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Homework Statement
Calculate RLC circuit responses to a step input?
I believe the step input needs to be 10V.
Homework Equations
Series RLC Characteristic Equation:
S^{2} + S(R/L) + 1/LC = 0
Putting the above equation into standard form:
S^{2} + 2αS + ω_{0}^{2} = 0, it follows that:
α = R/2L
ω_{0} = 1 / √LC
where: α is the Damping Coefficient and
ω_{0} is the Natural or Resonant Frequency
The roots of the Characteristic Equation, by using the Quadratic Formula, are:
S_{1},_{2} = α ± √(α^{2}  ω_{0}^{2})
For each type of damping condition, the voltage and current solutions take a
different form:
CAPACITOR VOLTAGE FOR A STEP INPUT TO A SERIES RLC CIRCUIT:
Overdamped: α> ω_{0}, Vc(t) = Vc(∞) + A_{1}e ^{S}_{1}^{t} + A_{2}e ^{S}_{2}^{t} Volts
Critically Damped: α= ω_{0}, Vc(t) = Vc(∞) + (A_{1} + A_{2}t)e ^{αt} Volts
Underdamped: α< ω_{0}, Vc(t) = Vc(∞) + (A_{1}cos ω_{d}t + A_{2} sin ω_{d}t)e ^{αt} Volts
ω_{d} = √(α^{2}  ω_{0}^{2})
INDUCTOR CURRENT FOR A STEP INPUT TO A SERIES RLC CIRCUIT:
Overdamped: α> ω_{0}, I_{L}(t) = I_{L} (∞) + B_{1}e ^{S}_{1}^{t} + B_{2}e ^{S}_{2}^{t} Amps
Critically Damped: α= ω_{0}, I_{L} (t) = I_{L} (∞) + (B_{1} + B_{2}t)e ^{αt} Amps
Underdamped: α< ω_{0}, I_{L} (t) = I_{L} (∞) + (B_{1}cos ω_{d}t + B_{2} sin ω_{d}t)e ^{αt} Amps
ω_{d} = √(α^{2}  ω_{0}^{2})
Can you please let me know if these formulas are correct to continue answering this question?
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