Circuit responses to a step input

[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 ^S₁^t + A₂e ^S₂^t Volts
Critically Damped: α= ω₀, Vc(t) = Vc(∞) + (A₁ + A₂t)e ^-αt Volts
Underdamped: α< ω₀, Vc(t) = Vc(∞) + (A₁cos ω_dt + A₂ sin ω_dt)e ^-αt Volts
ω_d = √(α² - ω₀²)

INDUCTOR CURRENT FOR A STEP INPUT TO A SERIES RLC CIRCUIT:
Overdamped: α> ω₀, I_L(t) = I_L (∞) + B₁e ^S₁^t + B₂e ^S₂^t Amps
Critically Damped: α= ω₀, I_L (t) = I_L (∞) + (B₁ + B₂t)e ^-αt Amps
Underdamped: α< ω₀, I_L (t) = I_L (∞) + (B₁cos ω_dt + B₂ sin ω_dt)e ^-αt Amps
ω_d = √(α² - ω₀²)

Can you please let me know if these formulas are correct to continue answering this question?

 

[gneill]

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'

 

[gneill]

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!