EJERCICIOS (CAPITULO 8)

EJERCICIOS (CAPITULO 8)

EJERCICIO 8.1

[a]

α =1/(2RC)=1/(2(1000)(2 × 10−6))= 250

ωo2 =1/LC=1/((12.5)(2 × 10−6))= 40,000

s1,2 = −250 ±= −250 ± 150[pic 1]

s1 = −100 rad/s         s2 = −400 rad/s

[b]

SOBREAMORTIGUADO

[c]         Queremos, ωd = 120 rad/s:

ωd =[pic 2]

α2 = ω2o− ω2 = d = 40,000 − (120)2 = 25,600

α = 160

1/(2RC)= 160; ·  R =1/(2(160)(2 × 10−6))= 1562.5Ω

[d]

s1, s2 = −160 ±= −160 ± j120 rad/s[pic 3]

[e]

 α ==1/(2RC)[pic 4]

R =1/(2(200)(2 × 10−6))= 1250Ω        

EJERCICIO 8.2

        [a]         

−α +o = −250[pic 5]

−α -o = −1000[pic 6]

                Sumando las ecuaciones anteriores. , − 2α = −1250

α = 625 rad/s

1/2RC=1/2R(0.1 × 10−6)= 625

R = 8kΩ

2 = 750[pic 7]

4(α2 − ω2o) = 562,500

 ωo = 500 rad/s

ω2o = 25 × 104 =1/LC

         L =1/((25 × 104)(0.1 × 10−6))= 40H

[b]

 iR = v(t)/R= −1e−250t + 4e−1000t mA,         t≥ 0+

iC = C(dv(t)/dt)= 0.2e−250t − 3.2e−1000t mA,         t≥ 0+

iL = −(iR + iC) = 0.8e−250t − 0.8e−1000t mA,         t≥ 0

EJERCICIO 8.3

        [a]

iR(0) =15/200= 75mA

iL(0) = −45mA

iC(0) = −iL(0) − iR(0) = 45 − 75 = −30mA

[b]

α =1/2RC=1/(2(200)(0.2 × 10−6))= 12,500

ω2o =1/LC=1/((50 × 10−3)(0.2 × 10−6))= 108

s1,2 = −12,500 ±√(1.5625 × 108 – 108) = −12,500 ± 7500

s1 = −5000 rad/s;                   s2 = −20,000 rad/s

v = A1e−5000t + A2e−20,000t

v(0) = A1 + A2 = 15

dv/dt(0) = −5000A1 − 20,000A2 =(−30 × 10−3) /(0.2 × 10−6 )= −15 × 104V/s

Resolviendo, A1 = 10;      A2 = 5

v = 10e−5000t + 5e−20,000t V, t≥ 0

[c]         

iC = C(dv/dt) = 0.2 × 10−6[−50,000e−5000t − 100,000e−20,000t]

= −10e−5000t − 20e−20,000t mA

iR = 50e−5000t + 25e−20,000t mA

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