# Loop equations and Laplace transformation

Refer to attached circuit diagram.

a) Write the loop equations for the circuit.

b) Take Laplace transform of the equations

c) Solve for the current through the 5 ohm resistor.

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#### Solution Preview

Please see the attached file.

Solution

From and ,

First loop:

Second loop:

Therefore the equations of the ...

#### Solution Summary

The solution writes the loop equations for the given circuit diagram, takes the Laplace transformations and then solves for the current.

Laplace transform, Fourier transform, Unit-pulse response, Impulse response, DTFT, Differential equation

Please see the attachment for all the questions with proper symbols/notations.

1. Compute the Laplace transform of e^(-10t)cos(3)u(t) .

2. Compute the z-transform of the discrete time signal defined by:

x[n] = δ[n] + 5δ[n - 1]

3. Compute the inverse Laplace transform of X(s) = (s+2) / (s^2+7s+12) .

4. Determine if the signal given is linear, time-invariant, causal, and/or memoryless.

y(t) = [sin(6t)]x(t)

5. A continuous time signal x(t) has the Fourier transform X(ω) = 1 / (jω+b), where b is a constant. Determine the Fourier transform for v(t) = x(5t - 4).

6. Compute the unit-pulse response h[n] for the discrete-time system

y[n + 2] -2y[n + 1] + y[n] = x[n] (for n= 0, 1, 2, 3)

7. Determine the inverse DTFT of X(Ω) = sin(Ω)cos(Ω).

8. Determine if the signal given is linear, time-invariant, causal, and/or memoryless.

y(t) = d^(x(t))

9. Determine if x(n) = cos ( PI n / 4) cos ( PI n / 4) is periodic; if periodic, calculate the period.

10. Compute the DTFT of the discrete-time signal x[n] = (0.8)^n u[n].

11. Compute the impulse response h(t) for (dy(t) / dt) - 3y(t) = x(t).

12. Compute the inverse Fourier transform for X(ω) = sin^2(3ω).

13. Determine if x(t) = cos(3t + π / 4) is periodic; if periodic, calculate the period.

14. Determine if the linear time-invariant continuous-time system defined is stable, marginally stable, or unstable.

(s - 1) / (s^2 + 4s + 5)

15. For a discrete-time signal x[n] with the DTFT X(Ω) = 1 / (e^jΩ + b), where b is an arbitrary constant, compute the DTFT V(Ω) of v[n] = x[n - 5].

16. For the RC circuit shown in the figure (in the attachment), find the input/output differential equation.

17. Solve the differential equation (dy(t) / dt) + 2y(t) = x(t); where x(t) = u(t), and y(0) = 4.

19. Compute the Fourier Transform for the convolution of sin(2t)*cos(2t) .

20. Use the Laplace Transform to compute the solution to the differential equation defined by

(dy / dt) + 2y = u(t) where y(0) = 0.