All questions of Controllability, observability and stability of discrete state space models for Electrical Engineering (EE) Exam

Feedback and System Gain

Feedback is a technique used in control systems to adjust the output based on the difference between the desired output and the actual output. It is used to improve the performance and stability of a system. However, when the feedback is too much or too little, it can affect the system gain and stability.

System Gain

System gain refers to the ratio of the output to the input of a system. It is an important parameter that determines the performance of a system. In control systems, the gain is used to adjust the output of the system to match the desired output. It is a measure of how much the system amplifies the input signal.

Effect of Feedback on System Gain

When feedback is applied to a system, it can affect the system gain in the following ways:

1. Increase in Gain: When feedback is applied to a system, it can increase the gain of the system. This is because the feedback signal is added to the input signal, which results in an amplified output.

2. Decrease in Gain: On the other hand, feedback can also decrease the gain of a system. This happens when the feedback signal is out of phase with the input signal. In this case, the feedback signal cancels out some of the input signal, resulting in a lower output.

Conclusion

In conclusion, feedback can have both positive and negative effects on the system gain. It is important to balance the feedback to ensure that the system gain is optimized for the desired performance. Too much feedback can lead to instability, while too little feedback can result in poor performance. By adjusting the feedback, the system gain can be optimized for the desired performance.

Effect of Roots of Transfer Function on Stability
Stability of a system is determined by the location of the poles of the transfer function in the s-domain. The roots of the transfer function, which correspond to the poles of the system, play a crucial role in determining the stability of the system.

False Statement
The statement that the roots of the transfer function do not have any effect on the stability of the system is false. In fact, the location of the roots, specifically the poles, directly impacts the stability of the system.

Roots and Stability
- The poles of the transfer function determine the behavior of the system in the time domain.
- If the poles of the transfer function have negative real parts, the system is stable.
- If the poles have positive real parts, the system is unstable.
- If the poles have zero real parts, the system is marginally stable.
- The imaginary part of the poles determines the oscillatory behavior of the system.

Effect of Roots on Stability
The roots of the transfer function are crucial in determining the stability of the system. The poles dictate whether the system is stable, unstable, or marginally stable. Therefore, it is incorrect to state that the roots of the transfer function have no effect on the stability of the system.

Transfer functions are mathematical representations of the relationship between the input and output of a system. They are commonly used in control systems engineering to analyze and design systems.

The transfer function of a system can be obtained from a signal flow graph. A signal flow graph is a graphical representation of a system that shows the flow of signals through different components or blocks of the system. It consists of nodes (representing variables or signals) and directed edges (representing the flow of signals between nodes).

A signal flow graph can be used to represent the interconnection between different components or blocks of a system. Each block or component in the signal flow graph is associated with a transfer function. The transfer function of the entire system can be obtained by combining the transfer functions of the individual blocks or components according to the interconnections shown in the signal flow graph.

In a signal flow graph, the transfer function of a block or component is obtained by considering the output of the block as a ratio of the sum of the inputs to the block. This is based on the principle of superposition, which states that the response of a linear system to a sum of inputs is equal to the sum of the responses to each individual input.

The transfer function can also be obtained from an analogous table, which is a tabular representation of the interconnections between different components or blocks of a system. The analogous table specifies the transfer function of each block or component and the interconnections between them.

However, the transfer function cannot be directly obtained from the output-input ratio or the standard block system. The output-input ratio represents the relationship between the output and input variables of a system, but it does not provide the mathematical representation of this relationship in the form of a transfer function. Similarly, the standard block system is a graphical representation of a system, but it does not explicitly specify the transfer functions of the individual blocks or components.

Therefore, the correct answer is option 'A' - Signal flow graph.

Charge in Thermal-Electrical Analogy

In thermal-electrical analogy, charge is considered analogous to temperature. This analogy is used to understand and analyze the behavior of electrical circuits by drawing parallels between electrical and thermal systems. It allows engineers and researchers to apply concepts and principles from one domain to the other, making it easier to comprehend complex electrical phenomena.

Explanation:

1. Thermal and Electrical Systems
Both thermal and electrical systems involve the flow of energy. In a thermal system, energy is transferred in the form of heat, while in an electrical system, energy is transferred in the form of electrical charge.

2. Analogy between Heat Flow and Charge
In the thermal-electrical analogy, heat flow is considered analogous to electrical charge. Heat flow represents the transfer of thermal energy from a region of higher temperature to a region of lower temperature, similar to how charge flows from a region of higher electrical potential to a region of lower electrical potential.

3. Reciprocal of Temperature
The reciprocal of temperature is defined as 1/T, where T is the absolute temperature. In the thermal-electrical analogy, the reciprocal of temperature is considered analogous to electrical charge. This analogy is based on the fact that temperature and charge both represent the driving force for energy transfer in their respective systems.

4. Similarities between Temperature and Charge
Both temperature and charge exhibit similar properties and behaviors. They both have a magnitude and can be positive or negative. They can also be transferred from one point to another. In addition, temperature and charge are both scalar quantities, meaning they have magnitude but no direction.

Conclusion:

In the thermal-electrical analogy, charge is considered analogous to temperature. This analogy allows engineers to apply their understanding of thermal systems to analyze and solve electrical problems. By drawing parallels between the two domains, it becomes easier to comprehend and predict the behavior of electrical circuits.

Thermocouple

- A thermocouple is a temperature sensor that directly converts temperature into voltage.
- It consists of two different wires or metals that are joined together at one end.
- When there is a temperature difference between the two junctions, a voltage is generated, which is known as the thermoelectric effect.
- The voltage generated is proportional to the temperature difference between the two junctions.
- This voltage can be measured and used to determine the temperature.
- The most common type of thermocouple is the type K thermocouple, which is made up of chromel and alumel wires.
- Other common types include type J (iron and constantan), type T (copper and constantan), and type E (chromel and constantan).
- Each type of thermocouple has a different temperature range and accuracy.
- Thermocouples are widely used in various industries and applications, including temperature measurement in HVAC systems, industrial processes, and scientific research.
- They are rugged, reliable, and can withstand high temperatures and harsh environments.
- However, they have limitations in terms of accuracy and sensitivity compared to other temperature sensors like RTDs (Resistance Temperature Detectors) and thermistors.
- Thermocouples require a reference junction, known as a cold junction, where the temperature is measured and compensated for.
- The cold junction is typically connected to a temperature controller or a cold junction compensator to accurately measure the temperature.
- The voltage generated by the thermocouple is typically very small, so it needs to be amplified before it can be used for temperature measurement.
- Thermocouples can be connected in series or parallel to increase the voltage output or improve accuracy.
- They are commonly used in conjunction with data acquisition systems or temperature controllers to monitor and control temperature in various applications.

Introduction:
The Linear Variable Differential Transformer (LVDT) is a type of electrical transformer used for measuring linear displacement. It is a common type of sensor used in various industries to measure displacement accurately and reliably.

Explanation:
LVDT is specifically designed to measure linear displacement, and it does not directly measure other physical quantities such as velocity, acceleration, or pressure. Let's discuss each option in detail:

a) Displacement:
The primary purpose of an LVDT is to measure displacement. It is designed to provide an accurate and linear output proportional to the position of the core within its range of motion. By measuring the secondary voltage output, the displacement of the LVDT's core can be determined.

b) Velocity:
Velocity is a measure of the rate of change of displacement with respect to time. While LVDT itself does not directly measure velocity, it is possible to calculate velocity by differentiating the displacement measurement obtained from the LVDT output with respect to time.

c) Acceleration:
Acceleration is a measure of the rate of change of velocity with respect to time. Similar to velocity, LVDT does not directly measure acceleration. However, by differentiating the displacement measurement obtained from the LVDT output twice with respect to time, acceleration can be calculated.

d) Pressure:
Pressure is the force per unit area exerted on a surface. LVDT is not designed to measure pressure directly. Pressure sensors or transducers are specifically designed for that purpose. These sensors use different principles, such as piezoelectric, strain gauge, or capacitive, to measure pressure accurately.

Conclusion:
The Linear Variable Differential Transformer (LVDT) is a displacement sensor and is not designed to measure pressure. While LVDT can indirectly provide data that can be used to calculate velocity and acceleration, it is important to note that these quantities are not measured directly by the LVDT.

Introduction:
The force-voltage analogy is a powerful tool used in electrical engineering to analyze and design electrical circuits. It establishes an analogy between mechanical systems and electrical circuits, where force is analogous to voltage, and velocity is analogous to current.

Analogy between mechanical and electrical systems:
To understand why velocity is analogous to current in the force-voltage analogy, let's consider the analogy between mechanical and electrical systems.

In mechanical systems, force is the driving factor that causes motion, and velocity is the rate of change of displacement with respect to time. Similarly, in electrical systems, voltage is the driving factor that causes current flow, and current is the rate of flow of electric charge with respect to time.

Explanation:
To explain why velocity is analogous to current in the force-voltage analogy, let's consider the equation that relates force, velocity, and mass in a mechanical system:

Force = mass × acceleration

Similarly, in an electrical system, we have Ohm's law that relates voltage, current, and resistance:

Voltage = current × resistance

Comparing these two equations, we can establish an analogy between the variables:

- Force is analogous to voltage (both are the driving factors)
- Mass is analogous to resistance (both oppose the flow)
- Acceleration is analogous to current (both represent the rate of change)

Therefore, in the force-voltage analogy, velocity is analogous to current because they both represent the rate of change of a quantity. Just as velocity is the rate of change of displacement, current is the rate of flow of electric charge.

Conclusion:
In the force-voltage analogy used in electrical engineering, velocity is analogous to current. This analogy establishes a relationship between mechanical systems and electrical circuits, where force is analogous to voltage and velocity is analogous to current. Understanding this analogy helps in analyzing and designing electrical circuits by leveraging the principles and techniques used in mechanical systems.

Introduction:
In electrical engineering, the force-voltage analogy is a method used to analyze electrical circuits by drawing an analogy between electrical quantities and mechanical quantities. This analogy helps in understanding the behavior of electrical circuits by relating them to more familiar mechanical systems. In this analogy, mass, which represents inertia in mechanical systems, is analogous to certain electrical quantities.

Analogy between Electrical and Mechanical Systems:
The force-voltage analogy draws similarities between electrical quantities and mechanical quantities as shown below:

- Force (F) is analogous to Voltage (V)
- Velocity (v) is analogous to Current (I)
- Mass (m) is analogous to Inductance (L)
- Damping (b) is analogous to Resistance (R)

Explanation of the Analogy:
The analogy between mass and inductance can be understood by considering their respective roles in their corresponding systems.

1. Mass in Mechanical Systems:
In mechanical systems, mass represents inertia and resists changes in velocity. It stores kinetic energy and opposes changes in motion. The greater the mass, the more force is required to accelerate or decelerate the object.

2. Inductance in Electrical Systems:
Similarly, in electrical systems, inductance represents inertia and resists changes in current. It stores magnetic energy and opposes changes in current flow. The greater the inductance, the more voltage is required to change the current.

Key Points:
- Mass resists changes in velocity in mechanical systems.
- Inductance resists changes in current in electrical systems.
- Both mass and inductance store energy.
- Both oppose changes in their respective quantities.

Analogy Justification:
Since both mass and inductance have similar roles in their respective systems, they are analogous in the force-voltage analogy. The analogy helps in understanding the behavior of inductors in electrical circuits by relating them to masses in mechanical systems.

Conclusion:
In the force-voltage analogy, mass is analogous to inductance. Both represent inertia, store energy, and resist changes in their respective quantities. This analogy is a useful tool in electrical engineering to analyze and understand the behavior of electrical circuits by drawing parallels with mechanical systems.

Explanation:

Hang on vs. Bang on:
- In control systems, "hang on" refers to a type of control system where the output remains at a constant value until the input signal exceeds a certain threshold.
- On the other hand, "bang on" is a term used to describe a control system where the output changes abruptly in response to changes in the input signal.

Bang-bang Control System:
- A bang-bang control system is a type of control system that switches between two discrete states based on the input signal.
- In a bang-bang control system, the output is either fully on or fully off, with no intermediate levels.
- This type of control system is commonly used in applications where precise control is not necessary, and simple on-off control is sufficient.

Hang out:
- "Hang out" is not a term commonly used in the context of control systems with logic switching.
- It does not describe a specific type of control system or behavior.

Therefore, the correct answer to the question is option C) Bang-bang, as it is the only term related to control systems with logic switching.

Underdamped second-order systems are commonly used to model various physical systems in engineering. These systems exhibit oscillatory behavior in their step response, which can be characterized by several parameters, including the peak overshoot.

The peak overshoot is defined as the maximum percentage overshoot of the response curve with respect to the final steady-state value. It is a measure of how much the response exceeds the desired value before settling down. The peak overshoot occurs when the response reaches its maximum amplitude during the transient period.

The options given in the question are related to different aspects of the step response, and we need to determine which one is explicitly indicative of the peak overshoot.

Let's analyze each option:

a) Settling time: The settling time is the time taken by the response to reach and stay within a certain tolerance band around the final steady-state value. It is not directly related to the peak overshoot and depends on the desired level of accuracy.

b) Rise time: The rise time is the time taken by the response to rise from a specified lower threshold to a specified upper threshold. It is also not directly related to the peak overshoot and depends on the desired time for the response to reach a certain level.

c) Natural frequency: The natural frequency of a system determines the rate at which it oscillates in the absence of damping. While the natural frequency indirectly affects the peak overshoot, it is not explicitly indicative of it.

d) Damping ratio: The damping ratio (ζ) quantifies the level of damping in the system. It is the ratio of the actual damping coefficient to the critical damping coefficient. The damping ratio is directly related to the peak overshoot. Higher damping ratios lead to lower peak overshoots, while lower damping ratios result in higher peak overshoots. Therefore, the damping ratio is explicitly indicative of the peak overshoot.

In summary, the correct answer is option 'D' - damping ratio. The damping ratio of an underdamped second-order system directly determines the peak overshoot in the step response. Higher damping ratios result in lower peak overshoots, while lower damping ratios lead to higher peak overshoots.

Explanation:

Assertion (A): The zeroes on the real axis near the origin are generally avoided in design.
- Zeroes on the real axis near the origin can lead to instability in the system.
- These zeroes can cause oscillations and ringing in the system response.

Reason (R): In a sluggish system introduction of zeroes can improve the transient response.
- Introducing zeroes in a system can help in improving the transient response by damping out oscillations.
- By strategically placing zeroes, the system response can be made faster and more stable.

Explanation:
- Both Assertion (A) and Reason (R) are true.
- The reason (R) provides a correct explanation for the assertion (A) because introducing zeroes in a sluggish system can indeed improve the transient response by damping out oscillations and making the system faster and more stable.
- Therefore, option B is the correct answer.

Introduction:
A thermocouple is a temperature sensor that generates a voltage signal based on the temperature difference between two junctions of dissimilar metals. It is widely used in various industries for temperature measurement and control applications.

Working Principle of a Thermocouple:
A thermocouple consists of two wires made of different metals, typically known as the positive and negative legs or the hot and cold junctions. When there is a temperature difference between these two junctions, it causes a temperature-dependent voltage to be generated across the thermocouple.

Output of a Thermocouple:
The output of a thermocouple is a D.C. voltage. This voltage is known as the thermoelectric voltage or the Seebeck voltage. It is generated due to the phenomenon known as the Seebeck effect, which occurs when there is a temperature gradient along the length of the thermocouple.

The magnitude of the thermoelectric voltage depends on the temperature difference between the two junctions and the type of metals used in the thermocouple. Different combinations of metals produce different thermoelectric voltages.

Characteristics of Thermocouple Output:
- Polarity: The thermoelectric voltage has a polarity, which means it has a positive and negative terminal. The polarity depends on the temperature of the hot junction relative to the cold junction.
- Linearity: The thermocouple output is generally linear over a certain temperature range. This linearity allows for accurate temperature measurement and control.
- Temperature Range: Thermocouples can measure a wide range of temperatures, from very low (-200°C) to very high (over 2000°C), depending on the type of thermocouple used.

Measurement and Conversion:
To measure the output voltage of a thermocouple, it is connected to a temperature measuring instrument called a thermocouple meter or a temperature controller. These instruments are designed to measure the thermoelectric voltage and convert it into a corresponding temperature reading.

Conclusion:
In summary, the output of a thermocouple is a D.C. voltage. The magnitude of this voltage is proportional to the temperature difference between the two junctions of the thermocouple.

Transportation lag refers to the delay or latency that occurs during the transfer of materials or information in a system. It is a common phenomenon in various systems, including transportation networks, communication systems, and industrial processes. The correct answer to this question is option 'B', which states that transportation lag is seen when the time taken to transfer is large. Let's explore this answer in detail.

Explanation:
Transportation lag can occur due to various factors, including the distance between the source and destination, the speed of the transportation medium, and the efficiency of the transfer process. When the time taken to transfer the materials or information is large, it leads to transportation lag. This lag can have several consequences and impacts on the overall system.

Impact of Transportation Lag:
1. Delays: Transportation lag can cause delays in the delivery of materials or information. This can have significant implications in various industries, such as manufacturing, logistics, and supply chain management. Delays can result in increased costs, decreased efficiency, and customer dissatisfaction.

2. Inefficiency: When there is a significant transportation lag, it indicates an inefficient transfer process. Inefficient systems can lead to wastage of resources, increased energy consumption, and reduced overall system performance.

3. Bottlenecks: Transportation lag can create bottlenecks in the system. Bottlenecks are points in the system where the flow of materials or information is restricted, leading to congestion and reduced throughput. Identifying and resolving bottlenecks is crucial for improving system performance.

4. Planning and Scheduling: Large transportation lag can make planning and scheduling more challenging. It becomes difficult to predict the arrival or completion time of transfers accurately. This can affect production schedules, project timelines, and overall operational efficiency.

5. Quality and Reliability: Transportation lag can also impact the quality and reliability of the transfer process. For example, in communication systems, high latency can lead to signal degradation or loss, affecting the quality of communication.

Conclusion:
Transportation lag is seen in systems when the time taken to transfer materials or information is large. It can result in delays, inefficiency, bottlenecks, and challenges in planning and scheduling. Understanding and addressing transportation lag is crucial for improving system performance, reducing costs, and enhancing customer satisfaction.