# Circuit responses to a step input

• Engineering
agata78

## 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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## Answers and Replies

Mentor
You can find excellent treatments of the series RLC circuit on the web (google: RLC circuit) and confirm your equations there. From a brief perusal, what you've shown looks like the right stuff.

You haven't actually stated what the question is, so it's hard to say if what you've written is what you'll need to answer it agata78
The question states:

SUBJECT TITLE: Calculate circuit responses to a step input in practical situation
Calculate the circuit response for the circuit in the figure.

However, at the back of the question paper it says:
To achieve a Merit you will demonstrate the ability to:

M2: Select/design and apply appropriate methods/techniques
Students have simulated the circuit response to a step input of 10 Volts for the circuit in the figure and printed the transient analysis of the simulation.

Does my previous post begin to answer the initial question? 'Calculate the circuit response for the circuit in the figure'

Mentor
The question states:

SUBJECT TITLE: Calculate circuit responses to a step input in practical situation
Calculate the circuit response for the circuit in the figure.

However, at the back of the question paper it says:
To achieve a Merit you will demonstrate the ability to:

M2: Select/design and apply appropriate methods/techniques
Students have simulated the circuit response to a step input of 10 Volts for the circuit in the figure and printed the transient analysis of the simulation.

Does my previous post begin to answer the initial question? 'Calculate the circuit response for the circuit in the figure'

Sure. You can take the given circuit and calculate its parameters, classify it as under-damped, over-damped, or critically-damped, do a plot of current versus time, and so on. Looks like a fairly open-ended assignment so feel free to get creative.

You might even download a (free) copy of a circuit simulation package like LTSpice, build the circuit in it and compare the simulated results with your calculations. Depends upon how much time you've got on your hands!