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agata78
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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:
S2 + S(R/L) + 1/LC = 0
Putting the above equation into standard form:
S2 + 2αS + ω02 = 0, it follows that:
α = R/2L
ω0 = 1 / √LC
where: α is the Damping Coefficient and
ω0 is the Natural or Resonant Frequency
The roots of the Characteristic Equation, by using the Quadratic Formula, are:
S1,2 = -α ± √(α2 - ω02)
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> ω0, Vc(t) = Vc(∞) + A1e S1t + A2e S2t Volts
Critically Damped: α= ω0, Vc(t) = Vc(∞) + (A1 + A2t)e -αt Volts
Underdamped: α< ω0, Vc(t) = Vc(∞) + (A1cos ωdt + A2 sin ωdt)e -αt Volts
ωd = √(α2 - ω02)
INDUCTOR CURRENT FOR A STEP INPUT TO A SERIES RLC CIRCUIT:
Overdamped: α> ω0, IL(t) = IL (∞) + B1e S1t + B2e S2t Amps
Critically Damped: α= ω0, IL (t) = IL (∞) + (B1 + B2t)e -αt Amps
Underdamped: α< ω0, IL (t) = IL (∞) + (B1cos ωdt + B2 sin ωdt)e -αt Amps
ωd = √(α2 - ω02)
Can you please let me know if these formulas are correct to continue answering this question?
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