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Engineering and Comp Sci Homework Help
Circuit responses to a step input
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[QUOTE="agata78, post: 4543542, member: 488283"] [h2]Homework Statement [/h2] Calculate RLC circuit responses to a step input? I believe the step input needs to be 10V. [h2]Homework Equations[/h2] Series RLC Characteristic Equation: S[SUP]2[/SUP] + S(R/L) + 1/LC = 0 Putting the above equation into standard form: S[SUP]2[/SUP] + 2αS + ω[SUB]0[/SUB][SUP]2[/SUP] = 0, it follows that: α = R/2L ω[SUB]0[/SUB] = 1 / √LC where: α is the Damping Coefficient and ω[SUB]0[/SUB] is the Natural or Resonant Frequency The roots of the Characteristic Equation, by using the Quadratic Formula, are: S[SUB]1[/SUB],[SUB]2[/SUB] = -α ± √(α[SUP]2[/SUP] - ω[SUB]0[/SUB][SUP]2[/SUP]) For each type of damping condition, the voltage and current solutions take a different form: [B]CAPACITOR VOLTAGE FOR A STEP INPUT TO A SERIES RLC CIRCUIT:[/B] Overdamped: α> ω[SUB]0[/SUB], Vc(t) = Vc(∞) + A[SUB]1[/SUB]e [SUP]S[/SUP][SUB]1[/SUB][SUP]t[/SUP] + A[SUB]2[/SUB]e [SUP]S[/SUP][SUP][/SUP][SUB]2[/SUB][SUP]t[/SUP] Volts Critically Damped: α= ω[SUB]0[/SUB], Vc(t) = Vc(∞) + (A[SUB]1[/SUB] + A[SUB]2[/SUB]t)e [SUP]-αt[/SUP] Volts Underdamped: α< ω[SUB]0[/SUB], Vc(t) = Vc(∞) + (A[SUB]1[/SUB]cos ω[SUB]d[/SUB]t + A[SUB]2[/SUB] sin ω[SUB]d[/SUB]t)e [SUP]-αt[/SUP] Volts ω[SUB]d[/SUB] = √(α[SUP]2[/SUP] - ω[SUB]0[/SUB][SUP]2[/SUP]) [B]INDUCTOR CURRENT FOR A STEP INPUT TO A SERIES RLC CIRCUIT:[/B] Overdamped: α> ω[SUB]0[/SUB], I[SUB]L[/SUB](t) = I[SUB]L[/SUB] (∞) + B[SUB]1[/SUB]e [SUP]S[/SUP][SUB]1[/SUB][SUP]t[/SUP] + B[SUB]2[/SUB]e [SUP]S[/SUP][SUB]2[/SUB][SUP]t[/SUP] Amps Critically Damped: α= ω[SUB]0[/SUB], I[SUB]L[/SUB] (t) = I[SUB]L[/SUB] (∞) + (B[SUB]1[/SUB] + B[SUB]2[/SUB]t)e [SUP]-αt[/SUP] Amps Underdamped: α< ω[SUB]0[/SUB], I[SUB]L[/SUB] (t) = I[SUB]L[/SUB] (∞) + (B[SUB]1[/SUB]cos ω[SUB]d[/SUB]t + B[SUB]2[/SUB] sin ω[SUB]d[/SUB]t)e [SUP]-αt[/SUP] Amps ω[SUB]d[/SUB] = √(α[SUP]2[/SUP] - ω[SUB]0[/SUB][SUP]2[/SUP]) Can you please let me know if these formulas are correct to continue answering this question? [/QUOTE]
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Circuit responses to a step input
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