#48 Transient Analysis #48 — Overdamped and critically damped parallel RLC

Below are all the lines written in the notebook together with the full narration transcript.

1. 1. Separate the two non-oscillatory cases

   

   The relation between α and ω₀ selects the parallel-RLC roots

   α>ω₀ → overdamped

   α=ω₀ → critically damped

   Both cases have real roots

   Both return to equilibrium without oscillation

   They differ in root pattern and settling speed

   Narration transcript

   From the previous lesson, we already know that the parallel R L C natural response is controlled by the relationship between alpha and omega zero. Today we focus only on the two non-oscillatory cases: overdamped and critically damped. In both cases, the response returns to equilibrium without ringing. The difference is in the root pattern and in how fast the response settles.

2. 2. Overdamped solution form

   

   Condition: α>ω₀

   Δ=√(α²−ω₀²) is real

   $s₁=-\alpha+\Delta$

   $s₂=-\alpha-\Delta$

   The roots are distinct, real, and negative

   $x_{\mathrm{n}}(t)=A₁e^{s₁t}+A₂e^{s₂t}$

   No oscillation: one fast and one slow exponential mode

   Narration transcript

   Start with the overdamped case, where alpha is greater than omega zero. The characteristic equation then gives two distinct real roots: s one equals negative alpha plus the square root of alpha squared minus omega zero squared, and s two equals negative alpha minus the same square root. Because both roots are real and negative, the natural response becomes a sum of two decaying exponentials. There is no oscillation. Instead, the waveform returns smoothly, with one fast mode and one slower dominant mode.

3. 3. Overdamped numerical example

   

   α=5 s⁻¹, ω₀=3 rad/s

   $\Delta=\surd (25-9)=4$

   $s₁=-5+4=-1 s⁻¹$

   $s₂=-5-4=-9 s⁻¹$

   $x(t)=1.125e^{-t}-0.125e^{-9t}$

   The checks x(0)=1 and x′(0)=0 pass

   e^−9t dies quickly; e^−t becomes dominant

   Narration transcript

   For a quick example, choose alpha equal to five and omega zero equal to three. Then the roots are negative one and negative nine. A normalized response can be written as one point one two five e to the minus t, minus zero point one two five e to the minus nine t. The e to the minus nine t term dies out very quickly. After the initial moment, the slower e to the minus t term dominates the shape. That is why overdamped responses often look sluggish even though two modes are present.

4. 4. Critically damped solution form

   

   Condition: α=ω₀

   s₁=s₂=−α [repeated root]

   A second independent solution is needed for two constants

   $x_{\mathrm{n}}(t)=(A₁+A₂t)e^{-\alphat}$

   The t factor is the signature of the repeated root

   Fastest ideal return without oscillation

   Narration transcript

   Now move to the critically damped case, where alpha equals omega zero. Here the two real roots collapse into one repeated root at s equals negative alpha. A repeated root changes the structure of the solution. The response is no longer just A one e to the s t. Instead, we need the form open parenthesis A one plus A two t close parenthesis e to the minus alpha t. This extra t factor is the signature of critical damping.

5. 5. Critically damped numerical example

   

   $\alpha=\omega₀=4 s⁻¹$

   Repeated root: s=−4

   $x(t)=(1+4t)e^{-4t}$

   $x(0)=1$

   $x′(t)=-16t e^{-4t}$

   $x′(0)=0$

   It falls rapidly to zero without overshoot or ringing

   Narration transcript

   As an example, let alpha equal omega zero equal four. With normalized initial conditions, one valid response is open parenthesis one plus four t close parenthesis e to the minus four t. This curve is still non-oscillatory, but it returns faster than the overdamped case. That is the main practical meaning of critical damping: it is the fastest return to equilibrium without overshoot or oscillation.

6. 6. Compare overdamped and critical response

   

   Overdamped: α>ω₀

   Two distinct real roots and two exponential modes

   The slow root can dominate for a long time

   Critical: α=ω₀

   Repeated root and (A₁+A₂t)e^−αt

   Critical damping is the fastest non-oscillatory boundary

   Next: underdamped and undamped response

   Narration transcript

   Let us summarize. When alpha is greater than omega zero, the circuit is overdamped and the natural response is a sum of two real exponential modes. When alpha equals omega zero, the circuit is critically damped and the repeated root produces the form open parenthesis A one plus A two t close parenthesis e to the minus alpha t. Both cases are non-oscillatory, but critical damping settles faster. Next lesson, we continue with the oscillatory cases: underdamped and undamped response.