All questions of Introduction to Digital Control for Electrical Engineering (EE) Exam

 For a good control system the speed of response and stability must be high and for the slow and sluggish response is not used and undesirable.

Transfer function is a mathematical representation of a system that describes the relationship between the Laplace output and Laplace input of the system. It is defined as the ratio of the Laplace output to the Laplace input, assuming zero initial conditions.

Transfer function represents the behavior of the system in the frequency domain. It is a powerful tool in analyzing and designing linear time-invariant (LTI) systems.

Transfer function is denoted by 'H(s)', where 's' is the complex frequency variable in the Laplace domain.

The general form of a transfer function is:
H(s) = Y(s) / X(s)

Where:
H(s) is the transfer function
Y(s) is the Laplace output
X(s) is the Laplace input

Transfer function is independent of the initial conditions of the system. It only considers the input and output relationship under steady-state conditions.

Hence, the correct answer is option 'c) 0', indicating that the transfer function does not consider the initial conditions.

The transfer function is a frequency-domain representation, and it provides valuable information about the system's stability, frequency response, and transient response. It allows engineers to analyze and design control systems and filters.

Key Points:
- Transfer function is the ratio of Laplace output to Laplace input.
- It represents the behavior of the system in the frequency domain.
- Transfer function is denoted by 'H(s)'.
- It is independent of the initial conditions of the system.
- The transfer function provides valuable information about the system's stability and frequency response.
- Option 'c) 0' is the correct answer.

(t - T0). The signal x(t - T0) is a time-shifted version of x(t) and has the same frequency as x(t).

Region of Convergence (ROC) in Z-transform:
The Region of Convergence (ROC) is a critical concept in the Z-transform domain and determines the values of z for which the Z-transform of a discrete-time signal is well-defined.

Absolutely Summable:
The correct statement is that the region of convergence of the Z-transform of x[n] consists of the values of z for which x[n] is absolutely summable. This means that the series formed by the magnitude of x[n] is convergent. In other words, the sum of the absolute values of x[n] over all n is finite.

Explanation:
The Z-transform of a discrete-time signal x[n] is given by X(z) = Σ(x[n] * z^(-n)). For the Z-transform to exist, the series formed by x[n] * z^(-n) must converge. This is ensured by the condition of absolute summability.

Importance of ROC:
The ROC also determines the stability and causality of the system represented by the Z-transform. Different ROCs correspond to different system behaviors, and understanding the ROC is crucial in analyzing and designing systems in the Z-domain.

Conclusion:
In summary, the region of convergence of the Z-transform of x[n] consists of the values of z for which x[n] is absolutely summable. This condition ensures the convergence of the Z-transform series and is essential for analyzing the properties of discrete-time signals and systems in the Z-domain.

Non-touching loops are loops in a network that do not share any common node. These loops are important in analyzing feedback systems and determining their stability.

Explanation:
In a network or a system, loops are formed by the branches or paths that connect various nodes. These loops can be interconnected and share common nodes. However, non-touching loops are those loops that do not have any common node between them.

To understand this concept, let's consider an example of a simple electrical network with three loops:

Loop 1: ABCD
Loop 2: DEFC
Loop 3: BEFD

In this example, Loop 1 and Loop 2 share the common node D, while Loop 1 and Loop 3 share the common node B. However, Loop 2 and Loop 3 do not have any common node. Therefore, Loop 2 and Loop 3 are non-touching loops.

Significance:
Non-touching loops are important in analyzing the stability of feedback systems. In feedback systems, loops are formed due to the presence of feedback paths. These loops can have a significant impact on the overall system behavior, including stability and performance.

By analyzing the non-touching loops, engineers can determine the extent to which each loop contributes to the overall system response. This analysis helps in designing and optimizing feedback systems to achieve desired performance and stability criteria.

Loop gain and stability:
In feedback systems, loop gain plays a crucial role in determining stability. Loop gain is the gain around the complete loop formed by the feedback paths. Non-touching loops contribute to the overall loop gain.

When non-touching loops are present in a system, their contributions to the loop gain are multiplied by each other. If the product of the gains of non-touching loops is less than 1, the system is stable. On the other hand, if the product is greater than 1, the system is unstable.

Therefore, analyzing non-touching loops helps in understanding the stability of feedback systems and designing appropriate compensators or control strategies to ensure stability.

In conclusion, non-touching loops in a network or system are loops that do not share any common node. These loops are important in analyzing the stability of feedback systems and determining the overall loop gain. By analyzing the non-touching loops, engineers can optimize feedback systems for desired performance and stability criteria.

Assertion (A): For the rational transfer function H(z) to be causal, stable and causally invertible, both the zeroes and the poles should lie within the unit circle in the z-plane.

Reason (R): For a rational system, ROC is bounded by poles.

Explanation:

To understand the assertion and reason given in the question, let's break it down into two parts and analyze each one separately.

Part 1: For the rational transfer function H(z) to be causal, stable and causally invertible, both the zeroes and the poles should lie within the unit circle in the z-plane.

Explanation of Assertion (A):
The statement in Assertion (A) is correct. In order for a rational transfer function, represented by H(z), to be causal, stable, and causally invertible, both the zeroes and the poles of the transfer function should lie within the unit circle in the z-plane.

Causal: A rational transfer function is causal if all the poles of the transfer function lie strictly inside the unit circle. This condition ensures that the output of the system depends only on the present and past inputs, not on future inputs.

Stable: A rational transfer function is stable if all the poles of the transfer function lie strictly inside the unit circle. Stability implies that the output of the system remains bounded for any bounded input.

Causally Invertible: A rational transfer function is causally invertible if all the zeroes of the transfer function lie strictly inside the unit circle. Causal invertibility ensures that the original input can be uniquely determined from the output of the system.

Explanation of Reason (R):
The statement in Reason (R) is also correct. For a rational system, the Region of Convergence (ROC) is bounded by the poles of the transfer function.

ROC: The ROC is the region in the z-plane where the z-transform of a sequence converges. In the case of a rational transfer function, the ROC can be bounded or unbounded.

For a rational system, the ROC is bounded by the poles of the transfer function. This means that the poles determine the boundary of the ROC. If any pole lies outside the unit circle, the ROC will be unbounded and the system will not be causal, stable, or causally invertible.

Conclusion:
Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). The poles of a rational transfer function determine the boundary of the ROC, and for the transfer function to be causal, stable, and causally invertible, both the zeroes and the poles should lie within the unit circle in the z-plane.

Assertion (A): An LTI discrete system represented by the difference equation y(n-2) - 5y(n-1) + 6y(n) = x(n) is unstable.

Reason (R): A system is unstable if the roots of the characteristic equation lie outside the unit circle.

Explanation:
To determine the stability of the LTI discrete system represented by the given difference equation, we need to analyze its characteristic equation.

The characteristic equation of the system can be obtained by setting y(n) = 0 and solving for the values of z, where z is the complex variable representing the z-transform domain.

The given difference equation is:

y(n-2) - 5y(n-1) + 6y(n) = x(n)

Let's rewrite this equation in terms of the shift operator z:

z^2Y(z) - 5zY(z) + 6Y(z) = X(z)

Dividing both sides by Y(z), we get:

z^2 - 5z + 6 = X(z)/Y(z)

The characteristic equation is obtained by setting the numerator X(z) equal to zero:

z^2 - 5z + 6 = 0

Now, let's solve this quadratic equation to find the roots of the characteristic equation:

(z - 3)(z - 2) = 0

The roots are z = 3 and z = 2.

Stability Analysis:

For a discrete LTI system to be stable, all the roots of the characteristic equation should lie inside the unit circle in the z-plane.

In the given case, the roots of the characteristic equation are z = 3 and z = 2.

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Conclusion:

Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).

The stability of the LTI discrete system represented by the given difference equation can be determined by analyzing the roots of the characteristic equation, and in this case, the roots lie inside the unit circle, indicating stability.

When deriving the transfer function of a linear element only initial conditions are assumed to be zero, loading cannot be assumed to be zero.

Feedback Control Systems: Sensitivity to Parameter Changes

Feedback control systems are used to control the output of a system by measuring the output and comparing it to a desired value. The error signal is then fed back to the input of the system to adjust it and minimize the error. In this process, there are two paths: the forward path and the feedback path.

Sensitivity to Parameter Changes

The sensitivity of a control system is defined as the degree to which the system output changes in response to a change in the system parameters. In other words, how much the system output changes in response to a change in the input or feedback path parameters.

For feedback control systems, there are two types of sensitivity to parameter changes: sensitivity to forward path parameter changes and sensitivity to feedback path parameter changes.

- Sensitivity to Forward Path Parameter Changes: The forward path is the path from the input to the output of the system. A change in the forward path parameters, such as the gain or transfer function of the system, can affect the system output. Feedback control systems are less sensitive to changes in the forward path parameters than to changes in the feedback path parameters. This is because the feedback loop provides a corrective signal that compensates for the changes in the forward path.
- Sensitivity to Feedback Path Parameter Changes: The feedback path is the path from the output to the input of the system. A change in the feedback path parameters, such as the gain or transfer function of the feedback loop, can affect the system output. Feedback control systems are more sensitive to changes in the feedback path parameters than to changes in the forward path parameters. This is because the feedback loop is responsible for the correction of the error signal.

Conclusion

In conclusion, feedback control systems are less sensitive to changes in the forward path parameters than to changes in the feedback path parameters. This is because the feedback loop provides a corrective signal that compensates for the changes in the forward path. Therefore, it is important to consider the sensitivity of a control system to parameter changes when designing and implementing feedback control systems.

Solution:
The characteristic equation of the control system is given by:
s6 2s5 8s4 12s3 20s2 16s 16=0
To determine the number of roots of the equation which lie on the imaginary axis of s-plane, we need to find the roots of the equation and then check which roots lie on the imaginary axis.

Finding Roots:
We can find the roots of the equation using any of the standard methods such as Routh-Hurwitz criterion, root locus method, or MATLAB. Here, we will use the Routh-Hurwitz criterion.

Routh-Hurwitz Criterion:
The Routh-Hurwitz criterion is a mathematical test that determines the stability of a control system by examining the locations of the roots of the characteristic equation in the complex plane. The criterion states that if all the coefficients of the characteristic equation are real and positive, then the control system is stable if and only if all the roots of the characteristic equation lie in the left-half of the complex plane.

The Routh-Hurwitz criterion provides a systematic way to determine the number of roots of a polynomial equation that lie on the imaginary axis of the complex plane. The number of roots on the imaginary axis is equal to the number of sign changes in the first column of the Routh array.

Applying Routh-Hurwitz Criterion:
The Routh array for the given characteristic equation is as follows:

s6 2s5 8s4 12s3 20s2 16s 16
s5 8s3 20s1 16s
0 16s 16
16

The first column of the Routh array has two sign changes. Therefore, the number of roots of the characteristic equation that lie on the imaginary axis of the s-plane is equal to the number of sign changes in the first column of the Routh array, which is 4.

Therefore, the correct answer is option 'C' (4).

The output of a first order hold between two consecutive sampling instants is a Ramp Function.

Explanation:
A first order hold is a fundamental component in signal processing systems that bridges the gap between continuous-time and discrete-time signals. It is commonly used in digital-to-analog converters (DACs) to reconstruct continuous-time signals from discrete samples.

To understand why the output of a first order hold is a Ramp Function, let's first consider its operation. A first order hold takes a discrete-time input signal and holds its value constant until the next sampling instant. At the next sampling instant, the output of the first order hold transitions smoothly to the new value based on the rate of change of the input signal.

Key Points:
- First order hold bridges the gap between continuous-time and discrete-time signals.
- It is used in digital-to-analog converters (DACs) to reconstruct continuous-time signals.
- The operation of a first order hold involves holding the input signal value until the next sampling instant.
- The output of the first order hold transitions smoothly to the new value based on the rate of change of the input signal.

When the input signal is a constant, the output of the first order hold will also be a constant. This is because there is no change in the input signal between consecutive sampling instants, so the first order hold simply holds the value.

However, when the input signal changes between consecutive sampling instants, the output of the first order hold will be a ramp function. This is because the first order hold smoothly transitions from the previous value to the new value, creating a linear change over time.

Key Points:
- When the input signal is a constant, the output of the first order hold is also a constant.
- When the input signal changes between consecutive sampling instants, the output of the first order hold is a ramp function.

Therefore, the correct answer to the given question is option 'C' - Ramp Function. The output of a first order hold between two consecutive sampling instants is a ramp function when the input signal changes.

Regenerative feedback in a system implies feedback with a positive sign. This means that the feedback signal is added to the input signal, resulting in an amplified output. Let's understand this in more detail:

Introduction to feedback
Feedback is an important concept in control systems. It involves taking a portion of the output signal and feeding it back to the input of the system. This allows the system to regulate its behavior and achieve desired performance.

Positive feedback vs Negative feedback
Feedback can be classified into two types: positive feedback and negative feedback.

Negative feedback: In negative feedback, the feedback signal is subtracted from the input signal. This has a stabilizing effect on the system, as it tends to reduce any deviations between the desired output and the actual output. Negative feedback is commonly used in control systems to improve stability, accuracy, and reduce errors.

Positive feedback: In positive feedback, the feedback signal is added to the input signal. This amplifies the input and can lead to an unstable behavior. Positive feedback is not commonly used in control systems, as it can result in oscillations or uncontrollable behavior.

Regenerative feedback
Regenerative feedback is a specific type of positive feedback where the feedback signal reinforces the input signal, causing an increase in the output. This can lead to continuous amplification and can result in oscillations or instability in the system.

Example of regenerative feedback
An example of regenerative feedback is the Schmitt trigger circuit. This circuit is used to convert a noisy or distorted input signal into a clean digital signal. It uses positive feedback to create hysteresis, which helps in noise immunity and signal conditioning.

The Schmitt trigger circuit consists of an operational amplifier and positive feedback resistors. When the input signal crosses a certain threshold, the output changes state, and the positive feedback reinforces this change, causing a rapid transition in the output. This positive feedback helps in noise rejection and provides a sharp output transition.

Conclusion
In summary, regenerative feedback implies feedback with a positive sign, where the feedback signal is added to the input signal. This can lead to amplification and instability in the system. Negative feedback, on the other hand, involves subtracting the feedback signal from the input signal and is commonly used to improve stability and accuracy in control systems.