Network Theory MCQ Quiz - Objective Question with Answer for Network Theory - Download Free PDF

Explanation:

Analysis of the Circuit to Determine the Value of Current (I):

In the given problem, we are required to determine the value of the current (I) in the circuit. However, based on the provided options and the correct answer being Option 2 (Indeterminate), it suggests that the circuit configuration or the data provided in the problem is insufficient to calculate the current. Let us analyze why this is the case and why the other options are incorrect.

Reason for Option 2 (Indeterminate) Being Correct:

To determine the current (I) in a circuit, certain essential information is required, such as:

• The complete circuit diagram, which includes the arrangement of components like resistors, voltage sources, and their respective values.
• Ohm's Law or Kirchhoff's Laws for analyzing the circuit.
• Boundary conditions or constraints, if any.

However, in this case, the problem does not provide sufficient details about the circuit configuration, such as the values of resistances, voltage sources, or the overall circuit layout. Without this critical information, it is impossible to apply the necessary electrical laws to calculate the current. Hence, the value of current (I) cannot be determined from the given data, making the correct answer Option 2: Indeterminate.

Important Information:

Let us analyze why the other options are incorrect:

Option 1: 2 A

This option suggests that the current in the circuit is 2 A. However, without knowing the specific values of resistances, voltage sources, or the circuit configuration, there is no basis to claim that the current is precisely 2 A. This is purely speculative and cannot be justified with the given information.

Option 3: -1 A

This option assumes that the current has a value of -1 A. The negative sign might indicate the direction of current flow, but again, without any data about the circuit elements or their arrangement, there is no justification for this value. Thus, this option is also incorrect.

Option 4: 1 A

This option states that the current in the circuit is 1 A. Like Option 1, this is an arbitrary value that cannot be verified in the absence of information about the circuit's resistances, voltage sources, or layout. Therefore, this option is not valid.

Option 5: (Not provided in the problem statement)

The problem does not explicitly provide details for Option 5. However, any assumption about the value of the current without proper data is inherently flawed and cannot be considered correct.

Conclusion:

The inability to determine the current (I) in the circuit arises from a lack of critical information regarding the circuit's components and their configuration. In such cases, it is crucial to recognize that the problem is indeterminate, as no definitive calculation can be performed. This highlights the importance of providing complete circuit details when solving electrical circuit problems.

Explanation:

Mutual Inductance Between Two Coils

Definition: Mutual inductance is a measure of the interaction between two coils, where the magnetic field generated by the current in one coil induces a voltage in the other coil. It depends on the geometry of the coils, the number of turns in each coil, and the relative positioning or orientation of the coils.

Problem Statement: The total inductance of two coils, A and B, when connected in series is given as either 0.5 H or 0.2 H, depending upon the relative directions of the current in the coils. The self-inductance of coil A is 0.2 H, and we need to determine the mutual inductance (M) between the two coils.

Solution:

When two coils are connected in series, the equivalent inductance (L_total) can be calculated using the following formula:

Case 1: Currents in the same direction

The total inductance is given by:

L_total = L_A + L_B + 2M

Where:

• L_A = Self-inductance of coil A
• L_B = Self-inductance of coil B
• M = Mutual inductance between the two coils

In this case, L_total = 0.5 H (as given in the problem).

Substituting the values:

0.5 = 0.2 + L_B + 2M

Case 2: Currents in opposite directions

The total inductance is given by:

L_total = L_A + L_B - 2M

In this case, L_total = 0.2 H (as given in the problem).

Substituting the values:

0.2 = 0.2 + L_B - 2M

Step-by-Step Calculation:

From Case 2:

0.2 = 0.2 + L_B - 2M

Rearranging:

L_B - 2M = 0

L_B = 2M

From Case 1:

0.5 = 0.2 + L_B + 2M

Substitute L_B = 2M:

0.5 = 0.2 + 2M + 2M

0.5 = 0.2 + 4M

Rearranging:

4M = 0.5 - 0.2

4M = 0.3

M = 0.3 ÷ 4

M = 0.075 H

Conclusion:

The mutual inductance between the two coils is 0.075 H, which corresponds to Option 4.

Important Information

To further analyze the other options:

• Option 1: 0.25 H - This value is incorrect. If mutual inductance were 0.25 H, the calculated total inductance values would not match the given problem statement.
• Option 2: 0.05 H - This value is too low and does not satisfy the equations for total inductance in both cases (same direction and opposite direction currents).
• Option 3: 0.15 H - This value is incorrect. Substituting M = 0.15 H into the equations for total inductance leads to inconsistencies with the given values of 0.5 H and 0.2 H.
• Option 4: 0.075 H - This is the correct answer, as demonstrated above.

Explanation:

Nodal Method of Circuit Analysis

Definition: The nodal method of circuit analysis, also known as nodal analysis, is a technique used to determine the voltage at various nodes in an electrical circuit. It is based on Kirchhoff's Current Law (KCL) and Ohm's Law. This method is particularly useful for analyzing circuits with multiple nodes and components.

Working Principle:

Nodal analysis relies on the following principles:

• Kirchhoff's Current Law (KCL): States that the algebraic sum of currents entering and leaving a node is zero. This law is used to write equations at each node in the circuit.
• Ohm's Law: Relates the voltage across a resistor to the current flowing through it and its resistance (V = IR). This law helps express currents in terms of voltages and resistances.

By combining KCL and Ohm's Law, nodal analysis allows us to systematically solve for the unknown node voltages in the circuit.

Steps for Nodal Analysis:

1. Identify all nodes: Assign a reference node (ground) and label the remaining nodes with variables representing their voltages.
2. Apply KCL at each non-reference node: Write equations expressing the sum of currents entering and leaving the node as zero.
3. Use Ohm's Law: Replace the currents in the KCL equations with expressions in terms of voltages and resistances.
4. Solve the system of equations: Solve the resulting simultaneous equations to find the node voltages.

Advantages:

• Simplifies complex circuit analysis by focusing on node voltages rather than branch currents.
• Efficient for circuits with multiple components connected in parallel.
• Provides a systematic approach to solving electrical circuits.

Disadvantages:

• May require solving a large system of simultaneous equations for circuits with many nodes.
• Not as intuitive as mesh analysis for circuits with many series components.

Applications:

• Nodal analysis is widely used in electrical engineering for circuit design and analysis.
• It is particularly useful for analyzing circuits with operational amplifiers, resistors, capacitors, and inductors.

Correct Option Analysis:

The correct option is:

Option 3: KCL and Ohm's Law.

This option accurately describes the basis of nodal analysis. Kirchhoff's Current Law (KCL) is used to write equations at each node, and Ohm's Law is used to express currents in terms of voltages and resistances. Together, these principles form the foundation of the nodal method.

Additional Information

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

Option 1: KCL and KVL.

This option is incorrect because nodal analysis does not directly use Kirchhoff's Voltage Law (KVL). While KVL is essential for other methods like mesh analysis, nodal analysis exclusively relies on KCL and Ohm's Law.

Option 2: KCL, KVL, and Ohm's Law.

This option is also incorrect because the nodal method does not require the use of KVL. Although Ohm's Law is involved, KVL is unnecessary for nodal analysis. Including KVL in this context adds confusion and is not accurate.

Option 4: KVL and Ohm's Law.

This option is incorrect because nodal analysis does not use Kirchhoff's Voltage Law (KVL). KVL is used in mesh analysis, not nodal analysis, making this option irrelevant to the nodal method.

Conclusion:

Nodal analysis is a powerful technique for solving electrical circuits, especially those with multiple nodes. By relying on Kirchhoff's Current Law (KCL) and Ohm's Law, it provides a systematic approach to determine node voltages. Understanding the distinction between nodal and mesh analysis, as well as the principles involved, is essential for effectively analyzing and designing electrical circuits.

Concept:

The voltage across an inductor is given by,

The energy stored in an inductor is given by,

Calculation:

Given:

L = 2 H,  \frac{di}{dt} = 3 \, \text{A/s} , t = 4 s

Using the formula for voltage:

Current after 4 seconds:

Now, energy stored in the inductor:

Final Answer:

Voltage across the inductor, V_L = 6~\text{V}

Energy stored in the inductor, W_L = 144~\text{J}

Correct Option: (4)

Explanation:

Factors Determining the Rating of a Resistor

Definition: The rating of a resistor is a critical parameter that defines the maximum amount of electrical power it can dissipate without being damaged. This rating is essential for ensuring the reliability and longevity of the resistor in various electrical and electronic circuits.

Correct Option:

The correct option is:

Power dissipation capacity

This factor is primarily used to determine the rating of a resistor. The power dissipation capacity of a resistor indicates the maximum power it can handle before it overheats and potentially fails. This is calculated using the formula P = V²/R, where P is the power in watts, V is the voltage across the resistor, and R is the resistance in ohms. The power rating is usually specified in watts (W) and is a crucial parameter when selecting a resistor for a particular application.

Additional Information

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

Option 1: Material used for construction

While the material used for constructing a resistor is important for determining its properties, such as temperature stability and resistance value, it is not the primary factor for determining the resistor's rating. The material affects characteristics like tolerance and temperature coefficient but does not directly define the power dissipation capacity.

Option 3: Temperature coefficient

The temperature coefficient of a resistor indicates how its resistance changes with temperature. While this is an important parameter for precision applications, it is not the primary factor for determining the power rating. The power dissipation capacity is more directly related to the resistor's ability to handle electrical power without overheating.

Option 4: Colour code

The colour code on a resistor is a method of indicating its resistance value and tolerance. It does not provide information about the power rating. The colour code is a useful tool for identifying resistors quickly, but it does not determine the maximum power dissipation capacity.

Conclusion:

Understanding the various factors that influence the rating of a resistor is essential for selecting the appropriate component for a given application. The power dissipation capacity is the primary factor used to determine the rating of a resistor, as it defines the maximum power the resistor can handle without being damaged. Other factors, such as the material used for construction, temperature coefficient, and colour code, provide additional information about the resistor's characteristics but do not directly determine its rating. By focusing on the power dissipation capacity, engineers and designers can ensure the reliable and safe operation of resistors in their circuits.

Concept

The power transfer efficiency is:

​

The current across any resistor is given by:

where, I = Current

V = Voltage

R = Resistance

Calculation

Let the voltage and internal resistance of the voltage source be V and R respectively.

Case 1: When the current of 2 A flows through 5 Ω resistance.

 .... (i)

Case 2: When the current of 1.6 A flows through 10 Ω resistance.

 .....(ii)

Solving equations (i) and (ii), we get:

2(5+R)=1.6(10+R)

10 + 2R = 16 + 1.6R

0.4R = 6

R = 15Ω

Putting the value of R = 15Ω in equation (i):

V = 40 volts

Case 3: Current when the load is 15Ω

η = 50%

Additional Information Condition for Maximum Power Transfer Theorem:

When the value of internal resistance is equal to load resistance, then the power transferred is maximum.

Under such conditions, the efficiency is equal to 50%.

Concept:

Thevenin's Theorem:

Any two terminal bilateral linear DC circuits can be replaced by an equivalent circuit consisting of a voltage source and a series resistor.

To find V_oc: Calculate the open-circuit voltage across load terminals. This open-circuit voltage is called Thevenin’s voltage (V_th).

To find I_sc: Short the load terminals and then calculate the current flowing through it. This current is called Norton current (or) short circuit current (i_sc).

To find R_th: Since there are Independent sources in the circuit, we can’t find Rth directly. We will calculate Rth using Voc and Isc and it is given by

Application:

Given: Load line equation = V + i = 100

To obtain open-circuit voltage (Vth) put i = 0 in load line equation 

⇒ Vth = 100 V

To obtain short-circuit current (isc) put V = 0 in load line equation

⇒ isc = 100 A

So, 

Equivalent circuit is

Current (i) = 100/2 = 50 A

Applying loop-law in the given circuit.

- V + i × R = 0

- V + I × 1 = 0

⇒ V = i

Given Load line equation is V + i = 100

Putting V = i 

then i + i = 100 

⇒ i = 50 A

Ohm’s law: Ohm’s law states that at a constant temperature, the current through a conductor between two points is directly proportional to the voltage across the two points.

Voltage = Current × Resistance

V = I × R

V = voltage, I = current and R = resistance

The SI unit of resistance is ohms and is denoted by Ω.

It helps to calculate the power, efficiency, current, voltage, and resistance of an element of an electrical circuit.

Limitations of ohms law:

• Ohm’s law is not applicable to unilateral networks. Unilateral networks allow the current to flow in one direction. Such types of networks consist of elements like a diode, transistor, etc.
• Ohm’s law is also not applicable to non – linear elements. Non-linear elements are those which do not have current exactly proportional to the applied voltage that means the resistance value of those elements’ changes for different values of voltage and current. An example of a non-linear element is thyristor.
• Ohm’s law is also not applicable to vacuum tubes.

Concept:

Ideal voltage source: An ideal voltage source have zero internal resistance.

Practical voltage source: A practical voltage source consists of an ideal voltage source (VS) in series with internal resistance (RS) as follows.

An ideal voltage source and a practical voltage source can be represented as shown in the figure.

Ideal current source: An ideal current source has infinite resistance. Infinite resistance is equivalent to zero conductance. So, an ideal current source has zero conductance.

Practical current source: A practical current source is equivalent to an ideal current source in parallel with high resistance or low conductance.

Ideal and practical current sources are represented as shown in the below figure.

• When an ideal voltage source and an ideal current source in series, the combination has an ideal current sources property.
• Current in the circuit is independent of any element connected in series to it.

Explanation:

In a series circuit, the current flows through all the elements is the same. Thus, any element connected in series with an ideal current source is redundant and it is equivalent to an ideal current source only.

In a parallel circuit, the voltage across all the elements is the same. Thus, any element connected in parallel with an ideal voltage source is redundant and it is equivalent to an ideal voltage source only.

Concept:

When resistances are connected in parallel, the equivalent resistance is given by

When resistances are connected in series, the equivalent resistance is given by

Calculation:

Given that R₁ = R₂ = R₃ = 6 Ω and all are connected in parallel.

⇒ R_eq = 2 Ω

When capacitors are connected in series across DC voltage:

• The charge of each capacitor is the same and the same current flows through each capacitor in the given time.
• The voltage across each capacitor is dependent on the capacitor value.

When capacitors are connected in parallel across DC voltage:

• The charge of each capacitor is different and the current flows through each capacitor in the given time are also different and depend on the value of the capacitor.
• The voltage across each capacitor is the same.

The circuit after removing the voltage source

The total resistance of the new circuit will be the equivalent resistance of the network.

R_eq = R_t = 3 + 2 + 2 = 7 Ω 

The equivalent resistance of the network is 7 Ω.

 Mistake PointsWhile finding the equivalent resistance of the network, don't consider the internal resistance of the voltage source. Please read the question carefully it is mentioned in the question as well.

There are two kinds of voltage or current sources:

Independent Source: It is an active element that provides a specified voltage or current that is completely independent of other circuit variables.

Dependent Source: It is an active element in which the source quantity is controlled by another voltage or current in the circuit.

Distributed Network:

• If the network element such as resistance, capacitance, and inductance are not physically separated, then it is called a Distributed network.
• Distributed systems assume that the electrical properties R, L, C, etc. are distributed across the entire circuit.
• These systems are applicable for high (microwave) frequency applications.

Lumped Network:

• If the network element can be separated physically from each other, then they are called a lumped network.
• Lumped means a case similar to combining all the parameters and considering it as a single unit.
• Lumped systems are those systems in which electrical properties like R, L, C, etc. are assumed to be located on a small space of the circuit.
• These systems are applicable to low-frequency applications.

Kirchoff's Laws:

• Kirchhoff’s laws are used for voltage and current calculations in electrical circuits.
• These laws can be understood from the results of the Maxwell equations in the low-frequency limit.
• They are applicable for DC and AC circuits at low frequencies where the electromagnetic radiation wavelengths are very large when we compare with other circuits. So they are only applicable for lumped parameter networks.

Kirchhoff's current law (KCL) is applicable to networks that are:

• Unilateral or bilateral 
• Active or passive 
• Linear or non-linear
• Lumped network

KCL (Kirchoff Current Law): According to Kirchhoff’s current law (KCL), the algebraic sum of the electric currents meeting at a common point is zero.

Mathematically we can express this as:

Where in represents the nth current

M is the total number of currents meeting at a common node.

KCL is based on the law of conservation of charge.

Kirchhoff’s Voltage Law (KVL):

It states that the sum of the voltages or electrical potential differences in a closed network is zero.