Problem 1:

(a) Find H(ω) = I(ω)/VS(ω) for the circuit above.

(b) Find the following values:

(1) Find H(0) and H(∞)

(2) Find the peak frequency ωp, and the peak amplitude |H(ωp)|

(3) Find the frequencies ωp ± σp at which the circuit amplitude is

(4) Find the phases 6 H(ωp - σp), 6 H(ωp), and 6 H(ωp + σp),

(c) Suppose that R = 1k?, L = 1mH, and C = 0.1µF. Sketch |H(ω)| and 6 H(ω) as functions of ω, showing the particular frequency points that you calculated in part (b).

Problem 2:

Find H(ω) = Vo(ω)/VS(ω). Sketch |H(ω)|, showing the values of |H(0)|, |H(∞)|, and |H(ω0)| where ω0 = p1/LC = 16radians sec.

Problem 3:

The current source in the circuit above delivers the current f(t) = 6 + 3 cos(10t) - 2 sin(20t) - cos(100000t).

Find the voltage y(t).

Problem 4:

A piston moves up and down; its height is given by zs(t) = cos(ωt). Dangling from the bottom of the piston there is a small mass, m, whose position is zo(t).

Newton's law specifies that f = mA where A = d2 zo dt2 is the acceleration of the mass. The force on the mass, f, is supplied by a spring of stiffness k that connects the mass to the piston, therefore f = k(zs(t) - zo(t))

Find the frequency response H(ω) = Zo(ω)/Zs(ω), and sketch |H(ω)| as a function of ω. Be sure to indicate the values of H(ω) at ω = 0, ω = p k/m, and ω → ∞.