Schedule

Chapter 7 · CLO4 · Weeks 12–13

Natural and Step Responses of RL and RC Circuits

Chapter 7 puts together everything in the course. When a switch moves in a circuit with one inductor or one capacitor, every current and voltage moves from an initial value to a final value along an exponential curve. Chapter 7 runs through weeks 12 and 13, and its quiz is at the start of the last class, week 13, session 2 (Thu 14 Jan). The final exam covers Chapters 4 to 7.

When

  • Week 12 · Tue 5 Jan, Thu 7 JanNatural responses of RL/RC circuits. Step responses of RL/RC circuits.
  • Week 13 · Tue 12 Jan, Thu 14 JanGeneral solution for step and natural responses. Problems.

Quiz Device-free quiz at the start of the last class, week 13, session 2 (Thu 14 Jan)

Materials

Open practice · AI allowedSolve the practice sheet onlineEach question tells you at once whether your answer, and your reason, are right.Start the sheet

What you will learn

  • The natural response: stored energy drains through resistors, and the inductor current or capacitor voltage decays as $\exp(-t/\tau)$.
  • The step response: a dc source drives the inductor current or capacitor voltage from its initial value to a new final value.
  • The time constant: $\tau = L/R$ for an inductor and $\tau = RC$ for a capacitor, where $R$ is the Thévenin resistance the element sees after switching. After one $\tau$, 37% of the change is still to come; after five, less than 1%.
  • One formula for both: $x(t) = x_f + (x_0 - x_f)\exp(-t/\tau)$. Find the initial value, the final value and $\tau$, and the answer follows.
  • What cannot jump: the inductor current and the capacitor voltage are the same just after switching as just before. Resistor currents and voltages can jump, unless the circuit forces them to follow $i_L$ or $v_C$.
  • Dependent sources: the resistance that sets $\tau$ then needs a test source, as in Chapter 4.

Where the earlier chapters come back

ChapterWhat Chapter 7 uses from it
Chapters 2 and 3: Kirchhoff’s laws, dividersThe circuit before and after switching is a resistive circuit
Chapter 4: Thévenin equivalents$\tau$ uses the Thévenin resistance the element sees; dependent sources need a test source
Chapter 6: Inductors and capacitorsThe continuity rules give the initial value, and the dc rules (short circuit, open circuit) give the final value

In your program

  • Electrical and Electronics: motor starting currents, relay and contactor timing, and the charging of the capacitors in every power supply are first-order responses.
  • Computer, Communication and Telecom: every logic signal charges the capacitance of the next gate through a resistance; that RC time constant limits how fast a chip can run.
  • Biomedical: in Nilsson’s simplified model, a pacemaker times each heartbeat with one resistor and one capacitor (S14, and Nilsson’s Practical Perspective for this chapter). Real pacemakers use a crystal-timed microcontroller, but the RC idea is the same.
  • Mechanical and Industrial: solenoid valves and relays take a few milliseconds to pull in and let go, set by $L/R$ (S15).
  • Surveying: sensors are read after their RC input filters have settled; waiting less than about five time constants gives a wrong reading.

As an engineer

  • Three numbers decide the answer: the initial value, the final value and $\tau$. Find each one from its own circuit: before switching, long after switching, and with the independent sources off.
  • Use the right variable: solve for the inductor current or the capacitor voltage first, because only these are guaranteed to be continuous. Get every other quantity from it.
  • Polarity first: an initial voltage or current that points against the reference is negative. Read the figure before you write $x(0)$.

In real life

  • A pacemaker (Nilsson’s simplified model): a capacitor charges toward the battery voltage; at 75% of it, a controller fires a pulse into the heart and starts again. Doubling $R$ halves the heart rate (S14).
  • A relay letting go: after the switch opens, the coil current decays through a diode; adding a resistor shortens $\tau$, so the relay releases sooner, but raises the voltage spike (S15).

How to study this chapter, and where AI fits

  • AI tools are allowed on the practice sheet. Use them to check your work, not to replace it.
  • In a 2025 study of Gemini 2.5 Pro on undergraduate circuit problems, misread source polarities caused 6 of its 17 wrong answers, and misread current directions 5 (a small sample, but a clear pattern; arXiv 2512.10159). In this chapter they show up as an initial voltage with the wrong sign, or a time constant built from the wrong resistance. S11 shows a polarity slip, and S12 a wrong resistance: find them before you trust any answer.
  • Check every answer yourself: $x(0)$ must equal the initial value, $x(\infty)$ the final value, and $\tau$ must come from the circuit after switching.
  • The quiz is device-free, and every question is a twin of a practice-sheet question. If you can do the sheet on your own, you will do well on the quiz.

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