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

Explanation:

Analysis of Thevenin and Norton Equivalent Circuits

Definition: Thevenin and Norton equivalent circuits are techniques used in electrical engineering to simplify complex networks into more manageable forms for analysis. Thevenin's theorem represents a network as an ideal voltage source (V_TH) in series with a resistance (R_TH), while Norton’s theorem represents the same network as an ideal current source (I_N) in parallel with a resistance (R_N).

Key Relationship:

The resistances R_TH (Thevenin resistance) and R_N (Norton resistance) are always equal in value:

R_TH = R_N

This equivalence arises because both Thevenin and Norton transformations describe the same electrical network, but in different forms. The resistance remains unchanged during the transformation, as it represents the inherent impedance of the circuit.

Correct Option Analysis:

The correct option is:

Option 4: R_TH = R_N

This option is correct because the Thevenin equivalent resistance (R_TH) and Norton equivalent resistance (R_N) of a given passive linear network are always equal. This fundamental relationship ensures consistency in circuit transformations and analysis.

Calculation:

Using Kirchhoff rule 

V = (I + I_o ) R 

⇒ I = V/R - I₀

at V =0 ⇒ I = - I_o

at I = 0 ⇒ V = I₀ R

Thus only correct graph is 

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 

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

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

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:

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

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 

Solution:

• The Super Position Theorem becomes important for the circuits having sources operating at different frequencies.
• In this case, since the impedances depend on frequency, we must have a different frequency-domain circuit for each frequency.
• Both thecircuit should be analyse separately and all the calculations for current, voltage and power should be done accordingly.

Important Points The power consumed by a element of the circuit can be added directly to know the total power consumed if the circuit is operating under different frequencies.

Mistake Points Do not ry to add current and voltage calculated with two different frequencies in phasor domain. Add them in time domain only to avoid the mistake.

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).