Circuito RL “Respuesta forzada”
𝑉𝑅 + 𝑉𝐿 = 𝐸 … (0)
𝑖(𝑡) 𝑅 + 𝐿
𝑑𝑖
=𝐸
𝑑𝑡
𝑖(𝑡) 479.8 + 0.1
𝐷→
t<0
No hay C.I
𝑑𝑖
= 1.44𝑉
𝑑𝑡
𝑑
𝑑𝑡
𝑖(𝑡)[𝑅 + 𝐿𝐷] = 𝐸 … (1)
𝑖(𝑡)[479.8 + 0.1𝐷] = 1.44
t=0
4.5 𝑉
𝐼=
= 6.52𝑚𝐴
690 𝛺
*Solución ec. Dif. No Homogénea
𝑖(𝑡) = 𝑖ℎ 𝑒 𝐷𝑡 + 𝑖𝑝 … (2)
*D=?
479.86 + 0.1𝐷 = 0
𝑉 = (6.52 𝑚𝐴)(220𝛺) = 1.375𝑉
𝑉 = (6.52 𝑚𝐴)(470𝛺) = 3.06𝑉
∴𝐷=−
𝑅
479.86𝛺
= −
= 4798.6
𝐿
0.1ℎ
*𝑖𝑝 =? 𝐷 = 0 𝑑𝑒 … (1)
𝑖𝑝 = (479.86 + 0) = 𝐸
∴ 𝑖𝑝 =
𝐸
1.375
=
= 2.86𝑚𝐴
𝑅 479.86
∗ 𝑖ℎ =? 𝑡 = 0 𝑖(0) = 0
𝑉𝐿 (0) = 0
𝑉220 = 1.375
𝑉𝑅 (0) = { 𝑉470= 3.06𝑉
𝑉330= 0
𝑖𝐿 (0) = 0
t>0
𝑖(0) = 𝑖ℎ 𝑒 0 + 𝑖𝑝 ∴ 0 = 𝑖 +
=−
𝐸
𝐸
→ 𝑖ℎ = −
𝑅
𝑅
1.375
= −2.86𝑚𝐴
479.86
*Sustituyendo en … (2)
𝐸 𝑅
𝐸
𝑖(𝑡) = − 𝑒 − 𝐿 𝑡 +
𝑅
𝑅
𝑖(𝑡) = −2.86𝑚𝐴 𝑒 −4798.6𝑡 + 2.86𝑚𝐴
𝑉𝑅 = R[i(t)] = −
𝐸
𝑅
𝑅
𝐸
𝐸
𝑅
𝑒 − 𝐿 𝑡 + = [1 − 𝑒 − 𝐿 𝑡 ] =
𝑅
𝑅
2.86𝑚𝐴[1 − 𝑒 −4798.6𝑡 ]
*Sustituyendo en … (0)
𝑅
𝐼=
4.5 𝑉
= 6.52𝑚𝐴
690 𝛺
𝑉220 = (6.52 𝑚𝐴)(220𝛺) = 1.375𝑉
𝑖(0) = 6.52𝑚𝐴
𝑍𝑡ℎ = 479.86𝛺
𝑉𝐿 = 𝐸𝑒 − 𝐿 𝑡 = 1.44𝑒 −4798.6𝑡
𝑉𝑅 (𝑡) = 𝑅[𝑖(𝑡)]
𝑅
𝑅
𝐸
𝑉𝑅 (𝑡) = 𝑅 ( ) (1 − 𝑒 − 𝐿 𝑡 ) = 𝐸(1 − 𝑒 − 𝐿 𝑡 )
𝑅
𝑉𝑅 (𝑡) = 1.44(1 − 𝑒 −4798.6𝑡 )
Circuito RC “Respuesta forzada”
𝑖(𝑡) = 𝐶
𝑅𝐶
𝑑𝑉
𝑑𝑣
= 0.1
𝑑𝑡
𝑑𝑡
𝑑𝑣(𝑡)
+ 𝑣(𝑡) = 𝐸
𝑑𝑡
𝐷→
𝑑
𝑑𝑡
𝑅𝐶𝐷𝑣(𝑡) + 𝑣(𝑡) = 𝐸
𝑣(𝑡) + 𝑅𝐶
t<0
No hay C.I
t=0
𝑑𝑣(𝑡)
=𝐸
𝑑𝑡
𝑣(𝑡) + 𝑅𝐶𝐷 = 𝐸
𝑣(𝑡) [𝑅𝐶𝐷 + 1] = 𝐸
*Solución ec. Dif. No Homogénea
𝑣(𝑡) = 𝑉ℎ 𝑒 𝐷𝑡 + 𝑉𝑝
1
* 𝐷 =? 𝑅𝐶𝐷 + 1 = 0
∴ 𝐷 = − 𝑅𝐶 = −0.097
*𝑉𝑝 =?
𝐷=0
𝑉𝑝 (𝑅𝐶𝐷 + 1) = 0.986
∴ 𝑉𝑝 = 𝐸 = 0.986𝑉
∗ 𝑉ℎ =? 𝑡 = 0 𝑣(0) = 0
𝑣(0) = 𝑉ℎ 𝑒 0 + 𝑉𝑝 ∴ 0 = 𝑉ℎ + 𝑉𝑝
(4.5𝑉)(132𝛺)
= 0.986𝑉
470𝛺 + 132𝛺
𝑉330 = 𝑉𝑅𝑒𝑞=
𝑖330 (0) = 𝑖𝐶 (0) =
𝑉330
0.986𝑉
=
= 2.98𝑚𝐴
330𝛺
330𝛺
t>0
𝑉ℎ = −𝐸 = −0.986𝑉
1
1
𝑉𝑐 (𝑡) = −𝐸𝑒 −𝑅𝐶𝑡 + 𝐸 = 𝐸 (1 − 𝑒 −𝑅𝐶𝑡 )
= 0.986(1 − 𝑒 −0.097𝑡 )
1
𝑉𝑅 (𝑡) = 𝐸 − 𝑉𝑐 (𝑡) = 𝐸 + 𝐸𝑒 −𝑅𝐶𝑡 − 𝐸
1
= 𝐸𝑒 −𝑅𝐶 𝑡
= 0.986𝑒 −0.097𝑡
𝑅𝑡ℎ = 103.05𝛺
𝑉𝑡ℎ = 0.986𝑉
𝑉𝑅 + 𝑉𝐶 = 𝐸 … (0)
𝑖(𝑡) 𝑅 +
𝑖(𝑡) 103.05 +
1
∫ 𝑖(𝑡)𝑑𝑡 = 𝐸
𝐶
1
∫ 𝑖(𝑡)𝑑𝑡 = 0.986𝑉
0.1