SysSig - Week 3 to 4 Rehash - 8.1 to 8.4

Define a natural response

• power is suddenly removed

Define a step response

• power is suddenly applied

Draw a basic parallel RLC circuit undergoing either responses

State the property measured when analyzing:

• Natural parallel

• Natural Series

• Step Parallel

• Step Series

• Nat Par = x(t) = v(t)

• Nat Ser = x(t) = i(t)

• Step Par = y(t) = iL (t)

• Step ser = y(t) = vc(t)

Derive the natural response of a parallel RLC circuit

• Vo = initial voltage of capacitor

• Io = inital current of inductor

• We find an equation that describes the change in voltage over tiem

  • v is the same for all elements

• ir + ic + iL = 0

  • v/R + C(dv/dt) + 1/L (vdt) + Io = 0

  • (1/R)v’ + Cv’’ + 1/L(v) = 0

  • v’’ + (1/RC)v’ + (1/LC) (v) = 0

• solve differential equation

Describe how to solve the natural response of an RLC circuit

• v’’ + (1/RC)v’ + (1/LC) (v) = 0

• overdamped, underdamped, critical

• v(0+) = Vo = voltage of capacitor before

• v’(0+) = 1/C * ic(0+)

  • ic(0+) = -(Vo/R) - Io = Current in capacitor

    • ic + iL + ir = 0 before switch opened

Draw a basic series RLC circuit undergoing either responses

Derive the second order DE when solving the natural response of a prallel RLC circuit

Find the solutions to the final voltage of a parallel RLC circuit undergoing a natural response

Explain the components of the solutions to the second order DE and describe when the discriminant is:

• = 0

• < 0

• > 0

For an overdamped parallel RLC response, describe how to find v(t)

For an underdamped parallel natural RLC response, describe how to find v(t)

Describe the characteristics of an underdamped response

For a critically daped parallel natural RLC response, describe how to find v(t)

Summarize the working process for the natural response of a parallel RLC circuit

Lay the groundwork for describing the step response of a paralllel RLC circuit

• initial conditions present

• DC current source

• as t tends to infinity

  • ic → open circuit as t → infinity

  • iL → short circuit as t → infinity

  • iR → 0 as t → infinity

  • v → 0 as t → infinity

• so we measure current iL of the inductor

Derive the second order DE for the step response of a parallel RLC circuit

• iL + iR + ic = I

  • 1/ L * dv + v/R + C (dv/dt) = I

  • 1/L * v + (1/R) v’ + Cv’’ = I

  • v’’ + (1/RC) v’ + (1/LC) v = 0

    • v = Li’, v’ = Li’’

• or v = Li’, v’ = Li’’

  • iL + v / R + Cv’ = I

    • iL + L/R i’ + (LC)i’’ = I

  • (1/LC) * iL + (1/RC) i’ + i’’ = I/LC

Describe the indirect approach to solving the second order DE of a the step response for a parallel RLC circuit

Describe the direct approach to finding v(t) for the step response of a parallel RLC circuit

Derive the formulas for i(t) for the natural response of a series RLC circuit

• mesh current method

• vr + vc + vL = 0

  • Ri + (1/C) di + Li’ = 0

  • Ri’ + (1/C) i + L i’’ = 0

    • i’’ + (1/LR) i’ + (1/LC) i = 0

Summarize the process for finding i(t) for the natural response of a series RLC circuit

For the step response of an RLC series circuit:

• derive the second order DE

• Derive the solutiions for that DE

Summarize the working process for solving vc(t) for the step response of a series RLC circuit