The document discusses linear time-invariant systems and their representations using differential equations and Laplace transforms. It provides equations for the Laplace transform of derivatives and inverse Laplace transforms. It also presents the differential equation and transfer function for a second-order RLC circuit system. The transfer function is expressed as a ratio of the output and input Laplace transforms, equal to 1 divided by the characteristic equation containing coefficients for the inductor, resistor and capacitor components.

3. •
•

4. •
𝑮 𝒔 =
𝑻𝒓𝒂𝒏𝒔𝒇𝒐𝒓𝒎𝒂𝒅𝒂 𝒅𝒆 𝒍𝒂 𝒓𝒆𝒔𝒑𝒖𝒆𝒔𝒕𝒂 𝒆𝒏 𝒆𝒔𝒕𝒂𝒅𝒐 𝒏𝒖𝒍𝒐
𝑻𝒓𝒂𝒏𝒔𝒇𝒐𝒓𝒎𝒂𝒅𝒂 𝒅𝒆 𝒍𝒂 𝒔𝒆ñ𝒂𝒍 𝒅𝒆 𝒆𝒏𝒕𝒓𝒂𝒅𝒂

6. •
𝑎𝑛𝑦(𝑛)
𝑡 + ⋯ + 𝑎1 ሶ
𝑦 𝑡 + 𝑎0𝑦 𝑡 = 𝑏0𝑢 𝑡 + 𝑏1 ሶ
𝑢 𝑡 + ⋯ 𝑏𝑚𝑢(𝑚)
𝑡
• 𝒚 𝒕
• 𝒖 𝒕
• 𝒂𝒏 𝒃𝑚

7. •
ℒ
𝑑𝑓(𝑡)
𝑑𝑡
= 𝑠ℒ 𝑓(𝑡) − 𝑓 0
ℒ
𝑑2
𝑓(𝑡)
𝑑𝑡2
= 𝑠2
ℒ 𝑓(𝑡) − 𝑠𝑓 0 −
𝑑𝑓(0)
𝑑𝑡
•
ℒ
𝑑𝑛
𝑓(𝑡)
𝑑𝑡𝑛
= 𝑠𝑛
ℒ 𝑓(𝑡) − ෍
𝑘=1
𝑛
𝑠𝑘−1
𝑑𝑛−𝑘
𝑓(0)
𝑑𝑡𝑛−𝑘

8. •
𝒕 < 𝟎
𝑎𝑛𝑠𝑛
𝑌(𝑠) + ⋯ + 𝑎1 𝑠𝑌(𝑠) + 𝑎0𝑌(𝑠) = 𝑏0𝑈(𝑠) + 𝑏1𝑠𝑈 𝑠 + ⋯ 𝑏𝑚𝑠𝑚
𝑈 𝑠
• 𝒀 𝒔
• 𝑼 𝒔
• 𝒂𝒏 𝒃𝒏

9. 𝒀(𝒔) 𝑼 𝒔
(𝑎𝑛𝑠𝑛 + ⋯ + 𝑎1 𝑠 + 𝑎0)𝑌(𝑠) = (𝑏0+𝑏1𝑠 + ⋯ 𝑏𝑚𝑠𝑚)𝑈 𝑠
𝐺 𝑠 =
𝑌(𝑠)
𝑈(𝑆)
𝐺 𝑠 =
𝑏0 + 𝑏1𝑠 + ⋯ 𝑏𝑚𝑠𝑚
𝑎𝑛𝑠𝑛 + ⋯ + 𝑎1 𝑠 + 𝑎0

10. 𝐺 𝑠 =
𝑌(𝑠)
𝑈(𝑆)
=
𝑏0 + 𝑏1𝑠 + ⋯ 𝑏𝑚𝑠𝑚
𝑎𝑛𝑠𝑛 + ⋯ + 𝑎1 𝑠 + 𝑎0
• 𝒏
•
𝑮 𝒔
𝑃 𝑠 = 𝑎𝑛𝑠𝑛
+ ⋯ + 𝑎1 𝑠 + 𝑎0

11. •
𝒗(𝒕) 𝒗𝑪(𝒕)
•
• 𝒗(𝒕)
• 𝑳
• 𝑹
• 𝑪

12. •
𝑣𝑖𝑛(𝑡) = 𝑅𝑖(𝑡) + 𝐿
𝑑𝑖(𝑡)
𝑑𝑡
+ 𝑣𝐶
•
𝑖 𝑡 = 𝐶
𝑑𝑣𝐶
𝑑𝑡

13. • 𝒊 𝒕
𝑣𝑖𝑛(𝑡) = 𝑅𝐶
𝑑𝑣𝐶
𝑑𝑡
+ 𝐿𝐶
𝑑2𝑣𝐶
𝑑𝑡2
+ 𝑣𝐶
•
𝐿𝐶
𝑑2
𝑣𝐶
𝑑𝑡2
+ 𝑅𝐶
𝑑𝑣𝐶
𝑑𝑡
+ 𝑣𝐶 = 𝑣𝑖𝑛(𝑡)

14. •
𝑢 𝑡 = 𝑣(𝑡)
•
𝑦 𝑡 = 𝑣𝑐(𝑡)

15. •
𝑢 𝑡 = 𝑅𝐶 ሶ
𝑦(𝑡) + 𝐿𝐶 ሷ
𝑦(𝑡) + 𝑦(𝑡)
•
𝑢 𝑡 = 𝐿𝐶 ሷ
𝑦(𝑡) + 𝑅𝐶 ሶ
𝑦(𝑡) + 𝑦(𝑡)

16. •
ℒ 𝑢 𝑡 = ℒ 𝐿𝐶 ሷ
𝑦(𝑡) + 𝑅𝐶 ሶ
𝑦(𝑡) + 𝑦(𝑡)
𝑈 𝑠 = 𝐿𝐶𝑠2
𝑌(𝑠) + 𝑅𝐶𝑠𝑌(𝑠) + 𝑌(𝑠)
•
𝑈 𝑠 = 𝐿𝐶𝑠2
+ 𝑅𝐶𝑠 + 1 𝑌(𝑠)

17. • 𝒀(𝒔)/𝑼(𝒔)
𝑌(𝑠)
𝑈(𝑠)
=
1
𝐿𝐶𝑠2 + 𝑅𝐶𝑠 + 1
•
𝐺 𝑠 =
1
𝐿𝐶
𝑠2 +
𝑅
𝐿
𝑠 +
1
𝐿𝐶

18. 𝑭 𝒙 𝒎
𝑏
𝑘
𝑥
𝐹

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