# Laplace tranforms, transient current series CR circuit

• Engineering
• DanRow93
In summary: The correct one should be i/s = C (sV(s) - vc(0) / s). In summary, by applying Kirchoff's voltage law and using Laplace Transforms, an expression for the transient circuit current can be deduced for a series CR circuit with a step voltage of 120v applied. The equations for the transient voltages across the resistor and capacitor can also be obtained by using the equations for voltage and current in the Laplace Domain. Specifically, the voltage source can be represented by V/s, the resistor by R, the inductor by Ls, and the capacitor by 1/sC.
DanRow93

## Homework Statement

A step voltage of 120v is applied to a series CR circuit. R = 20KΩ, C = 4µF

1. Deduce, using Kirchoff's voltage law and Laplace Transforms, an expression for the transient circuit current.

2. Using the equation obtained in 1. deduce the equations for the transient voltages across the resistor and capacitor.

## Homework Equations

V = vc + vr
vr = iR
i = C dvc/dt
vr = CR dvc/dt
V = vc + CR dvc/dt

## The Attempt at a Solution

I am first trying to find the transient circuit current, so I will use the equation i = C dvc/dt

Using Laplace Transforms gives i/s = C (sV(s) - vc(0))

vc(0) = 0

So i/s = C (sV(s))

i/s = (4x10^(-6))(sV(s))

i/(4x10^(-6)s) = sV(s)

I'm stuck here, I can't seem to find a Laplace Transform to put the equation into the (t) domain?

Or should I have started with the equation V = vc + CR dvc/dt

You can bypass the differential equation interpretation if you simply replace each circuit component with its Laplace Domain counterpart, then write the usual circuit equations using them. The voltage source is a step function, so that's trivial, and use the "Laplace Impedances" for the other components:

##V \rightarrow \frac{V}{s}##
##R \rightarrow R##
##L \rightarrow L s##
##C \rightarrow \frac{1}{sC}##

DanRow93 said:
Using Laplace Transforms gives i/s = C (sV(s) - vc(0))
This equation is wrong.

## 1. What is a Laplace transform?

A Laplace transform is a mathematical operation that converts a function of time into a function of complex frequency. It is used to simplify differential equations and analyze signals in the frequency domain.

## 2. What is a transient current series CR circuit?

A transient current series CR circuit is an electrical circuit that consists of a resistor (R) and a capacitor (C) connected in series. It is used to study the transient behavior of current in the circuit, which refers to how the current changes over time when the circuit is switched on or off.

## 3. How is a Laplace transform used to analyze a transient current series CR circuit?

A Laplace transform can be applied to the differential equations that describe the behavior of a transient current series CR circuit. This allows us to analyze the circuit in the frequency domain, where we can easily calculate the steady-state response and transient response of the circuit.

## 4. What is the time constant of a transient current series CR circuit?

The time constant of a transient current series CR circuit is the product of the resistance and capacitance in the circuit (RC). It represents the time it takes for the capacitor to charge or discharge to 63.2% of its maximum value.

## 5. How can the Laplace transform be used to calculate the voltage or current in a transient current series CR circuit?

The Laplace transform can be used to obtain the transfer function of the circuit, which relates the input voltage or current to the output voltage or current. By taking the inverse Laplace transform, we can then obtain the time-domain expression for the voltage or current in the circuit.

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