Thevenin's and Norton's Theorem

Thevenin’s theorem

Thevenin’s theorem states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source VTh in series with a resistor RTh, where VTh is the open-circuit voltage at the terminals and RTh is the input or equivalent resistance at the terminals when the independent sources are turned off.

Thevenin's Theorem states that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single voltage source and series resistance connected to a load. The qualification of “linear” is identical to that found in the Superposition Theorem, where all the underlying equations must be linear. If we're dealing with passive components, this is true. However, there are some components which are nonlinear: that is, their opposition to current changes with voltage and/or current. As such, we would call circuits containing these types of components, nonlinear circuits.

Thevenin's Theorem is especially useful in analyzing power systems and other circuits where one particular resistor in the circuit (called the “load” resistor) is subject to change, and re-calculation of the circuit is necessary with each trial value of load resistance, to determine voltage across it and current through it.

In finding the thevenin's resistance RTh, we need to consider two cases:
 
Case 1: If the network has no dependent sources, we turn off all independent sources. Rth is the inpit resistance of the network looking between terminals a and b, as shown below:           

Case 2: if the network has dependent sources. As with Superposition, dependent sources are not to be turned off because they are controlled by circuit variables.


Sample problem: Determine the thevenin’s equivalent  circuit between the terminals A&B For the circuit shown :

Answer:
First we are about to find the thevenin’s resistance. To find the thevenin’s resistance we remove the resistance R_L and open circuit the AB terminals. Then we remove the voltage source and short circuit it.

We can easily find the thevenin’s resistance now.

R^th = (5//10 + 3)

Now we have to find the thevenin’’s voltage. For this we remove the load resistance R_L.

 

Applying nodal equation method to point c,

(V-30)/5  + (V-0)/10  = 0

V = 20V

From figure 11.3 you can see that,

V_A = V_C

V_A = V^TH

Therefore V = V^TH

V^TH = 20V

NORTON'S THEOREM

In 1926, about 43 years after Thevenin published his theorem, E. L. Norton, an American engineer at Bell Telephone Laboratories, proposed a similar theorem.

Edward Lawry Norton was an accomplished Bell Labs engineer and scientist famous fordeveloping the concept of the Norton equivalent circuit.

Norton's theorem states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source I_N in parallel with a resistor R_N where I_N  is the short-circuit current  through the terminals and R_N is the input or equivalent resistance at the terminals when the independent sources are turned off.

Norton resistances are equal; that is,

To find the Norton current I_N, we determine the short-circuit current flowing from a to b in both circuits.

Original Circuit

Norton Equivalent Circuit

  Since the two circuits are equivalent. Thus,

 Dependent and independent sources are treated the same way as in Thevenin's theorem. Observe the close relationship between Norton's and Thevenin's theorems. Rn = Rth, and: 

 This is essentially source transformation. For this reason, source transformation is often called Thevenin-Norton Transformation.


Sample Problem:

Find the Norton’s equivalent circuit across A-B terminals for the circuit shown.

 Answer:

First we remove the 10Ω resistor and short circuit the terminals A&B. 

The current flowing through the short circuited terminals is called the Norton’s curren I_N.

To find the I_N we apply nodal equation for point C

(V – 30)/5 + V/10 + V/3 = 0

V = 180/19

Ohms law to 3Ω resistance

I = V/R

I_N = (180/19) / 3

I_N = 60/19

This is the Norton’s current.

Now we are about to find the Norton’s resistance. Note that this is equal to the thevenin’s resistance also.

R_N = (10//5) + 3

R_N = 19/3Ω

TO KNOW MORE ABOUT THEVENIN'S AND NORTON'S THEOREM: