← MES Jr. Grade
Past Questions — 2024
The stability of a control system can be determined from:
A)
Root locus
B)
Nyquist plot
C)
Bode plot
D)
All of the above
Stability analysis is a fundamental part of control systems engineering, and it can be performed using both time-domain and frequency-domain methods.
1. Root Locus (Option a): This is a graphical method that shows how the poles of the closed-loop transfer function move in the s-plane as a system parameter (usually the gain K) varies. If any part of the root locus enters the right-half of the s-plane, the system becomes unstable. Thus, it is used to determine absolute and relative stability.
2. Nyquist Plot (Option b): Based on the Nyquist Stability Criterion, this frequency-domain method uses the open-loop frequency response to determine the stability of the closed-loop system by observing the encirclements of the critical point (-1, j0). It is a powerful tool for determining absolute stability.
3. Bode Plot (Option c): This method represents the frequency response in terms of magnitude and phase. Stability is determined using the Gain Margin (GM) and Phase Margin (PM). If both GM and PM are positive, the system is stable.
Since all three methods are valid and widely used tools for determining the stability of a control system, the correct answer is 'All of the above'.
The Poynting vector represents:
A)
Electric field strength
B)
Magnetic flux density
C)
Power flow per unit area
D)
Energy density
The Poynting vector, denoted by $\mathbf{S}$, is defined as the cross product of the electric field intensity $\mathbf{E}$ and the magnetic field intensity $\mathbf{H}$: $\mathbf{S} = \mathbf{E} \times \mathbf{H}$.
1. **Physical Significance**: The Poynting vector represents the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field. In simpler terms, it describes the power flow per unit area (measured in Watts per square meter, $\text{W/m}^2$).
2. **Analysis of Options**:
- **Option a (Electric field strength)**: This is represented by $\mathbf{E}$, not the Poynting vector.
- **Option b (Magnetic flux density)**: This is represented by $\mathbf{B}$, not the Poynting vector.
- **Option c (Power flow per unit area)**: This is the correct definition of the magnitude and direction of the Poynting vector.
- **Option d (Energy density)**: Energy density is the energy stored per unit volume, given by $w = \frac{1}{2}(\epsilon E^2 + \mu H^2)$, which is a scalar quantity, whereas the Poynting vector is a vector representing flow.
What are universal gates? Implement AND and OR using the universal gates.
The unit of S-parameters is:
A)
Ohms
B)
Siemens
C)
Dimensionless
D)
Decibels
S-parameters, or Scattering parameters, are used in microwave engineering to describe the electrical behavior of linear electrical networks. They are defined as the ratio of the reflected or transmitted voltage wave amplitude to the incident voltage wave amplitude. Mathematically, an S-parameter is expressed as: S = (b / a), where 'a' is the amplitude of the incident wave and 'b' is the amplitude of the reflected or transmitted wave. Since it is a ratio of two similar quantities (voltage waves), the units cancel out, making S-parameters dimensionless.
Analysis of other options:
- a. Ohms: This is the unit of impedance (Z-parameters), not S-parameters.
- b. Siemens: This is the unit of admittance (Y-parameters), not S-parameters.
- d. Decibels: While S-parameters are often converted to decibels (dB) for easier representation of gain or loss (e.g., Return Loss), the S-parameter itself is a complex coefficient and is fundamentally dimensionless.
The complement of the expression (X + X')· Y is:
A)
Y'
B)
1
C)
0
D)
X
To find the complement of the expression (X + X') · Y, we follow these steps using Boolean Algebra laws:
1. **Simplify the inner expression**: According to the Complement Law in Boolean Algebra, (X + X') = 1. This is because if X is 0, X' is 1, and if X is 1, X' is 0; in either case, the OR operation results in 1.
2. **Substitute the simplified value**: The expression (X + X') · Y becomes (1) · Y.
3. **Apply the Identity Law**: According to the Identity Law, 1 · Y = Y.
4. **Find the complement**: The question asks for the complement of the expression. The complement of Y is denoted as Y'.
Therefore, the complement of (X + X') · Y is Y'.
Analysis of other options:
- Option b (1): This would be the complement if the original expression simplified to 0.
- Option c (0): This would be the complement if the original expression simplified to 1.
- Option d (X): This is unrelated to the simplified result of the expression.
In antenna theory, the ratio of maximum radiation intensity to average radiation intensity is called:
A)
Directivity
B)
Gain
C)
Efficiency
D)
Beamwidth
In antenna theory, Directivity (D) is defined as the ratio of the maximum radiation intensity (U_max) to the average radiation intensity (U_avg) over all directions. Mathematically, it is expressed as D = U_max / U_avg. Since the average radiation intensity is equal to the total radiated power (P_rad) divided by 4π, the formula is often written as D = 4πU_max / P_rad.
Why other options are incorrect:
- b. Gain: Gain is the product of directivity and the efficiency of the antenna (G = ηD). It accounts for the power losses within the antenna structure, whereas directivity is a purely geometric property of the radiation pattern.
- c. Efficiency: Efficiency (η) is the ratio of the power radiated by the antenna to the power delivered to the antenna. It is a scalar value, not a ratio of radiation intensities.
- d. Beamwidth: Beamwidth (typically Half Power Beam Width or HPBW) is the angular separation between the points where the radiation intensity is half of its maximum value. It is measured in degrees or radians, not as a ratio of intensities.
Explain the difference between waveguide modes TE, TM, and TEM with examples of where each occurs.
The radiation resistance of an antenna is 80Ω and loss-resistance is 10Ω. What is its directivity if the power gain is 20?
Find the reflection coefficient of the wave with SWR of 3.5.
A)
0.55
B)
0.23
C)
0.48
D)
0.68
To find the reflection coefficient (Γ) from the Standing Wave Ratio (SWR), we use the standard formula relating the two:
SWR = (1 + |Γ|) / (1 - |Γ|)
Given:
SWR = 3.5
Rearranging the formula to solve for |Γ|:
SWR * (1 - |Γ|) = 1 + |Γ|
SWR - SWR * |Γ| = 1 + |Γ|
SWR - 1 = |Γ| + SWR * |Γ|
SWR - 1 = |Γ| * (1 + SWR)
|Γ| = (SWR - 1) / (SWR + 1)
Substituting the given value:
|Γ| = (3.5 - 1) / (3.5 + 1)
|Γ| = 2.5 / 4.5
|Γ| = 25 / 45
|Γ| = 5 / 9
|Γ| ≈ 0.5555...
Rounding to two decimal places, we get 0.56. Looking at the provided options:
a. 0.55
b. 0.23
c. 0.48
d. 0.68
Option (a) 0.55 is the closest value to the calculated result of 0.555. In many competitive exams, slight rounding differences occur, but 0.55 is the only mathematically plausible choice among the options provided.
The S-matrix element S21 represents:
A)
Reflection coefficient at port 1
B)
Transmission from port 1 to port 2
C)
Transmission from port 2 to port 1
D)
Isolation between ports
In Microwave Engineering, the Scattering Matrix (S-matrix) is used to describe the electrical behavior of multi-port networks. The elements of the S-matrix are denoted as Sij, where 'i' represents the port where the wave is observed (output) and 'j' represents the port where the wave is incident (input).
1. S11: The wave is incident at port 1 and observed at port 1. This represents the reflection coefficient at port 1.
2. S21: The wave is incident at port 1 and observed at port 2. This represents the forward transmission coefficient from port 1 to port 2.
3. S12: The wave is incident at port 2 and observed at port 1. This represents the reverse transmission coefficient from port 2 to port 1.
4. S22: The wave is incident at port 2 and observed at port 2. This represents the reflection coefficient at port 2.
Therefore, S21 specifically describes the transmission from port 1 to port 2. Option (a) refers to S11, option (c) refers to S12, and option (d) refers to the general concept of isolation (which is achieved when S21 and S12 are very small).
An isolator allows microwave power to pass:
A)
In both directions
B)
In one direction only
C)
With attenuation in both directions
D)
With amplification in one direction
A microwave isolator is a non-reciprocal two-port device. Its primary function is to allow microwave power to flow in one direction (from the input port to the output port) with minimal attenuation, while absorbing or blocking power flowing in the opposite direction (from the output port back to the input port). This is typically achieved using ferrite materials and the principle of Faraday rotation.
- Option (a) is incorrect because a device that allows power in both directions is called a reciprocal device (like a cable or a standard attenuator).
- Option (c) is incorrect because while there is some insertion loss in the forward direction, the device is specifically designed to provide high attenuation in only one direction, not both.
- Option (d) is incorrect because an isolator is a passive device; it cannot provide amplification.
The excess-3 code for 597 is given by:
A)
100011001010
B)
100010100111
C)
010110010111
D)
010110101101
To convert a decimal number to Excess-3 code, follow these steps:
1. Separate the decimal number into its individual digits: 5, 9, and 7.
2. Add 3 to each individual digit:
- For 5: 5 + 3 = 8
- For 9: 9 + 3 = 12
- For 7: 7 + 3 = 10
3. Convert each of these resulting sums into their 4-bit binary equivalents:
- 8 in binary is 1000
- 12 in binary is 1100
- 10 in binary is 1010
4. Concatenate the binary groups together: 1000 1100 1010.
Comparing this result to the options:
- Option (a) 100011001010 matches our calculation.
- Option (b) 100010100111 is incorrect.
- Option (c) 010110010111 is incorrect.
- Option (d) 010110101101 is incorrect.
State and explain Gauss’s Law.
Determine the gain margin and phase crossover frequency of a system with open loop transfer function G(s) = 1 / [s(1+2s)(1+s)].
What is the characteristic of a good control system?
A)
Insensitive to the parameter variation but sensitive to the input commands
B)
Neither sensitive to parameter variations nor sensitive to input commands
C)
Insensitive to the input command
D)
Sensitive to parameter variation
In control system engineering, a 'good' or 'ideal' control system must possess two primary characteristics regarding sensitivity:
1. Sensitivity to Input Commands: The system must be highly responsive (sensitive) to the reference input or command signal. This ensures that the system tracks the desired set-point accurately and quickly.
2. Insensitivity to Parameter Variations: Internal parameters of a system (such as resistance, capacitance, or gain) can change over time due to aging, temperature fluctuations, or manufacturing tolerances. A robust control system should be insensitive to these changes so that the overall performance (stability and accuracy) remains constant despite internal variations.
Analyzing the options:
- Option (a) correctly identifies both requirements: high sensitivity to the input and low sensitivity to internal parameter changes.
- Option (b) is incorrect because a system that is not sensitive to input commands would fail to perform its intended task.
- Option (c) is incorrect for the same reason as (b).
- Option (d) is incorrect because sensitivity to parameter variation leads to instability and unpredictable behavior.
A Zener diode is operating in the breakdown region. The voltage across it will:
A)
Increase rapidly with current
B)
Decrease rapidly with current
C)
Remain almost constant with current
D)
Vary sinusoidally
A Zener diode is a specially designed PN junction diode that is intended to operate in the reverse breakdown region. When the reverse bias voltage reaches the Zener breakdown voltage (Vz), the diode enters the breakdown region. In this region, the I-V characteristic curve becomes nearly vertical. This means that even if the current flowing through the diode (Iz) increases significantly, the voltage across the diode (Vz) remains almost constant. This unique property is why Zener diodes are widely used as voltage regulators to provide a stable reference voltage despite fluctuations in load current or input voltage.
Analysis of other options:
- Option (a) is incorrect because a rapid increase in voltage with current would describe a standard resistor or a diode in the forward region, not a Zener in breakdown.
- Option (b) is incorrect as the voltage does not decrease when current increases in the breakdown region.
- Option (d) is incorrect because sinusoidal variation is a characteristic of an AC signal source, not the intrinsic V-I characteristic of a diode component.
What is meant by polarization of the wave? Explain the different types of wave polarization.
Find the force between 2 C and -1 C separated by a distance 1m in air (in newton).
A)
18 X 106
B)
18 X 109
C)
18 X 10-6
D)
18 X 108
To find the force between two point charges, we use Coulomb's Law: F = k * (|q1 * q2|) / r², where:
- k is the electrostatic constant (approximately 8.987 × 10⁹ N·m²/C², often rounded to 9 × 10⁹ N·m²/C² for competitive exams).
- q1 = 2 C
- q2 = -1 C
- r = 1 m
Step-by-step calculation:
1. Calculate the product of the magnitudes of the charges: |2 C * (-1 C)| = 2 C².
2. Square the distance: (1 m)² = 1 m².
3. Apply the formula: F = (9 × 10⁹ N·m²/C²) * (2 C²) / (1 m²).
4. F = 18 × 10⁹ N.
Since the charges have opposite signs, the force is attractive. The magnitude is 18 × 10⁹ Newtons. Comparing this to the options:
- Option (a) 18 X 10⁶ is incorrect (wrong power of 10).
- Option (b) 18 X 10⁹ matches our calculation.
- Option (c) 18 X 10⁻⁶ is incorrect (wrong power of 10).
- Option (d) 18 X 10⁸ is incorrect (wrong power of 10).
In a differential amplifier, the common-mode gain should be:
A)
High
B)
Low
C)
Equal to differential gain
D)
Zero
A differential amplifier is designed to amplify the difference between two input signals (differential mode) while rejecting signals that are common to both inputs (common-mode), such as noise or interference.
1. **Differential Gain (Ad):** This is the gain applied to the difference between the two inputs. For a useful amplifier, this should be high.
2. **Common-Mode Gain (Ac):** This is the gain applied to signals that appear identically on both inputs. In an ideal scenario, the amplifier should completely ignore these signals.
3. **Common-Mode Rejection Ratio (CMRR):** This is defined as the ratio of differential gain to common-mode gain (CMRR = Ad / Ac). To maximize the CMRR (which indicates a better quality amplifier), the common-mode gain (Ac) must be as small as possible.
Therefore, for optimal performance and noise rejection, the common-mode gain should be low. While 'Zero' (option d) is the ideal theoretical value, in practical engineering and circuit design, the goal is to keep it 'Low'. In the context of standard multiple-choice questions for electronics, 'Low' is the correct descriptive requirement for a functional differential amplifier.
What should be the nature of bandwidth for a good control system?
A)
Small
B)
Medium
C)
Large
D)
All of the mentioned
In control systems, bandwidth is defined as the range of frequencies over which the system tracks the input signal with acceptable accuracy (typically where the magnitude is within -3dB of the DC gain).
1. **Speed of Response**: There is a direct relationship between bandwidth and the rise time (speed) of a system. A larger bandwidth implies a faster response and a shorter rise time, allowing the system to respond quickly to changes in the input.
2. **Tracking Ability**: A system with a larger bandwidth can track higher-frequency input signals without significant attenuation or phase lag.
3. **Trade-off (Noise)**: While a very large bandwidth can make a system more susceptible to high-frequency noise, in the context of general 'performance' and 'speed' for a good control system, a larger bandwidth is desired to ensure agility and fast transient response.
Therefore, for a system to be considered 'good' in terms of responsiveness and tracking capability, a large bandwidth is preferred.
Calculate the skin depth of a conductor, having a conductivity of 200 units. The wave frequency is 10 GHz in air.
A)
355.8
B)
3.558
C)
35.58
D)
0.3558
To calculate the skin depth (δ) of a conductor, we use the standard formula:
δ = 1 / sqrt(π * f * μ * σ)
Given data:
1. Conductivity (σ) = 200 S/m (assuming units are Siemens per meter as per standard EM problems).
2. Frequency (f) = 10 GHz = 10 * 10^9 Hz.
3. Permeability (μ): Since the material is not specified as magnetic, we use the permeability of free space, μ₀ = 4π * 10^-7 H/m.
Step-by-step calculation:
1. Calculate the product inside the square root:
π * f * μ * σ = π * (10 * 10^9) * (4π * 10^-7) * 200
= π * 10^10 * 4π * 10^-7 * 200
= 4 * π^2 * 10^3 * 200
= 800 * π^2 * 1000
= 800,000 * π^2
≈ 800,000 * 9.8696
≈ 7,895,683.5
2. Calculate the square root:
sqrt(7,895,683.5) ≈ 2809.9
3. Calculate the skin depth (δ):
δ = 1 / 2809.9 ≈ 0.0003558 meters
4. Convert to millimeters (common unit for skin depth in such options):
δ = 0.0003558 * 1000 = 0.3558 mm
Comparing this result with the given options, the numerical value matches option (d).
Given that CMRR is 100 dB. Input common-mode voltage is 12 V. Differential voltage gain is 4000. Calculate output common-mode voltage.
A)
48V
B)
0.48V
C)
20V
D)
11V
Step-by-step calculation:
1. **Identify Given Values**:
- Common Mode Rejection Ratio (CMRR) = 100 dB
- Input Common-mode Voltage ($V_{cm}$) = 12 V
- Differential Voltage Gain ($A_d$) = 4000
2. **Convert CMRR from dB to linear scale**:
The formula for CMRR in decibels is: $\text{CMRR(dB)} = 20 \log_{10}(\text{CMRR}_{linear})$
$100 = 20 \log_{10}(\text{CMRR}_{linear})$
$5 = \log_{10}(\text{CMRR}_{linear})$
$\text{CMRR}_{linear} = 10^5 = 100,000$
3. **Calculate Common-mode Gain ($A_{cm}$)**:
The definition of CMRR is the ratio of differential gain to common-mode gain:
$\text{CMRR} = \frac{A_d}{A_{cm}}$
$A_{cm} = \frac{A_d}{\text{CMRR}_{linear}} = \frac{4000}{100,000} = 0.04$
4. **Calculate Output Common-mode Voltage ($V_{out(cm)}$)**:
The output common-mode voltage is the product of the common-mode gain and the input common-mode voltage:
$V_{out(cm)} = A_{cm} \times V_{cm}$
$V_{out(cm)} = 0.04 \times 12\text{ V} = 0.48\text{ V}$
Conclusion: The output common-mode voltage is 0.48V, which corresponds to option (b).
Looping on a K-map always results in the elimination of:
A)
Variables within the loop that appear only in their complemented form
B)
Variables that remain unchanged within the loop
C)
Variables within the loop that appear in both complemented and uncomplemented form
D)
Variables within the loop that appear only in their uncomplemented form
In a Karnaugh Map (K-map), the goal of 'looping' (grouping adjacent 1s for Sum-of-Products or 0s for Product-of-Sums) is to simplify a Boolean expression using the Adjacency Property. The fundamental Boolean identity used here is X + X' = 1 (or X · X' = 0). When we group cells in a K-map, we are looking for variables that change their state (from 0 to 1 or vice versa) across the grouped cells. If a variable appears as both complemented (0) and uncomplemented (1) within a single loop, it means that the output of the function is independent of that variable for that specific group. Therefore, that variable is eliminated from the resulting simplified term. Conversely, variables that remain constant (either always 0 or always 1) throughout the loop are retained in the final simplified expression.
- Option (a) is incorrect because if a variable is only complemented, it is constant and thus retained.
- Option (b) is incorrect because variables that remain unchanged are the ones that are kept in the expression.
- Option (d) is incorrect because if a variable is only uncomplemented, it is constant and thus retained.
The gain margin of a system is defined as the factor by which gain can be multiplied before the system becomes:
A)
Stable
B)
Marginally stable
C)
Unstable
D)
Oscillatory
Gain Margin (GM) is a key stability metric in control systems, typically analyzed using the Bode plot or Nyquist plot. It is defined as the amount of additional gain that can be added to the system before it reaches the threshold of instability.
1. **Stability Threshold**: A system is considered 'marginally stable' when its closed-loop poles lie exactly on the imaginary axis of the s-plane. In the frequency domain, this occurs when the open-loop gain is exactly 1 (0 dB) at the phase crossover frequency (where the phase angle is -180°).
2. **Definition of GM**: The Gain Margin is the reciprocal of the magnitude of the open-loop transfer function at the phase crossover frequency. Mathematically, if the magnitude at -180° is |G(jωpc)|, then GM = 1 / |G(jωpc)|.
3. **Interpretation**: If you multiply the current gain by the Gain Margin, the magnitude at the phase crossover frequency becomes exactly 1. At this point, the system is no longer strictly stable but has not yet become fully unstable; it is 'marginally stable' (often characterized by sustained oscillations).
Why other options are incorrect:
- (a) Stable: The system is already stable if it has a positive gain margin; the GM describes how much *more* gain it can take before losing that stability.
- (c) Unstable: While the system becomes unstable if the gain is increased *beyond* the gain margin, the specific point defined by the gain margin factor is the boundary, which is marginal stability.
- (d) Oscillatory: While marginal stability results in oscillations, 'marginally stable' is the formal technical term used in control theory to define the stability boundary.
Routh Hurwitz criterion gives:
A)
Number of roots in the right half of the s-plane
B)
Value of the roots
C)
Number of roots in the left half of the s-plane
D)
Number of roots in the top half of the s-plane
The Routh-Hurwitz stability criterion is a mathematical method used in control systems to determine the stability of a linear time-invariant (LTI) system by examining the characteristic equation.
1. **Purpose**: The primary goal of the Routh-Hurwitz criterion is to determine if any roots of the characteristic equation lie in the Right Half of the s-plane (RHP). If any root exists in the RHP, the system is unstable.
2. **Mechanism**: By constructing a Routh array from the coefficients of the characteristic polynomial, the criterion states that the number of sign changes in the first column of the Routh array is equal to the number of roots of the characteristic equation located in the right half of the s-plane.
3. **Analysis of Options**:
- **Option (a)**: Correct. The number of sign changes in the first column directly indicates the number of roots in the RHP.
- **Option (b)**: Incorrect. Routh-Hurwitz tells us *how many* roots are in the RHP, but it does not provide the *exact numerical values* of those roots. Finding the values would require solving the polynomial (e.g., using root-finding algorithms).
- **Option (c)**: Incorrect. While you can infer the number of roots in the left half by subtracting the RHP roots from the total degree of the polynomial, the criterion's direct output is the count of RHP roots.
- **Option (d)**: Incorrect. The s-plane is divided into left and right halves for stability analysis; the 'top half' (imaginary axis) does not determine stability in the context of the Routh-Hurwitz criterion.