APOL4

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{
𝑍𝐴 = 3 − 𝑗2
𝑍𝐵 = 8 + 𝑗10
𝑍𝑋𝐶2 = −𝑗10
 
𝑍𝐼𝑁 =
𝑍𝐴 × 𝑍𝐵
𝑍𝐴 + 𝑍𝐵
+ 𝑍𝑋𝐶2 → 𝑍𝐼𝑁 =
(3 − 𝑗2) × (8 + 𝑗10)
(3 − 𝑗2) + (8 + 𝑗10)
+ (−𝑗10) → 
→ 𝑍𝐼𝑁 =
(44 + 𝑗14)
(11 + 𝑗8)
+ (−𝑗10) → (3,22 − 1,070 ) + (−𝑗10) → 𝑍𝐼𝑁 = 3,22 − 𝑗11,07 
 
∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ 𝑍𝐼𝑁 = 3,22 − 𝑗11,07 [Ω] ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ 
 
𝑋𝐶1 =
1
𝜔𝐶 
 →
1
50 × 10 × 10−3 
→ 𝑋𝐶1 = −𝑗2 [Ω] 
𝑋𝐿1 = 𝜔𝐿 → 50 × 0.2 𝐻 → 𝑋𝐿1 = 𝑗10 [Ω] 
𝑍𝑋𝐶2 =
1
𝜔𝐶2
 → 
1
50×2×10−3 
 → 𝑍𝑋𝐶2 = −𝑗10 [Ω] 
 
 
 
𝑍𝐼𝑁 =
𝑍𝐴 × 𝑍𝐵
𝑍𝐴 + 𝑍𝐵
 → {
𝑍𝐴 = 50 + 𝑗20
𝑍𝐵 = 50 + 𝑗80
𝑉𝐺 = 110 < 0°
 
𝑍𝐼𝑁 =
𝑍𝐴 × 𝑍𝐵
𝑍𝐴 + 𝑍𝐵
 → 
(50 + 𝑗20) × (50 + 𝑗80)
(50 + 𝑗20) + (50 + 𝑗80)
 → 
𝑍𝐼𝑁 
900 + 𝑗5000
100 + 𝑗100
 → 𝑍𝐼𝑁 = 29 + 𝑗20,5 [Ω] → 
𝑍𝐼𝑁 = 35,92 < 34,79° 
 
𝐼𝐺 = 
𝑉𝐺 
𝑍𝐼𝑁
 → 
110 < 0°
35,92 < 34,79°
 → 𝐼𝐺 = 3,06 < −34.79° 
𝑖𝑔 (𝑡) = 3,06√2 𝑠𝑒𝑛 (𝜔. 𝑡 − 34,79°)[𝐴] 
 
𝐼1 = 
𝑉𝐺
𝑍𝐴
 → 
110 < 0°
50 + 𝑗20
 → 𝐼1 = 2,04 < −21,8° 
 
𝑖1 (𝑡) = 2,04√2 𝑠𝑒𝑛(𝜔. 𝑡 − 21,8°)[𝐴] 
 
𝐼2 =
𝑉𝐺
𝑍𝐵
 
110 < 0°
50 + 𝑗80
 → 𝐼2 = 1,16 < −57,99° 
 
𝑖2(𝑡) = 1,16√2 𝑠𝑒𝑛 (𝜔. 𝑡 − 57,99°)[𝐴] 
 
 
 
 
 
 
𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑖𝑎 𝑑𝑒 𝑅𝑒𝑠𝑠𝑜𝑛â𝑛𝑐𝑖𝑎 → 𝑓0 
 
Quando 𝑋𝐶 = 𝑋𝐿 → 
1
2𝜋𝑓𝐶
 = 2𝜋𝑓𝐿 → 𝑓0 = 
√1 
2𝜋×√𝐿𝐶
 → 
→ 
1
2𝜋 × √10−3 × 100 × 10−9 
 → 𝑓0 = 15915,49 [𝐻𝑧 ] 
 
𝐼 =
𝑉𝐺
𝑍𝐼𝑁
 → 𝑠𝑒 𝑋𝐶 = 𝑋𝐿 𝑒𝑛𝑡ã𝑜 𝑍𝐼𝑁 = 150 [Ω] → 
15
150
 → 𝐼 = 100𝑚𝐴 
 
 
 
 
 
[𝑍] = [
𝑍11 𝑍12
𝑍21 𝑍22
] → 𝑍11 = 
𝑉1
𝐼1
 𝑠𝑒 𝐼2 = 0 → 𝑍11 =
20𝐼1 + 40𝐼1 
𝐼!
→ 
𝑍11 = 60 [Ω] 
 
𝑍21 = 
𝑉2
𝐼1
 𝑠𝑒 𝐼2 = 0 → 𝑍21 =
40𝐼1
𝐼1
→ 𝑍21 = 40 [Ω] 
 
Z12 =
V1
I2
 se I1 = 0 → Z12 =
40I2
I2
 → Z12 = 40 [Ω] 
 
Z22 =
V2
I2
 se I1 = 0 → Z22 =
30I2 + 40I2
I2
 → Z22 = 70 [Ω] 
 
 
[𝑍] = [
𝑍11 𝑍12
𝑍21 𝑍22
] → [𝑍] = [
60Ω 40Ω
40Ω 70Ω
] 
V1
1/
(2𝜋 ×
√10−3 
 
V2
1/
(2𝜋 ×
√10−3 
 
I2
1/
(2𝜋 ×
√10−3 
 
I1
1/
(2𝜋 ×
√10−3 
 
 
 
 
 
 
𝑌𝑇 = 𝑌𝑅 + 𝑌𝐿 → 
1
80
+
1
𝑗60
 → 0,0125 (𝑠 ) − 𝑗0,0167 (𝑠) → 𝑌𝑇 = 0,0125 (𝑠 ) − 𝑗0,0167 (𝑠) 
𝑍𝐼𝑁 = 
1
𝑌𝑇
 → 
1
 0,0125 (𝑠 ) − 𝑗0,0167 (𝑠) 
 → 𝑍𝐼𝑁 = 48 < 53,18° 
 
𝐼 =
𝑉𝐺
𝑍𝐼𝑁
 → 
120 < 90°
48 < 53,18°
 → 𝐼 = 2,5 < 36,8° 
 
 
 
 
 
𝑍𝐼𝑁 = 4 − 𝑗3 → 𝑍𝐼𝑁 = 𝑍 < 𝜃 → 𝑍 = √𝑅2 + 𝑋𝐶
2 → 
→ 𝑍 = √42 + (3)2 → 𝑍 = 5 [Ω] 
𝜃 = 𝑎𝑟𝑐𝑡𝑔 
−𝑗3
4 
 → 𝜃 = −36,86° 
𝑍𝐼𝑁 = 𝑍 < 𝜃 → 𝑍𝐼𝑁 = 5 < −36,86° 
 
𝐼 =
𝑉𝐺
𝑍𝐼𝑁
→ 
10 < 0°
5 < −36,86° 
→ 𝐼 = 2 < 36,86° 
 
𝑖(𝑡) = 2√2 𝑠𝑒𝑛 (𝜔. 𝑡 + 36,86° )[𝐴] 
 
 
{
𝑅 = 4 Ω
𝑋𝐿 = 𝑗3Ω
𝑉𝐺 = 40 < 0° [𝑉]
 
𝑍𝐼𝑁 = 4 + 𝑗3 → 𝑍𝐼𝑁 = 5 < 36,86° 
 
𝐼 =
𝑉𝐺
𝑍𝐼𝑁
→ 
40 < 0°
5 < 36,86°
 → 𝐼 = 8 < −36,86° → 
 
𝑖(𝑡) = 8√2 𝑠𝑒𝑛 (𝜔. 𝑡 − 36,86°)[𝐴]