#49 Transient Analysis #49 — Underdamped and undamped parallel RLC

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

1. 1. Separate the two oscillatory cases

   

   The relation between α and ω₀ determines the characteristic roots

   0<α<ω₀ → underdamped

   α=0 → undamped

   Both cases have complex-conjugate roots

   Amplitude decays in the underdamped case

   Amplitude remains constant in the undamped case

   Narration transcript

   In the previous lesson, we covered the non-oscillatory side of second-order response. Now we move to the oscillatory cases. These appear when the roots are no longer two different real numbers. For the parallel R L C circuit, the key question is again the relationship between alpha and omega zero. Today we study the underdamped case and the ideal undamped limit.

2. 2. Underdamped solution form

   

   Condition:

   $\displaystyle 0<\alpha <\omega ₀$

   $\displaystyle s₁,₂=-\alpha \pm j\omega _{d}$

   $\displaystyle \omega _{d}=\sqrt{\omega ₀²-\alpha ²}$

   ω_d is the damped natural angular frequency

   $\displaystyle x_{n}=e^{-\alpha t}\left[B₁\cos \left(\omega _{d}t\right)+B₂\sin \left(\omega _{d}t\right)\right]$

   The e^−αt envelope shrinks successive peaks

   Oscillation continues while energy falls each cycle

   Narration transcript

   Start with the underdamped case, where alpha is less than omega zero. The characteristic roots become a complex conjugate pair: negative alpha plus or minus j omega d, where omega d equals the square root of omega zero squared minus alpha squared. The natural response therefore takes the form e to the minus alpha t times an oscillating term. In other words, the waveform oscillates, but its amplitude shrinks under an exponential envelope.

3. 3. Underdamped numerical example

   

   α=0.45 s⁻¹, ω₀=2.2 rad/s

   ω_d=√(2.2²−0.45²)≈2.1535 rad/s

   Envelope time constant:

   $\displaystyle \frac{1}{\alpha }\approx 2.222 s$

   Oscillation period:

   $\displaystyle T_{d}=\frac{2\pi }{\omega _{d}}\approx 2.918 s$

   $\displaystyle x_{n}=e^{-0.45t}\left[B₁\cos \left(2.1535t\right)+B₂\sin \left(2.1535t\right)\right]$

   Initial conditions determine B₁ and B₂

   The decay envelope and oscillation frequency play separate roles

   Narration transcript

   For example, choose alpha equal to zero point four five and omega zero equal to two point two. Then the response looks like e to the minus alpha t cosine omega d t. The key visual idea is the envelope. The oscillation continues to cross zero, but the peaks get smaller and smaller because energy is dissipated in the resistor. This is the classic damped oscillation shape.

4. 4. Undamped solution form

   

   Condition:

   $\displaystyle \alpha =0$

   $\displaystyle s₁,₂=\pm j\omega ₀$

   The exponential envelope becomes e⁰=1

   $\displaystyle x_{n}=B₁\cos \left(\omega ₀t\right)+B₂\sin \left(\omega ₀t\right)$

   The ideal circuit has no resistance or energy loss

   Energy moves continuously between L and C

   Amplitude never decays; oscillation persists at ω₀

   Narration transcript

   Now consider the ideal undamped case. Here alpha equals zero, which means no damping is present. In practice, this is the ideal L C limit. The roots sit exactly on the imaginary axis at plus or minus j omega zero. The natural response becomes a pure sinusoidal combination: B one cosine omega zero t plus B two sine omega zero t. There is no decaying envelope.

5. 5. Compare energy and waveform

   

   Underdamped:

   $\displaystyle 0<\alpha <\omega ₀$

   R converts stored energy into heat

   Envelope: ±Ke^−αt

   The oscillation frequency ω_d is below ω₀

   Undamped:

   $\displaystyle \alpha =0$

   There is no energy loss and the envelope is constant

   The oscillation frequency is exactly ω₀

   Narration transcript

   Compare the two oscillatory cases carefully. In the underdamped response, the circuit still oscillates, but the resistor removes energy over time, so the amplitude decays. In the undamped response, energy keeps exchanging between the capacitor and the inductor without loss, so the amplitude stays constant. Both are oscillatory. The difference is whether the envelope decays or remains flat.

6. 6. Summarize underdamped and undamped response

   

   Complex roots create oscillatory response

   Real part −α → envelope decay rate

   Imaginary part ω_d → oscillation rate

   As α approaches zero, ω_d approaches ω₀

   At α=0 the decaying envelope disappears

   Initial conditions determine B₁ and B₂

   Next: combine the natural and forced responses

   Narration transcript

   Let us summarize. When alpha is less than omega zero, the circuit is underdamped and the roots are complex conjugates. The response oscillates under an exponential envelope, and the damped frequency is omega d. When alpha is zero, the circuit becomes the ideal undamped L C case, with sustained sinusoidal oscillation. Next lesson, we move from natural response cases to the full second-order response with forcing.