All questions of Transient Analysis in AC & DC Circuits for Electrical Engineering (EE) Exam

Understanding Critically Damped Second-Order Systems
In control systems, a critically damped second-order system achieves the fastest response without oscillation. The time constant, denoted as τ, is crucial for determining the system's response dynamics.
Key Concepts of Critically Damped Systems
- Natural Frequency (ωn): This represents the frequency at which the system would oscillate if there were no damping.
- Damping Ratio: For critically damped systems, the damping ratio equals 1, leading to a specific response characteristic.
Time Constant τ
In critically damped systems, the time constant τ is defined as the time it takes for the system to respond to a step input. The relationship between the time constant and the natural frequency is given by:
- Formula for Time Constant: τ = 1 / ωn
This indicates that as the natural frequency increases, the time constant decreases, leading to a faster system response.
Why Option A is Correct
- The correct answer is 1 / ωn.
- This choice reflects the relationship between the time constant and the natural frequency of the critically damped system.
Comparison with Other Options
- Option B (2 / ωn): This would imply a slower response, not characteristic of critical damping.
- Option C (1 / (2ωn)): This suggests an alternative damping context, not applicable here.
- Option D (ωn): Represents the natural frequency itself, not a time constant.
In summary, for a critically damped second-order system, the time constant τ is indeed 1 / ωn, making option A the correct choice. Understanding this relationship is essential for analyzing system behavior in control engineering.

The given transfer function is:
G(s) = s^6 / (Ks^2 + s^6)


The damping ratio, denoted by ζ (zeta), is a parameter that characterizes the behavior of a second-order linear system. It determines the response of the system to a step input and provides information about the system's stability and transient response.


The damping ratio is defined as the ratio of the actual damping coefficient to the critical damping coefficient. Mathematically, it is expressed as:
ζ = c / c_critical

where c is the actual damping coefficient and c_critical is the critical damping coefficient.


To find the damping ratio of the given transfer function G(s), we need to determine the actual damping coefficient c and the critical damping coefficient c_critical.

1. Actual Damping Coefficient (c):
The actual damping coefficient can be determined by examining the denominator of the transfer function. In this case, the denominator is Ks^2 + s^6. Since the denominator is a quadratic polynomial, we can compare it with the general form of a second-order system's characteristic equation:
s^2 + 2ζω_ns + ω_n^2

Comparing the coefficients, we can see that the actual damping coefficient c is equal to 2ζω_n, where ω_n is the natural frequency.

2. Critical Damping Coefficient (c_critical):
The critical damping coefficient corresponds to the case where the system is critically damped. In this case, the damping ratio is equal to 1. For a critically damped system, the damping coefficient c_critical is given by:
c_critical = 2√(K)


Now, we can equate the actual damping coefficient c to the critical damping coefficient c_critical and solve for the damping ratio ζ.

2ζω_n = 2√(K)

Dividing both sides by 2ω_n, we get:
ζ = √(K) / ω_n

In the given transfer function, the numerator is s^6 and the denominator is Ks^2 + s^6. The natural frequency ω_n can be determined by finding the square root of the coefficient of s^2 in the denominator.

Since the damping ratio is given as 0.5, we can substitute this value into the equation for ζ and solve for K.

0.5 = √(K) / ω_n

Squaring both sides, we get:
0.25 = K / (K + 1)

Simplifying the equation, we have:
0.25(K + 1) = K

Expanding the equation, we get:
0.25K + 0.25 = K

Rearranging the terms, we get:
0.75K = 0.25

Dividing both sides by 0.75, we get:
K = 1/3

Therefore, the value of K for which the damping ratio is 0.5 is 1/3, which corresponds to option C.

Ω resistor, as shown in the circuit below:

Battery (+12V) ---[Switch]--- [Capacitor] ---[Resistor]--- Ground

When the switch is closed, the capacitor begins to charge. The time constant of this RC circuit is given by the product of the resistance and the capacitance (RC). In this case, the time constant is 1KΩ * C, where C is the capacitance in farads.

The time constant determines how quickly the capacitor charges. Specifically, it is the time it takes for the capacitor to charge to approximately 63.2% of its final voltage.

If we want to calculate the time it takes for the capacitor to charge to 99% of its final voltage, we can use the formula:

t = -RC * ln(1 - 0.99)

In this case, the final voltage is 12V, so we can plug in the values:

t = -1KΩ * C * ln(1 - 0.99)

To calculate the time it takes for the capacitor to charge to 99% of its final voltage, we need to know the capacitance value. Can you provide the capacitance value?

Explanation:

The given unit step response of a second-order system is:

y(t) = 1 - e^(-5t) - 5t * e^(-5t)

To analyze the system, we need to find the natural frequency and the damping ratio.

1. The underdamped natural frequency is 5 rad/s:

To find the natural frequency, we need to find the coefficient of the exponential term in the unit step response. In this case, the coefficient is -5.

Natural frequency (ωn) = sqrt(coefficient) = sqrt(-5) = √5

So, the given statement is correct.

2. The damping ratio is 1:

To find the damping ratio, we need to find the coefficient of the linear term in the unit step response. In this case, the coefficient is -5t.

The damping ratio (ζ) is given by the formula:

ζ = coefficient / (2 * natural frequency) = (-5) / (2 * √5) = -√5 / 2

The given statement is incorrect. The damping ratio is not 1.

3. The impulse response is 25te^(-5t):

To find the impulse response, we need to differentiate the unit step response with respect to time.

Differentiating the unit step response, we get:

y'(t) = 5e^(-5t) - 5e^(-5t) - 5te^(-5t) + 25te^(-5t)

Simplifying, we get:

y'(t) = 5te^(-5t)

Comparing this with the given impulse response, we can see that they are the same.

So, the given statement is correct.

Therefore, the correct statements are:

1. The underdamped natural frequency is 5 rad/s.
3. The impulse response is 25te^(-5t).

Hence, the correct answer is option D) 1, 2, and 3.

The given network function is s^2 + 10s + 24/s^2 + 8s + 15. This represents neither an RC impedance nor an RL impedance nor an LC impedance. Let's analyze the given network function in detail to understand why it does not represent any of these impedances.

1. RC Impedance:
An RC impedance is represented by a transfer function in the form of 1/(RCs + 1), where R is the resistance and C is the capacitance. In the given network function, we do not have a term in the form of RCs, so it cannot represent an RC impedance.

2. RL Impedance:
An RL impedance is represented by a transfer function in the form of RL/(RLs + 1), where R is the resistance and L is the inductance. Similarly, in the given network function, we do not have a term in the form of RLs, so it cannot represent an RL impedance.

3. LC Impedance:
An LC impedance is represented by a transfer function in the form of 1/(LCs^2 + RCs + 1), where L is the inductance, C is the capacitance, and R is the resistance. In the given network function, we do not have a term in the form of LCs^2, so it cannot represent an LC impedance.

4. None of the mentioned:
Based on the analysis above, we can conclude that the given network function does not fit into any of the mentioned categories, i.e., RC impedance, RL impedance, or LC impedance. Therefore, the correct answer is option 'D' - None of the mentioned.

In summary, the given network function does not represent any of the mentioned impedances (RC, RL, or LC). It is important to understand the different forms of transfer functions for these impedances to accurately identify and categorize them.

Time Constant in an R-L circuit refers to the time taken by the current to reach approximately 63.2% of its final value after a change in voltage or current is applied to the circuit.

The time constant is determined by the values of resistance (R) and inductance (L) in the circuit. It is denoted by the symbol τ (tau) and is calculated using the formula:

τ = L / R

where:
τ = time constant
L = inductance in henries (H)
R = resistance in ohms (Ω)

Explanation:
To understand why the time constant represents 63.2% of the final value, we need to consider the behavior of an R-L circuit when a voltage or current is applied.

- Initially, when a change in voltage or current is applied, the current in an inductor (L) cannot change instantaneously. It builds up gradually.
- As time progresses, the current increases, but it takes a certain amount of time for the current to reach its maximum value.
- The time constant represents the time it takes for the current to reach 63.2% of its final value.
- After one time constant (τ), the current in the circuit is approximately 63.2% of the final value.
- It takes multiple time constants for the current to reach its final value. However, the majority of the change occurs within the first few time constants.

In other words, the time constant is a measure of the rate at which the current in an R-L circuit approaches its final value. It indicates how quickly the current rises when a voltage or current is applied and how long it takes for the current to settle at its final value.

Hence, the correct answer is option C: 63.2% of the final value.

Understanding the Assertion and Reason
The assertion (A) states that a step function is the first derivative of a ramp function, and an impulse function is the first derivative of a step function. This statement is indeed true in the context of signal processing.
Key Points of Assertion (A):
- Ramp Function: A ramp function increases linearly over time. Its derivative represents the rate of change.
- Step Function: The step function is constant (zero) until a specific point, where it jumps to a value. The derivative of a ramp function is a step function, indicating an instantaneous change.
- Impulse Function: An impulse function is a theoretical function that represents an instantaneous spike in value. The derivative of a step function, which changes at a single point, is an impulse function.
Examining the Reason (R)
The reason (R) posits that the output time response follows the same sequence as that of the input functions based on the derived time response expression. This statement is also true but does not directly explain the assertion.
Key Points of Reason (R):
- Output Time Response: It is derived from the system's response to various input functions.
- Sequence of Functions: While the time response does follow the sequence of input functions, this does not clarify why the relationships between ramp, step, and impulse functions exist.
Conclusion
- Correctness of A and R: Both A and R are true statements. However, R does not explain A adequately.
- Option B: Thus, the correct answer is option 'B': Both A and R are true, but R is not the correct explanation of A.

To calculate the power factor of a series RL circuit, we need to understand the concept of conductance and susceptance.

1. Conductance:
Conductance, denoted by G, is the reciprocal of resistance. It represents the ease with which current can flow through a circuit. The unit of conductance is Siemens (S).

2. Susceptance:
Susceptance, denoted by B, is the reciprocal of reactance. It represents the ease with which an alternating current can flow through a circuit. The unit of susceptance is also Siemens (S).

Given:
Conductance (G) = 30 S
Susceptance (B) = 40 S

Now, the power factor (PF) of a series RL circuit can be calculated using the following formula:

PF = G / √(G^2 + B^2)

Substituting the given values into the formula:

PF = 30 / √(30^2 + 40^2)
= 30 / √(900 + 1600)
= 30 / √(2500)
= 30 / 50
= 0.6

Therefore, the power factor of the series RL circuit is 0.6.

Explanation:
In a series RL circuit, the power factor is determined by the ratio of the conductance to the square root of the sum of the squares of conductance and susceptance. Conductance represents the real or active power component of the circuit, while susceptance represents the reactive power component.

A power factor of 0.6 indicates that the circuit has a combination of both active and reactive power. It implies that the circuit is not purely resistive but has a certain amount of inductive or capacitive reactance.

A higher power factor (closer to 1) indicates a circuit with less reactive power and more active power. A lower power factor (closer to 0) indicates a circuit with more reactive power and less active power.

Therefore, in this case, the power factor is 0.6, indicating that the series RL circuit has both resistive and reactive components.

Reciprocity Theorem in Electrical Engineering
The Reciprocity Theorem in electrical engineering states that in a linear bilateral network, the ratio of voltage to current at one point in the network is equal to the ratio of current to voltage at another point in the network.

Ratio Considered in Reciprocity Theorem
In the Reciprocity Theorem, the ratio considered is the ratio of voltage to current. This means that the voltage at one point in the network divided by the current at that same point is equal to the current at another point in the network divided by the voltage at that point.
Therefore, the correct answer is option 'A) Voltage to current'.
By understanding and applying the Reciprocity Theorem in electrical network analysis, engineers can simplify calculations and determine the relationship between different points in a network more efficiently.
Overall, the Reciprocity Theorem plays a crucial role in analyzing electrical networks and understanding the interrelationship between voltage and current at different points within the network.