Question 1

System bandwidth and tracking: For a control system to follow arbitrary input signals accurately, what should be the characteristic of its bandwidth?

Accepted Answer

Introduction / Context:Bandwidth in control and communication systems describes the frequency range over which the system can respond effectively. A wider bandwidth indicates faster system response and better tracking of rapid input changes.Given Data / Assumptions:System is linear time-invariant.Bandwidth is defined as the frequency at which the gain falls by –3 dB from its low-frequency value.Concept / Approach:To follow arbitrary inputs, particularly those containing higher frequency components, the system must have a sufficiently wide bandwidth so that these components are not attenuated.Step-by-Step Solution:Consider a sinusoidal input with frequency ω.If ω is within system bandwidth, output closely follows input.If ω is outside bandwidth, output is distorted or attenuated.Therefore, large bandwidth ensures accurate following of arbitrary signals.Verification / Alternative check:Nyquist and frequency response analysis confirm: system tracking ability correlates with bandwidth. Higher bandwidth → better tracking, at the cost of possible noise sensitivity.Why Other Options Are Wrong:Small or very small bandwidth filters out higher-frequency components, making the system sluggish.Neither small nor large is meaningless as bandwidth must be adequate.Zero bandwidth would make the system static with no response.Common Pitfalls:Assuming infinite bandwidth is always good; in practice, noise rejection and physical limitations must be balanced.Final Answer:Large.

Question 2

In an AC circuit where the input frequency is such that the resistance R is equal to the capacitive reactance XC, what is the phase relationship between the output voltage v0 and the input voltage vi?

Accepted Answer

Introduction / Context:Phase relationships in RC circuits are a fundamental concept in AC circuit analysis. When the resistance R and the capacitive reactance XC are equal, the circuit reaches a specific balance point that defines the exact phase shift between input and output voltages.Given Data / Assumptions:Series RC circuit with input sinusoidal source.At the operating frequency, R = XC.v0 refers to the voltage across the capacitor or across the resistor depending on the circuit definition. Standard analysis assumes v0 across the capacitor.Concept / Approach:Impedance of resistor: Z_R = R∠0°. Impedance of capacitor: Z_C = −jXC. When R = XC, the total impedance becomes Z_total = R − jR. The phasor angle θ = tan⁻¹(−R/R) = −45°. Therefore, the total current leads the input voltage by 45°, and the voltage across the capacitor lags the input by 45°.Step-by-Step Solution:Z_total = R − jR = R(1 − j).Magnitude |Z_total| = √(R² + R²) = R√2.Phase angle = tan⁻¹(−R/R) = −45°.Current I leads source voltage by 45°. Capacitor voltage v0 lags current by 90°. Therefore v0 lags vi by 45° overall.Verification / Alternative check:Simulation or phasor diagrams confirm that at the condition R = XC, the circuit behaves as a 45° phase lag network.Why Other Options Are Wrong:90° lag: true only if pure capacitor (R = 0).Leads: incorrect for a capacitive branch output.In phase: occurs only if XC = 0 (pure resistor).Common Pitfalls:Mixing up whether v0 is across R or C; for capacitor output, lag occurs.Final Answer:v0 lags vi by 45°

Question 3

When a unit step input voltage is applied to a simple RC lag network, what is the time-domain behavior of the output response?

Accepted Answer

Introduction / Context:Lag networks (first-order RC circuits) are widely used in control and communication systems for shaping frequency response. Understanding their time-domain step response helps correlate time and frequency domain behavior.Given Data / Assumptions:Unit step input u(t) applied at t = 0.Lag network: RC with transfer function H(s) = 1 / (1 + sRC).Output across capacitor.Concept / Approach:The step response of a first-order low-pass network is exponential. It starts at 0 V and asymptotically approaches the final steady-state value equal to the input amplitude (here, 1 V).Step-by-Step Solution:Transfer function H(s) = 1 / (1 + sRC).Laplace of input: 1/s.Output: Y(s) = (1/s)(1 / (1 + sRC)).Inverse Laplace: y(t) = 1 − e^(−t/RC).Hence, y(t) rises exponentially from 0 to 1 as t → ∞.Verification / Alternative check:Compare with capacitor charging curve; matches physical intuition.Why Other Options Are Wrong:Constant from start: wrong, no system can instantaneously jump to final.Decreasing from 1 to 0: describes discharge, not step input.Oscillation: absent in first-order lag networks.Common Pitfalls:Confusing lag network with lead or resonant circuits; forgetting exponential charging law.Final Answer:increases exponentially from zero to the final value

Question 4

In a minimum-phase system, what is the location of poles and zeros in the s-plane that defines its minimum-phase characteristic?

Accepted Answer

Introduction / Context:Minimum-phase systems are important in control and signal processing because they achieve the smallest possible phase shift for a given magnitude response. They are stable and causal with all zeros located in the left half-plane.Given Data / Assumptions:Minimum-phase refers to transfer functions with all poles stable (LHP) and all zeros in the LHP.Focus is on zero locations since pole stability is assumed in any well-designed system.Concept / Approach:Minimum-phase condition means every zero contributes nonnegative phase lag consistent with stability. If a zero is in the right half-plane, the system becomes nonminimum-phase even if stable.Step-by-Step Solution:Check pole locations: for stability, poles in LHP are required but not sufficient for minimum phase.Zeros: all must also lie in LHP to avoid extra phase lag.Thus, correct characterization is 'all zeros in LHP'.Verification / Alternative check:Textbooks: A system with transfer function (s+1)/(s+2) is minimum-phase; (s−1)/(s+2) is stable but nonminimum-phase due to RHP zero.Why Other Options Are Wrong:'Poles in LHP' only ensures stability, not minimum phase.'Poles in RHP' is unstable.'All except one' permits RHP zeros, disqualifying minimum phase.'All poles and zeros LHP' is technically correct but the defining feature is zeros, not poles (already assumed stable).Common Pitfalls:Confusing stable with minimum-phase; forgetting that nonminimum-phase systems can be stable but with undesired phase properties.Final Answer:all zeros lie in the left half-plane

Question 5

In mechanical vibration and acoustics, mechanical impedance is defined as the ratio of which two quantities?

Accepted Answer

Introduction / Context:Mechanical impedance is the dynamic analog of electrical impedance, representing how a mechanical system resists motion when subjected to a harmonic force. It is crucial in vibration isolation, acoustics, and structural dynamics.Given Data / Assumptions:Force is harmonic and measured as RMS value.Velocity is steady-state sinusoidal, measured as RMS value.Displacement and acceleration are related but not directly in the impedance definition.Concept / Approach:Definition: Mechanical impedance Z_m = F_rms / v_rms. It has units of N·s/m, analogous to electrical ohms. It combines effects of mass (inertial reactance), damping (resistive), and stiffness (elastic reactance) on system response.Step-by-Step Solution:Let applied force = F0 sin(ωt).Resulting velocity v(t) = V0 sin(ωt + φ).Impedance magnitude Z_m = F0_rms / V0_rms.General form: Z_m = R + j(X_m), where R is damping, X_m = ωM − (1/ωK).Verification / Alternative check:Textbooks confirm: mechanical impedance = force / velocity.Why Other Options Are Wrong:Force/displacement ratio is stiffness, not impedance.Velocity/displacement ratio is frequency (ω), not impedance.Acceleration/force is inverse of mass, not impedance.Common Pitfalls:Mixing stiffness and impedance; forgetting impedance links force to velocity, not displacement.Final Answer:rms force to rms velocity

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