Question 1

Insulation resistance scaling: A cable has an insulation resistance of 10^10 Ω; if the length of the cable is doubled (same material and geometry), what will be the new insulation resistance?

Accepted Answer

Introduction / Context: Insulation resistance (IR) is a key safety and performance parameter for power and signal cables. Understanding how IR scales with cable dimensions helps in predicting field test results and designing runs to meet standards. This question focuses on the dependence of IR on the length of a uniform cable with unchanged materials and cross-sectional geometry. Given Data / Assumptions: Initial insulation resistance R_i = 10^10 Ω. Cable length is doubled; cross-section and insulation composition are unchanged. Temperature and moisture conditions remain the same (IR is temperature and humidity sensitive in practice). Concept / Approach: For a uniform dielectric layer between a conductor and a return (or sheath), the insulation resistance behaves like the resistance of a material block: R ∝ ρ * (length)/(area) for Cartesian geometry; for coaxial geometry the exact formula differs, but over long lengths the total insulation resistance of a cable section is inversely proportional to its length. When two identical cable lengths are connected in parallel (as with doubling the physical length between measuring terminals), leakage paths effectively add in parallel, halving the total IR. Step-by-Step Solution: Recognize scaling: R_new ∝ 1/length.Doubling length ⇒ R_new = R_i / 2.Compute: R_new = (10^10 Ω) / 2 = 0.5 x 10^10 Ω.Express optionally as 5 x 10^9 Ω (equivalent form). Verification / Alternative check: If three identical segments in series are measured end-to-end from core to sheath, each segment provides a parallel leakage path, so the meter ``sees' the parallel combination; adding a second identical segment halves the IR—consistent with the derived scaling law. Why Other Options Are Wrong: 2 x 10^10 Ω and 4 x 10^10 Ω: would imply IR increases with length, which is contrary to parallel-leakage behavior. 10^10 Ω: suggests no change with length, incorrect for uniform insulation. 5 x 10^9 Ω is numerically the same as 0.5 x 10^10 Ω but is not the option marked in the stem here; the keyed form chosen is 0.5 x 10^10 Ω. Common Pitfalls: Confusing series vs. parallel reasoning—here, leakage surfaces effectively add in parallel as length increases. Ignoring environmental effects; while important in practice, the theoretical scaling assumes constant conditions. Final Answer: 0.5 x 10^10 Ω

Question 2

Electron drift velocity in conductors: Compared with the speed of light, how large is the typical drift velocity of electrons under DC conditions in a metal?

Accepted Answer

Introduction / Context: Students often confuse the rapid propagation of electrical signals with the actual drift motion of electrons through a conductor. The drift velocity is the average net velocity of charge carriers due to an applied electric field. This question clarifies the scale of electron drift in metals compared to the speed of light, an essential idea for understanding conduction and signal transmission. Given Data / Assumptions: Ordinary metallic conductor (e.g., copper) at room temperature. Direct current (steady-state) conditions with practical current densities. Comparison is qualitative against the speed of light c ≈ 3 x 10^8 m/s. Concept / Approach: Drift velocity v_d is given by v_d = I / (n * q * A), where I is current, n is free-electron number density, q is electron charge, and A is cross-sectional area. For typical values (n ≈ 8.5 x 10^28 m^-3 for copper, I in amperes, and A in mm^2 to cm^2 range), v_d evaluates to millimeters or micrometers per second to a few centimeters per second—orders of magnitude lower than c. In contrast, the electromagnetic wave (signal) propagates near a significant fraction of c depending on medium permittivity, which is why light-bulbs appear to turn on virtually instantly, even though individual electrons move very slowly on average. Step-by-Step Solution: Use v_d = I / (n * q * A) for an order-of-magnitude estimate.Example: I = 1 A, A = 1 mm^2 = 1 x 10^-6 m^2, n ≈ 8.5 x 10^28 m^-3, q = 1.6 x 10^-19 C.Compute denominator: n * q * A ≈ 8.5 x 10^28 * 1.6 x 10^-19 * 1 x 10^-6 ≈ 1.36 x 10^4.Then v_d ≈ 1 / (1.36 x 10^4) ≈ 7.4 x 10^-5 m/s = 0.074 mm/s (tiny compared to c). Verification / Alternative check: Even at higher currents or smaller cross sections, v_d rarely exceeds a few cm/s in household wiring. Thus, it remains negligible compared with c by factors of 10^9 or more. Why Other Options Are Wrong: Equal, almost equal, half, or one-tenth of c: these refer to signal or field propagation speeds in media, not carrier drift, and are many orders of magnitude too large for electron drift in metals. Common Pitfalls: Conflating signal propagation (an electromagnetic phenomenon) with carrier drift (a material transport phenomenon). Ignoring that thermal velocities are high but random, while drift is the small net average superimposed on random motion. Final Answer: very small as compared to speed of light

Question 3

Parallel resonance (tank circuit): The statement “In a parallel resonant circuit, the total line current at resonance is maximum” is evaluated as true or false. Choose the correct judgement with reasoning.

Accepted Answer

Introduction / Context: Parallel resonance, also called anti-resonance, occurs when an inductor and capacitor are connected in parallel (often with losses), forming a tank circuit. At the resonant frequency, the reactive currents in the inductor and in the capacitor circulate between the branches. Understanding whether the source (line) current is maximized or minimized at resonance is crucial for filters and reactive power control. Given Data / Assumptions: Standard parallel RLC circuit where R models losses. Steady-state sinusoidal operation at or near resonance. Normal quality factor (finite Q), not an ideal lossless case. Concept / Approach: At parallel resonance, the net susceptance of the inductor and capacitor cancels (B_L + B_C = 0). As a result, the input admittance is minimized and the input impedance is maximized. While very large branch currents may circulate between L and C (because each branch sees near-resonant conditions), the vector sum at the source terminal is small because the reactive currents largely cancel. Therefore, the line current drawn from the source is minimized, not maximized, at resonance. This is the opposite of a series resonant circuit, where impedance is minimized and current is maximized at resonance. Step-by-Step Solution: Write input admittance: Y_in = G + j(B_L + B_C).At resonance: B_L + B_C = 0 ⇒ Y_in = G_min (purely real and small).Thus Z_in = 1 / Y_in is maximum, so source current I = V / Z_in is minimum.Branch currents can still be large but largely cancel at the junction. Verification / Alternative check: Measure current in a lab tank circuit: ammeter in the line shows a dip at resonance, while branch meters show peaks—directly confirming the theoretical claim of minimum line current at resonance. Why Other Options Are Wrong: “True” and conditional versions mistake the high circulating branch currents for line current; the source sees maximum impedance at resonance, not minimum. “False if resistance is zero” is misleading; in the ideal lossless limit, the line current goes to zero (not maximum) because net susceptance cancels exactly. Common Pitfalls: Confusing series vs. parallel resonance; they exhibit opposite input-current behaviors. Equating circulating reactive current with source current—these are not the same at resonance in a parallel network. Final Answer: False

Question 4

RMS and average value definitions: Do the terms “RMS” (root-mean-square) and “average value” apply only to sinusoidal waveforms?

Accepted Answer

Introduction / Context: Root-mean-square (RMS) and average (mean) values are general mathematical constructs used to quantify the magnitude of time-varying signals. In electrical engineering, they are crucial for relating AC signals to equivalent DC heating or computing offsets and energy. This question addresses a common misconception that these metrics are defined only for sinusoidal waveforms. Given Data / Assumptions: No special assumptions on waveform: it may be sinusoidal, non-sinusoidal periodic, or even aperiodic (provided the relevant integrals converge). Standard time averaging over one period for periodic signals or over an interval for arbitrary signals. Concept / Approach: The RMS value of a signal x(t) over a period T is defined as x_rms = sqrt((1/T) * ∫_0^T x(t)^2 dt). The average (mean) over a period is x_avg = (1/T) * ∫_0^T x(t) dt. Neither definition requires x(t) to be sinusoidal; they apply universally to any signal for which these integrals exist. For non-sinusoidal periodic waveforms such as square, triangular, or PWM waveforms, RMS and average values are routinely computed and used for power and control design. Even for random signals, RMS is used (e.g., noise RMS). Step-by-Step Solution: Recall definitions of RMS and average—no sinusoid prerequisite.Provide examples: RMS of a square wave of amplitude A is A; of a triangle is A/√3; average of a full-wave rectified sine is 2A/π.Conclude: The statement “apply only to sine waves” is false. Verification / Alternative check: Standards for power quality and instrumentation specify RMS measurement of arbitrary waveforms (true-RMS meters) precisely because non-sinusoidal loads are common. Average values are similarly computed for any waveform to determine offsets or rectified averages. Why Other Options Are Wrong: Any option asserting exclusivity to sine waves contradicts the general mathematical definitions. “Apply to DC only” is incorrect; DC is a trivial special case with RMS equal to the absolute value and average equal to the DC level. Common Pitfalls: Assuming instrument displays that are calibrated for sine waves define the mathematics; they are practical approximations, not restrictions of the definitions. Final Answer: False

Question 5

Power factor limits in AC circuits: Identify the maximum and minimum possible values of the power factor for practical AC circuits.

Accepted Answer

Introduction / Context: Power factor (PF) is the cosine of the phase angle between voltage and current in sinusoidal steady state, or more generally the ratio of real power to apparent power. It is a cornerstone metric for efficiency and power quality in AC systems. This question asks for the extreme limits of power factor for typical passive or general AC loads as used in engineering practice. Given Data / Assumptions: Standard definition PF = P / S for sinusoidal systems with P ≥ 0 for passive loads. We consider the magnitude behavior in typical utility contexts; sign (leading/lagging) indicates reactive direction but not the limit on magnitude. No active power reversal (generation) unless explicitly noted. Concept / Approach: For passive loads absorbing power, real power P is nonnegative and bounded by apparent power S. Hence 0 ≤ PF ≤ 1, with PF = 1 for purely resistive loads (no reactive component) and PF = 0 for purely reactive loads (inductor or capacitor only). Some texts allow negative PF when current leads or lags by more than 90° or when power flow reverses, but in standard distribution contexts we report magnitude between 0 and 1 and then state “leading” or “lagging.” Therefore, the correct pair of maximum and minimum values is 1 and 0. Step-by-Step Solution: Write PF = |cos φ| for sinusoidal passive loads.As φ varies from 0° (resistive) to ±90° (purely reactive), |cos φ| ranges from 1 down to 0.Thus maximum PF = 1, minimum PF = 0. Verification / Alternative check: In power system measurements, utilities target PF close to 1 via capacitor banks or filters. Measurements on unloaded induction motors (highly inductive) show PF near 0, approaching the lower limit in practice. Why Other Options Are Wrong: 2 and 0: PF cannot exceed 1 by definition (P ≤ S). 0 and −1 or 1 and −1: these mix sign conventions; the magnitude still does not exceed 1. 0.95 and 0: 0.95 is a common target but not the maximum theoretical limit. Common Pitfalls: Confusing sign (leading/lagging) with magnitude; magnitude is capped at 1. Assuming non-sinusoidal definitions change the upper bound; with suitable generalization, the magnitude remains ≤ 1. Final Answer: 1 and 0

Question 6

History of network synthesis: The synthesis approach specifically for “minimum functions” (in passive network theory) was proposed by which researcher?

Accepted Answer

Introduction / Context: Network synthesis matured through contributions from several pioneers. Among these, the concept of the “minimum function” occupies a special role in ladder realizations and transmission-line inspired techniques. This question asks you to identify the researcher associated with proposing the synthesis of minimum functions, a cornerstone step that influenced later canonical forms. Given Data / Assumptions: “Minimum function” here refers to a special positive-real function with interlacing real-axis (or imaginary-axis) poles and zeros, suited to economical passive realizations. We focus on classic 20th-century synthesis literature. Concept / Approach: R. E. Richards is credited with the development of methods in network synthesis that leverage transmission-line concepts, including Richards transformation and associated realizations. In this context, he proposed the synthesis of minimum functions, enabling practical ladder-type networks that satisfy the positive-real condition with minimal complexity. This work is distinct from Brune’s realization (which introduced ideal transformers to realize general positive-real functions) and from Bott–Duffin (which later showed that any positive-real function can be realized with passive elements without transformers). Bode’s contributions were predominantly in filter design and feedback, not specifically the minimum-function synthesis proposal asked here. Step-by-Step Solution: Identify the association: Minimum function → Richards.Differentiate from contemporaries: Brune → ideal transformer realization; Bott–Duffin → transformerless general PR realization.Conclude that the correct name linked with minimum-function synthesis is R. Richards. Verification / Alternative check: Historical texts in network synthesis outline this lineage: Brune (1920s), Cauer (1920s–30s), Richards (1940s), and Bott–Duffin (late 1940s). Richards’ minimum-function methods are typically treated alongside his famous Richards transformation in distributed filter realizations. Why Other Options Are Wrong: O'Brune: the surname is Brune (Otto Brune), not “O'Brune,” and his main contribution was the ideal-transformer realization of PR functions, not specifically minimum functions. Bott and Duffin: known for the general proof of realization without transformers, not the initial minimum-function synthesis. H. W. Bode: influential in filter and feedback theory, but not credited with the minimum-function synthesis asked here. None of the above: incorrect because Richards is indeed the correct attribution. Common Pitfalls: Conflating general PR realizations (Brune, Bott–Duffin) with the specific minimum-function synthesis (Richards). Final Answer: R. Richards

Question 7

RMS comparison of sin and cos waves: Given two sinusoidal waveforms A = 100 sin(ωt) and B = 100 cos(ωt), compare their root-mean-square (RMS) values.

Accepted Answer

Introduction / Context: RMS (root-mean-square) value quantifies the effective magnitude of a time-varying signal in terms of its ability to deliver power to a resistive load. For sinusoids, the RMS depends on amplitude, not on phase shift. This question checks whether phase-shifted sinusoids of the same amplitude share the same RMS value. Given Data / Assumptions: Two signals: A(t) = 100 sin(ωt) and B(t) = 100 cos(ωt). Both have the same amplitude 100 V (or A in general units). Steady-state sinusoidal waveforms, same frequency ω. Concept / Approach: For a sinusoid x(t) = X_m sin(ωt + φ), the RMS value is X_rms = X_m / √2. The phase φ does not affect the RMS because the squaring operation removes the phase dependence when averaging over a full period. Therefore, any two sinusoids at the same amplitude have identical RMS irrespective of being sin or cos or any phase shift between them. Step-by-Step Solution: Compute RMS for A: A_rms = 100 / √2.Compute RMS for B: B_rms = 100 / √2 (cos is sin phase-shifted by 90°).Compare: A_rms = B_rms ⇒ equal. Verification / Alternative check: Time-average over one period T: (1/T) ∫_0^T 100^2 sin^2(ωt) dt = 100^2/2, same as for cos^2(ωt). Square root yields 100/√2 for both. Simulation or measurement with a true-RMS meter yields identical readings for equal-amplitude sin and cos waveforms at the same frequency. Why Other Options Are Wrong: Claims that one RMS exceeds the other rely on phase, which cancels in RMS computation. “May or may not be equal” is incorrect for sinusoids with the same amplitude; they are always equal. Equality does not require identical phase—only identical amplitude and waveform type. Common Pitfalls: Confusing instantaneous values (which depend on phase) with RMS, a time-averaged quantity. Assuming different starting phases change energy content over a full cycle; they do not. Final Answer: rms values of the two waves are equal

Networks Analysis and Synthesis Questions