Thevenin’s Theorem

According to this theorem if a resistor of R ohms be connected between any two terminals of a linear bilateral network (the circuit which has the same characteristics in either direction such as transmission line then resulting steady state current through resistor will be  V   equal to where V is the p.d. across
                                                                                        RL+R
these two points and R is the resistance of the network measured between these two points.

For understanding this theorem let us consider a circuit (shown in) which consists of a source of e.m.f. E with internal resistance of r Ω connected to an external circuit consisting of resistance of R₁ and R₂ ohms in series.

So far as terminals AB across which a resistance of R₂ ohms is connected, the network acts as a source of voltage, say of V volts where V is the voltage across AB before connecting external resistor of R₁ ohms.

The resistance of network measured between terminals AB (with source of e.m.f removed leaving its internal resistance in the circuit) is given by the expression

 R =      1       =   R₂(R₁+r)
       1   +    1     R₁+r+R₂
      R₁+r   R₂


Now when external resistor of R₁ ohms is connected across terminals AB, the network behaves as a source of voltage V volts, the internal resistance of R ohms and current flowing through the resistance R₁ is given by the expression

              I =   V
                            R+RL
Since     V =  (    E     )  R₂ and R = R₂(R₁+r)
                    r+R₁+R₂                    R₁+R₂+r
                 
             ER₂
          r+R₁+R₂
     I = R₂(R₁+r) + RL =         ER₃         
        R₁+R₂+r          R₂(R₁+r)+RL(R₁+R₂+r)

The Thevenin’s theorem can be stated as below:

The current in any bilateral circuit element which may be called R₁) in a network is the same as would obtained if R₁ were supplied with a source of voltage V in series with an equivalent resistance R,V being the open circuit voltage at the terminals form which R₁ has been removed and R being the resistance that would be measured at these terminals after all sources have been removed and each has been replaced by its internal resistance.

This theorem is advantageous when we are to determine the current in a particular passive element of a linear bilateral network particularly when it is desired to find the current which flows through a resistor for different values of the later.

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