85-3 The source-free RC circuit The series combination of a resistor and capacitor has a greater practical importance than does the combination of a resistor and an inductor We shall assume an initial stored energy in the capacitor i(t adu U=0 Let v(t)=Ae-iyr dt R CAse i+-e 0 u(t) R R A(Cs1+)e=0→s R I RC D(0)=Vo ∴U(t)=AeRC U(0)=V0∴A=V o(t)=e RC and i(t) RC R R
The series combination of a resistor and capacitor has a greater practical importance than does the combination of a resistor and an inductor. §5-3 The source-free RC circuit We shall assume an initial stored energy in the capacitor: RC t t Ae − ( ) = 0 0 (0) =V A =V s t Let t Ae dt R d C 1 + = 0 ( ) = 0 1 1 1 + = s t s t e R A CAs e RC e s R A Cs s t 1 ) 0 1 ( 1 1 1 + = → = − t RC t RC e R V R t t V e and i t 1 0 1 0 ( ) ( ) ( ) − − = = = 0 V0 ( ) = − − + ( t ) R i(t) C
o(t o(t)=voe RC and i(t) e Rc R R LEttE RC U()=eand训=px R =1,→>D(z)=0.368 =3-57,→>U(3-5z)<<1 0.368 R↑,z个,pa=U2/R↓,t个 or:C个,r个,n=C2个,t个
t t e R V t V e and i t Let RC 1 0 1 0 ( ) ( ) : − − = = = t 0 V0 0 0.368V = = or C w C t R p R t C R , 2 1 : , , , , / , 2 2 t RC t RC e R V R t t V e and i t 1 0 1 0 ( ) ( ) ( ) − − = = = 0 0 3 5 , (3 5 ) 1 , ( ) 0.368 t V t V = − → − = → =
Many of the circuits for which we would like to find the natural response contain more than a single resistor and capacitor=> An equivalent circuit consisting of only one resistor and one capacitor R.R C T=RC=(r,+ R R 3 r,+ r (t)=voe R i1(0) R +r i1(t)=i1(0)e R,+ r1+r
Many of the circuits for which we would like to find the natural response contain more than a single resistor and capacitor. => An equivalent circuit consisting of only one resistor and one capacitor. t eq t V e C R R R R R C R V − = + = = + = 0 1 3 1 3 2 0 ( ) ( ) (0) t i t i e − + ( ) = (0 ) 1 1 1 3 3 1 3 1 3 2 0 1 (0 ) R R R R R R R R V i + + + = +
