Thevenin theorem in Electrical Network

0
786

THEVENIN THEOREM  :

This theorem state that any two terminal network containing voltage source and current source , can be replaced by an equivalent circuit consisting of an voltage source Vth in series with an impedance Rth,

Where Vth is an open circuited voltage between terminals of network and Rth is impedance calculated between terminal of network

THEVENIN4

                                          Electrical Network ( N1) 

by help of thevenin theorem ,above electrical network N1 can be represented as  

where Rth = thevenin impedance 

            Vth = thevenin voltage 

 

THEVEVNIN3

                              Thevenin equivalent circuit of above network 

 

PROOF OF THEVENIN THEOREM :

 

thevenin1

In above circuit , first  calculate current chart?cht=tx&chl=I %7B%7BL%7D%7D in impedance chart?cht=tx&chl=Z %7B%7BL%7D%7D by using mesh analysis and after that calculate current chart?cht=tx&chl=I %7B%7BL1%7D%7D in impedance chart?cht=tx&chl=Z %7B%7BL%7D%7D by using thevenin theorem , if these current will be same (chart?cht=tx&chl=I %7B%7BL%7D%7D%20%3DI %7B%7BL1%7D%7D)  then thevenin theorem will be proved 

 

STEP(1): calculation of load current chart?cht=tx&chl=I %7B%7BL%7D%7D%20 in load chart?cht=tx&chl=Z %7B%7BL%7D%7D%20 by using mesh analysis 

 

for Mesh1 ( contains Z1,Z2 and E ) : suppose in mesh1 current flow is I1 

for Mesh2 ( contains Z2,Z3 and chart?cht=tx&chl=Z %7B%7BL%7D%7D%20) suppose in mesh2 current flow is I2

 

Apply KVL in Mesh1:

E= Z1I1 +Z2(I1-I2)

E= (Z1+Z2)I1 -Z2I2                              

here I2 = chart?cht=tx&chl=I %7B%7BL%7D%7D%20 ( current flow in load chart?cht=tx&chl=Z %7B%7BL%7D%7D%20)

E= (Z1+Z2)I1 – Z2chart?cht=tx&chl=I %7B%7BL%7D%7D%20                                     eq(1) 

 

Apply  KVL in Mesh2:

Z3I2+chart?cht=tx&chl=Z %7B%7BL%7D%7D%20I2 +Z2 (I2 – I1) =0

I2= chart?cht=tx&chl=I %7B%7BL%7D%7D%20

(Z3+chart?cht=tx&chl=Z %7B%7BL%7D%7D%20+Z2)chart?cht=tx&chl=I %7B%7BL%7D%7D%20 – Z2I1  =0                            eq(2)

After solving eq(1) and eq(2) ,will get 

chart?cht=tx&chl=I %7B%7BL%7D%7D%20= EZ2/( Z1Z3+Z1chart?cht=tx&chl=Z %7B%7BL%7D%7D%20+Z1Z2+Z2Z3+Z2chart?cht=tx&chl=Z %7B%7BL%7D%7D%20)       eq(3)

 

STEP(2): calculation of load current (chart?cht=tx&chl=I %7B%7BL1%7D%7D%20 ) in load chart?cht=tx&chl=Z %7B%7BL%7D%7D%20 by using Thevenin theorem

(1) Calculation for Vth: remove the load chart?cht=tx&chl=Z %7B%7BL%7D%7D%20 and find the voltage across open circuited terminals as shown in figure below 

THEVENIN2 

Voltage across open circuited terminal will be 

    Vth   = EZ2 / ( Z1+Z2) 

(2) Calculation for Rth : for calculation of Rth first make voltage source E short circuited and calculate equivalent resistance  across open circuited terminal as figure shown below 

 

THEVENIN5

Rth  =    Z3 + Z1Z2 / (Z1+Z2) 

(3) Calculation of currentchart?cht=tx&chl=I %7B%7BL1%7D%7D%20  in chart?cht=tx&chl=Z %7B%7BL%7D%7D%20 :  in thevenin equivalent circuit as shown below 

 

THEVEVNIN3 1

chart?cht=tx&chl=I %7B%7BL1%7D%7D%20  =   Vth / ( Rth + chart?cht=tx&chl=Z %7B%7BL%7D%7D%20

after calculation 

chart?cht=tx&chl=I %7B%7BL1%7D%7D%20 = EZ2 /( Z1Z3+Z1chart?cht=tx&chl=Z %7B%7BL%7D%7D%20+Z1Z2+Z2Z3+Z2chart?cht=tx&chl=Z %7B%7BL%7D%7D%20)     eq(4)

from eq(3) and eq(4) it is clear that 

chart?cht=tx&chl=I %7B%7BL%7D%7D%20 = chart?cht=tx&chl=I %7B%7BL1%7D%7D%20   hence thevenin theorem proved