Past Questions

In network theory, a two-port network is said to be reciprocal if the ratio of the response at one port to the excitation at the other port remains the same when the positions of the excitation and response are interchanged. For different parameter sets, the condition for reciprocity is as follows: 1. Z-parameters: z12 = z21 2. Y-parameters: y12 = y21 3. ABCD-parameters: AD - BC = 1 4. h-parameters: The h-parameter equations are: V1 = h11*I1 + h12*V2 I2 = h21*I1 + h22*V2 For a reciprocal network, the condition is h12 = -h21. This is because the h-parameters are derived from the Z-parameters. Specifically, h12 = z12 / z22 and h21 = 1 / z21 (or more accurately, h21 = z21 / z22 in some contexts, but the standard definition leads to the sign difference due to the direction of current I2 in the h-parameter definition). In the standard h-parameter definition, I2 is defined as flowing out of the port. For reciprocity (where z12 = z21), the relationship between the reverse voltage gain (h12) and the forward current gain (h21) results in h12 = -h21. Comparing this to the options: - Option (a) h12 = h21 is incorrect. - Option (b) h12 = -h21 is the correct condition for reciprocity. - Option (c) is a variation of the ABCD reciprocity condition (AD-BC=1) but applied to h-parameters, which is incorrect. - Option (d) h11 = h22 is the condition for symmetry, not reciprocity.

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.

In solid-state physics, materials are classified based on the size of the forbidden energy gap (Eg) between the valence band (filled with electrons) and the conduction band (empty at 0K). 1. **Semiconductors**: These materials have a small, finite energy gap. Typically, this gap is on the order of 1 electron-volt (eV). For example, Silicon (Si) has a bandgap of approximately 1.1 eV and Germanium (Ge) has approximately 0.67 eV at room temperature. Therefore, '1 eV' is the correct order of magnitude. 2. **Insulators**: These materials have a very large energy gap, typically greater than 3-5 eV (e.g., Diamond is ~5.5 eV), making it very difficult for electrons to jump to the conduction band. 3. **Conductors (Metals)**: In metals, the valence and conduction bands overlap, meaning the energy gap is zero. Evaluating the options: - Option (a) 1 MeV (Mega electron-volt) is $10^6$ eV, which is the energy scale of nuclear physics, not semiconductor physics. - Option (b) 0.1 MeV is also $10^5$ eV, which is far too high. - Option (c) 1 eV is the characteristic energy gap for semiconductors. - Option (d) 5 eV is characteristic of an insulator.

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 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.

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.

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.

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.

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.

A continuous-time sinusoidal signal is typically represented as x(t) = A sin(ωt + φ) or x(t) = A cos(ωt + φ), where 'ω' is the angular frequency in radians per second. 1. Definition of Periodicity: A signal is periodic if x(t) = x(t + T) for all t, where T is the fundamental period. 2. For a sinusoid: sin(ω(t + T)) = sin(ωt + ωT). 3. For the signal to repeat itself, the phase shift ωT must be an integer multiple of 2π (the period of a basic sine/cosine function). 4. For the fundamental period (the smallest positive value of T), we set ωT = 2π. 5. Solving for T gives: T = 2π / ω. Comparing this to the options: - Option (a) T = 2π / ω is the correct mathematical derivation. - Option (b) T = 2π / 3ω is incorrect as it represents a signal with three times the frequency. - Option (c) T = π / ω is incorrect; this would be the period of a squared sinusoid or a signal with double the frequency. - Option (d) T = π / 2ω is incorrect.

A shift register is a sequential logic circuit used for the storage and transfer of data. The classification of shift registers is based on how data enters (input) and exits (output) the register: 1. **SIPO (Serial-In, Parallel-Out):** Data is fed into the register one bit at a time (serially) through a single input line. Once the register is filled, all stored bits are available simultaneously at the output pins (parallelly). Therefore, it converts serial data to parallel data. 2. **SISO (Serial-In, Serial-Out):** Data enters serially and exits serially. It acts as a delay line and does not perform data conversion. 3. **PISO (Parallel-In, Serial-Out):** Data is loaded into the register all at once (parallelly) and then shifted out one bit at a time (serially). This converts parallel data to serial data. 4. **PIPO (Parallel-In, Parallel-Out):** Data is loaded parallelly and retrieved parallelly. It is used primarily for temporary data storage (buffer) and does not perform conversion.

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)

Euler's formula is a fundamental identity in complex analysis and signal processing. It relates the complex exponential function to trigonometric functions. The general formula is given by: e^(ix) = cos(x) + i sin(x). In the context of electrical engineering and signals and systems, the imaginary unit 'i' is typically denoted as 'j' to avoid confusion with current (i). Therefore, substituting θ for x and j for i, we get: ejθ = cos θ + j sin θ. Evaluating the options: - Option (a) matches the identity exactly. - Option (b) represents ej(-θ) or the complex conjugate of ejθ. - Option (c) and (d) are incorrect because they introduce a coefficient of 2 to the angle in either the cosine or sine term, which violates the identity.

To understand diffusion current in a p-n junction, we must look at the movement of majority carriers due to concentration gradients. 1. **Diffusion Process**: Diffusion occurs when particles move from a region of higher concentration to a region of lower concentration. 2. **Hole Diffusion**: In a p-n junction, holes are the majority carriers in the p-side and minority carriers in the n-side. Therefore, holes naturally diffuse from the p-side to the n-side. 3. **Electron Diffusion**: Electrons are the majority carriers in the n-side and minority carriers in the p-side. Therefore, electrons naturally diffuse from the n-side to the p-side. 4. **Direction of Current**: By convention, the direction of electric current is the same as the direction of positive charge flow (holes) and opposite to the direction of negative charge flow (electrons). - Hole flow (p to n) $\rightarrow$ Current flows from p to n. - Electron flow (n to p) $\rightarrow$ Current flows from p to n. Since both the diffusion of holes and the diffusion of electrons contribute to a current flowing from the p-side to the n-side, the net diffusion current is from the p-side to the n-side. This diffusion current is what creates the depletion region and the built-in potential in an unbiased junction. Options c and d are incorrect because diffusion is a property of the concentration gradient, whereas biasing primarily affects the drift current and the width of the depletion region.

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.