Laplace Transform_03

Laplace Transform_03 - CY1B2 & SE1B2 Laplace Transform...

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CY1B2 & SE1B2 Laplace Transform by Dr. Virginie F. Ruiz Dept. of Cybernetics, room 184 V.F.Ruiz@reading.ac.uk VFR Dept. of Cybernetics LT. 1
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Structure of the course 5 lectures, 1 exam question. Signal, system and Laplace transform Laplace transform Inverse Laplace transform Solving differential equation, i.e. time response of system Transfer function, poles, zeros and stability, i.e. analysis of systems VFR Dept. of Cybernetics LT. 2
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Text books Bolton, ‘Laplace and z- transforms’, Mathematics for engineers Series, Longman Scientific & Technical, ISBN 0 582 228190. Stroud, ‘Laplace Transform’, Stanley Thornes, ISBN 0 8590 002 0. Whitehouse, ‘Circuit analysis’, Horwood Engineering Science Series, Horwood, ISBN 1-898563-40-3. VFR Dept. of Cybernetics LT. 3
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Signals, Systems and Laplace Transform An introduction VFR Dept. of Cybernetics LT. 4
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Laplace Transform (LT) VFR LT. 5 Dept. of Cybernetics Time domain s-domain LT Interest Solve differential equations describing signals and systems more easily. Determine systems’ characteristics (transfer function, poles, zeros, stability. .. i.e. analyse, design systems. ) ( t f ) ( s F
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Signals and Systems Signals are met in diverse fields Engineering voltages/currents in circuits Physics e.g. radiation Mech. Eng. e.g. vibration studies Astromomy e.g. 2D pulsars, distant stars Biomedicine e.g. EEG, EMG, ECG, MRI Seismology e.g Tectonic plate movement, earthquake prediction Oil exploration e.g. geophones Economics e.g. level of trading in stoke market Signals: A signal is the space or time variations in the physical state of an object. The quantification of this state is used to represent, store, transmit a message. The course is not about the message but about the properties of the signals. Systems: A system operates on a signal (the input) to modify, transform or re-express it in another form (output signal) which may be more desirable. VFR Dept. of Cybernetics LT. 6
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VFR LT. 7 Dept. of Cybernetics Periodic and aperiodic: a) if x ( t +T 0 ) = x ( t ) then the signal is periodic and continuous b) otherwise the signal is aperiodic. Causal signal models: If x ( t ) = 0 for all t < t 0 (initial time, e.g. 0) then we have a causal signal Special signal models Impulse signal rectangular pulse: • duration ε • amplitude 1/ ε • area: –A s ε becomes smaller: • zero everywhere except at a single point. ± δ ( t ) is called the unit impulse function or Delta-Dirac function. Delayed impulse: Impulse train : = 0 1 ) ( dt t δ ) ( 0 t t = n k kt t 0 ) ( 0 t 0 1 t 0 1/ ( t )
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VFR LT. 8 Dept. of Cybernetics t u ( t ) 1 0 Unit step signal denoted or as it is often referred to as Heaviside’ function Mathematical model: Delayed unit step Delayed step with amplitude k ) ( t u ) ( t H < = 0 1 0 0 ) ( t for t for t u < = 0 0 0 1 0 ) ( t t for t t for t t u t 1 0 t 0 < = 0 0 0 0 ) ( t t for k t t for t t ku
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VFR LT. 9 Dept. of Cybernetics t r ( t ) 0 Unit ramp signal slope 1 Mathematical model: Delayed unit ramp: Delayed ramp with slope b < = 0 0 0 ) ( t for t t for t r < =
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Laplace Transform_03 - CY1B2 &amp; SE1B2 Laplace Transform...

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