Question 1

In AC circuit terminology, the “effective” value of a periodic voltage—equal to the DC value that would deliver the same average power to a resistor—is called the ______ value.

Accepted Answer

Introduction: The effective value of an AC quantity is the figure used for power calculations in resistive loads. Recognizing that this is the RMS value is fundamental to using meters and datasheets correctly. Given Data / Assumptions: Resistive load for definition of equivalence. Periodic waveform; for pure sine, Vrms = Vpeak / sqrt(2). Average power equivalence to DC is the reference concept. Concept / Approach: RMS (root mean square) is computed by squaring the instantaneous value, averaging over a period, and taking the square root. This process yields the DC-equivalent heating effect in a resistor and is why most AC meters report RMS or RMS-calibrated values. Step-by-Step Explanation: 1) Define effective requirement: same average power as a DC source.2) Use RMS definition to translate any periodic waveform to an equivalent DC magnitude.3) For a sine wave: Vrms = Vpeak / 1.414 and Irms = Ipeak / 1.414.4) Use RMS values in power formulas: P = Vrms * Irms for resistive loads. Verification / Alternative check: For a purely resistive load, Pavg = (1/T) * ∫ v^2(t)/R dt = Vrms^2 / R, identical in form to DC, confirming RMS as the effective value. Why Other Options Are Wrong: Average: for a pure sine, the algebraic average over a cycle is zero. Peak or peak-to-peak: helpful for headroom, not for power equivalence. Crest: refers to crest factor (Vpeak/Vrms), not an absolute value definition. Common Pitfalls: Using peak values in power calculations or confusing RMS with average responding meters calibrated for sine waves. Final Answer: root mean square

Question 2

In audio and acoustics, which device converts incident sound (pressure) waves into corresponding electrical signals suitable for amplification or recording?

Accepted Answer

Introduction: Transducers convert energy from one form to another. In audio systems, capturing acoustic energy and turning it into an electrical signal is the first step for measurement, processing, transmission, or recording. Given Data / Assumptions: We need a device that responds to sound pressure and produces an electrical output. Representative applications: live sound, studio recording, telephony, voice recognition. No requirement for amplification or radiation—only conversion from acoustic to electrical. Concept / Approach: A microphone is an electroacoustic transducer. Condenser (capacitor) mics vary capacitance with diaphragm movement, dynamic mics use electromagnetic induction with a moving coil, and piezoelectric mics use crystal-induced charge—each producing a corresponding voltage/current that mirrors the sound waveform. Step-by-Step Solution: 1) Identify the input energy domain: acoustic (pressure waves in air).2) Identify the required output domain: electrical (voltage/current).3) Map candidate devices: only a microphone performs this acoustic-to-electrical conversion.4) Conclude that a microphone is the correct choice. Verification / Alternative check: Reverse device (loudspeaker) converts electrical energy to sound; the microphone is its conceptual counterpart, confirming roles in the signal chain. Why Other Options Are Wrong: Amplifier: increases signal amplitude; does not perform energy-domain conversion. Antenna: converts electrical signals to electromagnetic radiation and vice versa, not sound. Filter: shapes spectra; no energy-domain conversion. Loudspeaker: converts electrical to acoustic (opposite direction). Common Pitfalls: Assuming an amplifier “creates” signal from sound. Without a transducer, the amplifier has nothing to amplify. Final Answer: a microphone

Question 3

For a sinusoidal waveform with peak value Vp = 12 V, what is the average (rectified) value over a half cycle (i.e., the mean of |v(t)| for a sine wave with Vp = 12 V)?

Accepted Answer

Introduction: This problem tests understanding of average (also called rectified average) for a sinusoidal waveform. For AC power and measurements, three common measures are used: peak value Vp, root-mean-square (rms) value Vrms, and average of the rectified waveform Vavg. Knowing how these relate is essential for converting between specification formats and instrument readings. Given Data / Assumptions: A pure sinusoidal voltage with peak value Vp = 12 V. The requested average is the mean of the absolute value over one full period (equivalently, the average over one half cycle of the positive sine). No DC offset is present. Concept / Approach: For a sine wave v(t) = Vp * sin(ωt), the average of |v(t)| over a period is Vavg = (2/π) * Vp. This is different from Vrms = Vp / √2 and the algebraic average over a full cycle, which is zero for a centered sine. The key constant for the rectified average is 2/π ≈ 0.637. Step-by-Step Solution: Use Vavg = (2/π) * VpCompute numerical factor: 2/π ≈ 0.637Multiply by Vp: Vavg = 0.637 * 12 VVavg ≈ 7.644 VRounded to two decimals: 7.64 V Verification / Alternative check: Compare with Vrms: Vrms = 12 / √2 ≈ 8.49 V, which is higher than the rectified average as expected. The full-cycle algebraic average is 0 V, reinforcing that the problem specifically asks for rectified average, not algebraic average. Why Other Options Are Wrong: 3.82 V or 4.24 V: Do not correspond to standard constants times 12 V. 9.42 V: Too high; would exceed Vrms which contradicts the known relationships. 12.0 V: The peak value, not the average of the rectified sine. Common Pitfalls: Confusing average-rectified value with rms value. Using the algebraic average over the full period (which is zero) for a centered sine wave. Rounding constants too early; keep precision until the end. Final Answer: 7.64 V

Question 4

In basic wave propagation, if the frequency of a radio wave is increased while the propagation speed remains constant (approximately c in free space), how does its wavelength change?

Accepted Answer

Introduction: This question probes the inverse relationship between frequency and wavelength for electromagnetic waves such as radio signals. Understanding this relationship is crucial for antenna design, spectrum planning, and interpreting datasheets for RF systems. Given Data / Assumptions: Propagation in free space or a fixed medium with constant phase velocity v (≈ c in free space). Frequency f is increased. We seek the qualitative change in wavelength λ. Concept / Approach: The fundamental relation is v = f * λ. For constant v, increasing f must reduce λ so that their product remains constant. This is an exact inverse proportionality: λ = v / f. Step-by-Step Solution: Start with v = f * λHold v constant (medium unchanged).Solve for λ: λ = v / fIf f increases, the denominator increases, so λ decreases. Verification / Alternative check: Example: In free space, v ≈ 3.0 * 10^8 m/s. At 100 MHz, λ = 3 m; at 200 MHz, λ = 1.5 m. Doubling frequency halves wavelength, confirming the inverse relationship. Why Other Options Are Wrong: Increase / remain the same: Contradicts λ = v / f for constant v. Cannot tell: The relationship is deterministic given constant propagation speed. First increase then decrease: No such behavior in a fixed medium. Common Pitfalls: Forgetting that this relation assumes constant medium and neglects dispersion; in many RF cases, the approximation is valid. Confusing group velocity with phase velocity; here we consider the standard phase relation. Final Answer: decrease

Question 5

Which statement best describes the waveform components of an ideal triangle wave in steady state?

Accepted Answer

Introduction: A triangle wave is a fundamental non-sinusoidal waveform used in modulation, timing, and function generators. Recognizing its time-domain structure helps in understanding its frequency spectrum and how it is produced by integrator circuits. Given Data / Assumptions: Ideal triangle wave centered around zero with symmetric rise and fall times. Constant slope segments (linear ramps). No DC offset or asymmetry unless specified. Concept / Approach: An ideal triangle wave consists of a linear rise followed by a linear fall of equal magnitude but opposite slope, repeating every period. This symmetry leads to a spectrum with only odd harmonics whose amplitudes decrease with the square of harmonic number, but the time-domain hallmark is equal positive and negative ramps. Step-by-Step Solution: Consider a comparator–integrator oscillator: constant comparator output levels drive the integrator with constant current.Constant current through the timing capacitor causes voltage to change linearly: dv/dt = I/C (constant).When thresholds are crossed, polarity reverses and the slope changes sign but not magnitude.Thus, the output is equal-magnitude positive and negative ramps forming a symmetric triangle. Verification / Alternative check: On an oscilloscope, measuring rise time and fall time over the same vertical excursion shows equality for an ideal triangle, validating equal ramp magnitudes and durations. Why Other Options Are Wrong: Positive-only or negative-only ramps: Would imply a sawtooth or unipolar waveform. Unequal ramps: Produces an asymmetric triangle or sawtooth, not the ideal triangle. Sinusoid with harmonics only: Describes a Fourier series perspective but not the direct time-domain ramp description requested. Common Pitfalls: Confusing triangle with sawtooth (which has unequal ramp durations or slopes). Ignoring DC offset or duty-cycle variations that break symmetry in real circuits. Final Answer: positive and negative ramps of equal value

Question 6

On a standard oscilloscope, which selector control lets you use the graticule’s major and minor time divisions to determine a waveform’s period and frequency?

Accepted Answer

Introduction: Oscilloscopes display voltage versus time using a calibrated grid (graticule). Correctly choosing the horizontal time scale is crucial for measuring period, frequency, and timing relationships like pulse width and rise time. Given Data / Assumptions: A general-purpose analog or digital oscilloscope with a timebase control labeled in time per division. Graticule with major and minor divisions uniformly spaced. Stable triggered display of the waveform. Concept / Approach: The time/cm (time/div) control sets the horizontal scale, specifying how much time each division represents. By counting divisions that span one period of the waveform and multiplying by the time/div setting, you obtain the period; the reciprocal gives frequency. Step-by-Step Solution: Set time/div to a value that displays several cycles clearly.Count the number of horizontal divisions for one complete cycle.Compute period: T = (divisions) * (time/div).Compute frequency: f = 1 / T. Verification / Alternative check: Compare with automatic frequency readout (if available) or use cursors to measure time between peaks; both methods should agree with the manual time/div approach within instrument accuracy. Why Other Options Are Wrong: Focus and intensity: Adjust trace sharpness and brightness, not time scaling. Volts/cm: Vertical scale used for amplitude measurements. Trigger level: Stabilizes the display but does not set time per division. Common Pitfalls: Mistaking seconds/div for milliseconds/div or microseconds/div; always check units. Measuring an incomplete cycle due to incorrect triggering. Final Answer: time/cm control

Question 7

In pulse and rectangular-wave terminology (e.g., radar or digital timing), which term explicitly expresses the frequency of a rectangular pulse train?

Accepted Answer

Introduction: Rectangular waves and pulse trains are described by several timing terms. Correctly distinguishing between frequency, period, and related descriptors is essential in communications, radar, and digital electronics. Given Data / Assumptions: Uniform pulse train with constant spacing between pulses. Standard radar/digital nomenclature: PRF and PRT. We seek the term that denotes frequency specifically. Concept / Approach: PRF stands for Pulse Repetition Frequency, the number of pulses per second. It is the reciprocal of PRT (Pulse Repetition Time), which is the time interval between successive pulses. While Hz is the unit of frequency, PRF is the named parameter used to specify the frequency of a pulse train in context. Step-by-Step Solution: Define PRF: PRF = pulses per second = 1 / PRT.Define PRT: the period between pulse starts (seconds).Relate to frequency units: PRF is measured in hertz (Hz).Thus the correct term for pulse frequency is PRF. Verification / Alternative check: Datasheets for radar, lidar, and ultrasonic rangefinders list PRF as a key specification for update rate and ambiguity; it always denotes pulses per second, confirming the terminology. Why Other Options Are Wrong: Hz: A unit, not the contextual term; PRF is the named parameter. Period: Time quantity, not frequency. PRT: Reciprocal of PRF; denotes time, not frequency. Duty cycle: Fraction of time the signal is high; independent of PRF for a given pulse width. Common Pitfalls: Confusing units (Hz) with named parameters (PRF). Mixing PRF with carrier frequency in pulse-modulated systems; they are distinct. Final Answer: PRF

Question 8

On an oscilloscope, which adjustable control is used to align (shift) the trace up/down or left/right so that it coincides with a chosen reference line on the graticule?

Accepted Answer

Introduction: Accurate oscilloscope measurements depend on correctly placing the waveform on the graticule so that amplitude and timing can be read from the scale. The control that repositions the trace without changing its scale is essential for this alignment. Given Data / Assumptions: A standard oscilloscope with separate horizontal and vertical position controls. Graticule with labeled divisions used as reference. We want to align the trace with a reference line without altering scale. Concept / Approach: The position control shifts the trace origin. Vertical position moves the baseline up or down; horizontal position shifts it left or right. This does not affect volts/div or time/div calibration—only the trace offset. Step-by-Step Solution: Choose the desired reference line (e.g., center horizontal line).Adjust vertical position to align the baseline or a key level with that line.If needed, adjust horizontal position to center a trigger point or feature.Verify that scale settings remain unchanged. Verification / Alternative check: Most oscilloscopes indicate the position offset on-screen (digital) or via knob detents (analog). Returning the position control to center returns the trace to its default alignment, confirming the function. Why Other Options Are Wrong: Focus: Adjusts sharpness, not position. Intensity: Adjusts brightness. Volts/cm and time/cm: Change scale, not position. Common Pitfalls: Misusing scale knobs when only offset is needed, thereby altering measurements. Forgetting separate channel position controls when multiple traces are displayed. Final Answer: position control

Question 9

Household mains supply is typically specified as 230 V AC (rms). For such a sinusoidal source, what is the peak voltage Vp delivered to a resistive appliance?

Accepted Answer

Introduction: AC mains specifications such as 230 V refer to rms (root-mean-square) values, which indicate the DC equivalent heating effect. Converting between rms and peak values for a sine wave is a routine task in power electronics and safety calculations. Given Data / Assumptions: Sine wave AC at Vrms = 230 V. Resistive load; no distortion assumed. We seek the peak value Vp. Concept / Approach: For a sinusoidal voltage: Vrms = Vp / √2, hence Vp = Vrms * √2. This relation arises from averaging the square of a sinusoid over a full cycle. Step-by-Step Solution: Use Vp = Vrms * √2Compute √2 ≈ 1.414Vp = 230 V * 1.414 ≈ 325.22 VRounded appropriately: Vp ≈ 325 V Verification / Alternative check: Peak-to-peak value Vpp = 2 * Vp ≈ 650 V, which is consistent with insulation rating considerations. Many appliance and rectifier designs reference these peak values when sizing components. Why Other Options Are Wrong: 163 V: This is 230 / √2 (the peak of a 115 Vrms system, not 230 Vrms). 230 V: The rms value, not the peak. 480 V and 650 V: 650 V is closer to peak-to-peak, not peak; 480 V is unrelated. Common Pitfalls: Confusing peak with rms or peak-to-peak. Forgetting that real mains may deviate slightly; calculations assume nominal values. Final Answer: 325 V

Question 10

In AC waveform terminology, what is the name of the magnitude by which an alternating quantity departs from zero (i.e., the maximum excursion from the reference level)?

Accepted Answer

Introduction: Describing waveforms precisely requires identifying amplitude, frequency, phase, and other parameters. This question focuses on the definition of amplitude, a foundational concept in signals and systems, audio engineering, and power analysis. Given Data / Assumptions: Alternating waveform centered about a reference (often zero). We refer to the maximum excursion from the reference, not peak-to-peak. Terminology follows standard electrical engineering conventions. Concept / Approach: Amplitude is the magnitude of maximum deviation from the reference level (often zero). For a sine v(t) = Vp * sin(ωt), the amplitude is Vp. Peak-to-peak is 2 * amplitude, and rms relates via Vrms = Vp / √2 for a sine. Correctly distinguishing these avoids measurement mistakes. Step-by-Step Solution: Identify the reference level (e.g., zero axis or average value).Measure the maximum distance from this level to the extreme of the waveform.That distance is the amplitude (often called peak value).Do not confuse with peak-to-peak, which spans both extremes. Verification / Alternative check: On an oscilloscope, amplitude is half the peak-to-peak vertical span when the waveform is centered on the reference line. Datasheets routinely specify amplitude or peak-to-peak depending on the application; conversions are straightforward for symmetric waveforms. Why Other Options Are Wrong: Altitude/attitude: Not standard electrical terms for waveform magnitude. Polarity: Indicates sign, not magnitude. Crest factor: Ratio of peak value to rms; not the magnitude itself. Common Pitfalls: Reporting peak-to-peak when amplitude is requested (factor of 2 error). Using rms interchangeably with amplitude, which is only true for specific waveforms with known relationships. Final Answer: amplitude

Question 11

When measuring a 50/60 Hz sine wave with a standard AC voltmeter designed for sinusoidal inputs, which value does the instrument display by default?

Accepted Answer

Introduction: AC voltmeters and multimeters commonly report a single representative value of a periodic voltage. For sinusoidal inputs, the conventional measure is the rms value, because it directly reflects the heating (power) equivalent of a DC voltage across a resistive load. Given Data / Assumptions: Input is a sinusoidal voltage at power-line frequency. Meter is a standard AC voltmeter or averaging-rectifier meter calibrated for sine waves, or a true-RMS meter. No significant waveform distortion. Concept / Approach: RMS (root-mean-square) is defined such that a voltage Vrms produces the same power in a resistor as a DC voltage of the same magnitude. Traditional analog meters measure average of the rectified sine and scale it to rms; modern DMMs compute true rms directly. In either case, the displayed value for a pure sine is Vrms. Step-by-Step Solution: For sine: Vrms = Vp / √2 and Vavg(rectified) = (2/π) * Vp.Meters calibrated for sine use the known ratio between Vrms and rectified average.Thus, the reading presented corresponds to Vrms.For non-sinusoidal waveforms, only a true-RMS meter reports correct rms. Verification / Alternative check: Compare measurements with an oscilloscope: compute Vrms from measured Vp as Vp/√2; values should match the meter reading within accuracy specs for a sine wave. Why Other Options Are Wrong: Average: Some meters measure this internally but display rms for sine calibration. Peak / peak-to-peak: Not displayed by standard voltmeters; these are oscilloscope quantities. Crest factor: A ratio, not a voltage magnitude. Common Pitfalls: Using an averaging meter on non-sinusoidal waveforms; readings will be in error. Confusing rms with peak values when sizing components. Final Answer: rms

Question 12

Electric power is generated at large plants, transmitted over long distances, and then delivered to residences. Which option best describes the overall voltage level change from generation through transmission to residential supply?

Accepted Answer

Introduction: Power systems use transformers and transmission lines to move energy efficiently from generating stations to end users. Understanding why voltages are stepped up and then stepped down is a core concept in power engineering and safety. Given Data / Assumptions: Generation at a plant, long-distance transmission, and local distribution to homes. Use of transformers for stepping voltage levels up or down. Goal: minimize losses while ensuring safe delivery. Concept / Approach: Transmission losses are proportional to I^2 * R. For a given power P, increasing voltage V reduces current I since P = V * I. Therefore, power is transmitted at high voltage (and correspondingly lower current) to reduce losses and conductor size. Near consumers, transformers step voltage down to safe levels (e.g., 230 V rms) for residential use. Step-by-Step Solution: At the plant, generators feed step-up transformers to raise voltage to transmission levels (tens to hundreds of kV).High-voltage lines carry power over long distances with reduced current and lower I^2 * R losses.Substations use step-down transformers to medium voltage for distribution networks.Local transformers further reduce to low-voltage service (e.g., 230/120 V) for residential loads. Verification / Alternative check: Utility grid diagrams and standards consistently show transmission at high voltages (e.g., 110 kV, 220 kV, 400 kV) and distribution to homes at low voltage. This architecture is universal because it is efficient and safe. Why Other Options Are Wrong: High→High or Low→Low: Do not reflect the necessary step-down for residential safety and appliance ratings. Low→High: Opposite of the actual overall path to homes. Medium→Medium: Not representative of the full generation-to-home journey. Common Pitfalls: Confusing transmission voltage levels with distribution service voltages. Assuming higher voltage always means more power; power depends on both voltage and current. Final Answer: high voltage to low voltage

Question 13

Oscilloscope fundamentals (CRT-based): What primary quantities does a cathode-ray oscilloscope (CRO/CRT oscilloscope) visually display on its screen for a typical time-domain measurement? Select the best pair that reflects what is directly observed …

Accepted Answer

Introduction: An analog CRT oscilloscope (often called a CRO) is a time-domain instrument. It plots a voltage as a function of time using a vertical deflection system (voltage) and a horizontal timebase (time). From this plot, we directly read amplitude and time-related quantities such as period; frequency is then inferred as the reciprocal of the measured period. Given Data / Assumptions: Standard time-domain setup with channel input driving vertical plates. Horizontal sweep set by the timebase (time/div). Graticule used to measure vertical and horizontal distances. Concept / Approach: The oscilloscope presents instantaneous voltage on the vertical axis versus time on the horizontal axis. By counting horizontal divisions for one complete cycle and multiplying by the time/div setting, we directly obtain the period. Amplitude is read vertically in volts/div. Frequency is a derived value: f = 1 / T, not a directly displayed quantity. Step-by-Step Solution: Identify vertical quantity: voltage (scaled by volts/div).Identify horizontal quantity: time (scaled by time/div).Measure one cycle width in divisions → compute period: T = divisions * time/div.Optionally compute frequency: f = 1 / T (derived, not directly displayed). Verification / Alternative check: Any CRO user manual describes measurements of amplitude and period directly from the screen. Built-in cursors (in DSOs) still operate on time and voltage axes, reinforcing that these are the directly displayed axes. Why Other Options Are Wrong: current and frequency: Current is not directly displayed unless converted to voltage via a shunt; frequency is derived. rms voltage and current: RMS requires calculation; the scope shows instantaneous waveform. frequency and voltage: Frequency is computed from period; period is the directly measured time quantity. phase angle and impedance: These require multi-channel or additional computations; not direct screen axes. Common Pitfalls: Confusing what is directly displayed (time and voltage) with values derived after calculation (frequency, RMS). Assuming current is shown without a transducer; scopes measure voltage. Final Answer: voltage and period

Question 14

Pulse measurements on oscilloscopes: For defining the rise time of a pulse waveform, what standard percentage limits of the full-amplitude transition are used?

Accepted Answer

Introduction: Rise time is a key specification for digital signals, oscilloscopes, and amplifiers. It characterizes how quickly a waveform transitions from low to high. Using consistent percentage thresholds ensures repeatable measurements across instruments and datasheets. Given Data / Assumptions: A pulse with defined low and high steady levels (full amplitude = Vhigh − Vlow). Measurements made on a time-domain instrument. Industry-standard definition desired. Concept / Approach: By convention, the rise time tr of a pulse is measured between 10% and 90% of the full amplitude. This avoids the flat extremes where noise and overshoot may dominate and provides a robust, comparable metric for bandwidth-related performance. Step-by-Step Solution: Find amplitude: A = Vhigh − Vlow.Compute 10% and 90% levels: V10 = Vlow + 0.1A, V90 = Vlow + 0.9A.Measure times t10 and t90 when waveform crosses these levels.Rise time: tr = t90 − t10 (the standard 10–90% metric). Verification / Alternative check: Most oscilloscope manuals and high-speed design texts define rise time as 10–90%. Some applications use 20–80% or 5–95%, but 10–90% remains the default for general-purpose characterization and bandwidth estimation (e.g., BW ≈ 0.35 / tr for single-pole systems). Why Other Options Are Wrong: 0–50% or 0–100%: Include regions dominated by offset, slow tails, or overshoot; nonstandard. 5–95% or 20–80%: Sometimes used, but not the most widely accepted standard. Common Pitfalls: Measuring to or from the wrong baseline when pulses ride on DC offsets. Ignoring probe bandwidth and scope rise time, which can artificially lengthen the measured transition. Final Answer: 10 to 90 percent

Question 15

Reading period from the oscilloscope: A square wave occupies 3 major horizontal divisions per cycle on the screen, and the time scale is set to 50 ms per division (time/cm = 50 ms). What is the waveform period?

Accepted Answer

Introduction: Oscilloscope time measurements rely on counting divisions across the graticule and multiplying by the timebase setting (time per division). This provides a direct way to determine the period T of a repetitive waveform and, by inversion, its frequency f = 1/T. Given Data / Assumptions: Horizontal span per division: 50 ms/div (time/cm = 50 ms). Waveform spans 3 major divisions per cycle. Signal is stable and one complete cycle is clearly visible. Concept / Approach: The period equals the number of divisions per cycle multiplied by the time per division. Once T is known, frequency can be derived if needed. This is the most fundamental oscilloscope time measurement technique. Step-by-Step Solution: Identify divisions per cycle: N = 3 divisions.Time per division: Td = 50 ms/div.Compute period: T = N * Td = 3 * 50 ms = 150 ms.Optional frequency: f = 1 / T ≈ 6.67 Hz. Verification / Alternative check: Measuring two or more cycles and dividing by the number of cycles gives the same period and can reduce reading error if the trace is noisy or the trigger is marginal. Why Other Options Are Wrong: 50 ms / 100 ms / 200 ms / 75 ms: Each assumes a different number of divisions or time/div; none matches 3 * 50 ms. Common Pitfalls: Mistaking minor for major divisions; always confirm the scale marks. Reading from an oblique or distorted trace; adjust focus and intensity for clarity. Final Answer: 150 ms

Question 16

Sine-wave phase reference: At which phase angle does a standard sine wave reach its maximum positive value relative to its zero-phase reference?

Accepted Answer

Introduction: Phase angle descriptions help locate key points on a sinusoid. Knowing where maxima and minima occur is essential for phasor analysis, AC power, and signal timing in communication systems. Given Data / Assumptions: Reference sine wave: v(θ) = Vm * sin(θ). Phase angle θ measured in degrees from 0° to 360°. Conventional positive maximum sought. Concept / Approach: The sine function attains its maximum value +1 at θ = 90°. Therefore the waveform’s maximum positive amplitude occurs at 90°. At 270° (or −90°), the sine reaches −1 (negative peak). At 0° and 180°, the waveform crosses zero and reaches zero, respectively, but not the maximum. Step-by-Step Solution: Consider v(θ) = Vm * sin(θ).sin(0°) = 0 → zero crossing.sin(90°) = +1 → maximum positive value Vm.sin(270°) = −1 → maximum negative value −Vm. Verification / Alternative check: Graph of a sine wave or unit-circle evaluation confirms that the peak positive amplitude is at 90°. A derivative check shows dv/dθ = Vm * cos(θ) = 0 at 90°, indicating an extremum; second derivative test confirms it is a maximum. Why Other Options Are Wrong: 0° and 360°: Zero value (start/end of cycle), not maximum. −90°: Negative peak, not positive. 180°: Another zero crossing with negative slope. Common Pitfalls: Confusing sine with cosine; cosine has its maximum at 0°. Mixing degree and radian measures; 90° corresponds to π/2 radians. Final Answer: 90°

Question 17

Propagation time of electromagnetic waves: Approximately how long does it take an electromagnetic wave to travel from a transmitter to a receiver 1,000 miles away (assume free-space speed of light)?

Accepted Answer

Introduction: Radio and microwave systems rely on finite propagation speed—approximately the speed of light in free space. Estimating one-way delay is vital for ranging, synchronization, and understanding latency in long-distance links. Given Data / Assumptions: Distance D = 1,000 miles (straight-line, free space). Propagation speed c ≈ 186,000 miles/s (≈ 3.00 * 10^8 m/s). Neglect atmospheric or medium slowdowns. Concept / Approach: Travel time t equals distance divided by speed: t = D / c. Using miles keeps arithmetic simple for the given distance. The result is converted to milliseconds for readability. Step-by-Step Solution: Write t = D / c.Substitute: t = 1,000 miles / 186,000 miles/s.Compute: t ≈ 0.005376 s.Convert to milliseconds: t ≈ 5.376 ms ≈ 5.38 ms. Verification / Alternative check: Using SI: 1,000 miles ≈ 1,609 km; c ≈ 300,000 km/s → t ≈ 1,609 / 300,000 s ≈ 0.005363 s (close to the miles-based result). Minor differences stem from rounding c and unit conversions. Why Other Options Are Wrong: 10.8 ms / 53.8 ms / 108 ms: Correspond to longer distances or slower speeds than free-space c. 0.538 ms: Ten times too small; would imply ~10,000 miles in 5.38 ms. Common Pitfalls: Confusing one-way delay with round-trip time (RTT); RTT would be roughly twice the one-way value. Forgetting that propagation in cables or atmosphere is slightly slower than c, increasing delay. Final Answer: 5.38 ms

Question 18

High-frequency period calculation: What is the period (time for one cycle) of a 16 MHz sine wave?

Accepted Answer

Introduction: Converting between frequency and period is fundamental in signal processing, microcontroller timing, and communications. The period is the reciprocal of frequency: higher frequency means shorter period. Given Data / Assumptions: Frequency f = 16 MHz = 16 * 10^6 Hz. Ideal single-tone sine wave. We need T in seconds (or nanoseconds). Concept / Approach: The relationship is T = 1 / f. After computing T in seconds, convert to nanoseconds by multiplying by 10^9. This quick conversion is widely used for clock signals in digital electronics. Step-by-Step Solution: Compute T = 1 / f = 1 / (16 * 10^6) s.T = 0.0625 * 10^-6 s.Convert to nanoseconds: T = 62.5 ns.Thus, one cycle of a 16 MHz signal lasts 62.5 ns. Verification / Alternative check: Check by multiplication: 16 * 10^6 Hz * 62.5 * 10^-9 s = 1.0 (dimensionless), confirming consistency. Also, 8 MHz would be 125 ns; doubling frequency halves period, aligning with 16 MHz → 62.5 ns. Why Other Options Are Wrong: 196 ns / 31.25 ns / 19.9 ns / 6.25 ns: Do not equal 1 / 16 MHz; some correspond to other frequencies (e.g., 32 MHz → 31.25 ns). Common Pitfalls: Misplacing decimal points during unit conversion between microseconds and nanoseconds. Confusing MHz (10^6) with kHz (10^3) or GHz (10^9). Final Answer: 62.5 ns

Question 19

RMS-to-peak-to-peak conversion (sine wave): What is the peak-to-peak voltage of a sinusoidal AC waveform whose RMS value is 56 V?

Accepted Answer

Introduction: Converting between RMS, peak, and peak-to-peak values is essential for power calculations, component voltage ratings, and oscilloscope interpretation. For a pure sine wave, fixed ratios relate these quantities. Given Data / Assumptions: Sine wave with Vrms = 56 V. Standard sine relationships apply. No distortion or DC offset. Concept / Approach: For a sine wave: Vrms = Vp / √2, so Vp = Vrms * √2. Peak-to-peak is Vpp = 2 * Vp. Compute Vp, then double it to obtain Vpp. Keep sufficient precision for rounding to the nearest integer volt if options are whole numbers. Step-by-Step Solution: Compute peak: Vp = Vrms * √2 = 56 * 1.41421356 ≈ 79.2 V.Compute peak-to-peak: Vpp = 2 * Vp ≈ 158.4 V.Rounded to the nearest volt: Vpp ≈ 158 V.Therefore, the correct option is 158 V. Verification / Alternative check: Reverse check: If Vpp = 158 V ⇒ Vp ≈ 79 V, then Vrms ≈ 79 / √2 ≈ 55.9 V, which rounds to 56 V, confirming consistency. Why Other Options Are Wrong: 164 V: Would imply Vrms ≈ 58 V; too high. 82 V: That is approximately Vp, not Vpp. 79 V: Exactly the peak, not peak-to-peak. 112 V: Corresponds to 2 * Vrms, not a standard sine relation. Common Pitfalls: Confusing peak with peak-to-peak (factor of 2). Applying Vrms = Vavg (true only for DC) or mixing rectified-wave formulas. Final Answer: 158 V

Alternating Current vs Direct Current Questions