In an RC integrator used for pulse shaping, what happens if the capacitor becomes leaky (i.e., exhibits significant leakage resistance)?
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AThe time constant will be effectively reduced
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BThe waveshape of the output voltage across the capacitor is altered
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CThe amplitude of the output is reduced
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DAll of the above
Answer
Correct Answer: All of the above
Explanation
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