Norton’s Theorem: Statement, Calculation, Advantages, Limitations

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The Norton theorem has a resemblance to the Thevenin theorem in that it enables the simplification of any linear circuit into an equivalent circuit. In contrast to the utilization of a voltage source and a series resistance, the Norton equivalent circuit is comprised of a current source accompanied by a parallel resistance.

Norton's Theorem
Norton’s Theorem

Norton’s Theorem Statement

Norton’s theorem states that” With respect to the terminal pair AB, the network N may be replaced with a current sources IN in parallel with an internal impedances ZN or RN. The current source IN, called the Norton’s current is the current that would flow from A to B when the terminals A and B are shorted together and ZN or RN is the internal impedance of the network N as seen from terminal A and B with all the sources set to Zero i.e. with all the voltage sources shorted and all the current sources open circuited leaving behind their internal resistances or impedances.”

The parallel combination of IN and ZN or RN is known as Norton’s equivalent of the network N with respect to terminal pair AB.

Hence, if a load impedance RL or ZL be connected across terminal A & B of network N, the current IL delivered to the load may be calculated from the equivalent circuits by connecting this impedance across A&B and shall be given as

[when we compare Thevenin’s theorem and Norton’s theorem these two theorems are basically same as the knowledge of one equivalent circuit gives the other through a simple transformation of current source into voltage and vice-versa]

Procedure to obtain IN and RN(ZN)

  • Remove the portion of the network across which the Thevenin’s Equivalent circuit is to be found.
  • Mark the terminal AB of the remaining 2 terminal circuit.
  • Short the terminals AB. Obtain the short circuit current i.e IN flowing from A to B keeping all the sources of their normal values.
  • Claculate Zth=ZN(RN)

For the calculation of Norton’s Resistance

case I:

If circuit having only independent sources. In this case, set all the sources at Zero values i.e with all the voltage sources shorted and all the current sources open circuited leaving behind their internal resistances or impedances.

case II:

If circuit having independent and dependent sources both: In this case, we calculated IN ,then the internal impedance of the network N is obtained as RN.

case III:

If circuit having only dependent sources: In this case , we apply a voltage V at the terminal pair AB.A current I will flow due to application of V , then the internal impedance of the network N is obtained as

  • Draw the equivalent circuit across AB,  Norton’s current IN in parallel with ZN or RN.

Norton’s Theorem Worked Examples

Find the current across 10Ω resistor using Norton’s Theorem.

step1:

Removing the portion of the network 10Ω resistor across which the Norton’s theorem is to be applied and short terminal and obtain the short circuit current IN flowing through it keeping all the sources of their normal values.

step 2:

calculate the Rth or RN.

step 3:

Draw the equivalent circuit across the terminal 10Ω resistor with IN in parallel with ZN or RN.

where ,

IN=38/3 A

Rth=RN=ZN=3.75Ω

Application of Norton’s Theorems

  • Replaces the large part of a network i.e. complicated and uninteresting part by a very simple equivalent.
  • Helps in the rapid calculation of the voltage ,current and hence power which the original circuit is able to deliver to a load.
  • Helps to choose the best value of load resistance or impedance required to accomplish a maximum transfer of power.

Limitations of Norton’s Theorems

  • It is not applicable to the circuits consisting of unilateral elements like diode etc.
  • Not applicable for the circuits consisting of non-linear elements like diode ,transistor etc.
  • Not applicable for the circuits consisting of load series or parallel with controlled or dependent sources.
  • Not applicable to the circuits consisting of magnetic coupling between load and any other circuit element.

References

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