Q: For an RC differentiator (series C, shunt R), what must the instantaneous sum of the capacitor voltage and the resistor voltage equal?

Introduction / Context: KVL (Kirchhoff's Voltage Law) governs series circuits, including the classic RC differentiator configuration. Recognizing how the individual element voltages relate to the source is fundamental to predicting output behavior during rising and falling edges. Given Data / Assumptions: Series connection: input source → capacitor → resistor → return. Output is across R, but analysis concerns instantaneous voltages across C and R. Linear time-invariant behavior; no parasitics considered beyond RC. Concept / Approach: In any single-loop series circuit, KVL states that the algebraic sum of voltages around the loop is zero. Equivalently, the source voltage equals the sum of the drops across the series elements at every instant in time, regardless of transient or steady-state conditions. Step-by-Step Solution: Let v_in(t) be the source, v_C(t) the capacitor voltage, and v_R(t) the resistor voltage.KVL: v_in(t) - v_C(t) - v_R(t) = 0.Rearrange: v_C(t) + v_R(t) = v_in(t).Therefore, the instantaneous sum of the capacitor and resistor voltages equals the applied input voltage. Verification / Alternative check: SPICE simulations of a differentiator responding to a pulse show that at each time point, v_C + v_R tracks the excitation exactly (subject to numerical precision), confirming KVL in time domain. Why Other Options Are Wrong: Zero: Only true if the source were zero or defined with opposite polarity and summation convention. Less than the applied voltage: Violates KVL for a single series loop. Cannot be determined: It is precisely determined by KVL. Common Pitfalls: Dropping sign conventions and mixing polarities when summing voltages. Assuming energy storage elements 'hide' voltage; KVL still applies instant by instant. Final Answer: The applied input voltage

Q: In first-order circuits, when do we say a steady-state condition has been reached during a DC step response?

Introduction / Context: Steady state in basic RC/RL circuits typically refers to conditions after all transients have died out. For a DC step input, energy storage elements settle to constant voltages/currents and derivatives go to zero. This question targets that definition. Given Data / Assumptions: DC step input applied to a first-order circuit (e.g., charging a capacitor through a resistor). No leakage (unless stated), linear components, and ideal source. Concept / Approach: During a DC step, the forced response is a constant value, and the natural response decays exponentially with time constant tau. Steady state is reached as t → ∞, when the exponential term decays to zero. In a simple RC charge with a step from 0 to V_in, the capacitor voltage tends to V_in; thus output equals input (for the node across the capacitor) in steady state. Step-by-Step Solution: For a step from 0 to V_in: v_C(t) = V_in * (1 - exp(-t / tau)).As t → ∞, exp(-t / tau) → 0.Therefore, v_C(∞) = V_in, indicating steady state equals the input value.Similar logic applies for RL circuits where the inductor current settles to V_in / R in the long term. Verification / Alternative check: After ~5 tau, first-order responses are within ~1% of final value, commonly used as a practical steady-state criterion. Simulation or lab measurement confirms negligible change beyond this horizon. Why Other Options Are Wrong: Average value: Relevant to periodic waveforms, not to a DC step. 63% of input: That is the one-time-constant point, not steady state. Effective (rms) value: Pertains to AC; a DC final value equals the DC amplitude. Common Pitfalls: Confusing the 63% milestone with completion of the transient. Mixing AC metrics (rms, average over a cycle) with DC step responses. Final Answer: When the output voltage reaches the input voltage (for a DC step)

Q: In an RL differentiator, when the input pulse transitions from its low level to its high level, which statement is correct?

Introduction / Context: Inductors resist changes in current, which is central to how RL differentiators produce sharp voltage spikes at edges. Recognizing this physical property helps predict transient behavior in pulse circuits. Given Data / Assumptions: Series RL with output typically taken across the resistor (differentiator behavior for small tau). Input undergoes a rising step (pulse goes from low to high). Concept / Approach: Inductor voltage is v_L = L * di/dt. A finite inductance cannot instantaneously change current without requiring infinite voltage. Therefore, at the instant of a step, current through the inductor remains (momentarily) at its prior value, and the required voltage appears to support a rapid change in current thereafter. Step-by-Step Solution: At t = 0+, input steps upward.Current through L: i_L cannot jump; it begins changing from its initial value with slope di/dt = v_L / L.Large induced v_L appears briefly to force current to change toward the new steady state set by the source and R.Thus, the inductor prevents sudden change in current, not voltage. Verification / Alternative check: Ideal inductor i_L is continuous; measured waveforms show resistor voltage spikes ±V proportional to di/dt. Simulations or scope captures confirm that i_L ramps rather than jumps at the transition. Why Other Options Are Wrong: Prevents sudden change in voltage: That is the property of a capacitor, not an inductor. Instantly reaches 63%: 63% relates to one time constant, not an instantaneous effect. Voltage across inductor is zero: At an edge, v_L is typically largest, not zero. Common Pitfalls: Confusing capacitor and inductor edge properties. Assuming the 63% rule applies at t = 0 rather than at t = tau. Final Answer: The inductor prevents a sudden change in current

Q: An RC integrator is driven by a 24 V input pulse whose width equals five time constants (t = 5RC). To what voltage does the capacitor charge by the end of the pulse?

Introduction / Context: First-order charging behavior dictates how close an RC node gets to its final value within a given number of time constants. This question uses the canonical result at five time constants for a DC step/pulse of sufficient width. Given Data / Assumptions: RC integrator receiving a 24 V pulse. Pulse width equals 5 * tau where tau = R * C. Initial capacitor voltage assumed 0 V (uncharged) at the pulse start. Concept / Approach: For a step of amplitude V applied to an RC from rest, the capacitor voltage during charging is v_C(t) = V * (1 - exp(-t / tau)). At t = 5 * tau, the exponential term exp(-5) ≈ 0.006737, so the node is within about 0.67% of its final value V. Step-by-Step Solution: Use v_C(5tau) = 24 * (1 - exp(-5)).Compute exp(-5) ≈ 0.006737.So v_C ≈ 24 * (1 - 0.006737) = 24 * 0.993263 ≈ 23.838 V.Rounded to given choices, this is effectively 24 V. Verification / Alternative check: A common engineering rule: after 5 tau, a first-order system reaches ~99.3% of its final value. For 24 V, that is ≈ 23.84 V, which rounds to 24 V in typical multiple-choice contexts. Why Other Options Are Wrong: 15.12 V: Corresponds to ~63% of 24 V (one time constant), not five. 20.64 V: ~86% (two time constants), still far from 99.3%. 12 V: Exactly half the target; unrelated to standard time-constant milestones. Common Pitfalls: Using 1 - 1/5 instead of the exponential relation. Confusing 5 tau with 5% error rather than 0.67% error. Final Answer: 24 V

Q: To understand how a differentiator shapes its output, which parts of the pulse response must be analyzed?

Introduction / Context: Differentiators emphasize changes in the input signal. For pulse waveforms, the output is dominated by edge behavior, but the inter-edge interval also affects baseline and droop. This question probes comprehensive reasoning about the entire pulse cycle. Given Data / Assumptions: First-order RC or RL differentiator operating on rectangular pulses. Finite time constants and practical component values. Concept / Approach: The output to a pulse includes a positive spike at the rising edge (for standard polarity), a transient settling toward baseline between edges (with exponential decay governed by time constant), and a negative spike at the falling edge. Ignoring any portion yields an incomplete understanding of amplitude, symmetry, and baseline shift. Step-by-Step Solution: Rising edge: Large dv/dt produces a spike whose polarity follows the edge direction and whose initial magnitude reflects the abrupt step.Between edges: The output decays exponentially toward baseline; the decay rate depends on RC or L/R, affecting pulse-to-pulse overlap and droop.Falling edge: Opposite-polarity spike appears as the input steps back; its shape mirrors the rising-edge response in magnitude (for symmetric pulses). Verification / Alternative check: Time-domain convolution (input with differentiator impulse response) or Laplace-domain multiplication (H(s) with input step pairs) shows contributions at each interval, confirming all segments matter to the net waveform. Why Other Options Are Wrong: Considering only one edge or only the middle interval misses essential dynamics; all three regions determine output symmetry, amplitude, and baseline. Common Pitfalls: Assuming ideal impulses with zero-width edges, which hides practical finite-time constant effects. Neglecting recovery time between edges, leading to baseline shift and distortion. Final Answer: All of the above

Q: RC integrator design: A 15 V input pulse having a width equal to two time constants is applied to an RC integrator. To what voltage will the capacitor charge by the end of the pulse?

Introduction / Context: This problem checks time-domain charging behavior of a first-order RC integrator when it is driven by a finite-width rectangular pulse. Knowing how far a capacitor charges during a pulse is essential for pulse shaping, timing, and analog signal conditioning design. Given Data / Assumptions: Input pulse amplitude, Vin = 15 V. Pulse width equals two time constants, t = 2 * tau. Ideal RC integrator, no leakage or series parasitics beyond R and C. Capacitor initially uncharged just before the pulse edge (worst-case rise computation). Concept / Approach: For a step of amplitude Vin applied across an RC, the capacitor voltage follows vC(t) = Vin * (1 - e^(−t/tau)). For a finite-width pulse of duration t, the capacitor reaches vC(t) at the end of the pulse, provided the RC remains in its charging phase throughout that interval. Step-by-Step Solution: Given t = 2 * tauvC(t) = Vin * (1 - e^(−t/tau))vC(2 * tau) = 15 * (1 - e^(−2))e^(−2) ≈ 0.1353 ⇒ 1 − e^(−2) ≈ 0.8647vC ≈ 15 * 0.8647 ≈ 12.97 V ≈ 12.9 V Verification / Alternative check: As a rule of thumb, after 1 tau a capacitor reaches ~63.2% of its final value; after 2 tau it reaches ~86.5%. For 15 V, 0.865 * 15 ≈ 12.97 V, confirming the calculation. Why Other Options Are Wrong: 19.45 V and 15 V: Both exceed or equal the source; a passive RC cannot charge above Vin in this simple case. 8.6 V: Would correspond to approximately 1 tau, not 2 tau. Common Pitfalls: Confusing one time constant with two, or forgetting to apply the exponential factor e^(−t/tau). Always convert the pulse width to multiples of tau before applying the formula. Final Answer: 12.9 V

Question 1

Average output of an RC differentiator with repetitive pulses: What is the average (DC) value of the output waveform?

Accepted Answer

Introduction / Context: RC differentiators are used to emphasize rapid changes (edges) in input signals. When driven by repetitive pulses, the output consists of positive and negative spikes around transitions, with little average value over one period. Given Data / Assumptions: RC differentiator configuration. Input is a repetitive pulse train (periodic). Capacitor charges and discharges around transitions; steady segments yield near-zero output. Concept / Approach: A differentiator's output approximates the time-derivative of the input. Over each complete period of a periodic waveform, the average of its derivative is zero, since the waveform returns to the same value every cycle. Step-by-Step Reasoning: Let v_out ≈ RC * d(v_in)/dtAverage over one period T: (1/T) * ∫_0^T d(v_in)/dt dt = (1/T) * (v_in(T) - v_in(0))For periodic input, v_in(T) = v_in(0) ⇒ average derivative = 0Hence the average output ≈ 0 V Verification / Alternative check: On an oscilloscope, you typically see equal-area positive and negative spikes at rising and falling edges. Their net area per period is zero, confirming zero average. Why Other Options Are Wrong: Equal to the input or 63% of input: These refer to steady values in charging systems, not a differentiator's average. Cannot be determined: It can—by periodicity and calculus. Common Pitfalls: Confusing differentiators with integrators, or assuming a DC offset from asymmetrical pulses (standard symmetric pulses give zero average in ideal differentiators). Final Answer: is zero

Question 2

Capacitor behavior with long pulses: For a periodic pulse train where pulse width and the time between pulses are each ≥ 5 time constants, how does the capacitor charge and discharge each period?

Accepted Answer

Introduction / Context: Pulse-response in RC networks depends on the relation between time constants and pulse timing. If both the on-time and off-time are sufficiently long, the capacitor reaches its steady limits in each half of the cycle. Given Data / Assumptions: Pulse width ≥ 5τ and spacing ≥ 5τ (τ = RC). Standard first-order RC behavior. Ideal rectangular pulses. Concept / Approach: A first-order RC charges toward the applied level with the law v(t) = Vfinal * (1 - e^{-t/RC}). By t = 5τ, the exponential term e^{-5} ≈ 0.0067, meaning the capacitor is within 0.67% of its final value—effectively fully charged or discharged for practical purposes. Step-by-Step Reasoning: During pulse (≥ 5τ): v_C ≈ V_high (fully charged)Between pulses (≥ 5τ): v_C ≈ 0 or the low reference (fully discharged)Therefore, each period contains both full charge and full discharge events Verification / Alternative check: Using 1 - e^{-5} ≈ 0.9933 indicates a charge to 99.33% of the final value. Similarly, discharge leaves only 0.67%, which is practically zero. Why Other Options Are Wrong: Partial charge/discharge options contradict the ≥ 5τ condition. Common Pitfalls: Misinterpreting “fully” as exactly 100%. In engineering practice, ≥ 5τ is considered effectively full for time-domain design and timing calculations. Final Answer: fully charge and fully discharge during each period of the input waveform

Question 3

Edge-frequency estimate: If a pulse has a 10 µs rise time and 10 µs fall time, what is the highest significant frequency contained in the waveform?

Accepted Answer

Introduction / Context: Rise time is linked to the bandwidth of a system or equivalently to the highest significant frequency content in a pulse. A widely used rule-of-thumb relates the bandwidth to the rise time for first-order systems and many practical pulses. Given Data / Assumptions: Rise time Tr = 10 µs (and similar fall time). We are estimating the significant highest frequency component. Assume approximately first-order edge shaping. Concept / Approach: The common approximation is BW ≈ 0.35 / Tr, where Tr is the 10%–90% rise time (seconds) and BW is the -3 dB bandwidth (Hz). This corresponds to the frequency content needed to support that edge sharpness. Step-by-Step Solution: Tr = 10 µs = 10 × 10^-6 sBW ≈ 0.35 / Tr = 0.35 / 10e-6BW ≈ 35,000 Hz = 35 kHz Verification / Alternative check: For a 1 µs rise time, the estimate would be ~350 kHz; scaling Tr by 10 increases the rise time and reduces the bandwidth by 10, hence 35 kHz for 10 µs fits the rule. Why Other Options Are Wrong: 3.5 kHz or 10 kHz: Too low for a 10 µs edge. 100 kHz: Too high; would correspond to shorter rise times around 3.5 µs. Common Pitfalls: Confusing the 0.35 constant (rise time rule) with the 0.707 factor from -3 dB magnitude, or mis-converting microseconds to seconds. Always express Tr in seconds before applying the formula. Final Answer: 35 kHz

Question 4

Instantaneous voltage division in an RL integrator: At the instant of a rising pulse edge, how is the input voltage divided between R and L?

Accepted Answer

Introduction / Context: RL integrators respond to fast edges by momentarily placing most of the input across the inductor because inductors initially oppose changes in current. Understanding the instantaneous division right at the edge is key for predicting spike shapes and amplitudes. Given Data / Assumptions: RL integrator configuration (series R and L). Observe behavior at t = 0+ for a rising step (pulse edge). Ideal components; no core saturation or parasitics. Concept / Approach: At the instant of a rising edge, the current through an inductor cannot change instantaneously. Therefore i_L(0+) ≈ i_L(0-) and initially the current is essentially the pre-edge value. With nearly zero change in current at t = 0+, the resistor drop v_R = i * R is near zero, so the inductor must support almost the entire applied voltage. Step-by-Step Reasoning: At t = 0+: di/dt is large, but i itself has not yet increasedv_R(0+) = i(0+) * R ≈ 0v_L(0+) = Vin - v_R(0+) ≈ VinTherefore, the input is across L at the instant of the rising edge Verification / Alternative check: Time constants: i(t) rises with τ = L/R. Only after a finite time will current build and a significant voltage appear across R; initially it is negligible. Why Other Options Are Wrong: All across R or 63% splits: These describe later times, not the instant of the edge. Common Pitfalls: Confusing immediate (t = 0+) conditions with steady-state values or mixing RC intuition with RL behavior. Inductors initially behave like opens to current changes, forcing the applied voltage across themselves. Final Answer: all the input voltage is across the inductor

Question 5

In an RC integrator used for pulse shaping, what happens if the capacitor becomes leaky (i.e., exhibits significant leakage resistance)?

Accepted Answer

Introduction / Context: An RC integrator ideally uses a high-quality capacitor and a resistor to produce an output proportional to the time integral of the input (for appropriate RC/pulse-width ratios). Real capacitors, however, have leakage (modeled as a large but finite resistance in parallel with the capacitor). This question tests the practical impact of leakage on an RC integrator's performance. Given Data / Assumptions: An RC integrator with series R and shunt C. The capacitor exhibits leakage, represented by a finite parallel resistance R_leak across C. Pulse or varying input signal; linear, small-signal analysis assumptions. Concept / Approach: The ideal time constant is tau_ideal = R * C. With leakage, the effective impedance across the capacitor is Z_C || R_leak, so the effective time constant becomes tau_eff = R * C_eff, where C_eff is reduced at low frequencies because the parallel path R_leak bleeds charge. This both reduces the effective time constant and alters the transfer function magnitude/phase, distorting amplitude and waveshape. Step-by-Step Solution: Model leakage: Replace C by C in parallel with R_leak.At low frequencies and during hold intervals, current through R_leak discharges the capacitor faster than ideal.Effective time constant: tau_eff < tau_ideal because the parallel leakage reduces the storage effectiveness of C.Amplitude effect: For a given input, less stored charge means lower peak output and increased droop, so amplitude decreases.Waveshape effect: Additional discharge path reshapes the output (rounding, droop between pulses, and reduced integration quality). Verification / Alternative check: Consider the Thevenin equivalent looking into the capacitor node. With R_leak present, the DC gain is reduced because the node cannot hold charge; time-domain step responses show faster decay toward ground, confirming a reduced effective time constant and lower peaks. Why Other Options Are Wrong: 'The time constant will be effectively reduced': True. 'The waveshape of the output voltage across the capacitor is altered': True, due to droop and distortion. 'The amplitude of the output is reduced': True, peaks fall because charge leaks away. Common Pitfalls: Assuming leakage only affects DC and not transient behavior. Ignoring that even large R_leak values can matter when pulse repetition intervals are long relative to tau. Final Answer: All of the above

Question 6

In an RC integrator, if the RC time constant is increased while the input pulse width remains the same, how do charge and discharge behaviors change during and between pulses?

Accepted Answer

Introduction / Context: RC integrators rely on a proper relationship between the RC time constant and the pulse width of the input signal. This question examines how increasing the time constant affects the capacitor's charge and discharge behavior. Given Data / Assumptions: An RC integrator receiving repetitive pulses of fixed width and amplitude. The RC time constant tau = R * C is increased (by raising R, C, or both). Pulse width and repetition period remain unchanged. Concept / Approach: For a fixed pulse width T_p, the fraction of full charge achieved during a single pulse is 1 - exp(-T_p / tau). As tau increases, T_p / tau decreases, thus the term 1 - exp(-T_p / tau) becomes smaller, meaning the capacitor charges less during the pulse. Similarly, between pulses, discharge follows exp(-T_gap / tau). Larger tau produces a slower discharge (i.e., discharges less between pulses). Step-by-Step Solution: Charge fraction during the pulse: Q_frac = 1 - exp(-T_p / tau).Increase tau ⇒ T_p / tau decreases ⇒ Q_frac decreases ⇒ less charge gained during each pulse.Discharge fraction between pulses: D_frac = exp(-T_gap / tau).Increase tau ⇒ T_gap / tau decreases ⇒ D_frac increases toward 1 ⇒ voltage decays more slowly ⇒ less discharge between pulses.Net effect: Smaller ripple and a smoother (but lower-slope) integrated output. Verification / Alternative check: Numerical example: For T_p = 1 ms, tau = 1 ms gives charge fraction ≈ 1 - e^-1 ≈ 0.632. If tau = 5 ms, charge fraction ≈ 1 - e^-0.2 ≈ 0.181 (much less). The discharge fraction over a fixed gap also becomes slower with larger tau, confirming less discharge. Why Other Options Are Wrong: Charges more / discharges more: Opposite of the exponential behavior when tau is increased. Charges more / discharges less: Only half correct; the charge part is wrong. Charges less / discharges more: Discharge statement is incorrect for larger tau. Common Pitfalls: Confusing the effects of increasing tau with increasing pulse width. Assuming symmetrical effects without considering exponential time constants. Final Answer: The capacitor charges less during a pulse and discharges less between pulses

Question 7

In an RC differentiator, which statement best describes the capacitor's charging behavior?

Accepted Answer

Introduction / Context: An RC differentiator produces an output proportional to the rate of change of the input when the RC time constant is much smaller than the signal's time scale. Understanding how the capacitor charges in response to a step or edge is key to predicting the waveform spikes. Given Data / Assumptions: RC differentiator: series capacitor, shunt resistor, output taken across the resistor. Consider response to a step/rising edge to discuss charging behavior. Linear components; no saturation. Concept / Approach: For a step change, the current through the capacitor is i_C = C * dV_in/dt. Immediately after the step, a large current flows, and the node voltages adjust per KVL/KCL. The capacitor voltage changes exponentially toward its new value with time constant tau = R * C, even in the differentiator topology (because the network remains first order). Step-by-Step Solution: For a step input from 0 to V_s at t = 0+, the initial capacitor voltage cannot change instantaneously, so V_C(0+) = 0.The subsequent evolution follows V_C(t) = V_s * (1 - exp(-t / tau)) in the canonical charging scenario.Thus, the charge trajectory is exponential with time constant tau = R * C, not purely dependent on input amplitude alone.Therefore, the correct qualitative statement is that the capacitor charges exponentially with a rate determined by RC. Verification / Alternative check: Laplace-domain analysis of the differentiator transfer function H(s) = sRC / (1 + sRC) shows a first-order pole at s = -1/(RC). Time-domain responses to steps necessarily follow exponential laws governed by that pole. Why Other Options Are Wrong: Only input amplitude: Ignores the RC time constant that sets the rate. Charges when input is decreasing: For a falling edge, the capacitor discharges (or charges to opposite polarity); the statement is misleading without context. Charges to approximately one time constant: Time constant has units of time, not voltage. Common Pitfalls: Confusing differentiator small-signal behavior with exact edge dynamics. Interpreting 'time constant' as a voltage level instead of a time scale. Final Answer: The capacitor charges exponentially at a rate depending on the RC time constant

Question 8

For an RC differentiator (series C, shunt R), what must the instantaneous sum of the capacitor voltage and the resistor voltage equal?

Accepted Answer

Introduction / Context: KVL (Kirchhoff's Voltage Law) governs series circuits, including the classic RC differentiator configuration. Recognizing how the individual element voltages relate to the source is fundamental to predicting output behavior during rising and falling edges. Given Data / Assumptions: Series connection: input source → capacitor → resistor → return. Output is across R, but analysis concerns instantaneous voltages across C and R. Linear time-invariant behavior; no parasitics considered beyond RC. Concept / Approach: In any single-loop series circuit, KVL states that the algebraic sum of voltages around the loop is zero. Equivalently, the source voltage equals the sum of the drops across the series elements at every instant in time, regardless of transient or steady-state conditions. Step-by-Step Solution: Let v_in(t) be the source, v_C(t) the capacitor voltage, and v_R(t) the resistor voltage.KVL: v_in(t) - v_C(t) - v_R(t) = 0.Rearrange: v_C(t) + v_R(t) = v_in(t).Therefore, the instantaneous sum of the capacitor and resistor voltages equals the applied input voltage. Verification / Alternative check: SPICE simulations of a differentiator responding to a pulse show that at each time point, v_C + v_R tracks the excitation exactly (subject to numerical precision), confirming KVL in time domain. Why Other Options Are Wrong: Zero: Only true if the source were zero or defined with opposite polarity and summation convention. Less than the applied voltage: Violates KVL for a single series loop. Cannot be determined: It is precisely determined by KVL. Common Pitfalls: Dropping sign conventions and mixing polarities when summing voltages. Assuming energy storage elements 'hide' voltage; KVL still applies instant by instant. Final Answer: The applied input voltage

Question 9

In first-order circuits, when do we say a steady-state condition has been reached during a DC step response?

Accepted Answer

Introduction / Context: Steady state in basic RC/RL circuits typically refers to conditions after all transients have died out. For a DC step input, energy storage elements settle to constant voltages/currents and derivatives go to zero. This question targets that definition. Given Data / Assumptions: DC step input applied to a first-order circuit (e.g., charging a capacitor through a resistor). No leakage (unless stated), linear components, and ideal source. Concept / Approach: During a DC step, the forced response is a constant value, and the natural response decays exponentially with time constant tau. Steady state is reached as t → ∞, when the exponential term decays to zero. In a simple RC charge with a step from 0 to V_in, the capacitor voltage tends to V_in; thus output equals input (for the node across the capacitor) in steady state. Step-by-Step Solution: For a step from 0 to V_in: v_C(t) = V_in * (1 - exp(-t / tau)).As t → ∞, exp(-t / tau) → 0.Therefore, v_C(∞) = V_in, indicating steady state equals the input value.Similar logic applies for RL circuits where the inductor current settles to V_in / R in the long term. Verification / Alternative check: After ~5 tau, first-order responses are within ~1% of final value, commonly used as a practical steady-state criterion. Simulation or lab measurement confirms negligible change beyond this horizon. Why Other Options Are Wrong: Average value: Relevant to periodic waveforms, not to a DC step. 63% of input: That is the one-time-constant point, not steady state. Effective (rms) value: Pertains to AC; a DC final value equals the DC amplitude. Common Pitfalls: Confusing the 63% milestone with completion of the transient. Mixing AC metrics (rms, average over a cycle) with DC step responses. Final Answer: When the output voltage reaches the input voltage (for a DC step)

Question 10

In an RL differentiator, when the input pulse transitions from its low level to its high level, which statement is correct?

Accepted Answer

Introduction / Context: Inductors resist changes in current, which is central to how RL differentiators produce sharp voltage spikes at edges. Recognizing this physical property helps predict transient behavior in pulse circuits. Given Data / Assumptions: Series RL with output typically taken across the resistor (differentiator behavior for small tau). Input undergoes a rising step (pulse goes from low to high). Concept / Approach: Inductor voltage is v_L = L * di/dt. A finite inductance cannot instantaneously change current without requiring infinite voltage. Therefore, at the instant of a step, current through the inductor remains (momentarily) at its prior value, and the required voltage appears to support a rapid change in current thereafter. Step-by-Step Solution: At t = 0+, input steps upward.Current through L: i_L cannot jump; it begins changing from its initial value with slope di/dt = v_L / L.Large induced v_L appears briefly to force current to change toward the new steady state set by the source and R.Thus, the inductor prevents sudden change in current, not voltage. Verification / Alternative check: Ideal inductor i_L is continuous; measured waveforms show resistor voltage spikes ±V proportional to di/dt. Simulations or scope captures confirm that i_L ramps rather than jumps at the transition. Why Other Options Are Wrong: Prevents sudden change in voltage: That is the property of a capacitor, not an inductor. Instantly reaches 63%: 63% relates to one time constant, not an instantaneous effect. Voltage across inductor is zero: At an edge, v_L is typically largest, not zero. Common Pitfalls: Confusing capacitor and inductor edge properties. Assuming the 63% rule applies at t = 0 rather than at t = tau. Final Answer: The inductor prevents a sudden change in current

Question 11

An RC integrator is driven by a 24 V input pulse whose width equals five time constants (t = 5RC). To what voltage does the capacitor charge by the end of the pulse?

Accepted Answer

Introduction / Context: First-order charging behavior dictates how close an RC node gets to its final value within a given number of time constants. This question uses the canonical result at five time constants for a DC step/pulse of sufficient width. Given Data / Assumptions: RC integrator receiving a 24 V pulse. Pulse width equals 5 * tau where tau = R * C. Initial capacitor voltage assumed 0 V (uncharged) at the pulse start. Concept / Approach: For a step of amplitude V applied to an RC from rest, the capacitor voltage during charging is v_C(t) = V * (1 - exp(-t / tau)). At t = 5 * tau, the exponential term exp(-5) ≈ 0.006737, so the node is within about 0.67% of its final value V. Step-by-Step Solution: Use v_C(5tau) = 24 * (1 - exp(-5)).Compute exp(-5) ≈ 0.006737.So v_C ≈ 24 * (1 - 0.006737) = 24 * 0.993263 ≈ 23.838 V.Rounded to given choices, this is effectively 24 V. Verification / Alternative check: A common engineering rule: after 5 tau, a first-order system reaches ~99.3% of its final value. For 24 V, that is ≈ 23.84 V, which rounds to 24 V in typical multiple-choice contexts. Why Other Options Are Wrong: 15.12 V: Corresponds to ~63% of 24 V (one time constant), not five. 20.64 V: ~86% (two time constants), still far from 99.3%. 12 V: Exactly half the target; unrelated to standard time-constant milestones. Common Pitfalls: Using 1 - 1/5 instead of the exponential relation. Confusing 5 tau with 5% error rather than 0.67% error. Final Answer: 24 V

Question 12

To understand how a differentiator shapes its output, which parts of the pulse response must be analyzed?

Accepted Answer

Introduction / Context: Differentiators emphasize changes in the input signal. For pulse waveforms, the output is dominated by edge behavior, but the inter-edge interval also affects baseline and droop. This question probes comprehensive reasoning about the entire pulse cycle. Given Data / Assumptions: First-order RC or RL differentiator operating on rectangular pulses. Finite time constants and practical component values. Concept / Approach: The output to a pulse includes a positive spike at the rising edge (for standard polarity), a transient settling toward baseline between edges (with exponential decay governed by time constant), and a negative spike at the falling edge. Ignoring any portion yields an incomplete understanding of amplitude, symmetry, and baseline shift. Step-by-Step Solution: Rising edge: Large dv/dt produces a spike whose polarity follows the edge direction and whose initial magnitude reflects the abrupt step.Between edges: The output decays exponentially toward baseline; the decay rate depends on RC or L/R, affecting pulse-to-pulse overlap and droop.Falling edge: Opposite-polarity spike appears as the input steps back; its shape mirrors the rising-edge response in magnitude (for symmetric pulses). Verification / Alternative check: Time-domain convolution (input with differentiator impulse response) or Laplace-domain multiplication (H(s) with input step pairs) shows contributions at each interval, confirming all segments matter to the net waveform. Why Other Options Are Wrong: Considering only one edge or only the middle interval misses essential dynamics; all three regions determine output symmetry, amplitude, and baseline. Common Pitfalls: Assuming ideal impulses with zero-width edges, which hides practical finite-time constant effects. Neglecting recovery time between edges, leading to baseline shift and distortion. Final Answer: All of the above

Question 13

RC integrator design: A 15 V input pulse having a width equal to two time constants is applied to an RC integrator. To what voltage will the capacitor charge by the end of the pulse?

Accepted Answer

Introduction / Context: This problem checks time-domain charging behavior of a first-order RC integrator when it is driven by a finite-width rectangular pulse. Knowing how far a capacitor charges during a pulse is essential for pulse shaping, timing, and analog signal conditioning design. Given Data / Assumptions: Input pulse amplitude, Vin = 15 V. Pulse width equals two time constants, t = 2 * tau. Ideal RC integrator, no leakage or series parasitics beyond R and C. Capacitor initially uncharged just before the pulse edge (worst-case rise computation). Concept / Approach: For a step of amplitude Vin applied across an RC, the capacitor voltage follows vC(t) = Vin * (1 - e^(−t/tau)). For a finite-width pulse of duration t, the capacitor reaches vC(t) at the end of the pulse, provided the RC remains in its charging phase throughout that interval. Step-by-Step Solution: Given t = 2 * tauvC(t) = Vin * (1 - e^(−t/tau))vC(2 * tau) = 15 * (1 - e^(−2))e^(−2) ≈ 0.1353 ⇒ 1 − e^(−2) ≈ 0.8647vC ≈ 15 * 0.8647 ≈ 12.97 V ≈ 12.9 V Verification / Alternative check: As a rule of thumb, after 1 tau a capacitor reaches ~63.2% of its final value; after 2 tau it reaches ~86.5%. For 15 V, 0.865 * 15 ≈ 12.97 V, confirming the calculation. Why Other Options Are Wrong: 19.45 V and 15 V: Both exceed or equal the source; a passive RC cannot charge above Vin in this simple case. 8.6 V: Would correspond to approximately 1 tau, not 2 tau. Common Pitfalls: Confusing one time constant with two, or forgetting to apply the exponential factor e^(−t/tau). Always convert the pulse width to multiples of tau before applying the formula. Final Answer: 12.9 V

Question 14

RC integrator charging: A 12 V input pulse with a width equal to one time constant is applied to an RC integrator. What voltage does the capacitor reach at the end of the pulse?

Accepted Answer

Introduction / Context: This question targets the exponential charging law of capacitors in first-order RC networks, a foundational concept in electronics used for timing, filtering, and pulse shaping. It specifically examines the capacitor voltage after exactly one time constant of charging. Given Data / Assumptions: Input pulse amplitude, Vin = 12 V. Pulse width t = tau (one time constant). Ideal RC network, initially vC(0) = 0 V (uncharged). Concept / Approach: For a step input of amplitude Vin applied to an RC, the capacitor charges as vC(t) = Vin * (1 - e^(−t/tau)). At t = tau, the result is vC = Vin * (1 − e^(−1)) ≈ Vin * 0.6321. Step-by-Step Solution: vC(tau) = 12 * (1 − e^(−1))e^(−1) ≈ 0.3679 ⇒ 1 − e^(−1) ≈ 0.6321vC ≈ 12 * 0.6321 ≈ 7.585 V ≈ 7.56 V (rounded to match option) Verification / Alternative check: Rule of thumb: one tau corresponds to ~63.2% of the final value. 0.632 * 12 ≈ 7.58 V, consistent with our calculation. Why Other Options Are Wrong: 12 V: Would require infinite time (full charge), not one tau. 6.3 V: A rough 0.525 * 12 V number would be too low; the correct 63.2% gives ~7.58 V. 0 V: Only at t = 0; not after charging for tau. Common Pitfalls: Using linear intuition instead of the exponential model, or misremembering 63.2% as 50%. Always apply the exact exponential term for accuracy. Final Answer: 7.56 V

Question 15

Pulse behavior in real electronic systems: Compared to single, isolated pulses, how frequently are repetitive-pulse waveforms encountered in practical electronics?

Accepted Answer

Introduction / Context: Modern electronic systems—from microcontrollers and digital communication links to switching power supplies—are dominated by repetitive timing signals. Recognizing that repetitive-pulse waveforms are ubiquitous helps learners prioritize analysis tools such as duty cycle, rise/fall times, harmonics, and average/ rms calculations. Given Data / Assumptions: General observation across digital, power, and mixed-signal electronics. Repetitive pulses include clocks, PWM control signals, line codes, and recurring trigger pulses. Single pulses occur but are comparatively less common in continuous operation. Concept / Approach: Repetitive pulses define timing. Digital logic requires a stable clock; PWM modulates average power or voltage by repeating pulses; serial data streams encode information in recurring bit intervals. Therefore, system design, filtering, and measurement equipment (oscilloscopes, spectrum analyzers) are optimized for repetitive phenomena. Step-by-Step Reasoning: Identify common sources: system clocks, PWM in motor drives and converters, carrier-based modulation, repetitive sampling pulses.Evaluate usage: these run continuously during device operation—milliseconds to years—far outnumbering isolated, one-shot events.Measurement practice: scope triggering, averaging, and spectral analysis presume or exploit periodicity to stabilize displays and extract signal properties. Verification / Alternative check: Open any embedded system: the microcontroller clock (tens of MHz) and PWM outputs (kHz–MHz) are always present when powered. Single-shot pulses appear mainly in test sequences, resets, or transient events. Why Other Options Are Wrong: Less often or about as often: Understates the central role of periodic timing in practical circuits. Twice as often: Arbitrary numeric claim without basis; the key idea is that repetitive pulses are the norm. Common Pitfalls: Equating classroom one-shot demonstrations with real-world operation; overlooking that most systems run continuously and therefore continuously repeat their timing signals. Final Answer: more often than single pulses

Question 16

Single 30 V, 6 ms pulse into an RC integrator (R = 3.3 kΩ, C = 2 µF): To what voltage does the capacitor charge by the end of the pulse?

Accepted Answer

Introduction / Context: This computation illustrates exponential charging during a finite pulse. Designers frequently need to predict the capacitor's voltage after a given pulse width to set thresholds, timing margins, and subsequent stage behavior. Given Data / Assumptions: R = 3.3 kΩ, C = 2 µF ⇒ tau = R * C. Input pulse amplitude Vin = 30 V. Pulse width t = 6 ms. Assume vC(0) = 0 V (uncharged) for a conservative estimate. Concept / Approach: Capacitor charging under a step input follows vC(t) = Vin * (1 − e^(−t/tau)). The pulse ends at t = 6 ms; the capacitor voltage at that instant is the charged value reached during the pulse. Step-by-Step Solution: tau = R * C = 3.3e3 * 2e−6 = 6.6e−3 s = 6.6 mst/tau = 6 ms / 6.6 ms ≈ 0.9091vC = 30 * (1 − e^(−0.9091))e^(−0.9091) ≈ 0.402vC ≈ 30 * (1 − 0.402) = 30 * 0.598 ≈ 17.94 V ≈ 17.91 V Verification / Alternative check: Since 6 ms is slightly less than one time constant (6.6 ms), a value a bit under 63.2% of 30 V (i.e., 18.96 V) is expected. Our result, ~17.9 V, is consistent because the pulse ends before one full tau. Why Other Options Are Wrong: 30 V: Requires infinite time; not possible with a 6 ms pulse. 12.09 V or 10.3 V: Too low; they correspond to shorter effective charge times. Common Pitfalls: Forgetting to compute tau, or using linear instead of exponential charging. Always use vC(t) = Vin * (1 − e^(−t/tau)). Final Answer: 17.91 V

Question 17

Single 6 V, 600 µs pulse into an RC integrator (R = 150 kΩ, C = 0.002 µF): To what voltage will the capacitor charge by the end of the pulse?

Accepted Answer

Introduction / Context: Here we calculate the charged voltage of a capacitor at the end of a finite pulse—key to understanding RC integrator performance for edge shaping and pulse area control. Given Data / Assumptions: R = 150 kΩ. C = 0.002 µF = 2 nF. Pulse amplitude Vin = 6 V. Pulse width t = 600 µs. Initial capacitor voltage vC(0) = 0 V (uncharged). Concept / Approach: Use the exponential charging law vC(t) = Vin * (1 − e^(−t/tau)). Compute the time constant tau = R * C, then evaluate at t = 600 µs. Step-by-Step Solution: C = 0.002 µF = 0.002 × 10^−6 F = 2 × 10^−9 Ftau = R * C = 150,000 * 2e−9 = 3e−4 s = 300 µst/tau = 600 µs / 300 µs = 2vC = 6 * (1 − e^(−2))e^(−2) ≈ 0.1353 ⇒ vC ≈ 6 * 0.8647 ≈ 5.19 V ≈ 5.16 V Verification / Alternative check: At t = 2 tau, a capacitor reaches about 86.5% of its final value. 0.865 * 6 ≈ 5.19 V, matching our computation and the provided option. Why Other Options Are Wrong: 6 V: Requires t → ∞ for full charge; not at 600 µs. 3.78 V: Around 63% of 6 V (one tau) rather than two tau. 0 V: Only at the instant before the pulse begins. Common Pitfalls: Misconverting 0.002 µF to farads, or forgetting that the capacitor does not reach 100% in finite time. Always compute t/tau first. Final Answer: 5.16 V

Question 18

RC integrator topology: In a standard RC integrator circuit, across which component is the output voltage taken?

Accepted Answer

Introduction / Context: Identifying where to measure the output in canonical RC networks is fundamental. For an integrator, the capacitor voltage represents the time integral of the input (scaled by RC), so the output is taken across the capacitor. Given Data / Assumptions: Classic passive RC integrator: series resistor R, capacitor C to ground. Small-signal, linear operation; frequency content where integration is a good approximation. No active elements (e.g., op-amps) in this basic form. Concept / Approach: The impedance of a capacitor is Xc = 1/(2 * pi * f * C). At higher frequencies, the capacitor impedance is small, causing most of the input to appear across R (output small). At lower frequencies (within integrator operation range), the capacitor voltage follows the integral of the input's slope changes, hence observed across C. Step-by-Step Reasoning: Integrator connection: Vin → R → node → C → ground.Vout is at the node between R and C, i.e., across the capacitor.This configuration shapes fast edges into smaller, smoothed responses consistent with integration. Verification / Alternative check: Small-signal transfer function: Vout/Vin = 1 / (1 + j * 2 * pi * f * R * C). For frequencies where |j * 2 * pi * f * R * C| ≫ 1, Vout ≈ (1 / (j * 2 * pi * f * R * C)) * Vin, which is proportional to 1/f and represents integration behavior. Why Other Options Are Wrong: Resistor: That would be a differentiator's output node in the simple RC form. Diode or source: Not part of the standard passive RC integrator output definition. Common Pitfalls: Confusing integrator and differentiator topologies; remember: integrator → output across C, differentiator → output across R. Final Answer: capacitor

Question 19

RC differentiator classification: An RC differentiator behaves as which type of linear filter with respect to frequency response?

Accepted Answer

Introduction / Context: Understanding the frequency-domain identity of time-domain circuits helps bridge intuition between transient and steady-state behavior. An RC differentiator emphasizes rapid changes (high-frequency components) and attenuates slow variations (low-frequency components). Given Data / Assumptions: Passive RC differentiator: capacitor in series with input, resistor to ground, output across the resistor. Small-signal linear analysis. No active gain elements. Concept / Approach: The transfer function magnitude increases with frequency over the useful operating region. At low frequency, the capacitor's reactance is high, blocking current and yielding near-zero output—characteristic of high-pass filters. At high frequency, the capacitor reactance drops, passing more signal to the output. Step-by-Step Reasoning: Impedances: Zc = 1/(j * 2 * pi * f * C), Zr = R.Voltage divider: Vout/Vin = Zr / (Zr + Zc) = R / (R + 1/(j * 2 * pi * f * C)).As f → 0, |Zc| → ∞ ⇒ Vout ≈ 0 (rejects low frequency).As f increases, |Zc| decreases ⇒ Vout increases (passes high frequency). Verification / Alternative check: Time-domain view: A differentiator produces spikes at edges (high dV/dt), which are inherently high-frequency components; thus, it acts as a high-pass element for signals. Why Other Options Are Wrong: Low-pass: Corresponds to the RC integrator output across C. Band-pass / band-stop: Require at least second-order (or more complex) behavior; a single RC differentiator is first-order high-pass. Common Pitfalls: Confusing integrator and differentiator placements (output across C vs R). Remember: differentiator output is across R and behaves as a high-pass filter. Final Answer: high-pass filter

Question 20

In pulse and digital electronics, which type of circuit is specifically designed to produce very short-duration spikes (narrow pulse triggers) from an input signal or event?

Accepted Answer

Introduction: Short-duration spikes, often called trigger pulses or narrow pulses, are essential in timing, synchronization, counting, and digital control. This question checks recognition of the circuit class whose primary function is to create sharp, brief pulses on demand, rather than to average, integrate, or otherwise reshape signals into long-duty cycles. Given Data / Assumptions: The target pulse must be narrow (short time width) with a quick rise and fall. Applications include clocking flip-flops, triggering monostables, or providing synchronization edges. We are selecting among distinct functional blocks used in basic electronics. Concept / Approach: A trigger pulse generator is explicitly built to deliver narrow pulses. Implementations range from differentiator-based edge generators with diode shaping, to Schmitt-trigger RC networks, to monostable multivibrators (one-shots) set for small time constants. By contrast, integrators emphasize area/averaging, and DC converters emphasize producing a steady level rather than a spike. Step-by-Step Solution: 1) Identify the required output: a narrow pulse (spike) suitable for triggering.2) Map candidate circuits to function: integrators average, converters rectify/average, timing blocks may produce long intervals.3) Choose the class designed for short pulses: the trigger pulse generator.4) Conclude that the correct answer is the purpose-designed generator of narrow pulses. Verification / Alternative check: Consider a monostable multivibrator configured with a small RC constant; it outputs a short pulse when triggered. This is a canonical trigger pulse generator use-case and matches the requirement. Why Other Options Are Wrong: RL integrator: shapes/averages signals rather than producing sharp, adjustable trigger spikes. Timing circuit: generic term; often used for long intervals rather than intentionally minimal widths. Pulse waveform-to-dc converter: averages or rectifies a pulse train into a DC level, not spikes. Monolithic voltage regulator: provides regulated DC voltage, not pulses. Common Pitfalls: Equating any RC network with a pulse generator. Differentiators can produce edges, but a dedicated trigger pulse generator includes amplitude control, shaping, and consistent pulse width across conditions. Final Answer: A trigger pulse generator

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