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Unformatted text preview: 1 1 Summary of Lecture #2 8/25/2010 • Introduction to Laplace transform approach • Laplace transform domain circuits • Basic signals – Unit step function, u(t) – Ramp function, r(t) – Pulse function, p(t) – Delta function , δ (t) – Example • Laplace transform, F(s) = L (f(t)) – Definition – Region of convergence (ROC) – Abscissa of convergence – Pole or poles of F(s) • Tables 12.1 and 12.2 4 Time domain approach to circuit problems • Use KCL and KVL to set up a dif. eq. • Identify the initial and final conditons • Solve the dif. eq. Laplace transform approach to circuit problems • Laplace transform of the input signals, V in (s) = L [v in (t)] • Transfer function of the circuit, H(s) • Output signal in s-domain, V out (s) = H(s) V in (s) • Inverse Laplace trasnform, output signal in time-domain, v out (t) = L- 1 [V out (s)] 5 Input Signal circuit Output Signal Laplace transform of input signal Laplace transform of circuit Laplace transform of output signal Figure 12.2 Tables 12.1 and 12.2 Circuit problems and Laplace transform 6 Resistors Inductors Capacitors Time domain representation Frequency domain representation V =j ϖ L I V = R I v = Ri di v L dt = dv i C dt = 1 V I j C = ϖ In Laplace transform domain V = RI V = sL I 1 V I sC = j ϖ→ ϖ→ ϖ→ ϖ→ s 7 Unit step function ≥ < = t , 1 t , ) t ( u ≥ < =- o o o t t , 1 t t , ) t t ( u t u(t) 1 t u(t - t o ) 1 t o t u(t + + + t 1 ) 1 t 1 t 1 > 0 t o > 0 Unit step function Shifted step functions - ≥- < = + 1 1 1 t t , 1 t t , ) t t ( u ≤ =- 2 2 2 t t , t t , 1 ) t t ( u t u(t 2- t)...
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- Spring '06