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

Introduction:
In an R-L circuit connected to an alternating sinusoidal voltage, the transient current refers to the initial current that flows through the circuit when it is first closed. This transient current occurs before the system reaches its steady-state behavior. The size of this transient current primarily depends on the instant in the voltage cycle at which the circuit is closed.

Explanation:
When an R-L circuit is connected to an alternating sinusoidal voltage, the voltage across the circuit undergoes continuous changes over time. This sinusoidal voltage can be represented by a waveform that oscillates between positive and negative peaks. The transient current occurs during the initial phase of the circuit when the voltage is changing.

Effect of instant in the voltage cycle:
The instant in the voltage cycle at which the circuit is closed plays a crucial role in determining the size of the transient current. This is because the voltage waveform is not static, and its value at a specific instant determines the instantaneous current flow.

Explanation with examples:
1. If the circuit is closed at the peak of the positive half-cycle of the voltage waveform, the voltage is at its maximum value. At this instant, the rate of change of voltage is zero, resulting in a smaller transient current. This is because the inductor opposes changes in current, and when the voltage is at its maximum, the inductor tends to resist any further increase in current.

2. On the other hand, if the circuit is closed at the zero-crossing point of the voltage waveform, the rate of change of voltage is maximum. In this case, the inductor has a maximum opposition to the change in current, resulting in a larger transient current.

Conclusion:
Therefore, the instant in the voltage cycle at which the circuit is closed determines the magnitude of the transient current in an R-L circuit connected to an alternating sinusoidal voltage. Closing the circuit at the peak of the positive half-cycle results in a smaller transient current, while closing it at the zero-crossing point leads to a larger transient current. It is important to consider this factor when analyzing the behavior of R-L circuits in transient conditions.

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.

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.

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.

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?

Calculation of Current in the Circuit at t = 0.6 sec

Given Data:
Resistance (R) = 20 Ω
Inductance (L) = 8 H
DC voltage source = 120 V
Time (t) = 0.6 sec

Formula Used:
The current in a series RL circuit can be calculated using the formula:
i(t) = (V/R) * (1 - e^(-Rt/L))

Calculation:
Substitute the given values into the formula:
i(0.6) = (120/20) * (1 - e^(-20*0.6/8))
i(0.6) = 6 * (1 - e^(-1.5))

Now, calculate the value of e^(-1.5):
e^(-1.5) ≈ 0.2231

Substitute the value back into the formula:
i(0.6) = 6 * (1 - 0.2231)
i(0.6) = 6 * 0.7769
i(0.6) ≈ 4.66 A

Therefore, the current in the circuit at t = 0.6 sec is approximately 4.66 A, which corresponds to option 'C'.

Explanation:

Network Function Analysis:
The given network function is in the form of a transfer function representing a system in the Laplace domain. It is a ratio of polynomials in the Laplace variable 's'.

RC, RL, and LC Impedance:
The network function (s^2 + 4s) / ((s + 1)(s + 2)(s + 3)) does not directly represent an RC, RL, or LC impedance. It is a transfer function that describes the input-output relationship of a system, rather than an impedance characteristic.

Impedance Representation:
Impedance characteristics are typically represented by expressions involving complex variables such as s or jω, where ω is the frequency. The given network function does not have the form of an impedance expression involving resistors, capacitors, or inductors.

Conclusion:
Therefore, the correct answer is option 'D' - None of the mentioned. The given network function is a transfer function and does not directly correspond to an RC, RL, or LC impedance. It describes the system's dynamics in the Laplace domain rather than an impedance characteristic.

To determine the approximate time taken for a step response to reach 98% of its final value for the given system, we need to analyze the system's transfer function.

Transfer function:
The transfer function of the given system is 2/s, where s represents the Laplace variable.

Step response:
A step response is the output of a system when a unit step input is applied.

Time taken for a step response to reach 98% of its final value:
To determine the time taken for a step response to reach 98% of its final value, we need to find the time at which the output reaches 0.98 times the final value.

Approximation using time constant:
In a first-order system, the time constant is defined as the time taken for the output to reach 63.2% of its final value. For a step response, the time constant (τ) is given by τ = 1/(ζωn), where ζ is the damping ratio and ωn is the natural frequency.

In this case, the transfer function is 2/s, which represents a first-order system. Since the damping ratio and natural frequency are not given, we'll assume a critically damped system (ζ = 1) for simplicity.

Time constant (τ) = 1/(1 × ωn) = 1/ωn

Approximation using time constant:
To approximate the time taken for the step response to reach 98% of its final value, we can use the 2% settling time formula for a second-order system.

Approximate settling time = 4/ζωn

For a critically damped system (ζ = 1), the approximate settling time is 4/ωn.

Since the system in question is a first-order system, we can approximate the settling time by doubling the time constant (τ).

Approximate settling time ≈ 2τ

In this case, the time constant (τ) is 1/ωn.

Approximate settling time ≈ 2 × (1/ωn) = 2/ωn

Therefore, the approximate time taken for the step response to reach 98% of its final value is 2/ωn.

Comparing options:
From the given options, option 'C' states that the approximate time taken is 4s.

Since we do not have any information about the natural frequency (ωn) of the system, we cannot determine the exact time taken. However, option 'C' can be considered a reasonable approximation based on the assumption of a critically damped system.

Therefore, option 'C' is the correct answer.

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.

RL Impedance Representation:
RL impedance refers to the impedance in a circuit that consists of a resistor and an inductor. The given network function can be represented as (s + 1)(s + 4)/s(s + 2)(s + 5).

Analysis:
1. The network function contains terms in the form of (s + a) where 'a' is a constant. These terms indicate poles in the system.
2. The terms in the denominator indicate the roots of the characteristic equation that represent the natural response of the system.
3. The presence of (s + 1)(s + 4) in the numerator suggests the presence of poles at s = -1 and s = -4.
4. Poles at negative real values imply the existence of energy storage elements like inductors in the system.
5. The absence of any term in the form of (s + b) where 'b' is a constant in the numerator indicates the absence of zeros in the system.

Conclusion:
Based on the analysis, the network function (s + 1)(s + 4)/s(s + 2)(s + 5) can be associated with an RL impedance. The presence of poles at negative real values (-1 and -4) suggests the involvement of energy storage elements like inductors, which aligns with the characteristics of RL impedance.