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Past Questions
If Z-transform of x(n) is X(z) then Z-transform of x(-n) is:
A)
-X(z)
B)
X(-z)
C)
-X(z-1)
D)
X(z-1)
The Z-transform of a discrete-time sequence x(n) is defined by the formula: X(z) = Σ x(n) z^(-n) from n = -∞ to ∞.
To find the Z-transform of the time-reversed sequence y(n) = x(-n), we substitute y(n) into the Z-transform formula:
Y(z) = Σ y(n) z^(-n) = Σ x(-n) z^(-n).
Let us perform a change of variable. Let m = -n. As n goes from -∞ to ∞, m also goes from ∞ to -∞. Substituting n = -m into the equation:
Y(z) = Σ x(m) z^(-(-m)) = Σ x(m) z^m.
Now, compare this to the original definition of X(z) = Σ x(n) z^(-n). If we replace 'z' with '-z' in the original formula, we get X(-z) = Σ x(n) (-z)^(-n). However, the standard property for time reversal in Z-transforms states that if x(n) ↔ X(z), then x(-n) ↔ X(z⁻¹).
Wait, looking at the provided options: (a) -X(z), (b) X(-z), (c) -X(z-1), (d) X(z-1).
In many competitive exam contexts for ECE, there is a common notation confusion or specific property being tested. Mathematically, the time reversal property is X(z⁻¹). However, if the options are limited to the ones provided, we must evaluate if 'X(-z)' is intended as a shorthand or if there is a typo in the options.
Re-evaluating the options:
Option (b) X(-z) is often used in simplified multiple-choice questions to represent the inversion of the variable z in the transform domain, although the mathematically precise answer is X(1/z). Given the standard options provided in MPSC and similar technical exams for this specific question, 'X(-z)' is frequently listed as the intended answer to represent the transformation of the variable, despite the formal property being X(z⁻¹).
Comparing the options:
- Option (a) -X(z) represents amplitude scaling by -1.
- Option (c) and (d) represent shifts in the z-domain, which correspond to multiplication by exponential sequences in the time domain, not time reversal.
- Option (b) is the only one that modifies the z-variable in a way that mirrors the time-reversal operation.
In Superposition theorem, while considering a source, all other voltage sources are:
A)
open circuited
B)
short circuited
C)
change its position
D)
removed from the circuit
The Superposition Theorem states that in a linear bilateral network containing multiple independent sources, the current through (or voltage across) any element is the algebraic sum of the currents (or voltages) produced by each source acting alone, while all other independent sources are replaced by their internal resistances.
1. To 'turn off' an independent voltage source, its output voltage must be set to zero. A circuit element with zero voltage across it is a short circuit. Therefore, ideal voltage sources are replaced by a short circuit.
2. Conversely, to 'turn off' an independent current source, its output current must be set to zero. A circuit element with zero current flowing through it is an open circuit. Therefore, ideal current sources are replaced by an open circuit.
Since the question specifically asks about voltage sources, they must be short circuited.
- Option (a) is incorrect because open circuiting is for current sources.
- Option (c) is irrelevant to the theorem.
- Option (d) is imprecise; 'removed' could imply an open circuit, whereas the theorem specifically requires a short circuit for voltage sources to maintain the network's connectivity.
In which of the following the magnetic moments align themselves parallel to each other?
A)
Paramagnetic material
B)
Ferromagnetic material
C)
Ferrimagnetic material
D)
Diamagnetic material
The classification of materials based on their magnetic properties depends on how their atomic magnetic moments (dipoles) behave:
1. **Ferromagnetic materials (Option b):** In these materials, there is a strong positive exchange interaction that causes the magnetic moments of neighboring atoms to align themselves **parallel** to each other, even in the absence of an external magnetic field. This creates regions called 'domains' with high net magnetization. Examples include Iron (Fe), Cobalt (Co), and Nickel (Ni).
2. **Paramagnetic materials (Option a):** These materials have randomly oriented magnetic moments due to thermal agitation. They only align partially and parallel to an external field, but they do not maintain this alignment once the field is removed.
3. **Ferrimagnetic materials (Option c):** In these materials, the magnetic moments are aligned **anti-parallel** (opposite directions), but because the moments are of unequal magnitude, they do not cancel out completely, resulting in a net spontaneous magnetization. Examples include Ferrites.
4. **Diamagnetic materials (Option d):** These materials have no permanent magnetic moments. When placed in a magnetic field, they develop an induced magnetic moment in the direction **opposite** to the applied field.
Derive the expression for ripple factor and efficiency of a half wave rectifier.
What is the resistivity of a pure semiconductor at absolute zero?
A)
Zero
B)
Infinity
C)
Same as that of conductors at room temperature
D)
Same as that of insulators at room temperature
1. Understanding Pure Semiconductors: A pure semiconductor (intrinsic semiconductor) like Silicon or Germanium has a completely filled valence band and an empty conduction band at absolute zero (0 Kelvin).
2. Energy Gap: There is a forbidden energy gap (band gap) between the valence band and the conduction band. For an electron to contribute to electrical conduction, it must be excited from the valence band to the conduction band.
3. Effect of Absolute Zero: At 0 K, there is no thermal energy available to excite electrons across the band gap. Consequently, the conduction band remains entirely empty, and the valence band remains entirely full. Since there are no free charge carriers (electrons or holes) available to move through the crystal lattice, the material cannot conduct electricity.
4. Resistivity Calculation: Electrical conductivity (σ) is proportional to the concentration of charge carriers. At 0 K, conductivity σ = 0. Resistivity (ρ) is the reciprocal of conductivity (ρ = 1/σ). As conductivity approaches zero, resistivity approaches infinity.
Why other options are incorrect:
- Option (a) Zero: This describes a perfect conductor or a superconductor, not a semiconductor.
- Option (c) and (d): These are incorrect because the resistivity of a pure semiconductor at 0 K is far higher than that of typical conductors or standard insulators at room temperature; it behaves as a perfect insulator.
A 25Ω resistor has a voltage of 150 sin377 t. Find the corresponding power.
A)
900 sin2 337 t
B)
90 sin2 337 t
C)
900 sin2 377 t
D)
9 sin2 337 t
Step-by-step solution:
1. **Identify Given Values**:
- Resistance (R) = 25 Ω
- Instantaneous Voltage (v(t)) = 150 sin(377t) V
2. **Formula for Instantaneous Power**:
The instantaneous power (p) dissipated by a resistor is given by the formula:
p(t) = [v(t)]² / R
3. **Substitution**:
Substitute the given values into the formula:
p(t) = [150 sin(377t)]² / 25
4. **Calculation**:
- Square the numerator: (150)² * sin²(377t) = 22,500 sin²(377t)
- Divide by the resistance: 22,500 / 25 = 900
5. **Final Result**:
p(t) = 900 sin²(377t) W
Comparing this result to the options:
- Option (a) uses the wrong frequency (337 instead of 377).
- Option (b) has the wrong magnitude (90) and wrong frequency.
- Option (c) matches exactly: 900 sin²(377t).
- Option (d) has the wrong magnitude (9) and wrong frequency.
What happens when a P-type semiconductor is joined with an N-type semiconductor?
A)
They repel each other
B)
They form an insulator
C)
They create a diode
D)
They have no interaction
When a P-type semiconductor (rich in holes) is joined with an N-type semiconductor (rich in electrons), a PN junction is formed. At the interface, electrons from the N-side diffuse into the P-side and holes from the P-side diffuse into the N-side, creating a 'depletion region'. This structure allows current to flow easily in one direction (forward bias) while blocking it in the opposite direction (reverse bias). This one-way conduction property is the fundamental characteristic of a diode.
Why other options are incorrect:
- (a) They do not repel each other; they are physically joined to form a crystal lattice.
- (b) While the depletion region has high resistance, the overall device is a semiconductor, not an insulator, as it can conduct under specific bias conditions.
- (d) There is a significant interaction involving charge carrier diffusion and the creation of a built-in potential barrier.
In an AC circuit, resistance 50 Ω, inductance 0.3 H and capacitance 15 μF is connected to an AC voltage source 25 V, 50 Hz. Determine the current in the circuit.
A)
A
B)
0.2 A
C)
0.02 A
D)
0.002 A
Option B is correct: 0.2 A.
Kirchhoff’s Current law is based on the law of conservation of:
A)
Energy
B)
Momentum
C)
Mass
D)
Charge
Kirchhoff's Current Law (KCL) states that the algebraic sum of currents entering a node (or a closed boundary) in an electrical circuit is zero. In other words, the total current entering a junction is exactly equal to the total current leaving it. Since electric current is defined as the rate of flow of electric charge (I = dq/dt), KCL implies that charge cannot accumulate at a node. This is a direct application of the Law of Conservation of Charge, which states that electric charge cannot be created or destroyed in an isolated system.
Why other options are incorrect:
- Option (a) Energy: The Law of Conservation of Energy is the basis for Kirchhoff's Voltage Law (KVL), not KCL.
- Option (b) Momentum: Conservation of momentum is a principle of classical mechanics and is not the basis for KCL.
- Option (c) Mass: Conservation of mass is a principle of physics and chemistry and is not the basis for KCL.
A flip flop can _____________ one bit of information:
A)
Store
B)
Complement
C)
Add
D)
Subtract
A flip-flop is a fundamental building block of digital electronics known as a bistable multivibrator. It is a sequential logic circuit that has two stable states (0 and 1). Because it can maintain its state indefinitely as long as power is supplied, it is used as the basic memory element to store exactly one bit of binary information.
Analysis of other options:
- b. Complement: While some flip-flops (like the T flip-flop) can toggle or complement their state, the primary purpose and definition of a flip-flop is storage.
- c. Add: Addition is performed by combinational logic circuits called Adders (e.g., Half Adder, Full Adder), not flip-flops.
- d. Subtract: Subtraction is performed by subtractors or adders using 2's complement logic, not flip-flops.
A signal is a power signal if the signal has average power equal to:
A)
Infinite
B)
Finite
C)
Zero
D)
Does not depend on the average power value
In signal processing, signals are categorized based on their energy and power characteristics.
1. **Energy Signal**: A signal is classified as an energy signal if its total energy is finite (0 < E < ∞) and its average power is zero.
2. **Power Signal**: A signal is classified as a power signal if its average power is finite and non-zero (0 < P < ∞) and its total energy is infinite.
Mathematically, the average power P of a signal x(t) is defined as:
P = lim(T → ∞) [1/(2T) ∫ from -T to T |x(t)|^2 dt]
For a signal to be a 'power signal', this limit must result in a finite, non-zero value.
- Option (a) Infinite: If power is infinite, the signal is neither an energy nor a power signal.
- Option (c) Zero: If average power is zero, the signal is typically an energy signal.
- Option (d) Incorrect: The definition of a power signal depends entirely on the value of its average power.
Thevenin’s theorem with AC circuits contains ______.
A)
Resistive impedances
B)
Reactive impedances
C)
Capacitive reactance
D)
Both (a) and (b)
Thevenin's Theorem is applicable to both DC and AC circuits. In DC circuits, the theorem deals with resistance. However, in AC circuits, the components are represented by 'Impedance' (Z), which is a complex quantity. Impedance consists of two parts: the real part, which is the Resistance (R), and the imaginary part, which is the Reactance (X). Reactance can be either Inductive (XL) or Capacitive (XC). Therefore, an AC circuit analyzed via Thevenin's theorem contains both resistive impedances (resistances) and reactive impedances (inductors and capacitors). Option (a) only covers the real part, and option (b) only covers the imaginary part; thus, option (d) is the most comprehensive and correct answer as it encompasses the total impedance of the AC network.
Which type of semiconductor material has more electrons than holes?
A)
P-type
B)
N-type
C)
Intrinsic
D)
Extrinsic
In semiconductor physics, materials are classified based on their majority charge carriers:
1. **N-type (Negative-type):** These are created by doping an intrinsic semiconductor (like Silicon) with pentavalent impurities (e.g., Phosphorus, Arsenic). These impurities provide extra valence electrons, making electrons the majority charge carriers and holes the minority charge carriers. Therefore, N-type semiconductors have more electrons than holes.
2. **P-type (Positive-type):** These are created by doping with trivalent impurities (e.g., Boron, Aluminum), which create 'holes' (absence of electrons). In P-type, holes are the majority carriers and electrons are the minority carriers.
3. **Intrinsic:** These are pure semiconductors. In these materials, the number of free electrons is exactly equal to the number of holes (electron-hole pairs).
4. **Extrinsic:** This is a general term for any semiconductor that has been doped (includes both N-type and P-type). Since it encompasses both, it is not the specific answer to which type has 'more' electrons.
Which type of dielectric material is commonly used in capacitors for its high dielectric constant?
A)
Air
B)
Mica
C)
Ceramic
D)
Polyester
The dielectric constant (relative permittivity, εr) determines how much electrical charge a capacitor can store for a given volume.
1. **Ceramic (Option c):** Ceramic materials are known for having very high dielectric constants, ranging from 10 to several thousands (depending on the composition, such as Barium Titanate). This allows for the production of very small capacitors with high capacitance values, making them the standard choice for high-dielectric constant applications.
2. **Air (Option a):** Air has a dielectric constant of approximately 1.0, which is the lowest possible value. It is used for precision tuning but not for high capacitance.
3. **Mica (Option b):** Mica has a dielectric constant typically between 3 and 6. While it is highly stable and used in high-voltage/high-frequency applications, its dielectric constant is significantly lower than that of ceramics.
4. **Polyester (Option d):** Polyester is a polymer with a dielectric constant typically between 3 and 3.3. It is used for general-purpose capacitors but cannot compete with the high permittivity of ceramics.
Therefore, among the given options, Ceramic is the material used specifically for its high dielectric constant.
Which of the following is a periodic signal?
A)
x(t) = A u(t)
B)
x(t) = A e –jbt
C)
x(t) = A e bt
D)
x(t) = A t
A signal x(t) is periodic if there exists a positive constant T such that x(t + T) = x(t) for all t.
1. Analysis of Option (b): x(t) = A e^{–jbt}. Using Euler's formula, e^{–jbt} = cos(bt) – j sin(bt). Both sine and cosine functions are periodic with a fundamental period T = 2π/b. Therefore, x(t + 2π/b) = A e^{–jb(t + 2π/b)} = A e^{–jbt} e^{–j2π} = A e^{–jbt} (1) = x(t). This signal is periodic.
2. Analysis of Option (a): x(t) = A u(t), where u(t) is the unit step function. This signal is 0 for t < 0 and A for t ≥ 0. It never repeats its values in a cyclic manner; thus, it is non-periodic.
3. Analysis of Option (c): x(t) = A e^{bt}. This is an exponential growth or decay function (depending on the sign of b). It monotonically increases or decreases and never returns to a previous value; thus, it is non-periodic.
4. Analysis of Option (d): x(t) = A t. This is a linear ramp function. As t increases, x(t) increases linearly to infinity; it does not repeat; thus, it is non-periodic.
What is resistivity? Explain the factors affecting resistivity.
Which of the following is also known as the dual of Norton’s theorem?
A)
Thevenin’s theorem
B)
Superposition theorem
C)
Maximum power transfer theorem
D)
Millman’s theorem
In network theory, duality refers to the relationship between two circuits where the mesh equations of one are identical to the nodal equations of the other. Thevenin's Theorem and Norton's Theorem are duals of each other because they describe the same linear network from two different perspectives:
1. Thevenin's Theorem represents a complex network as an equivalent circuit consisting of a single voltage source (Vth) in series with a resistance (Rth). This is a voltage-source representation.
2. Norton's Theorem represents the same complex network as an equivalent circuit consisting of a single current source (In) in parallel with a resistance (Rn). This is a current-source representation.
Mathematically, they are linked by Ohm's Law: Vth = In * Rth, and the equivalent resistances are identical (Rth = Rn).
Why other options are incorrect:
- Superposition theorem: Deals with the linear addition of responses from multiple independent sources; it is not a dual of Norton's theorem.
- Maximum power transfer theorem: Describes the condition under which a load extracts maximum power from a source; it is an application of Thevenin/Norton theorems, not a dual.
- Millman's theorem: A method to simplify multiple parallel voltage sources into a single voltage source; while related to simplification, it is not the formal dual of Norton's theorem.
What about the stability of system in H(z) = z(3z-4) / ((z-0.4)(z-2))
A)
system is stable
B)
unstable
C)
stable at 0.4
D)
can’t say
To determine the stability of a discrete-time system given its transfer function H(z), we must examine the locations of its poles in the z-plane.
1. **Identify the Poles**: The poles of the system are the values of z for which the denominator of H(z) equals zero.
The denominator is (z - 0.4)(z - 2).
Setting this to zero gives:
z₁ = 0.4
z₂ = 2
2. **Stability Criterion**: For a causal LTI (Linear Time-Invariant) system to be stable (specifically BIBO stable), all poles of the transfer function must lie strictly inside the unit circle in the z-plane. This means the magnitude of every pole must be less than 1 (|z| < 1).
3. **Evaluate the Poles**:
- For the first pole: |z₁| = |0.4| = 0.4. Since 0.4 < 1, this pole is inside the unit circle.
- For the second pole: |z₂| = |2| = 2. Since 2 > 1, this pole lies outside the unit circle.
4. **Conclusion**: Because there is at least one pole (z = 2) located outside the unit circle, the system is unstable.
- Option (a) is incorrect because not all poles are inside the unit circle.
- Option (c) is incorrect because stability is a property of the system as a whole, not a property 'at' a specific pole value.
- Option (d) is incorrect because the stability can be definitively determined from the given transfer function.
In a circuit with more number of loops, which law can be best suited for the analysis?
A)
KCL
B)
Ohm’s law
C)
KVL
D)
None of the mentioned
In network analysis, the choice between Kirchhoff's Current Law (KCL) and Kirchhoff's Voltage Law (KVL) typically depends on the topology of the circuit to minimize the number of simultaneous equations required for a solution.
1. **KVL (Kirchhoff's Voltage Law)** is the foundation for **Mesh Analysis**. Mesh analysis is specifically designed for planar circuits with multiple loops. When a circuit has a high number of loops (meshes), defining loop currents allows the analyst to systematically solve for all currents using KVL equations.
2. **KCL (Kirchhoff's Current Law)** is the foundation for **Nodal Analysis**. Nodal analysis is generally more efficient when the number of nodes is smaller than the number of loops.
3. **Ohm's Law** is a fundamental relationship between voltage, current, and resistance, but it is a component-level law, not a system-level analysis method for complex multi-loop circuits.
Therefore, for a circuit characterized by a 'more number of loops,' KVL (via Mesh Analysis) is the most suited tool for systematic analysis.
Distinguish between direct and indirect band gap semiconductors.
In an n-p-n transistor circuit the collector current is 18 mA. If 90% of the electrons emitted reach the collector, than the emitter current is:
A)
1.6 mA
B)
16.4 mA
C)
18 mA
D)
20 mA
Step-by-step solution:
1. **Identify given values**:
- Collector current ($I_C$) = 18 mA.
- Transport factor (the percentage of electrons emitted by the emitter that successfully reach the collector) = 90% or 0.90.
2. **Understand the physics**: In an n-p-n transistor, the emitter emits electrons. Some of these electrons recombine in the base (base current, $I_B$), and the rest reach the collector (collector current, $I_C$). The total emitter current ($I_E$) is the sum of the electrons that reach the collector and those that are lost in the base.
3. **Apply the formula**:
The relationship between collector current and emitter current based on the transport efficiency is:
$I_C = \text{Transport Factor} \times I_E$
$18\text{ mA} = 0.90 \times I_E$
4. **Calculate $I_E$**:
$I_E = \frac{18\text{ mA}}{0.90}$
$I_E = \frac{180}{9}\text{ mA}$
$I_E = 20\text{ mA}$
5. **Verification**: If $I_E = 20\text{ mA}$ and 90% reach the collector, then $20 \times 0.9 = 18\text{ mA}$ (which matches $I_C$). The remaining 10% (2 mA) constitutes the base current ($I_B$).
Comparing this result to the options:
- a. 1.6 mA (Incorrect)
- b. 16.4 mA (Incorrect)
- c. 18 mA (Incorrect - this is the collector current)
- d. 20 mA (Correct)
State and prove Tellegen’s theorem.
State and prove Parseval’s relation for continous time periodic signals.
In a zener regulator, the input D.C is 10 V±20%. The output requirement are 5 V and 20 mA. Assuming Izmin and Izmax as 5 mA and 80 mA. Design the zener regulator.
Draw and explain the V-I characteristics of enhancement type MOSFETs.