Signals and Systems

Laplace Transform and Its Applications

Student: Ali Abbasi (ID: 9531503)

Course Instructors: Dr. M Shahraki, Dr. M Mehrjoo

Course Number: 20 14 255

Faculty of Electrical and Computer Engineering,

University of Sistan and Baluchestan, Zahedan, Iran

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Frequency Range

Note: This slide has been used in assignment 9 to present applications of Fourier analysis.

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Audio Systems with Multiple Speakers

Picture Source: https://www.bhphotovideo.com/explora/amp/audio/tips-and-solutions/what-about-all-those-speaker-specs

Note: This slide has been used in assignment 9 to present applications of Fourier analysis.

Filters Circuits

Picture Source: Santiago, J. (2013). Circuit Analysis for Dummies: John Wiley & Sons. (with modifications)

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Low-pass Filter Analysis

5/16Picture Source: Santiago, J. (2013). Circuit Analysis for Dummies: John Wiley & Sons. (with modifications)

𝑣𝑖𝑛 𝑡 = 𝑣𝑅(𝑡) + 𝑣𝑜𝑢𝑡(𝑡)

𝑣𝑖𝑛 𝑡 = 𝑅𝐶𝑑𝑣𝑜𝑢𝑡(𝑡)

𝑑𝑡+1

𝐶න−∞

𝑡

𝐶𝑑𝑣𝑜𝑢𝑡(𝜏)

𝑑𝑡𝑑𝜏

𝑣𝑖𝑛 𝑡 = 𝑅𝑖(𝑡) +1

𝐶න−∞

𝑡

𝑖(𝜏) 𝑑𝜏

𝑣𝑖𝑛 𝑡 = 𝑅𝐶𝑑𝑣𝑜𝑢𝑡(𝑡)

𝑑𝑡+ 𝑣𝑜𝑢𝑡(𝑡)

𝑣𝑅(𝑡)

𝑣𝑜𝑢𝑡(𝑡)𝑣𝑖𝑛(𝑡)

𝑅

𝐶 𝐴𝑙𝑙 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠 𝑎𝑟𝑒 𝑧𝑒𝑟𝑜.

Low-pass Filter Analysis (cont.)

6/16

𝑣𝑖𝑛 𝑡 = 𝑅𝐶𝑑𝑣𝑜𝑢𝑡(𝑡)

𝑑𝑡+ 𝑣𝑜𝑢𝑡(𝑡)

𝑣𝑜𝑢𝑡 𝑡 = 𝑣𝑖𝑛(𝑡)(1 − 𝑒−1𝑅𝐶

𝑡)

𝑉𝑖𝑛 𝑠 = 𝑅𝐶𝑠𝑉𝑜𝑢𝑡(𝑠) + 𝑉𝑜𝑢𝑡(𝑠)

𝑉𝑖𝑛 𝑠 = 𝑉𝑜𝑢𝑡(𝑠)(𝑅𝐶𝑠 + 1)

𝑉𝑜𝑢𝑡 𝑠 =1

𝑅𝐶(

1

𝑠 +1𝑅𝐶

)𝑉𝑖𝑛(𝑠)

𝐿𝑎𝑝𝑙𝑎𝑐𝑒

𝐿𝑎𝑝𝑙𝑎𝑐𝑒−1

Devices in S-Domain

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𝑫𝒆𝒗𝒊𝒄𝒆 𝑻𝒊𝒎𝒆 − 𝑫𝒐𝒎𝒂𝒊𝒏

𝑺 − 𝑫𝒐𝒎𝒂𝒊𝒏

𝑽𝒐𝒍𝒕𝒂𝒈𝒆 𝑪𝒖𝒓𝒓𝒆𝒏𝒕𝑰𝒎𝒑𝒆𝒅𝒂𝒏𝒄𝒆(𝒘𝒊𝒕𝒉 𝒛𝒆𝒓𝒐𝒊𝒏𝒊𝒕𝒊𝒂𝒍

𝒄𝒐𝒏𝒅𝒊𝒕𝒊𝒐𝒏)

𝐼𝑉𝑆 𝑣𝑆(𝑡) 𝑉𝑆(𝑠) − −

𝐼𝐶𝑆 𝑖𝑆(𝑡) − 𝐼𝑆(𝑠) −

𝑉𝐶𝑉𝑆 𝑣2 𝑡 = 𝜇𝑣1 𝑡 𝑉2 𝑠 = 𝜇𝑉1 𝑠 − −

𝑉𝐶𝐶𝑆 𝑖2 𝑡 = 𝑔𝑣1 𝑡 − 𝐼2 𝑠 = 𝑔𝑉1 𝑠 −

𝐶𝐶𝑉𝑆 𝑣2 𝑡 = 𝑟𝑖1 𝑡 𝑉2 𝑠 = 𝑟𝐼1 𝑠 − −

𝐶𝐶𝐶𝑆 𝑖2 𝑡 = 𝛽𝑖1 𝑡 − 𝐼2 𝑠 = 𝛽𝐼1 𝑠 −

𝑅𝑒𝑠𝑖𝑠𝑡𝑜𝑟 𝑣𝑅 𝑡 = 𝑅𝑖𝑅(𝑡) 𝑉𝑅 𝑠 = 𝑅𝐼𝑅(𝑠)𝐼𝑅 𝑠 =

1

𝑅𝑉𝑅(𝑠)

𝑍𝑅 𝑠 = 𝑅

𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑜𝑟𝑣𝐶 𝑡 = න

0

𝑡

𝑖𝐶(𝜏) 𝑑𝜏 𝑉𝐶 𝑠 =1

𝑠𝐶𝐼𝐶 𝑠 +

𝑣𝑐(0)

𝑠

𝐼𝐶 𝑠 = 𝑠𝐶 𝑉𝐶 𝑠 − 𝐶𝑣𝑐(0) 𝑍𝐶 𝑠 =1

𝑠𝐶

𝐼𝑛𝑑𝑢𝑐𝑡𝑜𝑟𝑣𝐿 𝑡 = 𝐿

𝑑𝑖𝐿(𝑡)

𝑑𝑡

𝑉𝐿 𝑠 = 𝑠𝐿𝐼𝐿 𝑠 − 𝐿𝑖𝐿(0)𝐼𝐿 𝑠 =

1

𝑠𝐿𝑉𝐿 𝑠 +

𝑖𝐿(0)

𝑠

𝑍𝐿 𝑠 = 𝑠𝐿

S-Domain Thévenin’s and Norton’s Equivalent for Passive Elements

8/16Picture Source: Santiago, J. (2013). Circuit Analysis for Dummies: John Wiley & Sons. (with modifications)

Band-pass Filter: S-Domain Analysis

9/16Picture Source: Santiago, J. (2013). Circuit Analysis for Dummies: John Wiley & Sons. (with modifications)

𝑉𝑖𝑛(𝑠) 𝑉𝑜𝑢𝑡(𝑠)𝑅1

𝑠𝐶

𝑠𝐿

𝑉𝑜𝑢𝑡 𝑠 =𝑅

𝑠𝐿 +1𝑠𝐶

+ 𝑅𝑉𝑖𝑛 𝑠

𝑇 𝑠 =𝑉𝑜𝑢𝑡(𝑠)

𝑉𝑖𝑛(𝑠)=

𝑅

𝐿

𝑠

𝑠2 +𝑅𝐿

𝑠 +1𝐿𝐶

𝐴𝑙𝑙 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠 𝑎𝑟𝑒 𝑧𝑒𝑟𝑜.

Band-pass Filter: S-Domain Analysis (cont.)

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1

𝐿𝐶− 𝜔2 = ±

𝑅

𝐿𝜔 → 𝜔2 ±

𝑅

𝐿𝜔 −

1

𝐿𝐶= 0 →

𝜔𝐶1 = −𝑅

2𝐿+

𝑅

2𝐿

2

+1

𝐿𝐶

𝜔𝐶2 = +𝑅

2𝐿+

𝑅

2𝐿

2

+1

𝐿𝐶

1

𝐿𝐶− 𝜔2 = 0 → 𝜔0 =

1

𝐿𝐶

𝐵𝑎𝑛𝑑_𝑤𝑖𝑑𝑡ℎ = 𝜔𝐶2 − 𝜔𝐶1 =𝑅

𝐿𝑎𝑛𝑑 𝑄_𝑓𝑎𝑐𝑡𝑜𝑟 =

𝜔0

𝐵𝑎𝑛𝑑𝑊𝑖𝑑𝑡ℎ=1/ 𝐿𝐶

𝑅/𝐿=1

𝑅

𝐿

𝐶

𝑠 = 𝑗𝜔 → 𝑇 𝑗𝜔 =𝑉𝑜𝑢𝑡(𝑗𝜔)

𝑉𝑖𝑛(𝑗𝜔)=

𝑅

𝐿

𝑗𝜔

𝑗𝜔 2 +𝑅𝐿

𝑠 +1𝐿𝐶

=𝑅

𝐿

𝑗𝜔

1𝐿𝐶

− 𝜔2 +𝑅𝐿

𝑗𝜔

Band-pass Filter as a System

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• For a band-pass filter (all initial conditions are zero):

𝑇 𝑠 =𝐴02𝜁𝜔0𝑠

𝑠2 + 2𝜁𝜔0𝑠 + 𝜔02 =

𝐴0𝜔0𝑄 𝑠

𝑠2 +𝜔0𝑄 𝑠 + 𝜔0

2=

𝐴0𝐵𝑠

𝑠2 + 𝐵𝑠 + 𝜔02

𝐴0:𝑀𝑖𝑑𝑏𝑎𝑛𝑑 𝐺𝑎𝑖𝑛𝜔0: 𝑅𝑒𝑠𝑜𝑛𝑎𝑛𝑡 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦𝜁: 𝐷𝑎𝑚𝑝𝑖𝑛𝑔 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑄: 𝑄𝑢𝑎𝑙𝑖𝑡𝑦 𝐹𝑎𝑐𝑡𝑜𝑟

Control Systems and S-Domain

12/16Picture Source: “Applications of Laplace Transform in Control Systems.”, “Mobile Tutor” channel on YouTube.

Control Systems and S-Domain (cont.)

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• Gain Factor K:

• Nyquist Diagram

𝑇 𝑠 =1

𝐴𝑠2 + 𝐵𝑠 + 𝐶

• Help in solving differential equations of higher orders and evaluating system output.

Control Systems and S-Domain (cont.)

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Picture Source: 2010218 (Linear System and Control) at University of Sistan and Baluchestan, Semester 3981, Midterm Exam, by Dr.

Saeed Tavakoli Afshari.

Control Systems and S-Domain (cont.)

15/16

• Faster forecasting of the outputs of a system.

Picture Source: (1) https://en.m.wikipedia.org/wiki/Aircraft_flight_control_system (2) https://lancmoms.com/adaptive-cruise-control/

The END!