Norton's Theorem MCQ Quiz - Objective Question with Answer for Norton's Theorem - Download Free PDF

Explanation:

Norton’s Theorem

Definition: Norton’s Theorem is a fundamental concept in electrical circuit analysis. It states that any linear electrical network with voltage or current sources and resistances can be replaced by an equivalent circuit composed of a single current source in parallel with a single resistor connected to a load. This theorem is particularly useful for simplifying complex circuits to make analysis more manageable.

Correct Option Analysis:

The correct option is:

Option 4: An equivalent circuit composed of a single current source, parallel resistance, and parallel load.

This option accurately reflects the essence of Norton’s Theorem. According to the theorem, any linear network can be replaced by an equivalent circuit consisting of:

1. A single current source (known as Norton’s equivalent current, denoted as I_N).
2. A single resistance (known as Norton’s equivalent resistance, denoted as R_N) connected in parallel with the current source.
3. A load resistance connected in parallel with the equivalent circuit.

Steps to Apply Norton’s Theorem:

1. Identify the portion of the circuit: Select the part of the circuit where you want to calculate the load current or voltage, and remove the load resistance temporarily.
2. Calculate Norton’s Equivalent Current (I_N): Short-circuit the terminals where the load resistance was connected and calculate the current flowing through the short circuit. This current is I_N.
3. Calculate Norton’s Equivalent Resistance (R_N): Turn off all independent sources (replace voltage sources with short circuits and current sources with open circuits) in the original circuit, and calculate the equivalent resistance seen from the open terminals. This resistance is R_N.
4. Reconstruct the Norton Equivalent Circuit: Replace the original network with an equivalent circuit consisting of I_N in parallel with R_N, and reconnect the load resistance to this equivalent circuit.
5. Analyze the Equivalent Circuit: Use parallel circuit analysis to calculate the current through or voltage across the load resistance.

Advantages of Norton’s Theorem:

• It simplifies complex circuits, making it easier to analyze the behavior of the circuit with different load resistances.
• It is particularly useful for determining the current through or voltage across a specific load resistor in a circuit with multiple components.
• The theorem is applicable to both AC and DC circuits as long as the circuit is linear.

Disadvantages of Norton’s Theorem:

• It is limited to linear circuits and cannot be applied to circuits with non-linear elements such as diodes and transistors.
• The process of turning off independent sources and calculating equivalent resistance may become cumbersome for very large and complex circuits.

Applications:

• Used in electrical circuit analysis to simplify the study of load variations.
• Widely applied in power systems and electronics to understand the behavior of networks under different loading conditions.
• Useful in network theorems for solving problems in both academic and practical engineering scenarios.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: An equivalent circuit composed of a single current source, series resistance, and series load.

This option is incorrect because it does not align with the principles of Norton’s Theorem. Norton’s Theorem specifies that the equivalent circuit consists of a current source in parallel with a resistance. A series configuration of resistance and load is not applicable in the context of Norton’s equivalent circuit.

Option 2: An equivalent circuit composed of a single voltage source, parallel resistance, and parallel load.

This option describes a configuration that is related to Thevenin’s Theorem, not Norton’s Theorem. Thevenin’s Theorem states that any linear electrical network can be replaced by an equivalent circuit consisting of a single voltage source in series with a resistance. The presence of a voltage source and parallel components makes this option inconsistent with Norton’s Theorem.

Option 3: An equivalent circuit composed of a single voltage source, series resistance, and series load.

Similar to option 2, this description corresponds to Thevenin’s equivalent circuit. Thevenin’s Theorem involves a voltage source in series with a resistance, whereas Norton’s Theorem involves a current source in parallel with a resistance. Hence, this option is also incorrect.

Option 5: (Not mentioned in the problem context).

Since there is no description provided for Option 5, it is not relevant to the question and does not align with the principles of Norton’s Theorem.

Conclusion:

Norton’s Theorem is a powerful tool for simplifying the analysis of electrical circuits, especially when focusing on the behavior of a specific load. The correct representation of Norton’s equivalent circuit involves a single current source in parallel with a single resistance and a parallel load. This configuration facilitates efficient circuit analysis and provides insights into the impact of load variations on the overall circuit behavior. By contrast, the other options either describe configurations unrelated to Norton’s Theorem or pertain to Thevenin’s Theorem, highlighting the importance of understanding the distinctions between these two fundamental network theorems.

Norton Theorem

Norton's theorem states that it is possible to simplify any linear circuit, into an equivalent circuit with a single current source and a parallel resistance.

Current divider rule

When two resistances are connected in parallel, the current is divided as:

\(I_1={R_2\over R_1+R_2}\times I\)

\(I_2={R_1\over R_1+R_2}\times I\)

Calculation

The current through 5Ω resistance is given by:

\(I_L={15\over 15+5}\times 10\)

I_L = 7.5 A

Norton Theorem

Norton's theorem states that it is possible to simplify any linear circuit, into an equivalent circuit with a single current source and a parallel resistance.

When two identical Norton circuits are connected in parallel:

I = 2I_N

\(R={R_N\over 2}\)

Calculation

Given,  I_N = 3A 

R_N = 4Ω 

\(I=2\times 3=6\space A\)

\(R={4\over 2}=2\space \Omega \)

Explanation:

Norton's Theorem

Definition: Norton's theorem is a fundamental principle used in electrical engineering to simplify complex linear electrical networks. It states that any two-terminal linear network containing resistances, voltage sources, and current sources can be replaced by an equivalent current source in parallel with an equivalent resistance. This theorem is particularly useful for analyzing power systems and simplifying circuit analysis.

Working Principle: According to Norton's theorem, any linear electrical network can be reduced to a simple equivalent circuit consisting of a single current source (Norton equivalent current, I_N) in parallel with a single resistance (Norton equivalent resistance, R_N). The steps to determine the Norton equivalent circuit are as follows:

• Step 1: Identify the portion of the circuit for which the Norton equivalent is to be found. Remove the load resistor if present.
• Step 2: Calculate the Norton equivalent current (I_N) by finding the current that flows through the terminals when they are short-circuited.
• Step 3: Calculate the Norton equivalent resistance (R_N) by deactivating all independent sources (replace voltage sources with short circuits and current sources with open circuits) and determining the resistance seen from the open terminals.
• Step 4: Construct the Norton equivalent circuit with the calculated I_N in parallel with R_N.

Advantages:

• Simplifies the analysis of complex circuits by reducing them to a basic equivalent circuit.
• Makes it easier to analyze power transfer and the behavior of the circuit with different loads.
• Useful for both AC and DC circuit analysis.

Disadvantages:

• Applicable only to linear and bilateral networks.
• Requires the calculation of equivalent current and resistance, which may not always be straightforward.

Applications: Norton's theorem is commonly used in various fields of electrical engineering, including power systems, electronic circuit design, and network analysis. It is particularly useful for simplifying the analysis of circuits with multiple sources and resistances.

Correct Option Analysis:

The correct option is:

Option 1: Current Source

Norton's theorem helps determine the equivalent of a current source. The theorem states that any two-terminal linear network can be replaced by an equivalent current source in parallel with an equivalent resistance. This equivalent current source is known as the Norton equivalent current (I_N).

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: Capacitance

This option is incorrect. Norton's theorem does not deal with capacitance. Instead, it focuses on simplifying linear electrical networks by determining an equivalent current source and resistance. Capacitance is a different electrical property related to the ability of a system to store charge.

Option 3: Voltage Source

This option is incorrect. While there is a related theorem called Thevenin's theorem, which deals with determining the equivalent voltage source and series resistance for a linear network, Norton's theorem specifically focuses on an equivalent current source in parallel with a resistance.

Option 4: Resistance

This option is partially correct but not complete. Norton's theorem does determine an equivalent resistance (Norton equivalent resistance, R_N), but it primarily helps in finding the equivalent current source. The main focus of Norton's theorem is on the equivalent current source, not just the resistance.

Conclusion:

Understanding Norton's theorem is essential for simplifying the analysis of complex electrical circuits. The theorem allows us to replace a complicated network with an equivalent current source and parallel resistance, making it easier to analyze and understand the circuit's behavior. By focusing on the equivalent current source, Norton's theorem provides a powerful tool for electrical engineers to streamline circuit analysis and design.

Source Transformation

It is a technique in which a voltage source with series resistance is converted into a current source having resistance in parallel to it.

The circuit having a voltage source is the Thevenin circuit while the circuit having a current source is known as the Norton circuit.

∴ The source transformation of Thevenin theorem is known as the Norton theorem.

Thevenin Theorem

Thevenin's theorem states that it is possible to simplify any linear circuit, irrespective of how complex it is, to an equivalent circuit with a single voltage source and a series resistance.

Norton Theorem

Norton's theorem states that any linear circuit can be simplified to an equivalent circuit consisting of a single current source and parallel resistance that is connected to a load.

Concept:

Norton's Theorem:

In any linear, bidirectional circuit having more than one independent source, having more the active and passive element it can be replaced by a single equivalent current source IN in parallel with an equivalent resistance RN. 

Where 

IN = Norton or short circuit current

RN = Norton's resistance

Procedure in order to find Norton’s equivalent circuit, when only the sources of independent type are present.

•  Consider the circuit diagram by opening the terminals with respect to which, Norton’s equivalent circuit is to be found.
•  Find Norton’s current IN by shorting the two opened terminals of the circuit.
•  Find Norton’s resistance RN across the open terminals of the circuit, eliminating the independent sources present in it. 
•  Norton’s resistance RN will be the same as that of Thevenin’s resistance RTh.
•  Draw Norton’s equivalent circuit by connecting a Norton’s current IN in parallel with Norton’s resistance RN.

Explanation:

Given circuit is 

To find Norton current through terminal ab, ab terminal is short circuited, so no current will flow through the 5 Ω resistor.

Now the circuit will look like

\(6=\frac{V_x -16}{4}+\frac{V_x}{8+8}\)

16 × 6 = 4(V_x - 16) + V_x

5 V_x = 10 × 16

V_x = 32 V

I_N = V_x / 16 = 32 / 16 = 2 A

To find Norton's resistance, source should be replaced with internal resistance, so

• Current source is open circuited.
• Voltage source is short circuited.

The circuit will become

R_N = 5 || (8 + 4+ 8) = 5 || 20 = (5 × 20) / (5 + 20) = 4 Ω 

So the Norton equivalent circuit is

Norton Theorem

Norton's theorem states that any linear circuit can be simplified to an equivalent circuit consisting of a single current source and parallel resistance that is connected to a load.

Calculation

The Norton current is the short circuit current across the load terminal AB.

The short circuit path will make a 15Ω short circuit. So, 5Ω and 5Ω become in parallel.

\(I_N=10\times {5\over 5+5}\)

I_N = 5 A

Concept:

• Norton’s Theorem states that “Any linear circuit containing several energy sources and resistances can be replaced by a single constant current source in parallel with a single resistor “.
• It is an analytical method used to change a complex circuit into a simple equivalent circuit consisting of a single resistance in parallel with a current source.

Steps to follow for Norton’s Theorem:

• Calculate Norton’s current source by removing the load resistor from the original circuit and replacing it with a short circuit.
• Calculating the current through a shorted wire.
• Calculating the Norton resistance by removing all power sources in the original circuit (voltage sources shorted and current sources open).
• Calculating total resistance between the open connection points.
• Draw the Norton equivalent circuit, with the Norton current source in parallel with the Norton resistance.
• The load resistor re-attaches between the two open points of the equivalent circuit.

Calculation:

• To calculate Norton’s current I_N remove the load resistor from the original circuit and replacing it with a short circuit.

Then,

\({I_N} = \frac{{360}}{{30}} = 12\;A\)

Concept:

Norton’s Theorem states that “Any linear circuit containing several energy sources and resistances can be replaced by a single constant current source in parallel with a single resistor “.

Below is Norton Equivalent of the above circuit

I_N = Norton Current

R_N = Norton Resistance

R_L = Load Resistance

Steps to follow for Norton’s Theorem:

• Calculate Norton’s current source by removing the load resistor from the original circuit and replacing it with a short circuit.
• Calculating the current through a shorted wire.
• Calculating the Norton resistance by removing all power sources in the original circuit (voltage sources shorted and current sources open).
• Calculating total resistance between the open connection points.
• Draw the Norton equivalent circuit, with the Norton current source in parallel with the Norton resistance.
• The load resistor re-attaches between the two open points of the equivalent circuit.

Calculation:

Given RL = RN 

R_L = R_N  (see above circuit)

So the current through R_L is \({I_n \over 2}\)

Norton Theorem

Norton’s theorem states that any linear circuit can be simplified to an equivalent circuit consisting of a single current source and parallel resistance that is connected to a load. 

where, I_N = Nortan current

R_N = Norton resistance

R_L = Load resistance

Calculation

Given, R_L = 2RN 

Applying CDR, the load current is given by

\(I_L={R_N\over R_N+2RN}\times I_N\)

\(I_L={I_N\over 3}\)

Concept:

• Norton’s Theorem states that “Any linear circuit containing several energy sources and resistances can be replaced by a single constant current source in parallel with a single resistor".
• It is an analytical method used to change a complex circuit into a simple equivalent circuit consisting of a single resistance in parallel with a current source.

Steps to follow for Norton’s Theorem:

• Calculate Norton’s current source by removing the load resistor from the original circuit and replacing it with a short circuit.
• Calculating the current through a shorted wire.
• Calculating the Norton resistance by removing all power sources in the original circuit (voltage sources shorted and current sources open).
• Calculating total resistance between the open connection points.
• Draw the Norton equivalent circuit, with the Norton current source in parallel with the Norton resistance.
• The load resistor re-attaches between the two open points of the equivalent circuit.

Calculation:

To find Norton's Current the circuit can be redrawn as given below:

Here 20 Ω will be shorted.

The required Norton's current will be = \(\frac{50-0}{5} =\: 10 A\)

Concept:

Norton's Theorem:

In any linear, bidirectional circuit having more than one independent source, having more the active and passive element it can be replaced by a single equivalent current source IN in parallel with an equivalent resistance RN. 

Where 

IN = Norton or short circuit current

RN = Norton's resistance

Procedure in order to find Norton’s equivalent circuit, when only the sources of independent type are present.

•  Consider the circuit diagram by opening the terminals with respect to which, Norton’s equivalent circuit is to be found.
•  Find Norton’s current I_N by shorting the two opened terminals of the circuit.
•  Find Norton’s resistance R_N across the open terminals of the circuit, eliminating the independent sources present in it. 
•  Norton’s resistance R_N will be the same as that of Thevenin’s resistance R_Th.
•  Draw Norton’s equivalent circuit by connecting a Norton’s current I_N in parallel with Norton’s resistance R_N.

Explanation:

Norton’s resistance between terminals a – b

Open circuit the current source.

10 Ω and 20 Ω are in series, similarly, 80 Ω and 40 Ω are in series.

30 Ω and 120 Ω are in parallel.

Rab = (10 + 20) || (80 + 40)

R_ab = 30 || 120

\({R_{ab}} = \frac{{30 \times 120}}{{150}}\)

R_ab = 24Ω 

The correct answer is option 4):(linear networks)

Concept:

• Norton's Theorem states that “Any linear circuit containing several energy sources and resistances can be replaced by a single constant current source in parallel with a single resistor “.
• In any linear, bidirectional circuit having more than one independent source, having more active and passive elements it can be replaced by a single equivalent current source I_N in parallel with an equivalent resistance R_N.

Where I_N = Norton or short circuit current

R_N = Norton's resistance

• Procedure in order to find Norton’s equivalent circuit, when only the sources of independent type are present.
• Consider the circuit diagram by opening the terminals with respect to which, Norton’s equivalent circuit is to be found.
• Find Norton’s current I_N by shorting the two opened terminals of the circuit.
• Find Norton’s resistance R_N across the open terminals of the circuit, eliminating the independent sources present in it. Norton’s resistance R_N will be the same as that of Thevenin’s resistance R_Th.
• Draw Norton’s equivalent circuit by connecting Norton’s current I_N in parallel with Norton’s resistance R_N.

Thevenin’s theorem: Any two terminal bilateral linear DC circuits can be replaced by an equivalent circuit consisting of a voltage source (V_th) and a series resistor(R_th).

Norton’s Theorem: Any two terminal bilateral linear DC circuits can be replaced by an equivalent circuit consisting of a current source (I_SC) and a parallel resistor (R_th).

Norton’s theorem is the converse of Thevenin’s theorem.

Therefore, both Thevenin resistance and Norton resistance are same for a given circuit.

To find Rth: 

To calculate Thevenin’s Resistance we should replace all independent current sources by Open circuit and Independent voltage sources by Short circuit (keep dependent sources as it is).

Concept :

• Norton’s Theorem states that “Any linear circuit containing several energy sources and resistances can be replaced by a single constant current source in parallel with a single resistor “.
• It is an analytical method used to change a complex circuit into a simple equivalent circuit consisting of a single resistance in parallel with a current source.

Steps to follow for Norton’s Theorem:

• Calculate Norton’s current source by removing the load resistor from the original circuit and replacing it with a short circuit.
• Calculating the current through a shorted wire.
• Calculating the Norton resistance by removing all power sources in the original circuit (voltage sources shorted and current sources open).
• Calculating total resistance between the open connection points.
• Draw the Norton equivalent circuit, with the Norton current source in parallel with the Norton resistance.
• The load resistor re-attaches between the two open points of the equivalent circuit.