Norton's Theorem - Statement, Diagram, Formula, Problems

Q: What is Norton's Theorem?

Norton's theorem is used to find the equivalent and simple circuit of a complex electrical circuit. In this, various circuit resistances are replaced by equivalent resistance and a single current source.

Norton's Theorem Statement

Norton's theorem states that any 2-terminal linear and bilateral network or circuit having multiple independent and dependent sources can be represented in a simplified equivalent circuit known as Norton's equivalent circuit.

Norton's Theorem Circuit Diagram

Norton's equivalent circuit consists of Norton's current source, I_N in parallel with Norton's resistance, R_N. The parallel cthe ombination of current source and resistor is a practical current source. Hence, we can say that Norton's equivalent circuit is nothing but a practical current source.

Procedure of Norton's Theorem

It will take more time than the normal methods for finding the response of an element if the network/ circuit is having multiple sources and resistances. That time, we can use Norton's theorem to find the response easily. Now, let's see the steps for finding the response of an element when multiple sources and resistances are present in the network/ circuit by using Norton's theorem.

Step 1: Remove the element, where we are supposed to find the response from the given circuit. After the removal of the element, the terminals will be open.
Step 2: Find the current flowing through the terminals of the circuit obtained in Step 1 after shorting them. This current is known as short circuit current or Norton's equivalent current or Norton's current, I_Nin short.
Step 3: Replace all the independent sources with their internal resistances in the circuit obtained in Step 1.
Step 4: Find the equivalent resistance across the open-circuited terminals of the circuit obtained in Step 3 indirect methods if there are no dependent sources. This equivalent resistance is known as Norton's equivalent resistance or Norton's resistance, R_N in short.
Step 5: If dependent sources are present, then we can find the equivalent resistance across the open-circuited terminals of the circuit obtained in Step 3 by using the Test source method. In the test source method, we will connect a 1V source (or 1A source) across the open terminals and will calculate another parameter current (or voltage). We will get the value of Norton's resistance, R_N by taking the ratio of voltage and current across the 2 terminals.
Step 6: Draw Norton's equivalent circuit by connecting Norton's current, I_Nin parallel with Norton's resistance, R_N.

Step 7: Connect the element, where we are supposed to find the response at the open terminals of Norton's equivalent circuit obtained in Step 6.

Step 8: Find the response of that element by using laws or basic rules.

Norton's Theorem Formula

For the above-given circuit, Norton's Theorem formula would be:

I = I_N(R_N/R+R_N) & V= I_N(RR_N/R+R_N).

Problems on Norton's Theorem

Question 1: Find the current I of the following electric circuit using Norton's theorem.

Answer: 2/3 A

Question 2: Find the voltage, V of the following electric circuit by using Norton's theorem.

Answer: 1 volt

In this article, we discussed the statement of Norton's theorem and how to apply this theorem to DC circuits. Similarly, we can apply Norton's theorem for finding the response of an element when the circuit consists of multiple AC sources or the combination of DC and AC sources and multiple impedances/admittances.

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FAQs on Norton's Theorem

What is Norton's Theorem?

Norton's theorem is used to find the equivalent and simple circuit of a complex electrical circuit. In this, various circuit resistances are replaced by equivalent resistance and a single current source.

Can we apply Norton’s theorem to find equivalent resistance directly in the circuits, if dependent sources and multiple resistances are present?

No. We can’t directly find Norton’s equivalent resistance in the circuits if dependent sources are present. Here, we will use the Test source method. In this method, we will connect a 1V source (or 1A source) across the open terminals and will calculate another parameter current (or voltage). By taking the ratio of voltage and current across the two terminals, we will get the value of Norton’s resistance, R_N.

In how many methods, we can calculate Norton’s equivalent resistance, for the circuit, which is having multiple independent & dependent DC sources and multiple resistances?

There are two methods for finding Norton’s equivalent resistance, R_N. In the first method, initially, we will calculate open-circuit voltage, V_OC, and Norton’s current, I_N, and then will get the value of R_N by taking the ratio of V_OC and I_N. In the second method, we will use the test source method after replacing the independent DC sources with their internal resistances.

Whether we can apply Norton’s theorem to the AC circuits?

Yes. We can apply Norton’s theorem to the AC circuits just like the way we applied Norton’s theorem to DC circuits. If the circuit consists of multiple AC sources or the combination of DC and AC sources and multiple impedances/ admittances, then we can use Norton’s theorem.

What will be Norton’s equivalent circuit, if the circuit consists of multiple dependent sources and multiple resistances?

If the circuit consists of multiple dependent sources and multiple resistances, then the open-circuit voltage, V_OC will be 0 V and the Norton’s current, I_N will be 0 A. But we will get a non-zero value for Norton’s equivalent resistance, R_N. Hence, Norton’s equivalent circuit for this type of circuit is simply Norton’s equivalent resistance, R_N.

In how many methods, we can calculate Norton’s equivalent resistance, for the circuit, which is having multiple independent DC sources and multiple resistances?

There are two methods for finding Norton’s equivalent resistance, R_N. In the first method, we can directly find Norton’s equivalent resistance across the open terminals after replacing the independent DC sources with their internal resistances. In the second method, initially, we will calculate open-circuit voltage, VOC, and Norton’s current, IN, and then will get the value of RN by taking the ratio of VOC and IN.

What is a Norton's Theorem?