The signal generator is modeled as an ideal voltage source where vs (t) = 5 u(t)

Question

The signal generator is modeled as an ideal voltage source where vs (t) = 5 u(t) V in series with an source resistance RS = 50 ohm. The inductor will be modeled as an ideal L = 80 mH inductor in series with a wire resistance Rwire = 30 ohm. An LCR meter was used to determine these values. The capacitance is C = 0.1 microF.
Calculate roots of the characteristic equation, and determine the general form of the natural response of the capacitor voltage vC,n (t) for t ? 0 (i.e., overdamped, critically damped, and underdamped) when the resistor Rvar has values of: a) 4700 ohm, b) 1708.8544 ohm, and c) 330 ohm.

Done.

(a) s1 = -2170.93, s2 = -57579.07, OVERDAMPED

(b) s1 = s2 = -11180.34, CRITICALLY DAMPED

(c) s1 = -288.03, s2 = -4836.97, UNDERDAMPED

Find vC (0), d vC /d t|t = 0, and vC (?) given no initial energy is stored in the inductor and/or capacitor.

Please help with this part :-)

Determine expressions for the total capacitor voltage vC (t) for t ? 0 when the resistor Rvar has values of: a) 4700 ohm, b) 1708.8544 ohm, and c) 330 ohm.

Again, I could use help on this. Is the method for finding all three the same? If so, I only need one shown, I can figure out the rest.

Explanation / Answer

I will explain the first case in details : KVL yields (Rvar = 4700): Vs = (Rs + Rwire + Rvar)i + Vc + Ldi/dt let Rs + Rwire + Rvar = R since i = C dVc/dt s you get : LC d2Vc/dt2 + RC dVc/dt + Vc = Vs dividing by LC gives : d2Vc/dt2 + (R/L) dVc/dt + (1/LC)Vc = Vs/LC roots : => s2 + (R/L)s + (1/LC) = 0 with the given values of R,L,and C you get : s2 + 59750s + 125x106 = 0 => discriminant = 59750^2 - 4(125x106) > 0 so it's overdamped => s1,2 = [-59750

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