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 e(t) = E 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 e(t) constant equal to E. What are the energies stored in L and C in these conditions?

2. Demonstrate that the circuit is controlled by a differential equation of the form of:

d²u/dt² + 2λdu/dt + ω₀²u(∞) and give λ and ω₀ as a function of, R₁, R_2, 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 u (t) 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 = 200 Ω, R₂ = 5kΩ 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(314 t).

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 = 25μF.