ch6

Chapter 6: Applications of The Laplace Transform to Circuit Analysis Chapter 6.1: Laplace Circuit Solution: ƒ We wish to find the current: e(t) ƒ Ohm law : R i(t) L inductor + _ resistor di(t) R.i(t) + L = e(t) dt ƒ Determine i(t) by using Laplace Transform Chapter [2] 6.2: Circuit Element Models : 1) Resistor Model: Resistor u(t) U(s) u(t) = Ri(t) ⇒ U(s) = RI (s) Chapter [3] 2) Inductor Models: u(t) di u(t) = L (t) ⇒ U(s) = L(sI (s) − i(0)) dt (sL= cảm kháng toán tử) U(s) U(s) = sLI (s) − Li(0) Chapter U(s) U ( s ) i (0) I (s) = + sL s [4] 3) Capacitor Models: t u(t ) = ∫ i( x)dx + u(0) C0 (1/sC= dung kháng toán tử) U (s) = Chapter u (0) I (s) + sC s I(s) = sCU (s) −Cu (0) [5] 4) a) Source Models: Independent sources e S (t) → E S (s) jS (t) b) → J S (s) Dependent sources e D (t) = k i C (t) → E D (s) = k I C (s) jD (t) = k u C (t) → J D (s) = k U C (s) Chapter [6] ™ Example1: The s-domain Circuit Transform the circuit to s domain , assuming zero initial conditions ? u(t) ⇒ s 1H ⇒ sL = s 1 F ⇒ = sC s Chapter [7] 6.3: Transfomed Circuit Laws : 1) Transformed Ohm Law: U(s) = Z(s).I(s) I(s) = Y(s).U(s) Chapter - Y(s) Z(s) : trở kháng , tổng trở toán tử (Ω) Y(s) : dẫn nạp , tổng dẫn toán tử (S) (Ω) Với : Y = Z I(s U(s) ) Z(s) + [8] Properties of Z(s) and Y(s) : ™ Z(s) Y(s) tuân theo phép biến đổi tương đương điện trở điện dẫn ™Ví dụ : Xác định trở kháng toán tử tương đương : 0,5s Z = 0,5s + 2+ 0,5s Z = 0,5s + s +1 Chapter [9] 2) Transformed KCL and KVL : ƒ Luaät KCL : ∑ ± I ( s) = k node ƒ Luật KVL : ∑±U (s) = (Xét dấu mạch điện trở) k loop ƒ Do luật Ohm Kirchhoff viết cho mạch toán tử tương tự viết cho mạch phức nên ta áp dụng phương pháp phân tích mạch điện trở học cho sơ đồ toán tử tìm ảnh Laplace Chapter [10] ™ Example1: s-domain Circuit Analysis Find i1(t) in the circuit, assuming zero initial conditions ? Z= s s (5 + s) +5+ s 3( s+5) Z = s2 +5s+3 I1 = 1/ s 1+Z ( s) ( s2 +5s+3) = s[s2 +8s+18] −1 Chapter i1(t) = L {I1(s)} [11] 6.4: Transfer Function : input x(t) X(s) Circuit Transfomed Circuit ™ H(s) is the ratio of the output Y(s) to the input X(s), assuming all initial conditions are zero Chapter y(t) output Y(s) Y(s) H(s) = X(s) [12] ™ Determine H(s) : in two ways i Using the circuit analysis methods ii Using the propotional theorem to apply the ladder network Chapter [13] ™ Unit Impulse Response h(t) : input δ(t) Circuit h(t) output if x(t) = δ (t ) ⇒ h(t) = L −1{H(s)} = unit impulse response ƒ If we know h(t) → H(s) → Y(s) = H(s).X(s) and we can obtain the response y(t) of any input x(t) using y(t)=£-1{Y(s)} ƒ If we know h(t): can obtain the response y(t) of any input x(t) using the convolution integral in time-domain (See in the course named “Signal and System” ) Chapter [14] ™Example1: Transfer Function ƒ We have: and ƒ And the transfer function : ƒ The unit impulse function : Chapter [15]