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1、7-7 THE LAPLACE TRANSFORM (拉氏变换),Pierre Simon Laplace (1791-1872),Definition of The 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).,It is also known as one-sided Laplace transform.,f(t)：原函数；
2、F(S)：f(t)在S域中的象函数。,The inverse Laplace transform:,Properties of The Laplace Transform,Linearity （线性定理）,Time differentiation （微分定理）,Time integration （积分定理）,Time shift （时域位移定理）,Initial and final values （初值定理与终值定理）,In order for the final-value theorem to hold, all poles of F(S) must be located in the l
3、eft half of the S plane, the only exception is the case in which F(S) has a simple pole at S=0.,Convolution （卷积定理）,Frequency shift （频域位移定理）,Scaling （尺度变化定理）,Laplace transforms of some functions,The unit-step function 1(t),The unit-impulse function (t),Table 7-4 summarizes the Laplace transforms of s
4、ome common functions.,The roots of P(S)=0 are called the zeros of P(S), while the roots of Q(S)=0 are the poles of Q(S). （零点和极点）,The Inverse Laplace Transform,Steps to find the inverse Laplace transform:,Decompose F(S) into simple terms using partial fraction expansion. Find the inverse of each tem
5、by matching entries in Table 7-4.,Simple Poles,How to find the expansion coefficients Kn ?,Method a. （The residue method）,Method b.,Repeated Poles,Complex Poles,解：,Or：,7-8 APPLICATION TO CIRCUITS (运算电路 ),Transform a circuit in the time domain to the S domain,Resistor,The voltage-current relationship
6、 for a resistor: LU=LRi，LU=U(S)，Li=I(S) U(S)=RI(S)，or I(S)=GU(S),Capasitor, IC(S)=SCUC(S)CuC(0),Inductor,UL(S)=SLIL(S)LiL(0),The initial condition is modeled as a voltage or current source.,Coupled inductor,KCL、KVL in the S domain,The impedance in the S domain,The admittance in the S domain,Z(S)I(S)
7、=Useg(S) ，Y(S)U(S)=Iseg(S),Steps in applying the Laplace transform:,Transform the circuit from the time domain to the S domain. Solve the circuit using nodal analysis, mesh analysis, source transformation, superposition, or any circuit analysis technique with which we are familiar. Take the inverse
8、transform of the solution and thus obtain the solution in the time domain.,解：,图(b)：,解：,图(b)：,例：如图电路，R1=30，R2=R35，L1=0.1H，C=1000F，E=140V，开关闭合已久，求开关打开后的uk(t)和uC(t)。,解：,图(b)：,例：如图电路，R=10，0.9，L=0.05H，iL(0)=0，e(t)=100sin(2000t+60)V，求电感中的过渡电流。,解：,图(b)：,图(c)：,图(d)：,例：如图电路，R=1，C1=1F，C2=2F，uC1(0)=6V，uC2(0)=0，t=0时K闭合，开关动作后的uC1，uC2，i。,解：,图(b)：,
