Sistemas circuitos electricos

✼✳✶✸ ❈✐r❝✉✐t♦s ❡❧é❝tr✐❝♦s✳ ❯♥✐❞❛❞ ✼ ❙✐st❡♠❛s ❞❡ ❊❝✉❛❝✐♦♥❡s ❞❡ ♣r✐♠❡r ♦r❞❡♥ ❧✐♥❡❛❧❡s ❈✐r❝✉✐t♦s ❡❧é❝tr✐❝♦s✳ ▲❛s ❡❝✉❛❝✐♦♥❡s q✉❡ ❞❡s❝r✐❜❡♥ ❧❛s r❡❧❛❝✐♦♥❡s ✈♦❧t❛❥❡✲❝♦rr✐❡♥t❡ ♣❛r❛ ✉♥❛ r❡s✐st❡♥❝✐❛✱ ✉♥ ✐♥❞✉❝t♦r ② ✉♥ ❝♦♥✲ ❞❡♥s❛❞♦r ❥✉♥t♦ ❝♦♥ ❧❛s ❧❡②❡s ❞❡ ❑✐r❝❤❤♦✛ q✉❡ r❡str✐♥❣❡♥ ❡❧ ❝♦♠♣♦rt❛♠✐❡♥t♦ ❞❡ ❡st❛s ❝❛♥t✐❞❛❞❡s ❝✉❛♥❞♦ ❧♦s ❡❧❡♠❡♥t♦s s❡ ❝♦♥❡❝t❛♥ ❡♥ ❢♦r♠❛ ❡❧é❝tr✐❝❛ ❛ ✉♥ ❝✐r❝✉✐t♦✳ ❊❥❡♠♣❧♦ ❊❧ ❝✐r❝✉✐t♦ ❘▲❈ ❡♥ s❡r✐❡ ❞❡ ❧❛ ✜❣✉r❛ t✐❡♥❡ ✉♥❛ ❢✉❡♥t❡ ❞❡ ✈♦❧t❛❥❡ ❞❛❞❛ ♣♦r E(t) = sen(100t) ✈♦❧t✐♦s ✭❱✮✱ ✉♥❛ r❡s✐st❡♥❝✐❛ ❞❡ 0,02 ♦❤♠s ✭Ω✮✱ ✉♥ ✐♥❞✉❝t♦r ❞❡ 0,001 ❤❡♥r✐♦s ✭❍✮ ② ✉♥ ❝♦♥❞❡♥s❛❞♦r ❞❡ ✷ ❢❛r❛❞✐♦s ✭❋✮✳ ❙✐ ❧❛ ❝♦rr✐❡♥t❡ ② ❧❛ ❝❛r❣❛ ✐♥✐❝✐❛❧❡s ❡♥ ❡❧ ❝♦♥❞❡♥s❛❞♦r s♦♥ ✐❣✉❛❧❡s ❛ ❝❡r♦✱ ❞❡t❡r♠✐♥❛r ❧❛ ❝♦rr✐❡♥t❡ ❡♥ ❡❧ ❝✐r❝✉✐t♦ ♣❛r❛ t > 0✳ ❙♦❧✉❝✐ó♥ t❡♥❡♠♦s q✉❡ L = 0,001 ❍✱ R = 0,02 Ω✱ C = 2 ❋ ② E(t) = sen(100t)✳ ❙❡❣ú♥ ❧❛ ❧❡② ❞❡ ❝♦rr✐❡♥t❡ ❞❡ ❑✐r❝❤❤♦✛✱ ❧❛ ♠✐s♠❛ ❝♦rr✐❡♥t❡ ■ ♣❛s❛ ♣♦r ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❡❧ ❝✐r❝✉✐t♦✳ ▲❛ ❝♦rr✐❡♥t❡ q✉❡ ♣❛s❛ ♣♦r ❡❧ ❝♦♥❞❡♥s❛❞♦r ❡s ✐❣✉❛❧ ❛ ❧❛ r❛③ó♥ ✐♥st❛♥tá♥❡❛ ❞❡ ❝❛♠❜✐♦ ❞❡ s✉ ❝❛r❣❛ q✿ I= dq dt ✭✶✮ ♦❜s❡r✈❛♠♦s q✉❡ ❧❛ ❝❛í❞❛ ❞❡ ✈♦❧t❛❥❡ ❛ tr❛✈és ❞❡❧ ❝♦♥❞❡♥s❛❞♦r ✭❊❈✮✱ ❧❛ r❡s✐st❡♥❝✐❛ ✭❊❘✮ ② ❡❧ ✐♥❞✉❝t♦r ✭❊▲✮ s❡ ❡①♣r❡s❛♥ ❝♦♠♦ q dI ✭✷✮ Ec = , ER = RI, EL = L c Dt P♦r t❛♥t♦✱ ❧❛ ❧❡② ❞❡❧ ✈♦❧t❛❥❡ ❞❡ ❑✐r❝❤❤♦✛ EL + ER + EC = E s❡ ♣✉❡❞❡ ❡①♣r❡s❛r ❝♦♠♦ L dL 1 + RI + q = E(t) dt C ✭✸✮ ❊♥ ❧❛ ♠❛②♦r ♣❛rt❡ ❞❡ ❧❛s ❛♣❧✐❝❛❝✐♦♥❡s ♥♦s ✐♥t❡r❡s❛rá ❞❡t❡r♠✐♥❛r ❧❛ ❝♦rr✐❡♥t❡ I(t) ✳ ❙✐ ❞❡r✐✈❛♠♦s ✭3✮ dq ❝♦♥ r❡s♣❡❝t♦ ❞❡ t ② s✉st✐t✉✐♠♦s ■ ❡♥ ✈❡③ ❞❡ ✱ ♦❜t❡♥❡♠♦s dt L dI 1 dE d2 I +R + I= dt2 dt C dt ✭✹✮ ❆❧ s✉st✐t✉✐r ❧♦s ✈❛❧♦r❡s ❞❛❞♦s t❡♥❡♠♦s (0,001) ♦✱ ❡♥ ❢♦r♠❛ ❡q✉✐✈❛❧❡♥t❡✱ ❋❛❝✉❧t❛❞ ❞❡ ❈✐❡♥❝✐❛s ❯◆❆▼ ❊❝✉❛❝✐♦♥❡s ❉✐❢❡r❡♥❝✐❛❧❡s ✶ d2 I dI + (0,02) + (0,5)I = 100 cos(100t) dt2 dt d2 I dI + 20 + 500I = 100,000 cos(100t) dt2 dt ✭✺✮ Pr♦❢✳ ❊st❡❜❛♥ ❘✉❜é♥ ❍✉rt❛❞♦ ❈r✉③ ✶ ❯♥✐❞❛❞ ✼ ❙✐st❡♠❛s ❞❡ ❊❝✉❛❝✐♦♥❡s ❞❡ ♣r✐♠❡r ♦r❞❡♥ ❧✐♥❡❛❧❡s ✼✳✶✸ ❈✐r❝✉✐t♦s ❡❧é❝tr✐❝♦s✳ ▲❛ ❡❝✉❛❝✐ó♥ ❤♦♠♦❣é♥❡❛ ❛s♦❝✐❛❞❛ ❝♦♥ ✭✺✮ t✐❡♥❡ ❧❛ ❡❝✉❛❝✐ó♥ ❛✉①✐❧✐❛r r2 + 20r + 500 = (r + 10)2 + (20)2 = 0 ❝✉②❛s r❛í❝❡s s♦♥ 10 ± 20i P♦r ❧♦ t❛♥t♦✱ ❧❛ s♦❧✉❝✐ó♥ ❞❡ ❧❛ ❡❝✉❛❝✐ó♥ ❤♦♠♦❣é♥❡❛ ❡s Ih = C1 e−10t cos(20t) + C2 e−10t sen(20t) ✭✻✮ P❛r❛ ❞❡t❡r♠✐♥❛r ✉♥❛ s♦❧✉❝✐ó♥ ♣❛rt✐❝✉❧❛r ❞❡ ✭5✮✱ ♣♦❞❡♠♦s ✉s❛r ❡❧ ♠ét♦❞♦ ❞❡ ❝♦❡✜❝✐❡♥t❡s ✐♥❞❡t❡r♠✐✲ ♥❛❞♦s✳ ❍❛❝❡♠♦s Ip = A cos(100t) + B sen(100t) ② r❡❛❧✐③❛♠♦s ❡❧ ♣r♦❝❡❞✐♠✐❡♥t♦ ✈✐st♦ ❛♥t❡r✐♦r♠❡♥t❡ ♣❛r❛ ♦❜t❡♥❡r ✜♥❛❧♠❡♥t❡✱ ❝♦♥ tr❡s ❞❡❝✐♠❛❧❡s✱ A = −10,080, B = 2,122 P♦r ❧♦ t❛♥t♦✱ ✉♥❛ s♦❧✉❝✐ó♥ ♣❛rt✐❝✉❧❛r ❞❡ ✭5✮ ❡stá ❞❛❞❛ ♣♦r Ip = −10,080 cos(100t) + 2,122 sen(100t) ✭✼✮ ❈♦♠♦ I = Ih + Ip ✈❡♠♦s ❞❡ ✭❄❄✮ ② ✭❄❄✮ q✉❡ I(t) = e−10t (C1 cos(20t) + C2 sen(20t)) − 10,080 cos(100t) + 2,122 sen(100t) ✭✽✮ P❛r❛ ❞❡t❡r♠✐♥❛r ❧❛s ❝♦♥st❛♥t❡s C1 ② C2 ✱ ♥❡❝❡s✐t❛♠♦s ❧♦s ✈❛❧♦r❡s I(0) ❡ I ′ (0) ❙❛❜❡♠♦s q✉❡ I(0) = q(0) = 0✳ P❛r❛ ❞❡t❡r♠✐♥❛r I ′ (0) s✉st✐t✉✐♠♦s ❧♦s ✈❛❧♦r❡s ♣❛r❛ ▲✱ ❘ ② ❈ ❡♥ ❧❛ ❡❝✉❛❝✐ó♥ ✭3✮ ❡ ✐❣✉❛❧❛♠♦s ❧♦s ❞♦s ❧❛❞♦s ❡♥ t = 0✱ ♣❛r❛ ♦❜t❡♥❡r (0,001)I ′ (0) + (0,02)I(0) + (0,5)q(0) = sen(0) ❈♦♠♦ I(0) = q(0) = 0✱ ✈❡♠♦s q✉❡ I ′ (0) ✳ P♦r ú❧t✐♠♦✱ ✉s❛♠♦s I(0) ❡♥ ✭❄❄✮ ② ❧❛s ❝♦♥❞✐❝✐♦♥❡s ✐♥✐❝✐❛❧❡s I(0)I ′ (0) = 0 ♣❛r❛ ♦❜t❡♥❡r ❡❧ s✐st❡♠❛ I(0) = C1 − 10,080 = 0 I ′ (0) = −10C1 + 20C2 + 212,2 = 0 ❆❧ r❡s♦❧✈❡r ❡st❡ s✐st❡♠❛ t❡♥❡♠♦s q✉❡ C1 = 10,080 ② C2 = 5,570✳ P♦r ❧♦ t❛♥t♦✱ ❧❛ ❝♦rr✐❡♥t❡ ❡♥ ❡❧ ❝✐r❝✉✐t♦ ❘▲❈ ❡♥ s❡r✐❡ ❡s I(t) = e−10t (10,080 cos(20t) − 5,570 sen(20t)) − 10,080 cos(100t) + 2,122 sen(100t) ✭✾✮ ❖❜s❡r✈❡ q✉❡✱ ❝♦♠♦ ❡♥ ❡❧ ❝❛s♦ ❞❡ ❧❛s ✈✐❜r❛❝✐♦♥❡s ♠❡❝á♥✐❝❛s ❢♦r③❛❞❛s✱ ❧❛ ❝♦rr✐❡♥t❡ ❡♥ ✭❄❄✮ t✐❡♥❡ ❞♦s ❝♦♠♣♦♥❡♥t❡s✿ Ih ✱ ✉♥❛ ❝♦rr✐❡♥t❡ tr❛♥s✐t♦r✐❛ q✉❡ t✐❡♥❞❡ ❛ ❝❡r♦ ❝✉❛♥❞♦ t → ∞ ② ❧❛ ♦tr❛ ❝♦♠♣♦♥❡♥t❡ Ip = −10,080 cos(100t) + 2,122 sen(100t) ✉♥❛ ❝♦rr✐❡♥t❡ ❞❡ ❡st❛❞♦ ❡st❛❝✐♦♥❛r✐♦ s❡♥♦✐❞❛❧ q✉❡ ♣❡r♠❛♥❡❝❡✳ ❋❛❝✉❧t❛❞ ❞❡ ❈✐❡♥❝✐❛s ❯◆❆▼ ❊❝✉❛❝✐♦♥❡s ❉✐❢❡r❡♥❝✐❛❧❡s ✶ Pr♦❢✳ ❊st❡❜❛♥ ❘✉❜é♥ ❍✉rt❛❞♦ ❈r✉③ ✷