Maximum power transfer theorem for ac network

Maximum Power Transfer Theorem for AC Networks.

1. CIRCUIT THEORY
MAXIMUM POWER TRANSFER THEOREM
FOR
AC NETWORK
Wednesday, August 21, 2019 1syed hasan saeed
SYED HASAN SAEED

2. MAXIMUM POWER TRANSFER THEOREM FOR AC NETWORK
REFERENCE BOOKS
•Introductory Circuit Analysis, Robert L. Boylested, Pearson Education,
Prentice Hall.
•Networks And Systems, AshfaqHusain, KhannaBook Publishing Co (P)
Ltd. Delhi.
•Networks And Systems, A Sudhakr, ShyammohanS Palli, Tata McGraw
Hill, New Delhi.
•Network Analysis, M.E. Van Valkenburg, PHI Learning Private limited,
New Delhi.
•Circuit Analysis Principle and Applications, Allan H. Robbins &Wilhelm
C. Miller, DELMAR CENGAGE Learning, Indian Reprint.
Wednesday, August 21, 2019 syed hasan saeed 2

3. Wednesday, August 21, 2019 syed hasan saeed 3
V
th
Z
th
Linear, Active,
Bilateral
Network
a
b
b
a
I
L
I
L
Thevenin’s Equivalent
Network
Fig. (1a)
Fig. (1b)
Z
L Z
L
MAXIMUM POWER TRANSFER THEOREM FOR AC NETWORK
Consider the fig. (1a) and (1b). Where V
th and Z
thare Thvenin’s Equivalent
Voltage, Thevenin’s Equivalent Impedance respectively and Z
Lis load Impedance
Now find the value of load impedance when it receives maximum power from the
network.

4. MAXIMUM POWER TRANSFER THEOREM FOR AC NETWORK
PROOF:
Wednesday, August 21, 2019 syed hasan saeed 42
Lth
2
Lth
th
L
LthLth
th
L
Lth
th
L
LLL
ththth
)X (X )R (R
V
I
)X (X j )R (R
V
I
Z Z
V
I
jX R Z
jX R Z








Power Transferred to the load is-(1)-------
)X (X )R (R
R V
R I P
2
Lth
2
Lth
L
2
th
L
2
LL



5. MAXIMUM POWER TRANSFER THEOREM FOR AC NETWORK
For any network V
th, R
thand X
thare fixed.
CASE 1: When R
Lis constant and X
Lis varying:
Wednesday, August 21, 2019 syed hasan saeed 5 
 
 
(2)----------- X- X
0 X 2(XR V-
0
)X (X )R (R
X 2(XR V-[0] )X (X )R (R

dX
dP
0
)X (X )R (R
R V
dX
d

dX
dP
thL
ththL
2
th
2
2
Lth
2
Lth
LthL
2
th
2
Lth
2
Lth
L
L
2
Lth
2
Lth
L
2
th
LL
L















Hence, the magnitude of load reactance must be equal Thevenin’s reactance but
opposite phase difference. This is the first condition. The equation (1) becomes)3(
)R (R
R V
P
2
Lth
L
2
th
L




6. MAXIMUM POWER TRANSFER THEOREM FOR AC NETWORK
CASE 2: When X
Lis constant and R
Lis varying
Wednesday, August 21, 2019 syed hasan saeed 6 
 
 
(4)------- R )X (X R
0)R 2(RR V-][V )X (X )R (R
0
)X (X )R (R
)R 2(RR V-][V )X (X )R (R

dX
dP
0
)X (X )R (R
R V
dR
d

dR
dP
L
2
Lth
2
th
LthL
2
th
2
th
2
Lth
2
Lth
2
2
Lth
2
Lth
LthL
2
th
2
th
2
Lth
2
Lth
L
L
2
Lth
2
Lth
L
2
th
LL
L















PutX
th=-X
Linequation(4)
R
L=R
th---------------(5)
Hence,formaximumPowerTransfer,theloadresistancemustbeequaltothe
Thevenin’sEquivalentResistance.Theequation(3)becomes)6(
4R
V
P
th
2
th
max L


7. MAXIMUM POWER TRANSFER THEOREM FOR AC NETWORK
CASE 3: Apply equation (2) and (5) simultaneously then
Wednesday, August 21, 2019 syed hasan saeed 7

thththLLL Z X j R X j R Z
Hence,MaximumPowerTransferTheoremstatesthe“MaximumPoweris
transferredfromACnetworktotheloadwhenLoadImpedanceisequaltothe
complexconjugateofThevenin’sEquivalentImpedance”

8. MAXIMUM POWER TRANSFER THEOREM FOR AC NETWORK
EXAMPLE:FindtheThevenin’sEquivalentcircuitofthenetworkviewedfrom
terminala-bandtheLoadImpedanceforMaximumPowerTransfer.Alsofind
themagnitudeofMaximumPowerTransfertotheLoad.
Wednesday, August 21, 2019 syed hasan saeed 8V 010
0

8 ohm
6 ohm
2 ohm
a
bFig.2a
STEP 1: Remove 2 ohm resistor, set the voltage source to zero i.eshort
circuit and redraw the circuit as shown in fig.2b
STEP 2: Calculation of Z
thparallelin are Zand Z
ohm j6- jXZ
ohm j8jXZ
21
C2
L1


8 ohm
6 ohm
b
a
Fig.2b1Z 2Z

9. MAXIMUM POWER TRANSFER THEOREM FOR AC NETWORK
Wednesday, August 21, 2019 syed hasan saeed 9








0
0
0
th
0
00
21
21
th
9024
902
048
Z
902
)906)(90(8
Z Z
ZZ
z
STEP 3: Calculation of Thevenin’s Equivalent voltage V
th
8 ohm
6 ohm
a
bV 010
0

V
th
Apply KVL in mesh (1)
IV18030V
)90-(-j6)(5IZV
A905
902
010
I
j2I010
0 j6I j8I-010
0
th
0
2th
0
0
0
0
0








Fig. 3
-j6 ohm
j8 ohm

10. MAXIMUM POWER TRANSFER THEOREM FOR AC NETWORK
STPE 4: Draw the Thevenin’s Equivalent Circuit
Hence the circuit is Purely Inductive. Therefore, Power will be zero.
Wednesday, August 21, 2019 syed hasan saeed 10
V
th
Z
thV18030
0
 
0
9024
a
b
Fig. 4
For Maximum Power Transfer 

90 24 Z Z
0
thL

11. Wednesday, August 21, 2019 syed hasan saeed 11
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