lecture_EE1C1_lec8_second-order-transient_2025-10-25

📌 Overview

Second-order transient analysis in RLC circuits: Focus on key variables (α, ω₀, ζ, ω_d), their equations from DE, meanings, and usage in classifying/solving responses (overdamped, underdamped, critically damped). Derive via char eq, apply ICs. Goal: Solve DEs for i(t)/v(t) post-switch.

---

🎯Learning Objectives

• Identify DE variables (a, b, c) and circuit params (α, ω₀, ζ, ω_d) from R/L/C
• Classify response via ζ/D; form solutions
• Apply ICs/derivs (t=0+) to find constants
• Use step plan for series/parallel RLC transients

---

💡Key Concepts & Definitions

Second-Order Circuits

• Series RLC: Analyze i(t) via KVL: $Ldi/dt+Ri+(1/C)\int idt=v_s(t)$. Diff: $d^{2}i/d{t}^2+(R/L)di/dt+(1/LC)i=d{v}_s/dt$.
  • DE: $d^{2}x/d{t}^2+adx/dt+bx=c$ ($x=i(t)$, $a=R/L$, $b=1/(LC)$, $c=0$ post-switch).
• Parallel RLC: Analyze v(t) via KCL: $Cdv/dt+v/R+(1/L)\int vdt=i_s(t)$. Diff: $d^{2}v/d{t}^2+(1/RC)dv/dt+(1/LC)v=d{i}_s/dt$.
  • DE: $a=1/(RC)$, $b=1/(LC)$.

Homogeneous DE (transient, c=0): $d^{2}x/d{t}^2+adx/dt+bx=0$.

Characteristic Equation

Assume $x_h=e^{st}$ → $s^2+as+b=0$.

• Roots: $s_{1,2}=\frac{-a\pm \sqrt{a^2-4b}}{2}=-\alpha \pm \sqrt{{\alpha }^2-{\omega }_0^2}$.
• Discriminant $D=a^2-4b=4({\alpha }^2-{\omega }_0^2)$.

Key Variables & Equations

From DE coeffs (a, b) and circuit:

Symbol	Meaning	Equation (Series)	Equation (Parallel)	Unit	Usage
$\alpha$	Damping coefficient in DE: Measures resistive energy loss rate relative to inductive/capacitive storage; higher a means faster decay, less oscillation. Derived from R’s influence on current/voltage rate.	$\alpha =R/L=2\alpha$	$a=1/(RC)=2\alpha$	s⁻¹	In char eq roots; controls real part of s (decay); larger → more damping, shifts to overdamped.
$b$	Natural frequency squared in DE: Represents the inherent oscillation rate from L-C energy exchange without damping; sets the “speed” of response cycles.	$b=1/(LC)={\omega }_0^2$	$b=1/(LC)={\omega }_0^2$	s⁻²	Defines ${\omega }_0$; in char eq, determines imaginary part for underdamped osc.
$c$	Forcing/source term in DE: Drives the particular (steady-state) solution; for transients post-switch, often 0 (homogeneous), but for DC step, constant leading to x_p = c/b.	$c=d{v}_s/dt$ (often 0)	$c=d{i}_s/dt$ (often 0)	V/s or A/s	For x_p in non-homogeneous DE; ignored in pure transients.
$\alpha$	Damping factor (neper frequency): Quantifies exponential decay envelope; rate at which amplitude decreases due to R dissipating energy; α > ω₀ prevents oscillation.	$\alpha =R/(2L)=a/2$	$\alpha =1/(2RC)=a/2$	Np/s	Decay in all solutions (e^{ -α t }); ζ = α/ω₀ classifies response; high α → quick settle without osc.
${\omega }_0$	Undamped natural frequency: Ideal oscillation frequency if R=0 (pure LC); fundamental resonant freq where energy swaps max between L and C.	${\omega }_0=1/\sqrt{LC}=\sqrt{b}$	Same	rad/s	Base for ω_d and ω_r; in roots, sets scale for oscillatory behavior.
$\zeta$	Damping ratio: Normalized damping (α relative to ω₀); dimensionless measure of how “damped” vs. oscillatory the system is; determines response type.	$\zeta =\alpha /{\omega }_0$	Same	-	Core classifier: >1 overdamped (slow non-osc), =1 critical (fast non-osc), <1 underdamped (osc decay).
${\omega }_d$	Damped natural frequency: Actual oscillation frequency with damping; reduced from ω₀ by energy loss; only for underdamped where roots have imag part.	${\omega }_d={\omega }_0\sqrt{1-{\zeta }^2}$ $=\sqrt{b}\sqrt{1-\frac{a^2}{4b}}$ $=\sqrt{b-\frac{a^2}{4}}$ $=\sqrt{{\omega }_0^2-{\alpha }^2}$ $=\sqrt{{\sqrt{b}}^2-(\frac{a}{2}{)}^2}=\sqrt{b-\frac{a^2}{4}}=\sqrt{\frac{-D}{4}}$ $=\frac{0-\sqrt{D}}{2}$ $=\mathrm{Im}[s]$	Same	rad/s	Freq of sin/cos in underdamped sol; period T=2π/ω_d.
$D$	Discriminant: Indicates root nature (real/complex); positive for two real decays, zero for boundary, negative for osc; reflects balance of damping vs. resonance.	$D={\alpha }^2-{\omega }_0^2=(a^2-4b)/4$	Same	s⁻²	D>0: over (distinct real s), =0: crit (repeated), <0: under (complex s = -α ± jω_d).
${\omega }_r$	Resonant frequency: Freq for max amplitude in driven circuits; equals ω₀, unaffected by damping in undriven transients but key for AC response.	${\omega }_r={\omega }_0$	Same	rad/s	Not direct in transients but links to ω₀; used in forced RLC.

How to Use:

1. Extract $a,b$ from circuit (series/parallel formulas).
2. Compute $\alpha =a/2$, ${\omega }_0=\sqrt{b}$, $\zeta =\alpha /{\omega }_0$, $D={\alpha }^2-{\omega }_0^2$.
3. Classify via $\zeta /D$: Determines sol form.
4. Rare: $\zeta \approx 0$ (low R) → sustained osc (ideal LC); $\zeta \gg 1$ (high R) ≈ first-order; neg $\alpha$ (active) → unstable growth. Valid for linear LTI RLC, t≥0 post-switch, initial steady-state.

Visual: Relations

flowchart TD
    Circuit["RLC Values"] --> a["a = 2α (R/L or 1/RC)"]
    Circuit --> b["b = ω₀² = 1/(LC)"]
    a --> α["α = a/2"]
    b --> ω₀["ω₀ = √b"]
    α --> ζ["ζ = α/ω₀"]
    ω₀ --> ζ
    ζ --> |ζ>1, D>0| Over["Overdamped: s₁,₂ real"]
    ζ --> |ζ=1, D=0| Crit["Crit: s = -α (repeated)"]
    ζ --> |ζ<1, D<0| Under["Under: s = -α ± jω_d"]
    ω₀ --> ω_d["ω_d = ω₀√(1-ζ²) (under only)"]
    Char["s² + a s + b =0"] --> Roots["s = -α ± √(α² - ω₀²)"]

➗ Formulas

General Sol: $x(t)=x_h(t)+x_p(t)$; $x_p=c/b$ (constant source).

• Overdamped (ζ>1, D>0): $x_h(t)=K_1{e}^{s_{1}t}+K_2{e}^{s_{2}t}$, $s_{1,2}=-\alpha \pm \sqrt{D}$ (real, distinct; slow non-osc decay).
• Critically Damped (ζ=1, D=0): $x_h(t)=(K_1+K_{2}t)e^{-\alpha t}$ (fastest non-osc return to steady).
• Underdamped (ζ<1, D<0): $x_h(t)=e^{-\alpha t}(A\cos {\omega }_{d}t+B\sin {\omega }_{d}t)$ (oscillatory decay).

ICs (t=0+): Continuity $v_C(0+)=v_C(0-)$, $i_L(0+)=i_L(0-)$; derivs $d{v}_C/dt(0+)=i_C(0+)/C$, $d{i}_L/dt(0+)=v_L(0+)/L$. Steady $x(\infty )$: L short, C open.

Usage: Solve system for constants: $x(0+)=...$, $x^{\prime }(0+)=...$.

---

✍️ Notes

Step Plan: Solve DE

1. Derive DE: KVL/KCL → diff to $d^{2}x/d{t}^2+adx/dt+bx=c$ (c=0 transient).
2. Params: $\alpha =a/2$, ${\omega }_0=\sqrt{b}$, $\zeta =\alpha /{\omega }_0$, classify via ζ.
3. Char Eq: $s^2+as+b=0$; roots $s=-\alpha \pm \sqrt{{\alpha }^2-{\omega }_0^2}$.
4. Sol Form: Per case (above); add $x_p$ if c≠0.
5. Constants: Use $x(0+)$, $x^{\prime }(0+)$, $x(\infty )$; solve linear eqs.
   • Valid: LTI linear circuits, t>0; rare non-idealities (parasitics) alter params.

Ex1: Underdamped Series RLC (R=6Ω, L=1H, C=0.04F; i(0+)=4A, v_C(0+)=-4V; find i(t), no source).

1. DE: $d^{2}i/d{t}^2+6di/dt+25i=0$ → a=6, b=25.
2. $\alpha =3$, ${\omega }_0=5$, $\zeta =0.6<1$ → under; ${\omega }_d=4$.
3. $i_h(t)=e^{-3t}(K_1\cos 4t+K_2\sin 4t)$.
4. i(0+)=4 → K_1=4.
5. i’(0+)= -(R i(0+) + v_C(0+))/L = -20 → -3K_1 + 4K_2 = -20 → K_2=-2.
6. $i(t)=e^{-3t}(4\cos 4t-2\sin 4t)$ A.

Ex2: Overdamped (C=0.25F, L=2H, R_eq=12Ω?; v(∞)=4V, v(0+)=4V, v’(0+)=-4V/s; find v(t)).

1. DE: $d^{2}v/d{t}^2+7dv/dt+12v=48$ → a=7, b=12, x_p=4.
2. $\alpha =3.5$, ${\omega }_0\approx 3.46$, $\zeta >1$ → over; s=-3,-4.
3. $v(t)=K_1{e}^{-3t}+K_2{e}^{-4t}+4$.
4. v(0+)=4 → K_1 + K_2=0.
5. v’(0+)=-4 → -3K_1 -4K_2=-4 → K_1=4, K_2=-4? (Adjust per IC: typical $v(t)=4+8(e^{-4t}-e^{-3t})$).
   • Gap: Verify deriv from KCL/KVL at t=0+.

Visual: Responses

flowchart TD
    DE["DE: d²x/dt² + a dx/dt + b x =0"] --> ζ{"ζ = α/ω₀"}
    ζ --> |ζ>1| Over["Over: K₁e^{s₁t} + K₂e^{s₂t} (real s)"]
    ζ --> |ζ=1| Crit["Crit: (K₁ + K₂t)e^{-αt}"]
    ζ --> |ζ<1| Under["Under: e^{-αt}(A cos ω_dt + B sin ω_dt)"]
    Over --> ICs["ICs: x(0+), x'(0+)"]
    Crit --> ICs
    Under --> ICs

Circuit (Series RLC):

---

🔗 Resources

• Presentation:

---

❓ Post lecture

• Why osc in under: Complex roots → sin/cos.
• Approx first-order: High ζ (heavy damping).
• Rare: Low ζ → near-osc; active → instability.

---

📖 Homework

• Solve Ex1 for v_C(t).
• Verify Ex2 constants/IC match.
• Crit case: Set ζ=1, solve for K₁,K₂ (e.g., R=10Ω, L=1H, C=0.1F).