Bài giảng Signal processi - IV. Fourier Representation of Signals

TRƯỜNG ĐẠI HỌC SƯ PHẠM KỸ THUẬT 
TP. HỒ CHÍ MINH 
ELECTRICAL AND ELECTRONIC 
PRINCIPLES 
WEEK 7, 8 
Cuong Q. Ngo 
Last classes 
• Trigonometric Fourier series 
• Plot function - MATLAB 
• Circuit Application (Applying Fourier series) 
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Contents (Today class) 
• Laplace transform 
• Circuit element models 
• Circuit analysis 
• Transient analysis 
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1. Laplace Transform 
• The Laplace transform is an integral transformation of a 
function f (t) from the time domain into the complex frequency 
domain, giving F (s). 
• Where s is a complex variable: 𝑠 = 𝜎 + 𝑗𝜔 
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1. Laplace Transform 
• Determine the Laplace transform of 
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1. Laplace Transform 
• Laplace transform 
pairs (f(t) = 0, for t < 0) 
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1. Laplace Transform 
• Laplace transform 
pairs (f(t) = 0, for t < 0) 
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1. Laplace Transform 
• Inverse Laplace transform 
1. Decompose F(s) into simple terms using partial fraction 
expansion. 
2. Find the inverse of each term by matching entries in the 
previous table. 
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1. Laplace Transform 
• Find the inverse Laplace transform of 
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1. Laplace Transform 
• Answer: 
𝑓 𝑡 = 3𝑢 𝑡 − 5𝑒−𝑡𝑢 𝑡 + 3 sin 2𝑡 𝑢(𝑡) 
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1. Laplace Transform 
• Find f(t) given that 
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1. Laplace Transform 
• Answer: 
𝑓 𝑡 = (2 − 8𝑒−2𝑡 + 7𝑒−3𝑡)𝑢(𝑡) 
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2. Circuit element models 
• For a resistor 
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2. Circuit element models 
• Representation of an inductor: 
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2. Circuit element models 
• Representation of a capacitor: 
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2. Circuit element models 
• Impedance of elements in the s-domain (zero initial condition) 
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3. Circuit analysis 
• Steps in applying the Laplace transform 
1. Transform the circuit from the time domain to the s-domain. 
2. Solve the circuit using nodal analysis, mesh analysis, 
sourcetransformation, superposition, or any circuit analysis 
techniquewith which we are familiar. 
3. Take the inverse transform of the solution and thus obtain 
thesolution in the time domain. 
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3. Circuit analysis 
• Find vo(t) in the circuit, assuming zero intial condition 
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3. Circuit analysis 
• Answer: 
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3. Circuit analysis 
• The switch closes the circuit at t = 0. Find i(t) for t > 0 
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i(t) 
3. Circuit analysis 
• Answer: 
𝑖 𝑡 =
𝑉
𝑅
1 − 𝑒−𝑡
𝑅
𝐿 
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4. Trasient analysis (MultiSim) 
• Multisim computes the circuit’s response as a function of 
time. 
• Each input cycle is divided into intervals, and a DC analysis 
is performed for each time point in the cycle. 
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4. Transient analysis 
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