What is the Norton theorem?

In the current circuit law, the Norton theorem (also known as the Mayer-Norton theorem) is a simplification that can be applied to networks generated by a constant line of opposition, power sources, and current assets. In two network terminals, it can be switched on with a current supply and single resistor respectively.

What is the principle of Norton theorem?

Norton theory and it’s also known as, the Thevenin theorem, are widely used to simplify the circuit analysis and to look at the initial state of the circuit and the steady-state response to the situation.

Norton Theory was independently derived in 1926 by Siemens & Halske researcher Hans Ferdinand Mayer (1895-1980) and Bell Labs engineer Edward Lawry Norton (1898-1983).

Working principle of Norton’s theorem

To find the equivalent, the Norton current I_no is calculated as the current flowing on the terminals into a short circuit (zero resistance between A and B). The Norton resistance R_no is located by means of calculating the output voltage produced with no resistance connected on the terminals; equivalently, that is the load resistance among the terminals with all (independent) voltage sources short-circuited and independent constant current sources open-circuited. This is an equivalent circuit to calculating the Thevenin resistance.

The voltage at the terminals is calculated for a source of a 1 A. Take a look at the current on the terminals. This voltage divided via the 1 A current is the Norton impedance R_no. This method needs to be used if the circuit contains independent sources, but it can be used in all instances even when there aren’t any dependent sources.

Method of finding Norton theorem

There are 2 methods for finding Norton’s equivalent circuit. Based on the type of sources that can be present in the network.

Method 1

Follow these steps in order to locate Norton’s equivalent circuit, whilst only the sources of impartial type are present.

Step 1 − Consider the circuit diagram by way of opening the terminals with respect to which, Norton’s equivalent circuit is to be found.

Step 2 − Find Norton’s current I_N by shorting the 2 opened terminals of the above circuit.

Step 3 − Find Norton’s resistance R_N throughout the open terminals of the circuit considered in Step1 by eliminating the independent sources present in it. Norton’s resistance R_N will be the same as that of Thevenin’s resistance R_Th.

Step 4 − Draw Norton’s equivalent circuit via connecting a Norton’s current I_N in parallel with Norton’s resistance R_N.

Method 2

Observe these steps in order to find Norton’s equivalent circuit, while the sources of both independent type and dependent type sources are present.

Step 1 − Consider the circuit diagram by way of opening the terminals with respect to which Norton’s equivalent circuit is to be found.

Step 2 − Find the open-circuit voltage V_OC across the open terminals of the above circuit.

Step 3 − Find Norton’s current I_N by way of shorting the two opened terminals of the above circuit.

Step 4 − Find Norton’s resistance R_N by using the following formula.

$R_N=\frac{V_{OC}}{I_N}$

Step-5-  Draw Norton’s equivalent circuit by connecting a Norton’s current I_N in parallel with Norton’s resistance R_N.

Example

To find Norton’s equivalent of the above circuit we first need to remove the center 40 Ω load resistor and short circuit the terminals A and B to give us the following circuit.

The total short circuit current can now be calculated as:

With A-B shorted out:

$$\begin{gathered} I_1=\frac{10 \mathrm{V}}{10 \mathrm{\Omega }}=1 \mathrm{amp} \\ {I}_2=\frac{20 \mathrm{V}}{20 \mathrm{\Omega }}=1 \mathrm{amp} \end{gathered}$$

Therefore, $I_{\mathrm{short}-\mathrm{circuit}}=I_1+I_2=2 \mathrm{amp}$.

If short-out the two voltage sources and open circuit terminals A and B, the two resistors are now effectively connected together in parallel. The value of the inner resistor R_s is found by calculating the total load resistance on terminals A and B giving us the following circuit:

Consider the $10 \mathrm{\Omega }$ resistor in parallel with the $20 \mathrm{\Omega }$ resistor

Find the equivalent load resistance:

$$\begin{gathered} R_T=\frac{R_1\times R_2}{R_1+R_2} \\ {R}_T=\frac{20\times 10}{20+10} \\ {R}_T=6.67 \mathrm{\Omega } \end{gathered}$$

Again, the 2 resistors are connected in parallel across terminals A and B which gives us a total resistance of:

$$\begin{gathered} R_T=\frac{R_1\times R_2}{R_1+R_2} \\ {R}_T=\frac{6.67\times 40}{6.67+40} \\ {R}_T=5.72 \mathrm{\Omega } \end{gathered}$$

The voltage across the terminals A and B with the load resistor connected is given as:

$V=I\times R=2\times 5.72=11.44 \mathrm{V}$

Then the current flowing in the $40 \mathrm{\Omega }$ load resistor can be found as:

$I=\frac{V}{R}=\frac{11.44}{40}=0.286 \mathrm{amp}$

Hence, using Norton’s theorem the value of current calculated is 0.286 A.

Limitations and applications of Norton theorem

Limitations

• Like Thevenin’s theorem, Norton’s theorem is also applicable to 2-terminal, linear, active networks only.
• It is not valid to networks that have unilateral or non-linear elements like diode and transistors.
• The power dissipation across Norton’s equivalent circuit is not the same as the energy dissipation in the real circuit.
• This theorem isn't always valid for circuits that have magnetic locking or coupling to the load.

Application

• Are used to reduce a complex circuit into a simple circuit.
• Norton's theory helps solve problems in common generators with uneven emf and unequal impedances.
• Norton’s theory could be used in exchange for Thevenin's theory by modification of appropriate sources.

Common Mistakes

Remember that this theorem applies to both AC and DC regions. It works in AC circuits for impedance and resistance statistics. But in the case of DC circuits, it operates for resistance levels.

Context and Applications

In each of the expert exams for undergraduate and graduate publications, this topic is huge and is mainly used for:

• Bachelor of technology in the electrical and electronic department
• Bachelor of Science in physics
• Master of Science in physics

Related Concepts

• Ohm's Law
• Millman's theorem
• Source transformation
• Superposition theorem
• Thevenin's theorem

Practice Problems

Question-1 The Norton current is the____________.

(a) Short circuit current

(b) open circuit current

(c) open circuit and short circuit current

(d) none of these

Correct option- (a)

Explanation- Norton current is obtained by summarizing the specified terminals. Therefore, short circuit current. It is not a current circuit because when the specified terminals rotate it means that the current is equal to zero.

Question-2  Norton resistance is found by____________.

(a) short circuit current

(b) open circuit current

(c) both

(d) none of these

Correct option- (a)

Explanation- Current suitable sources have endless internal resistance which is why it behaves like an open circuit while suitable energy sources have zero internal resistance which is why it behaves like a short circuit. Therefore, to find resistance to Norton, all power sources are shorted and all current sources are turned on.

Question-3 Norton’s theorem is true for____________.

(a) non-linear network

(b) linear networks

(c) both

(d) none of these

Correct option- (b)

Explanation- The Norton theorem only applies to linear circuit elements and not to non-BJT elements, semiconductors, etc.

Question-4 In Norton’s theorem Isc is______________.

(a) Sum of two current sources

(b) A single current source

(c) infinite current source

(d) none of these

Correct option- (b)

Explanation- Norton's theory states that the combination of power sources, current sources, and dissimilar elements is equivalent to a single current IN and a single corresponding RN.

Question-5 Norton theory and it’s also known as, the ______________.

(a) Thevenin theorem

(b) superposition theorem

(c) Millman's theorem

(d) KVL and KCL

Correct option- (a)

Explanation- Thevenin theory is also known as Norton's dual theory because in the Norton theorem we find short circuit current which is double of open-circuit voltage - what we find in Thevenin's theory.