Tuesday, 30 July 2013

Thevenin’s And Norton's theorem

Thevenin’s Statement: A linear network consisting of a number of voltage sources and resistances can be replaced by an equivalent network having a single voltage source called Thevenin’s voltage ( Vth) and a single resistance called Thevenin’s resistance (Rth) .

Norton's statement
A linear network consisting of a number of voltage sources and resistances can be replaced by an equivalent network having a single current source called Norton current ( IN) and a single resistance called Norton’s resistance (Rn) .

There are three cases in Thevenin (Norton )equivalent depending on the sources present in the circuit

1) Circuit consisting of all independent sources

2) Circuit consisting of atleast on dependent source and atleast one independent source

3) Circuit consisting of only dependent sources and no independent sources

1) Circuit consisting of all independent sources


Solution
We want to create a Thevenin equivalent circuit of the circuit to the left of the terminals a-b.now separate RL from the circuit and find Rth from terminals a-b
Rth
For finding Rth. Short circuit the current sources, open circuit the voltage sources
Now the circuit appears as follows
From the above figure, it can be seen that the Thevenin resistance RTH is a parallel combination of a 3Ω resistor and a 6Ωresistor, in series with a 2Ω resistor.
Rth = ( ( 6 * 3 ) / ( 6 + 3 )) + 2 = 4Ω
Finding Vth
As current through 2Ω resistance is zero Vth at terminals at a-b appear across 3Ω resistor
Applying source transformation technique
Now redrawing the circuit
Applying nodal analysis
(Vth – 18)/6  + Vth / 3= 0
Solving above equation,we get Vth = 6 v
Now the equivalent circuit is 
2) Circuit consisting of atleast on dependent source and atleast one independent source
Find thevenin equivalent of above circuit across terminal ab
Solution
For VTH
Applying nodal analysis at node V1
V1-20  +   V1   +  V1 - VTH ­   = 0    ----------------------------- 1
    40        200        100

At node V1
i1 = V1/200   ----------------------------------------------------2

Applying nodal analysis at node VTH
VTH –V1  - 1.5 i1 =0  -------------------------------------------3
   100
Solving above equation we get
VTH  = 700 / 18 = 38.89 V
For RTH
From the above the circuit
In = IX + 1.5 I1 ------------------------------------------ 1
Applying nodal analysis at V1
V1-20  +   V1   +  V1 - 0­   = 0    ------------------------ 2
   40       200        100
Solving above equation we get
V1  =  (100 / 8) V
At node V1
i1 = V1/200   --------------------------------------------  3
i1 = 1 / 16
IX  = ( V1 – 0 ) / 100
IX  =  V1 / 100 = (1 / 8)A
Substituting the values of IX and i1 we get
In = 0.21875 A
RTH  =  VTH / In
Substituting the values of  VTH , In we get
RTH= 177.78Ω

3) Circuit consisting of only dependent sources and no independent sources


Find thevenin equivalent to the above circuit across ab
Solution
Connect a known current source or voltage source to the above circuit because without a independent source the circuit becomes inactive. So to make it active we are connecting a known current source or voltage source
From the above circuit VTH = V1
Applying nodal analysis at node V1
V1  +  V1 – VX  - 1  = 0
100            50

3V1 – 2VX = 100 ---------------------------------1
Applying nodal analysis at node VX

VX   +  (VX - V1)  + 0.1V1 = 0
200            50

16V1 + 5VX = 0
VX = (16/5)V1  ----------------------------------- 2
Substituting equation 2 in equation 1 we get
V1 = 10.6 V
RTH = VTH / 1
RTH = 10.6Ω

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