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Series vs Parallel RLC Comparison MicroSim

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Description

This MicroSim compares the step responses of series and parallel RLC circuits side by side. Both circuits share the same R, L, and C values but have different damping characteristics because their neper frequencies differ:

  • Series: \(\alpha_s = \frac{R}{2L}\) — increasing R increases damping
  • Parallel: \(\alpha_p = \frac{1}{2RC}\) — increasing R decreases damping

Both share the same natural frequency \(\omega_0 = \frac{1}{\sqrt{LC}}\).

Damping Conditions:

Condition Response
\(\zeta < 1\) (α < ω₀) Underdamped — oscillates around final value
\(\zeta = 1\) (α = ω₀) Critically damped — fastest rise without overshoot
\(\zeta > 1\) (α > ω₀) Overdamped — slow exponential rise, no oscillation

Interactive Features:

  • R Slider (1-200 Ω): Changes damping — note opposite effects on series vs parallel
  • L Slider (1-100 mH): Changes natural frequency and series damping
  • C Slider (1-100 µF): Changes natural frequency and parallel damping
  • Info Panel: Live display of α, ζ, ω₀, and damping type for both configurations

How to Use

  1. With defaults (R=20Ω, L=100mH, C=100µF), observe that series is underdamped while parallel may be overdamped (or vice versa)
  2. Drag the R slider and watch how series and parallel responses change in opposite directions
  3. Find the R value where both circuits are critically damped — they will generally differ
  4. Change L and C to shift the natural frequency ω₀ (shared by both)

Physics

Series RLC (capacitor voltage, voltage source input):

\[ V_c(t) = V_s \left(1 - \frac{1}{\sqrt{1-\zeta_s^2}} e^{-\alpha_s t} \sin(\omega_d t + \varphi)\right) \quad \text{(underdamped)} \]

Parallel RLC (voltage across parallel combination, current source input):

\[ v(t) = I_s R \left(1 - \frac{1}{\sqrt{1-\zeta_p^2}} e^{-\alpha_p t} \sin(\omega_d t + \varphi)\right) \quad \text{(underdamped)} \]

References

  • Chapter 7: Second-Order RLC Circuits
  • Chapter 6: Transient Analysis RC/RL
  • RL Charging MicroSim - First-order RL counterpart
  • RC Charging MicroSim - First-order RC counterpart