Question 1

AC power terminology — is apparent power expressed in watts (W)?

Accepted Answer

Introduction / Context: Correct AC power units are critical for specifying transformers, UPS systems, and inverters. Apparent power, real power, and reactive power use different units and convey different physical meanings. Given Data / Assumptions: Sinusoidal steady-state conditions. Three standard power quantities: real (P), reactive (Q), and apparent (S). Concept / Approach: Real power P is measured in watts (W). Reactive power Q is measured in volt-ampere reactive (var). Apparent power S is measured in volt-amperes (VA). Apparent power reflects the product of RMS voltage and RMS current irrespective of phase angle. Step-by-Step Solution: Define S = V_rms * I_rms (unit VA).Define P = V_rms * I_rms * cos(phi) (unit W).Define Q = V_rms * I_rms * sin(phi) (unit var).Power triangle relation: S^2 = P^2 + Q^2. Verification / Alternative check: Equipment nameplates often list kVA for transformers (apparent power rating) and kW for loads (real power). This practical distinction validates the unit separation. Why Other Options Are Wrong: Choosing “True” conflates apparent power with real power; only P uses watts. Common Pitfalls: Using kW when sizing generators for loads with poor power factor; the correct initial sizing metric is kVA, then ensure adequate kW capacity given expected PF. Final Answer: False

Question 2

Filters — do they pass all frequencies?

Accepted Answer

Introduction / Context: Filters are designed to selectively pass certain frequency ranges while attenuating others. This item checks a basic definition that underlies audio, RF, and signal-processing practice. Given Data / Assumptions: Filter categories include low-pass, high-pass, band-pass, and band-stop (notch). Real filters have finite roll-off and non-ideal ripple or phase characteristics. Concept / Approach: By definition, a filter does not pass all frequencies equally; it shapes the spectrum. The passband is defined where gain is near unity (or specified level), and the stopband where signals are significantly attenuated. Step-by-Step Solution: Identify a low-pass example: passes frequencies below a cutoff fc; attenuates above fc.High-pass: passes above fc; attenuates below.Band-pass: passes between f1 and f2; attenuates outside.Band-stop: attenuates a narrow range while passing others. Verification / Alternative check: Examine a magnitude response plot (|H(jω)| vs. ω). You will see defined pass/stop regions, not a flat “all-pass” transmission unless a specific all-pass filter is intentionally designed (which still alters phase). Why Other Options Are Wrong: “True” would describe an ideal wire or unity buffer, not a filter. Even an all-pass filter modifies phase rather than passing all frequencies unchanged in every respect. Common Pitfalls: Equating “filter” with “removes noise” generically; correct design requires selecting the passband/stopband to match the signal and noise spectra. Final Answer: False

Question 3

RC networks — is the total impedance set by combining resistance and capacitive reactance?

Accepted Answer

Introduction / Context: This statement tests whether the learner understands that both resistance and reactance contribute to a complex impedance in RC circuits, shaping magnitude and phase together. Given Data / Assumptions: Series or parallel RC configuration under sinusoidal steady-state. Resistor contributes real part; capacitor contributes negative imaginary part. Concept / Approach: Impedance is a complex quantity Z = R + jX. In RC circuits, X = −1 / (ωC) (capacitive), so Z results from vector (phasor) combination of R and Xc. The magnitude |Z| and angle θ determine currents and phase shift. Step-by-Step Solution: Series RC: Z = R − j * (1 / (ω * C)); |Z| = sqrt( R^2 + (1 / (ω * C))^2 ).Parallel RC: Admittance Y = 1/R + j * ω * C; then Z = 1 / Y, again reflecting both R and C.Therefore, impedance is determined by combining resistance and capacitive reactance according to circuit topology. Verification / Alternative check: Measure with an LCR meter: changing frequency alters Xc and thus the measured |Z| and phase, proving that both R and C determine the net impedance. Why Other Options Are Wrong: “False” would ignore the reactive contribution and predict incorrect currents, voltages, and power factor. Common Pitfalls: Treating magnitudes algebraically without vector addition; impedance must be combined using complex arithmetic, not simple arithmetic sums of magnitudes except in special cases. Final Answer: True

Question 4

Imaginary numbers — are they merely numbers that “exist only in the mind of the mathematician”?

Accepted Answer

Introduction / Context: This statement addresses a common misconception. Imaginary and complex numbers are indispensable in engineering and physics, not just abstract inventions without real-world application. Given Data / Assumptions: Imaginary unit j (or i in mathematics) satisfies j^2 = −1. Complex number z = a + j * b with real part a and imaginary part b. Concept / Approach: Imaginary numbers, combined with real numbers to form complex numbers, provide a powerful representation for oscillations, phasors, impedances, control poles/zeros, and quantum amplitudes. Their consequences are measurable and routinely verified in experiments and circuits. Step-by-Step Solution: AC circuit analysis: capacitor impedance Zc = 1 / (j * ω * C), inductor impedance Zl = j * ω * L.Signals: e^{jωt} compactly represents sinusoids via Euler’s identity, enabling linear system solutions.Control/communications: complex poles/zeros determine stability and resonance; constellations (QPSK, QAM) use I/Q axes (real/imaginary) for modulation. Verification / Alternative check: Laboratory measurements of phase shifts, resonance peaks, and frequency responses align with complex-number predictions, demonstrating their physical relevance. Why Other Options Are Wrong: Marking “True” denies a century of validated engineering methods that rely on complex numbers to match observed behavior. Common Pitfalls: Interpreting “imaginary” as “unreal.” The term is historical; the mathematics maps directly to orthogonal components that correspond to measurable sine and cosine projections. Final Answer: False

Question 5

RC circuits — does the total current always lead the source voltage?

Accepted Answer

Introduction / Context: This item checks phase intuition: in networks containing only resistors and capacitors (no inductors), is the net current ahead (leading) or behind (lagging) the source voltage? Given Data / Assumptions: RC network (series or parallel) with sinusoidal excitation. No inductive elements present. Concept / Approach: Capacitive current leads voltage by 90 degrees for an ideal capacitor. The presence of resistance reduces the lead, but cannot reverse it to a lag because there is no inductive reactance to pull the phase negative. Therefore, the total current in an RC-only network leads the source voltage by some angle between 0 and +90 degrees. Step-by-Step Solution: Series RC: current I is common and leads source voltage by angle θ = arctan(1 / (ω * R * C)).Parallel RC: total current Itot = IR + IC; IR is in phase with V, IC leads V by 90 degrees; vector sum leads V by 0 < θ < 90 degrees.Thus, in both topologies, without L the net effect is a leading current. Verification / Alternative check: Compute the input admittance Y = G + jB for an RC network; susceptance B = +ωC > 0, so the current phasor leads the voltage phasor by angle φ = arctan(B/G) ≥ 0. Why Other Options Are Wrong: “False” would suggest zero or lagging phase for some RC arrangement, which would require inductive behavior not present in R–C-only circuits. Common Pitfalls: Confusing “output current in a branch” with “total line current.” Individual branch currents may have different phases, but the total seen by the source still leads for RC-only networks. Final Answer: True

Question 6

Complex numbers — does the polar form consist of a real part and a j part?

Accepted Answer

Introduction / Context: Complex numbers can be expressed in rectangular (Cartesian) or polar form. Correctly identifying each representation is vital for phasor arithmetic and impedance calculations. Given Data / Assumptions: Rectangular form: z = a + j * b. Polar form: z = M ∠ θ (or z = M * e^{jθ}). Concept / Approach: The statement describes the rectangular form (real part a and imaginary part b). Polar form instead uses magnitude M = sqrt(a^2 + b^2) and angle θ = atan2(b, a). Both forms are equivalent but emphasize different operations: addition favors rectangular; multiplication/division and phase analysis favor polar. Step-by-Step Solution: Given z = a + j * b, compute M = sqrt(a^2 + b^2).Compute θ = atan2(b, a) in degrees or radians.Write polar: z = M ∠ θ; rectangular is recovered by a = M * cos(θ), b = M * sin(θ). Verification / Alternative check: Convert a numerical example: z = 3 + j4 → M = 5, θ ≈ 53.13°. Rectangular uses real/imag parts; polar uses 5 ∠ 53.13°, confirming distinct descriptions. Why Other Options Are Wrong: Choosing “True” mixes the two forms and will cause errors in multiplying/dividing phasors and converting impedances. Common Pitfalls: Forgetting angle units when converting; always state degrees or radians explicitly to avoid phase mistakes. Final Answer: False

Question 7

Circuit phase angle — is it the angle between total current and the applied source voltage?

Accepted Answer

Introduction / Context: The circuit phase angle summarizes how reactive a load is. It links directly to power factor and the relationship among apparent, real, and reactive power in AC systems. Given Data / Assumptions: Single-frequency sinusoidal excitation. Definition of phase angle φ within a linear circuit. Concept / Approach: By standard convention, the circuit (or load) phase angle φ is the angle by which the current leads or lags the applied voltage. Positive φ typically indicates current leading (capacitive), negative φ indicates current lagging (inductive). Power factor is cos(φ). Step-by-Step Solution: Represent voltage and current as phasors V and I.Define φ = angle(I) − angle(V).Use φ to compute P = V_rms * I_rms * cos(φ), Q = V_rms * I_rms * sin(φ), S = V_rms * I_rms. Verification / Alternative check: On an oscilloscope with Lissajous or dual-trace display, measure time shift between v(t) and i(t); convert to angle at the operating frequency to obtain φ, which is precisely the angle between current and voltage phasors. Why Other Options Are Wrong: Answering “False” would contradict textbook definitions used in power engineering and circuit analysis. Common Pitfalls: Confusing the sign convention; always specify whether φ is defined as current relative to voltage or vice versa and maintain consistency when computing power factor. Final Answer: True

Question 8

Power factor (PF) in AC circuits — does it indicate how much of the apparent power (S) is actually real or true power (P)?  Consider a sinusoidal steady-state circuit where apparent power S = V_rms * I_rms and power factor PF = cos(phi).

Accepted Answer

Introduction / Context: Power factor in alternating current systems relates the apparent power flowing in a circuit to the useful or true power consumed by the load. A correct understanding of power factor enables better sizing of equipment and reduction of losses in transmission and distribution networks. Given Data / Assumptions: S refers to apparent power measured in volt-ampere. P refers to true or real power measured in watt. PF = cos(phi) where phi is the phase angle between voltage and current under sinusoidal steady state. Concept / Approach: By definition PF = P / S. Therefore PF numerically indicates the fraction of the total apparent power that is converted into real work or heat within the load. Reactive components cause current to lead or lag, increasing S without contributing to P, which reduces PF below 1.0. Step-by-Step Solution: Compute apparent power: S = V_rms * I_rms.Compute real power: P = V_rms * I_rms * cos(phi).Relate the two: PF = P / S = cos(phi).Interpretation: PF shows how much of S appears as useful P. Verification / Alternative check: If PF = 0.8, then P = 0.8 * S. Boosting PF with capacitors or synchronous condensers moves PF toward unity, increasing the ratio of P to S for the same current. Why Other Options Are Wrong: “Only true for purely resistive loads” is incorrect; the definition holds for any sinusoidal load. “True only when reactive power is zero” is misleading; PF defines the ratio even when reactive power is present. “True only at unity PF” is wrong; unity is just a special case where P = S. Common Pitfalls: Confusing power factor with efficiency. Efficiency is output power over input power, while PF is P over S and captures phase displacement and waveform effects (for sinusoidal case, displacement only). Final Answer: True

Question 9

Phase relations in capacitors — does the capacitor voltage always lead the current by 90 degrees in steady-state AC?

Accepted Answer

Introduction / Context: Understanding phase relationships between voltage and current for reactive components is essential in AC circuit design, power factor correction, and filter development. The question checks the specific lead-lag behavior of an ideal capacitor. Given Data / Assumptions: Ideal capacitor in sinusoidal steady state. Angular frequency is omega and current is nonzero. No nonideal parasitics unless discussed as a caveat. Concept / Approach: For an ideal capacitor, current leads voltage by 90 degrees. Equivalently, voltage lags current by 90 degrees. This comes directly from i = C * dv/dt, which shows current proportional to the derivative of voltage. Step-by-Step Solution: For v(t) = V_m * sin(omega t), dv/dt = omega * V_m * cos(omega t).Thus i(t) = C * dv/dt = C * omega * V_m * cos(omega t) = C * omega * V_m * sin(omega t + 90 degrees).Therefore current leads voltage by 90 degrees, not the other way around.So the statement that “capacitor voltage leads current” is incorrect. Verification / Alternative check: Phasor form: I = j * omega * C * V. The operator j indicates that the current phasor is rotated +90 degrees relative to the voltage phasor, confirming current leads voltage in a capacitor. Why Other Options Are Wrong: Resonance does not reverse the fundamental capacitor relation. Series resistance slightly alters phase from exactly 90 degrees but does not cause voltage to lead current. The DC option is irrelevant because a capacitor blocks steady DC after transients. Common Pitfalls: Memorization errors between inductors and capacitors. A helpful mnemonic: “ICE” and “ELI.” In a capacitor (C), current (I) leads voltage (E). In an inductor (L), voltage (E) leads current (I). Final Answer: False

Question 10

Ideal capacitor power consideration in AC circuits: Evaluate the statement: “In a purely capacitive AC circuit (ideal capacitor, no resistive losses), the energy converted to heat is proportional to the circuit current.” Choose the most accurate ass…

Accepted Answer

Introduction / Context: Reactive components (ideal capacitors and inductors) in AC circuits exchange energy with the source but, ideally, do not dissipate real power as heat. This distinction between reactive power and real power is foundational for power factor, filter design, and AC analysis. Given Data / Assumptions: Purely capacitive AC circuit with an ideal capacitor. No series resistance or dielectric loss is considered (lossless assumption). Steady sinusoidal operation. Concept / Approach: Instantaneous power in a pure capacitor alternates between positive and negative as the capacitor charges and discharges. The average over a full cycle is zero because the current leads the voltage by 90 degrees. Real power P_avg = V_rms * I_rms * cos(phi). For a pure capacitor, phi = +90 degrees, so cos(phi) = 0 and P_avg = 0 W. Therefore, no average energy is converted to heat in the ideal case, even though current can be large. Any heating in real components comes from non-ideal losses (equivalent series resistance or dielectric losses), not from the ideal reactive behavior itself. Step-by-Step Solution: Recall that for a pure capacitor, current leads voltage by 90 degrees.Real power formula: P_avg = V_rms * I_rms * cos(phi).Set phi = 90 degrees ⇒ cos(phi) = 0 ⇒ P_avg = 0 W.Therefore, the statement about heat proportional to current is incorrect in the ideal case. Verification / Alternative check: Compute instantaneous power p(t) = v(t) * i(t) for v = Vp sin(ωt) and i = C dv/dt = ωC Vp cos(ωt). Over one period, the average of sin(ωt)cos(ωt) is zero, confirming zero average real power. Why Other Options Are Wrong: Valid / applies only at resonance / true for low frequencies only: These either contradict the zero real power result or introduce conditions (like resonance) that still do not create real power in an ideal, lossless capacitor. Common Pitfalls: Equating high reactive current with heating; ignoring that only resistive components dissipate real power; forgetting the role of phase angle in P_avg calculation. Final Answer: Invalid

Question 11

In alternating-current (AC) analysis of a series RC circuit driven by a sinusoidal source, does the current lag or lead the applied source voltage? Choose the most accurate statement.

Accepted Answer

Introduction / Context: The phase relationship between current and voltage is a core concept in AC circuit analysis. In reactive networks, capacitors and inductors shift phase, causing current to lead or lag voltage depending on the component. This question focuses on the classic series RC circuit to test whether you recall which quantity leads and by how much. Given Data / Assumptions: A sinusoidal steady-state source is applied. Components are ideal: one resistor R in series with one capacitor C. Angular frequency is omega, capacitive reactance is Xc = 1 / (omega * C). No measurement loading effects are considered. Concept / Approach: The impedance of a series RC circuit is Z = R - jXc. The current I is the same through both elements and equals V / Z. Because the impedance has a negative imaginary part (capacitive), the current phasor leads the voltage phasor by an angle phi where tan(phi) = Xc / R. A helpful memory aid is “ICE” (current leads voltage in a Capacitive circuit; current lags in an Inductive circuit). Step-by-Step Solution: Write the impedance: Z = R - jXc.Determine the phase: phi = arctan(Xc / R) (positive for capacitive, indicating current leads).Conclude the relationship: current leads source voltage by phi, not lags. Verification / Alternative check: At very high frequency, Xc approaches 0, so Z approaches R and the phase approaches 0 degrees, consistent with small lead. At very low frequency, Xc is very large, the current is small, and the capacitor dominates, with current attempting to lead by up to 90 degrees in the ideal limit. Why Other Options Are Wrong: “Current lags” is characteristic of inductive circuits, not capacitive ones. “Only at low frequencies” is misleading; the lead exists at all frequencies, though the angle varies. DC with ripple is irrelevant; pure DC has no steady AC phase, and ripple still follows the AC rule: capacitive currents lead. Common Pitfalls: Confusing inductive and capacitive behavior (“ELI the ICE man” mnemonic helps). Another mistake is assuming the phase is fixed; it depends on R and Xc. Final Answer: Incorrect — in a series RC circuit, the current leads the source voltage by arctan(Xc/R).

Question 12

Series RC phasor sum — a resistor shows 6 V (rms) and a capacitor shows 8 V (rms) in the same series RC branch. Using vector (phasor) addition, what is the magnitude of the source voltage?

Accepted Answer

Introduction / Context: In AC circuits containing reactance, voltage drops do not add arithmetically. Instead, they combine as vectors (phasors) because they are out of phase. This question tests your ability to use the right kind of addition for a series RC circuit where resistor and capacitor voltages are in quadrature (90 degrees apart). Given Data / Assumptions: Series RC circuit under sinusoidal steady state. Measured magnitudes: V_R = 6 V (in phase with current), V_C = 8 V (lags current by 90 degrees). The source voltage equals the phasor sum of V_R and V_C. Concept / Approach: In a series RC circuit, current is common. The resistor voltage is V_R = I * R and is in phase with the current; the capacitor voltage is V_C = I * Xc, but since Xc introduces -90 degrees, V_C is 90 degrees behind the current. Therefore, V_R and V_C are orthogonal, making the source magnitude Vs the Pythagorean sum: Vs = sqrt(V_R^2 + V_C^2). Step-by-Step Solution: Identify orthogonal components: V_R along the real axis, V_C along the negative imaginary axis.Compute magnitude: Vs = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 V.Conclude the statement is correct: the source voltage magnitude is 10 V. Verification / Alternative check: If you convert to impedance and current, the same result emerges. For some I, V_R = I*R and V_C = I*Xc; geometry dictates a right triangle, yielding Vs = 10 V when the legs are 6 V and 8 V. Why Other Options Are Wrong: Arithmetic sum (14 V) ignores the 90-degree phase difference. “2 V” is non-physical for these magnitudes; phasor subtraction is not appropriate here. Frequency is not needed once V_R and V_C are given; their vector relationship is fixed. Common Pitfalls: Adding AC drops like DC series voltages or forgetting that resistor and capacitor voltages are orthogonal. Final Answer: Correct — Vs = 10 V.

Question 13

Resistance versus impedance — in AC analysis of an RC network, is resistance the same as impedance? Choose the most accurate statement.

Accepted Answer

Introduction / Context: In AC circuits, impedance generalizes resistance to include the effects of reactance from capacitors and inductors. Confusing resistance with impedance leads to incorrect magnitudes and phase angles. This question checks basic vocabulary and the mathematical form of impedance for an RC network. Given Data / Assumptions: Sinusoidal steady-state analysis. Components are ideal, with a resistor R and capacitor C. Capacitive reactance Xc = 1 / (omega * C). Concept / Approach: Resistance is a real scalar measured in ohms and relates voltage and current without phase shift: V = I * R. Impedance is a complex quantity, Z = R + jX, that relates phasor voltage and current: V = I * Z. For a capacitor, the reactance is negative imaginary (Xc is treated as -|Xc|), so in series RC we write Z = R - jXc. The magnitude is |Z| = sqrt(R^2 + Xc^2), and the phase angle is phi = arctan(Xc / R), with current leading voltage in a capacitive circuit. Step-by-Step Solution: Write Z for series RC: Z = R - jXc.Compute magnitude and angle: |Z| = sqrt(R^2 + Xc^2); phi = arctan(Xc / R).Recognize resistance equals impedance only if Xc = 0 (not true for a real capacitor at finite frequency). Verification / Alternative check: At extremely high frequencies Xc approaches 0, so Z approaches R (still not literally equal except in the ideal limit). At DC, Xc tends to infinity, the circuit blocks current, and Z is not equal to R. Why Other Options Are Wrong: “Correct — interchangeable” ignores the imaginary component from reactance. “Valid only at resonance” does not apply to a lone RC branch; resonance requires both L and C. “Valid only at high frequency” is an approximation, not an identity. Common Pitfalls: Using R instead of |Z| in current calculations or neglecting phase when combining AC voltages. Final Answer: Incorrect — impedance includes both R and Xc and is not the same as resistance.

Question 14

Parallel RC phase behavior — as the capacitance C increases in a parallel RC circuit excited by a sinusoid, how does the overall phase angle between the source voltage and total current change?

Accepted Answer

Introduction / Context: Phase angle in parallel circuits depends on the relative contributions of conductance and susceptance. In a parallel RC network, changing the capacitor value alters the reactive branch current dramatically, which changes the total current phasor and the angle it makes relative to the impressed voltage. Given Data / Assumptions: Sinusoidal steady-state excitation. Parallel RC: one branch is a resistor R (conductance G = 1/R), the other is a capacitor C (susceptance Bc = omega * C). Phase angle theta is the angle of total current relative to voltage (voltage is the reference in admittance analysis). Concept / Approach: Total admittance is Y = G + jBc, where Bc = omega * C. As C increases, Bc increases proportionally, so the reactive component of current grows relative to the resistive component. The phase angle of the total current is theta = arctan(Bc / G). Therefore, increasing C increases Bc, which increases |theta| toward +90 degrees (capacitive behavior). Step-by-Step Solution: Write Y: Y = G + jBc with Bc = omega * C.Compute angle: theta = arctan(Bc / G).Increase C → increase Bc → larger theta magnitude → more capacitive phase. Verification / Alternative check: Consider extreme cases. If C is very small, Bc ≈ 0 and the circuit is nearly resistive (theta ≈ 0). If C is very large, Bc dominates and theta approaches +90 degrees, indicating strongly capacitive current. Why Other Options Are Wrong: “More inductive” contradicts the nature of a capacitor. “No change” ignores the dependence of Bc on C. “Reverses sign / pure resistor” has no basis; adding capacitance increases capacitive character. Common Pitfalls: Confusing series and parallel behavior or using reactance Xc = 1/(omega*C) directly in parallel when admittance is more convenient. Final Answer: Correct — increasing C increases the capacitive phase angle magnitude.

Question 15

RC low-pass filter terminology — for a single-pole RC low-pass network, is the bandwidth equal to the -3 dB cutoff frequency f_c?

Accepted Answer

Introduction / Context: Bandwidth is a fundamental specification in filter design and signal processing. For simple, first-order filters, the definition is straightforward and often used in practical datasheets. This item ensures you understand how bandwidth relates to the -3 dB cutoff frequency in a single-pole RC low-pass network. Given Data / Assumptions: First-order RC low-pass filter (one resistor, one capacitor). Cutoff frequency f_c defined at the magnitude response where output power is half the midband (voltage down by 3 dB): |H(f_c)| = 1/sqrt(2). “Bandwidth” refers to the range of frequencies for which the filter passes signals with minimal attenuation. Concept / Approach: By convention, for a first-order low-pass, the passband is taken from DC (0 Hz) up to the -3 dB point f_c. Since that interval length is simply f_c − 0, the bandwidth equals f_c. This contrasts with band-pass filters, where bandwidth is f_2 − f_1 between the two half-power frequencies. Step-by-Step Solution: Identify filter type: first-order low-pass (transfer function magnitude decreases beyond f_c).Define bandwidth: from 0 Hz to f_c.Compute bandwidth: BW = f_c − 0 = f_c. Verification / Alternative check: The -3 dB definition implies that below f_c the attenuation is less than or equal to 3 dB; since there is only one cutoff, this region’s width equals f_c. Many textbooks and manufacturer notes explicitly state BW = f_c for first-order low-pass sections. Why Other Options Are Wrong: “Twice f_c” has no basis for a single-pole low-pass. “Defined only for high-pass” is incorrect; bandwidth is a universal metric. “Requires Q” applies to resonant second-order sections, not a first-order RC. Common Pitfalls: Confusing band-pass bandwidth (difference between two cutoffs) with low-pass bandwidth (single cutoff from DC). Final Answer: Correct — bandwidth equals f_c for a first-order RC low-pass filter.

Question 16

Reactance and susceptance equivalence — a capacitive reactance of 10 kΩ corresponds to what capacitive susceptance in siemens or microsiemens?

Accepted Answer

Introduction / Context: Reactance and susceptance are reciprocal concepts used in impedance and admittance analysis. For capacitors, engineers often flip between series (reactance) and parallel (susceptance) viewpoints. This question checks your fluency with the reciprocal relationship for a specific numeric value. Given Data / Assumptions: Capacitive reactance magnitude Xc = 10 kΩ. We are converting to capacitive susceptance Bc in siemens (S). Ideal sinusoidal steady-state conditions. Concept / Approach: Admittance Y is the reciprocal of impedance Z. For a purely capacitive element, the magnitude relationships simplify: |Z| = Xc and |Y| = Bc, and therefore Bc = 1 / Xc. Once Xc is given, its reciprocal yields susceptance in siemens. Converting to microsiemens helps match typical instrumentation scales. Step-by-Step Solution: Start from Bc = 1 / Xc.Insert Xc = 10 kΩ = 10,000 Ω.Compute: Bc = 1 / 10,000 = 0.0001 S = 100 × 10^-6 S = 100 µS. Verification / Alternative check: Cross-check by dimensional reasoning: ohms and siemens are reciprocals. If Xc is large (10 kΩ), its admittance must be small (100 µS), which is consistent. Why Other Options Are Wrong: 10 µS is off by a factor of 10; there is no extra factor of 10 in the reciprocal. 1 mS equals 1000 µS, which contradicts the reciprocal of 10 kΩ. Frequency is not needed here; Xc has already been provided, and Bc = 1/Xc. Common Pitfalls: Forgetting to convert kilo-ohms to ohms before taking the reciprocal or mixing up mS and µS. Final Answer: Correct — 100 µS.

Question 17

RC high-pass behavior — as input frequency decreases, does the output voltage magnitude of a simple RC high-pass filter increase or decrease?

Accepted Answer

Introduction / Context: High-pass filters are ubiquitous in coupling networks, audio systems, and sensor interfaces. Their defining characteristic is that they pass high-frequency content while attenuating low-frequency components and DC. This question probes your qualitative understanding of how output changes with frequency. Given Data / Assumptions: Single-pole RC high-pass filter (series capacitor, shunt resistor or equivalent topology). Sinusoidal steady-state behavior. No significant load interaction beyond the intended resistor. Concept / Approach: In a high-pass RC network, the capacitor’s reactance Xc = 1 / (omega * C) is large at low frequencies and small at high frequencies. The transfer function magnitude |H(jomega)| increases with frequency, approaching unity at very high frequency and approaching zero at very low frequency. Therefore, as frequency decreases, output voltage magnitude falls, not rises. Step-by-Step Solution: Recognize the high-pass role: blocks DC and very slow variations.At low frequency: Xc is large → series impedance high → output across the resistor is small.At high frequency: Xc is small → signal passes → output increases toward input magnitude. Verification / Alternative check: Check the -3 dB cutoff f_c = 1 / (2 * pi * R * C). Below f_c, output rolls off at about -20 dB/decade (first order). Above f_c, output asymptotically approaches full scale. Why Other Options Are Wrong: “Output increases as frequency decreases” contradicts high-pass behavior. “Only when load is inductive” is irrelevant to the basic trend; standard topology already determines the response. “Undetermined” is not correct; the qualitative direction is known without exact values. Common Pitfalls: Mixing up high-pass with low-pass or overlooking the role of the series capacitor as a frequency-dependent coupling element. Final Answer: Incorrect — the output of an RC high-pass decreases as frequency decreases.

Question 18

Choosing the output node — when a series RC network is used as a low-pass filter, from which component should the output be taken to obtain low-pass behavior?

Accepted Answer

Introduction / Context: First-order RC filters are formed by selecting which element develops the output. The same two components can implement either a high-pass or a low-pass, depending on where the output is taken. This question checks if you can identify the correct node for a low-pass configuration in a series RC network. Given Data / Assumptions: Series RC branch driven by a source. Sinusoidal steady-state analysis. Ideal components and light loading (or buffered output). Concept / Approach: In a series RC, the resistor and capacitor form a voltage divider whose division ratio varies with frequency. The capacitor’s reactance is high at low frequency and low at high frequency. Taking the output across the capacitor provides more voltage at low frequencies and less at high frequencies, which is the definition of low-pass behavior. Conversely, taking the output across the resistor yields a high-pass response. Step-by-Step Solution: Form the divider: Vout = Vin * Z_out / (R + Z_C).If Z_out = Z_C, then Vout = Vin * (Z_C / (R + Z_C)).As frequency increases, |Z_C| decreases, so Vout decreases → low-pass response. Verification / Alternative check: At DC (omega = 0), |Z_C| is very large, so almost all the source voltage appears across the capacitor (Vout near Vin). At very high frequency, |Z_C| is very small, so most voltage is across the resistor and the capacitor output tends toward zero. This exactly matches a low-pass characteristic. Why Other Options Are Wrong: Across the resistor is a high-pass configuration. “Either node is identical” is incorrect; the transfer function depends on the node. A parallel RC is not required; a series RC suffices for first-order low-pass. Common Pitfalls: Forgetting that “output across C = low-pass” and “output across R = high-pass” in the series RC topology. Final Answer: Correct — take the output across the capacitor.

Question 19

Combining R and Xc — in a series RC with R = 100 Ω and capacitive reactance Xc = 100 Ω, what is the magnitude of the total impedance Z?

Accepted Answer

Introduction / Context: Calculating impedance magnitude in series RC and RL circuits requires vector addition, not simple arithmetic. This question verifies that you can combine resistance and reactance correctly using the Pythagorean relationship that arises from orthogonal phasor components. Given Data / Assumptions: Series RC circuit, sinusoidal steady state. R = 100 Ω, Xc = 100 Ω. Ideal components and no additional series elements. Concept / Approach: Impedance is complex: Z = R - jXc for series RC. The real part is R and the imaginary part is -Xc. The magnitude of a complex number a + jb is sqrt(a^2 + b^2). Therefore, the impedance magnitude is |Z| = sqrt(R^2 + Xc^2). This is analogous to the right triangle formed by resistive and reactive voltage drops or currents in phasor diagrams. Step-by-Step Solution: Write Z = 100 - j100.Compute |Z| = sqrt(100^2 + 100^2) = sqrt(10,000 + 10,000) = sqrt(20,000).Evaluate sqrt(20,000) ≈ 141.4 Ω (since 141.4^2 ≈ 20,000). Verification / Alternative check: If R equals Xc in magnitude, the impedance magnitude is R * sqrt(2). Here, 100 * 1.414 ≈ 141.4 Ω, matching the calculation. The phase angle is phi = arctan(Xc / R) = arctan(1) = 45 degrees (current leads), which is consistent but not required to get |Z|. Why Other Options Are Wrong: Adding magnitudes (200 Ω) ignores orthogonality; reactance is imaginary and must be combined vectorially. “100 Ω because reactance cancels” is false; cancellation does not occur between real and imaginary parts. “Undetermined” is incorrect; |Z| is computable without knowing the angle. Common Pitfalls: Adding R and Xc directly or trying to subtract them numerically, both of which disregard the complex nature of impedance. Final Answer: Correct — the impedance magnitude is approximately 141.4 Ω.

Question 20

Parallel RC circuits and phase behavior with frequency  Evaluate the statement: “If the excitation frequency increases in a parallel RC circuit, the phase angle between the source voltage and the total current decreases.” State whether this claim is…

Accepted Answer

Introduction / Context: In alternating-current (AC) circuit analysis, understanding how phase angle varies with frequency is fundamental for filters, sensor interfaces, and power systems. A parallel RC network exhibits a frequency-dependent current lead relative to the applied voltage. This question tests whether you know the direction of that change as frequency increases. Given Data / Assumptions: Topology: parallel RC circuit (a resistor R in parallel with a capacitor C). Source: sinusoidal voltage excitation. Ideal components (ignore capacitor ESR and wiring parasitics unless noted). Phase angle referenced between source voltage and total line current. Concept / Approach: The parallel RC is best analyzed using admittance Y. The total admittance is Y = G + jB where G = 1/R and B = ωC. The total current phasor leads the voltage by angle θ such that tan(θ) = B / G = (ωC) / (1/R) = ωCR. Because B increases linearly with frequency, the current’s lead angle increases with frequency. Step-by-Step Solution: Write Y = G + jB with G = 1/R and B = ωC.Compute the phase: θ = arctan(B/G) = arctan(ωCR).As frequency f (and ω = 2πf) increases, ωCR increases monotonically.Therefore θ increases (the total current leads the voltage by a larger angle). Verification / Alternative check: At very low frequency (ω → 0), B → 0 and θ → 0°, so the circuit behaves resistively. At very high frequency, B ≫ G and θ approaches 90°, dominated by capacitive current. This confirms that θ increases with frequency, not decreases. Why Other Options Are Wrong: Correct: contradicts admittance-phase behavior; θ does not decrease with frequency.Valid only at resonance: RC networks do not resonate without L; no resonance applies.Applies only when R = XC: that is a series condition for a specific phase in series RC; not relevant here.Zero ESR clause: ESR slightly perturbs G, but the trend (θ increasing with f) remains. Common Pitfalls: Confusing series vs. parallel behavior; mixing voltage-lead with current-lead language; assuming capacitor current behavior from series circuits applies identically to parallel circuits. Final Answer: Incorrect

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