Bonjour,
Ex 1)
1) u(t) = 50√2sin(314t)
⇒ U = 50√2e^(314jt)
a) Z = R ⇒ I = U/R = 50√2/6000 x e^(314jt)
soit i(t) = 50√2/6000 x sin(314t)
b) Z = 1/jCω = -j/Cω = 1/ωC x e^(-jπ/2)
⇒ I = U/[e^(-jπ/2)/ωC] = 50√2e^(314jt) x ωC/e^(-jπ/2)
= 50√2 x 314x200.10⁻⁹ x e^(314jt + jπ/2)
soit i(t) = 50√2/314x200.10⁻⁹ x sin(314t + π/2)
c) Z = jLω = ωLe^(jπ/2)
⇒ I = U/ωLe^(jπ/2) = 50√2e^(314jt)/314x1,2xe^(jπ/2)
= 50√2/314x1,2 x e^(314jt - jπ/2)
soit i(t) = 50√2/314x1,2 x sin(314t - π/2)
Ex 2)
a) Z = R + jLω
avec R = 100 Ω, L = 0,5 H et ω = 2πf = 314 rad/s⁻¹
|Z| = √[R² + (Lω)²] = √[100² + (0,5x314)²] ≈ 186 Ω
cos(φ) = R/|Z| ≈ 100/186 = 0,537
et sin(φ) = Lω/|Z| ≈ 157/186 = 0,843
⇒ φ ≈ 57,5° = 1 rad
soit Z = 186e^(j)
b) Y = 1/Z = 1/(R + jLω) = (R - jLω)/(R + jLω)(R + jLω) = (R - jLω)/(R² + (Lω)²)
= 1/|Z|²x (R - jLω)
ou plus simplement avec la forme expo : Y = 1/186 x e^(-j)
c) I = U/Z = U x Y
= 50√2e^(314jt) x 1/186 x e^(-j)
= 50√2/186 x e^(314jt - j)
≈ 0,38e^(314t - 1)j
ou i(t) = 0,38sin(314t - 1)
UR = R x I = 38e^(314t - 1)j = 38sin(314t - 1)
et UL = jLω x I = Lωe^(π/2) x 0,38sin(314t - 1)
= 59,7sin(314t + π/2 - 1)
