# Problem finding the output voltage using Laplace transform

• diredragon
In summary: That would be the "hard way".In summary, the circuit shown is a boxcar integrator that uses a digital signal to control a switch. The input signal is multiplied by a series of impulses with a narrow width and a period of Ts. To find Xs(jω), multiply the input signal by the impulses and take the Fourier transform. To calculate G(jΩ), set t = nTs - 0.3Ts and repeat the process. The Laplace transform is not necessary for this problem.
diredragon

## Homework Statement

The input signal of the circuit shown below is ##x(t)=2\sin (ω_ot + \pi/6)##. The switch in the circuit is controlled with a digital signal of the form ##s(t)=\sum_{k=-\infty}^{+\infty} (u(t+ε-kT_s) - u(t-ε-kT_s))##, ##\frac{2\pi}{T_s}=800\pi##, ##ε\to 0##, so that when the signal equals 1 the switch is in the position 1 (closed).
a) Calculate ##X_s(jω)##
b) If ##g[n]=x_s(nT_s + 0.3T_s)## calculate ##G(jΩ)##.
Note: ##u(t)## is a step funtion, defined as ##u(0)=\frac{1}{2}##

## Homework Equations

3. The Attempt at a Solution [/B]
I am not sure how to start and what to do with this digital signal that's controlling the switch. It looks like a sum of delta impulses with very narrow width of ##ε## and a period of ##T_s##. My first thought was that this was some sort of sampling circuit and that i should do a convolution of the two signals to find the ##X_s(jω)##. Something like this:
##x_s(t)=x(t)*s(t)##
##X_s(jω)=\mathscr{L}\{x(t)\} \mathscr{L}\{s(t)\}##. Is this the correct approach?

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• Problem_Circuit.jpg
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Delta2
To start, you need to multiply not convolve. Do you see why? Then you need to consider the action of the second op amp circuit.

diredragon
This ckt is what is called a "boxcar integrator". The input signal is sampled and held until the next sample. So you can forget about the size of ε.

(a) One approach would be to recognize that the output is a series of pulses of width Ts, then recognizing that this output is equivalent to an "ideal" (multiply by δ(t-nT)) sampling of x(t), convolved with a single pulse of width Ts. The latter process is easy since integration with a δ funcion is mathematical pig heaven. Then take the Fourier integral Xs(f) of the output.

(b) let t = nTs - 0.3Ts instead of sin(wt ...) and compute Xs(f) again.

Don't use lapace transform at all.

diredragon

## 1. How is the Laplace transform used to find the output voltage?

The Laplace transform is a mathematical tool that converts a function of time into a function of complex frequency. By applying the Laplace transform to the input voltage and the transfer function of the circuit, the output voltage can be found by simply multiplying the transformed input voltage with the transformed transfer function.

## 2. What is the role of initial conditions in finding the output voltage using Laplace transform?

Initial conditions, such as the initial voltage or current in the circuit, are taken into account when using the Laplace transform to find the output voltage. These initial conditions are represented as constants in the transformed equations and are used to determine the complete solution for the output voltage.

## 3. Can the Laplace transform be used for any type of circuit?

Yes, the Laplace transform can be used to analyze the behavior of linear time-invariant (LTI) circuits, which are the most commonly used circuits in electrical engineering. However, for non-linear circuits, the Laplace transform may not be applicable.

## 4. Is there a specific step-by-step process for finding the output voltage using Laplace transform?

Yes, there is a standard process that is followed for using the Laplace transform to find the output voltage. This includes transforming the input voltage and transfer function, combining them using algebraic operations, and then taking the inverse Laplace transform to obtain the output voltage in the time domain.

## 5. What are the advantages of using Laplace transform to find the output voltage?

The Laplace transform provides a powerful tool for analyzing the behavior of circuits, as it allows for the use of complex frequency analysis. It also simplifies the mathematical calculations involved in finding the output voltage, as it reduces complex differential equations to algebraic equations. Additionally, the Laplace transform can be used to find the complete solution for the output voltage, including both the transient and steady-state behavior, making it a valuable tool for circuit analysis.

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