EE 2115-01 - Lec 25 - 2026_10_23

Overview

Hiren Trada explains how an RC circuit with output across the resistor responds to short pulses and periodic signals, showing how the time constant controls droop, discharge tails, and waveform preservation. He connects the circuit's high-pass behavior and DC blocking to amplifier coupling, then derives practical resistance-matching guidance for cascaded circuits.

Key takeaways

For a resistor-output RC circuit and a pulse much shorter than τ, the droop is approximately V₀Tp/τ because the capacitor has little time to charge.
After a short pulse ends, the capacitor discharges with opposite polarity, creating a negative resistor-output tail whose duration depends on the RC time constant.
A series RC circuit with output across the resistor blocks steady DC and passes changing signals, making it a high-pass filter; output across the capacitor instead gives low-pass behavior.
A larger RC time constant relative to the signal period can reduce waveform amplitude loss, but repeated pulses still settle into a steady-state output shaped by the duty cycle.
For better voltage transfer between cascaded stages, use low output resistance in the driving stage and high input resistance in the receiving stage.

Chapters

0:00 Short RC Pulses Produce Droop at the Resistor Output

For output measured across the resistor, a low-to-high step initially gives the resistor nearly the full input voltage while the capacitor voltage starts at zero.
If a pulse ends before the capacitor charges substantially, the resistor's decaying exponential is interrupted and the capacitor voltage remains small.
The voltage shortfall is called droop; after the input falls, the capacitor discharges with opposite polarity and creates a negative resistor-output tail.

11:19 Pulse Duration and the RC Time Constant Set the Discharge Tail

A 1 nF capacitor and 10 kΩ resistor are used to compare a longer 50 μs pulse with a much shorter 1 μs pulse.
The short-pulse droop follows the charging relation V₀(1 − e^(−t/τ)), evaluated over the pulse width; for a short pulse, the missing resistor voltage is approximately V₀Tp/τ.
The capacitor releases its stored voltage in the opposite direction after the input falls, and a large time constant can produce a long tail.

13:12 Repeated Pulses Settle into Steady State and Remove DC Offset

With a 1 V pulse train and a short pulse period, successive cycles arrive before the capacitor fully discharges, so the initial output peaks decrease until steady state.
For a 50% duty-cycle input switching between 0 V and 1 V, the input averages 0.5 V, but the steady resistor output is centered around 0 V.
At a 33% duty cycle, the longer off-time changes the output shape and leaves a DC component, illustrating that duty cycle affects the steady-state waveform.

20:29 RC High-Pass Behavior: The Capacitor Blocks DC

Using i = C dV/dt and Ohm's law, Trada relates resistor output to the derivative of the capacitor voltage and the input.
A constant DC voltage has zero time derivative, explaining mathematically why the capacitor blocks DC while allowing changing signals through.
Taking output across the resistor makes the series RC circuit a high-pass filter; taking output across the capacitor makes it a low-pass filter.

23:04 Choosing the Time Constant to Preserve AC Waveforms

The time constant τ = RC governs charging and discharging; a larger τ relative to the signal period helps preserve the input waveform across the resistor.
A sinusoidal example compares a 10 μs period with a 10 μs time constant, showing startup droop before the output settles closer to the input.
A resistor-output high-pass circuit rejects the input's DC component while passing AC, with some amplitude reduction and phase shift.

28:55 Periodic Signals, Harmonics, and Pulse-Train Filtering

Periodic signals repeat at a frequency, and their harmonic components occur at integer multiples of the fundamental, such as 10, 20, and 30 Hz.
A 0-to-1 V pulse train with a 0.5 V DC offset loses that offset at the resistor output after reaching steady state.
Increasing the capacitor from 1 nF to 2 nF with a 10 kΩ resistor increases τ and improves preservation of the pulse waveform's amplitude.

31:35 Capacitive Coupling Separates AC Signals from Amplifier Bias

A transistor amplifier can receive DC bias from a 20 kΩ and 20 kΩ divider, producing a 2.5 V Thevenin voltage from a 5 V supply and a 10 kΩ Thevenin resistance.
A coupling capacitor passes an AC sensor signal into the amplifier while preventing DC from the source from entering through that path.
Amplifier outputs can combine the amplified AC signal with DC bias, so an output capacitor and load resistance can filter out the DC component.

38:12 Output Coupling Capacitors Remove Amplifier DC Bias

An AC input applied to a DC-biased circuit can produce an output containing both AC and DC components.
An output capacitor paired with the next circuit's load resistance forms an RC filter that blocks DC and passes the desired AC signal.
The same coupling principle applies between stages such as preamplifiers, amplifiers, comparators, and other signal-processing circuits.

43:54 Thevenin Equivalents Simplify RC Loading Analysis

When additional resistances load an RC network, replace a voltage divider with its Thevenin equivalent: RTH = R1 ∥ R2 and the corresponding VTH.
For a divider output across R1, the voltage is scaled by R1/(R1 + R2), so the surrounding circuit changes the voltage delivered to the load.
Resistance in the next circuit can represent an amplifier's input resistance, not just a discrete resistor.

49:17 Match Circuit Resistances and Review Course Announcements

To transfer more voltage between cascaded stages, make the first circuit's output resistance low and the second circuit's input resistance high.
The same loading principle applies across a sensor, preamplifier, amplifier, and speaker chain.
The course midterm 2 date is set for November 9; midterm 1 grades are expected by Monday, with solutions and midterm 2 guidance planned for Wednesday.

Transcript excerpt

0:01 Good afternoon everyone. Let's continue. RC circuits, time constants and application of pulse input. Okay. So we had stopped here last class. Let's make sure zoom can All right. Anyone have any questions before we move on? Okay. So, just a brief recap. We're talking about RC circuits. To gain a better

0:48 understanding, uh we divided up the step input pulse um as two step um inputs. Step pulse means it goes up and it goes down. Uh we divided it into two parts. So we can look at uh both ends. The high so low to high transition and the high to low transition which you see here and here. And the output we are taking across the resistor. So let's look at a very short pulse now. So previously we had a rectangular pulse that had a low frequency. So the pulse width was wide. Here this might be this could possibly be a um rectangular pulse

1:40 that has low fre uh high frequency and the pulse width is short. So depending on the value of the capacitor and the resistor we would have a certain value for time constant. when we apply a short pulse, it also only high being applied to the circuit for a very short period of time. Now, if the time constant of the circuit is large or larger than the time period of the input and the capacitor does not have sufficient time to reach the charge. So here we are taking the output across the resistor. And if you zoom in you can see that the resistor output goes maximum

2:28 right capacitor starts with zero resistor output is maximum. So I'm going to write VR at maximum which would be the applied uh applied input voltage. And at the same time of course

This is the start of the transcript. Save this video free for the whole transcript, searchable and time-coded.

Related notes

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Hiren Trada.