ½VTH). 4. Percentage of efficiency,η% = 50%.

Where:

9η ,'( $ 9 ,

'( ,$ 9

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



Solution:

, '('( ,

:, ,,

a) , /

#

:, ; b) , /

#

:, ; c) , #/

# # #

:, ## ;

d) , /

# #

:, # ; e) , /

#

:, ;

RTH RL I VTH VL=IRL %ηηηη PL 75Ω 0 0.133A 10V 0V 0% 0W 75Ω 25Ω 0.1A 10V 2.5V 25% 0.25W 75Ω 50Ω 0.08A 10V 4V 40% 0.32W 75Ω 75Ω 0.067A 10V 5.0V 50% 0.336W 75Ω 100Ω 0.057A 10V 5.7V 57% 0.325W 75Ω 125Ω 0.05A 10V 6.5V 65% 0.312W

Figure 31

0, 0

25, 0.25

50, 0.32 75, 0.336 100, 0.325 125, 0.312

00.05

0.10.15

0.20.25

0.30.35

0.4

0 20 40 60 80 100 120 140

Load

Pow

er (W

)

Load Resistance (Ω)

Load Power (PL) vs Load Resistance(RL)



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).

Figure 33

'( Ω

Using VDR find VTH

'( $

Find Thevenin’s resistance (RTH) when voltage source is short circuit

Figure 34

'( ++

'( $

'( Ω

Draw the Thevenin’s equivalent circuit with the value of VTH and RTH

Figure 35

Maximum power transfer occur when R=RTH. So, the value of R is 10Ω.

,

, Maximum power transfer,

:,< = , ;

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



Step 2: V inactive, I active. So voltage source is short circuit.

Figure 38

Using CDR

>

$ $

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

Figure 40

' 11 ' /

@ '

@ $

@ $

Step 2: V1 inactive, V2 active

Figure 41

' 11 ' /

@@ '

#

@@ $ # #

@@ $ #

Step 3: Total current flow through each resistor

IR1 ⇒ I1’=0.429A I1’’=0.571A So @@ @ # IR2 ⇒ I2’=0.286A I2’’=0.714A So ? @@ @ # IR3 ⇒ I3’=0.143A I3’’=0.143A So ? @@ @



TUTORIAL 4 Find the current through each resistor for the networking below using Superposition Theorem. b)

b)

c)

R1

R2

R3

V1

V2

4kΩ

2kΩ

3kΩ

30V

25V

d)

e)

UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS · 2012-03-19 · 1.1 Mesh analysis ... Find...

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