87-4 The complex forcing function j(at+o) 々eJ(+8 9) N cos(ar +o) jlm sin(ar +9 jIm sin( at+p) Vm cos(at +9) Im cos(at +p jVm sin(at +9)- jIm sin(at+o Vm cos(at+9)+ jvm sin(at+9)-Im cos(at +p)+jIm sin(at +) A real, an imaginary, or a complex forcing function will produce a real, an imaginary, or complex response, respectively. Instead of applying a real forcing function to obtain the desired real response, we apply a complex forcing function whose real part is the given real forcing function; we obtain a complex response whose real part is the desired real response
§7-4 The complex forcing function − + V cos(t +) m jV sin(t +) m jI sin(t +) m I cos(t +) N m V cos(t +) m jV sin(t +) m I cos(t +) m jI sin(t +) m Vm cos(t +) + jV sin(t +) m I m cos(t +) + jI sin(t +) m A real, an imaginary, or a complex forcing function will produce a real, an imaginary, or complex response, respectively. Instead of applying a real forcing function to obtain the desired real response, we apply a complex forcing function whose real part is the given real forcing function; we obtain a complex response whose real part is the desired real response. j( t ) m V e + j( t ) m I e +
R i Jar cos att/sinat y cos ot L cos at= Relet The complex source is ve! The complex response isI, e/( ortp KVL: Ri+L dtUs RI ej(at+)+JaLIm ej(at+p)=V ear RIme/ +jalim ep=m Imn p(r+jOL)=v tan or e/p t R R+jOL R+0L R2+2L2 p=-tan R i(t)=Im cos(at+p) cos(at-tan R+0 R
e t j t j t = cos + sin cos Re[ ] j t t e = The complex source is . j t m V e The complex response is . j(t+ ) m I e s dt di KVL: Ri + L = j t m j t m j t RIm e j LI e V e + = ( + ) ( + ) m j m j RIm e + j LI e =V ( tan ) 2 2 2 1 R L j j m m m e R L V R j L V or I e − − + = + = m j I m e (R + jL) =V R L 1 tan− = − 2 2 2 R L V I m m + = ( ) cos( ) cos( tan ) 1 2 2 2 R L t R L V i t I t m m − − + = + = V t s = m cos − + R i L