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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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