Consider the overdamped continuous-time second-order system with transfer function
Shown in Figure E8.3.
a. Decompose H(s) into the sum of two first-order transfer functions H(s) = H1(s) + H2(s) where H1(s)=A1/(τ1s + 1) and H2(s)=A2/(τ2s + 1) (see above figure) and express the constants A1
and A2
in terms of the time constants τ1
and τ2.
b. The equivalent realizations of the same second-order system are simulated using explicit Euler integration with step size T. Find the fractional error in the frequency response functions H(ejωT), H1(ejωT), and H2(ejωT). Leave your answers in terms of τ1, τ2, and T.
c. Resolve the fractional errors into real and imaginary components, that is,
For τ1
= 1 s, τ2
= 10 s, and T = 0.05 s, plot eM, eM1, eM2 vs. ωT on a single graph and eA, eA1, eA2, vs. ωT on a different graph. Comment on the results.
d. Find exact expressions for the fractional gain error in H(ejωT), H1(ejωT), and H2(ejωT). Plot the fractional gain error in H(ejωT) vs. ωT and eM vs. ωT on the same graph. Repeat for H1(ejωT) and eM1 and then for H2(ejωT) and eM2.
e. Find exact expressions for the phase error in H(ejωT), H1(ejωT), and H2(ejωT). Plot the phase error in H(ejωT) vs. ωT and eA
vs. ωT on the same graph. Repeat for H1(ejωT) and eA1
and then for H2(ejωT) and eA2.
f. Simulate the two configurations shown in Figure E8.3 when u(t) = sin 50t, t ≥ 0 using an explicit Euler integrator with step size T = 0.01 s. Plot the continuous-time input and output and the simulated response on the same graph for each configuration. Do the results agree with the graphs obtained in parts (d) and (e)?