Reference no: EM131190678
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, 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 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:
vo(t) = 180sin(314 t).
1. Find the amplitude of the current i(t) and its phase relative to vo(t) using a phasor diagram.
2. On the same phasor diagram, deduce the amplitude of the input voltage vs(t) and its phase relative to i(t).
3. Give the instantaneous values of vs(t) and i(t).
4. Verify your results using a mathematical reasoning.
5. Plot on the same graph vo(t) and vs(t).
In this circuit R = 200Ω, L = 0.1 H and C = 25μF.
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