J. W. Nilsson & S. A. Riedel, Electric Circuits, 12th ed., Pearson, 2023 — second-order RLC response chapters.
Series/parallel RLC natural and step response, initial conditions, damping and systematic circuit solution.
DY-02
Course overview
RLC, second-order dynamics & resonance
DY-01 showed that one independent storage state creates one first-order mode. DY-02 adds a second independent state and derives the consequences rather than memorizing response categories. Starting from Kirchhoff laws and the constitutive relations of C and L, the course builds second-order equations, natural roots, damping, driven response, resonance and quality factor, then reconnects time-domain and frequency-domain descriptions.
5 chapters5 available · 0 plannedintermediate
Prerequisites
- DY-01 — first-order dynamics
- AC-02 — poles & frequency response
- Elementary differential calculus
By the end of the course
- Derive a second-order RLC equation from Kirchhoff laws.
- Interpret natural roots and damping regimes.
- Solve natural and driven responses.
- Relate resonance, bandwidth and quality factor.
- Reconcile time-domain, frequency-domain and energy viewpoints.
Syllabus
Five readings from two storage states to resonance and quality factor.
DY-01 showed that one independent storage state creates one first-order mode. DY-02 adds a second independent state and derives the consequences rather than memorizing response categories. Starting from Kirchhoff laws and the constitutive relations of C and L, the course builds second-order equations, natural roots, damping, driven response, resonance and quality factor, then reconnects time-domain and frequency-domain descriptions.
- 01→
Second-order circuit equations
Derive why a circuit with two independent energy-storage states can require two initial conditions and a second-order differential equation. Build the driven series-RLC capacitor-voltage equation directly from KVL, write the equivalent two-state first-order system, derive its characteristic polynomial, define the undamped natural angular frequency and damping coefficient from circuit parameters, prove the natural energy balance, and explain why component count alone does not determine dynamic order.
170–220 min · Intermediate · available - 02→
Natural response, roots & damping regimes
Solve the source-free second-order equation from its characteristic roots instead of memorizing waveform labels. Derive s₁,₂=−α±√(α²−ω₀²), classify over-, critical- and under-damping from the discriminant or damping ratio ζ, derive the correct time-domain form and constants from physical initial conditions, distinguish the undamped natural frequency ω₀ from the damped natural frequency ωd, derive the series-RLC critical resistance, and show why increasing resistance beyond critical damping can actually slow the dominant non-oscillatory mode.
175–225 min · Intermediate · available - 03→
Driven response & step response
Add forcing to the second-order dynamics without losing the natural-mode structure. Separate zero-input and zero-state response, derive the series-RLC capacitor-voltage transfer equation, solve the canonical DC-step response in over-, critical- and underdamped cases, derive peak time and percent overshoot only for the canonical underdamped all-pole step response, replace vague settling heuristics by explicit tolerance/envelope bounds, and show how nonzero initial energy changes the total waveform without changing the system poles.
180–230 min · Intermediate · available - 04→
Resonance, bandwidth & quality factor
Derive resonance from the actual transfer quantity rather than treating it as a universal property of the denominator. For the ideal series RLC, derive Z(jω), current resonance at ω0=1/√(LC), resistor-output band-pass response, exact half-power frequencies and Δω=R/L, Q=ω0L/R=1/(ω0RC)=1/(2ζ), voltage magnification across L/C, and the energy meaning of Q. Then contrast this with capacitor-voltage peaking, whose peak exists only for ζ<1/√2 and occurs at a different frequency.
185–235 min · Intermediate · available - 05→
Time-domain & frequency-domain synthesis
Close the second-order course by showing that poles, ring-down, damping, step overshoot, resonance bandwidth and Q are different measurements of the same ideal two-state dynamics only when their assumptions match. Derive Δω=2α for the ideal series-RLC current response, Q=ω0/(2α), the ring-down envelope time 1/α, logarithmic decrement, independent time/frequency parameter extraction, and consistency checks that reveal loading, zeros, parasitics or model failure.
185–235 min · Intermediate · available
Reference backbone
References used
The course follows standard undergraduate circuit and differential-equation treatments. Circuit derivations are anchored in Kirchhoff laws and energy storage; the canonical second-order form is then cross-checked against MIT 6.002/6.071 and MIT 18.03 treatments of modes, damping and frequency response.
C. K. Alexander & M. N. O. Sadiku, Fundamentals of Electric Circuits — second-order circuits and resonance chapters.
RLC equations, damping cases, resonance, bandwidth and quality factor.
A. Agarwal & J. H. Lang, Foundations of Analog and Digital Electronic Circuits, Morgan Kaufmann, 2005 — Chapters 12–14.
State/memory viewpoint, second-order dynamics, frequency response and Q.
MIT OpenCourseWare 6.002 — second-order systems, damping, frequency response, filters and Q materials.
Second-order systems, damping, sinusoidal steady state, filters and quality factor.
MIT OpenCourseWare 6.071J — Transient Response of Second Order Circuits lecture notes.
Series RLC, critical/over/underdamped responses and practical circuit examples.
MIT OpenCourseWare 18.03 — Modes, damping conditions, frequency response and RLC applications.
Canonical second-order ODE structure, characteristic roots and damping interpretation independent of circuit-specific notation.
AoN DY-01, AC-01 & AC-02.
Prerequisite storage-element laws, first-order state/pole reasoning, impedance and transfer-function notation.