Nbspeeen222 - determine the final value ut noted uinfin

Question

Assignment

Exercise 1: damping rate

We study the response of the circuit of figure 1 to a step e(t). Figure 2 shows the output of the circuit as function of the time. The step signal during all this exercise.

1. Give the permanent steady state equivalent of the circuit. Determine the final value u(t) noted u(∞) when t → +∞ and the value of constant equal to E. What are the energies stored in and in these conditions?

2. Demonstrate that the circuit is controlled by a differential equation of the form of: d²u/dt² + 2λ.du/dt + ω_o²u = ω_o²u(∞) and give λ and ω_o as a function of R₁, R₂, L, C.

3. Using an oscilloscope (settings: 200μs/div, 1V/div), we got the waveforms of u(t)

a. Determine the pseudo-period T of u(t)

b. Determine the value of the logarithm decrement given by the expression

δ = 1/k ln((u(t) - u(∞))/(u(t-kt) - u(∞))) where is an integer

5. The mathematical form of is u(t) = E + e^-λt(Acos(ωt) + Bsin(ωt).

Determine the relationship between δ, λ and T. Deduce the numerical value of λ.

6. Determine the value of C when R₁ = 200Ω, R₂ = 5 kΩ and L = 100mH.

Exercise 2: phasor diagram in AC circuit

The instantaneous value of the output signal of the following circuit is given by:

v_o(t) = 180sin(314t)

1. Find the amplitude of the current i(t) and its phase relative to v_o(t) using a phasor diagram.

2. On the same phasor diagram, deduce the amplitude of the input voltage v_s(t) and its phase relative to i(t).

3. Give the instantaneous values of v_s(t)and i(t).

4. Verify your results using a mathematical reasoning.

5. Plot on the same graph v_o(t) and v_s(t).

In this circuit R = 200Ω, L = 0.1 H and C = 25 μF.