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

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.

Solution:

To solve this problem, we need to use Kirchhoff's Voltage Law (KVL) and the concept of series circuits. Let's go through the detailed steps to find the voltage drop across R2.

Step 1: Understanding Series Circuits

In a series circuit, the current flowing through each component is the same, but the voltage drop across each component can be different. The total resistance in a series circuit is the sum of the individual resistances.

Given:

• R1 = 5 Ω
• R2 = 10 Ω
• V (total voltage) = 15 V

Step 2: Calculate the Total Resistance

The total resistance (Rtotal) in a series circuit is the sum of the resistances of the individual resistors:

R_total = R₁ + R₂

Rtotal = 5 Ω + 10 Ω = 15 Ω

Step 3: Calculate the Total Current

Using Ohm's Law, we can calculate the total current (I) flowing through the circuit:

V = I × Rtotal 

15 V = I × 15 Ω

I = 15 V / 15 Ω

I = 1 A

Step 4: Calculate the Voltage Drop Across R2

Now that we have the current flowing through the circuit, we can calculate the voltage drop across R2 using Ohm's Law:

V_R2 = I × R₂

VR2 = 1 A × 10 Ω

VR2  = 10 V

3ϕ balanced system

In a balanced three-phase system, the sum of the three phase voltage phasors is zero. This is a fundamental property of balanced three-phase systems. 

Explanation:

• In a balanced three-phase system, the three phase voltages have equal magnitudes and are 120 degrees apart from each other. 
• When the three phasors are added vectorially, they will cancel each other out, resulting in a sum of zero. 

Concept

Phase difference is used to describe the difference in degrees or radians when two or more alternating quantities reach their maximum or zero values.

Step 1: Consider any one term (either sin or cos) as a reference phasor.

Step 2: Plot the given quantity with respect to the reference phasor.

Step 3: Start moving in the anticlockwise direction. The quantity that comes first is said to be leading with respect to other quantity.

Explanation

If two AC waveforms have a phase difference of 0°, they are said to be in phase.

Concept

The inverse Laplace Transform of 

\({k\over s+a}=ke^{-at}\)

where, k = Constant

Calculation

Given, \(\rm I(s)=\frac{1.5}{s+4}\)

On comparison, k = 1.5 and a = 4

\({1.5\over s+4}=1.5e^{-4t}\)

Concept

The power transfer efficiency is:

​\(η={I^2R_L\over VI}\times 100\)

\(η={IR_L\over V}\times 100\)

The current across any resistor is given by:

\(I={V\over R}\)

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.

\(2={V\over 5+R}\) .... (i)

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

\(1.6={V\over 10+R}\) .....(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

\(I={V\over R+R_L}\)

\(I={40\over 15+15}={4\over 3}A\)

\(η={{4\over 3}\times 15\over 40}\times 100\)

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

\({{\rm{R}}_{{\rm{th}}}} = \frac{{{{\rm{V}}_{{\rm{oc}}}}}}{{{{\rm{i}}_{{\rm{sc}}}}}}\)  

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, \({R_{th}} = \frac{{{V_{th}}}}{{{i_{sc}}}} = \frac{{100}}{{100}} = 1{\rm{\Omega }}\)

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

\(\frac{1}{{{R_{eq}}}} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}} + \ldots + \frac{1}{{{R_n}}}\)

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

\({R_{eq}} = {R_1} + {R_2} + \ldots + {R_n}\)

Calculation:

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

\(\frac{1}{{{R_{eq}}}} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6}\)

⇒ 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:

\(\mathop \sum \limits_{n = 1}^M {i_n} = 0\)

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.