EE 2115-01 - Lec 24 - 2026_10_21

Overview

Hiren Trada analyzes how a series RC circuit responds to step and rectangular inputs, using KVL and the time constant τ = RC to explain capacitor charging, resistor voltage decay, and steady state. He then represents a rectangular pulse as the difference of two step inputs and traces how capacitor discharge reverses the resistor voltage polarity.

Key takeaways

For a series RC circuit driven by a step V0, KVL requires VR + VC = V0: VC rises exponentially as VR falls.
The time constant τ = RC sets the charging and discharging timescale; doubling either resistance or capacitance doubles τ.
An uncharged capacitor initially behaves as a short for the step transition, with VC(0+) = 0, VR(0+) = V0, and current V0/R; after DC steady state it behaves as an open circuit.
A rectangular pulse is the superposition of a positive step at its leading edge and a negative step at its trailing edge, so its RC response combines two exponential responses.
When a charged capacitor discharges through a resistor after the input is removed, current reverses direction and the resistor voltage changes polarity.

Chapters

0:00 RC Step Response: KVL Links Capacitor Charging to Resistor Decay

When an input step V0 is applied, capacitor voltage VC rises exponentially while resistor voltage VR falls exponentially.
Kirchhoff's voltage law gives Vin = VR + VC throughout the series circuit.
At t = 0, an initially uncharged capacitor has VC = 0, so the full input voltage appears across the resistor.

8:00 Series RC Connections, Output Choice, and the Time Constant

Swapping the physical order of R and C in a series circuit does not change its electrical behavior, but the output differs depending on whether it is measured across R or C.
The RC time constant is τ = RC; increasing R or C makes charging and discharging slower.
At approximately 0.69τ, the capacitor voltage reaches half the step input, and the resistor voltage is also half the input.

15:00 Capacitor Initial Conditions, Steady State, and Applications

Capacitor voltage cannot change instantaneously because an instantaneous voltage jump would require infinite current; the analogous inductor constraint applies to inductor current.
For an initially uncharged capacitor, VC(0+) = 0 and VR(0+) = V0; after a long time, VC = V0, VR = 0, and circuit current is zero.
A fully charged capacitor in a DC steady state behaves as an open circuit; Trada cites capacitive touch screens and motor or ceiling-fan circuits as applications.

17:00 Build a Rectangular Pulse from Two Step Inputs

A finite rectangular pulse can be represented as a positive step at its start minus a second positive step at its end.
For an output across the resistor, each positive step produces a decaying exponential response, and subtracting the delayed response creates the pulse output.
While the input remains flat between its two transitions, the circuit experiences a constant, DC-level input.

22:00 Pulse Transition Points A Through D and RC Charging

At point A, immediately before the pulse, Vin, VC, VR, and current are all zero for the assumed unenergized circuit.
At point B, the pulse begins: the capacitor voltage remains at zero instantaneously, the resistor takes the full V0, and the initial current is V0/R.
As charging proceeds toward point C, VC approaches V0 while VR and current decay toward zero.

26:30 Capacitor Discharge Reverses the Resistor Voltage

When the pulse ends, the charged capacitor acts as a source for the resistor and provides the discharge path.
The capacitor voltage cannot jump at the transition, but the resistor voltage and current reverse direction as discharge begins.
The reversed resistor polarity produces the negative decaying exponential associated with the second, subtracted step response.

Transcript excerpt

0:01 All right, good afternoon. Uh we are going to continue discussion of RC circuits to gain a better understanding of of how it can work as a filter and filtering out uh signals with varying frequencies. So, we're going to use um not use, but we're going to analyze the response of a RC circuit to a pulse or rectangular input. Okay. All right. So, what we have seen so far is that if you have a RC circuit,

0:50 meaning a resistor and a capacitor, applying a voltage to it is going to cause a current to start flowing in the circuit, and the capacitor will start storing energy. Um we call that process charging. So, capacitor starts charging the moment V in is applied. So, here we have the applied input voltage. Here is the resistor voltage, and here is the capacitor voltage. KVL always applies, and so if there is one source in the circuit and you have three components including the source, then the source is supplying the energy for the rest of the circuit. So, the voltage across that voltage source must equal the voltage sources, I mean not voltages

1:37 across the resistor and the capacitor. V in should be equal to VR plus VC. So, we all KVL is always uh in effect. Now, if we have a step input and we look at the voltage across the capacitor, voltage across the capacitor cannot change instantaneously. It takes a finite amount of time for it to reach a full charge, meaning accumulate all of the energy. And we've seen and derived the expression, which is exponential in nature. So, as the capacitor starts charging, we'll see current flowing through the circuit. When current initially starts flowing through the circuit, it will be at its maximum.

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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.