8.4 The Natural Response of a Series/Parallel RLC Circuit

8.4 The Natural Response of a Series/ParallelRLC Circuitinitial conditions⎧i(0)= I0⎨⎩v(0)= V0Step 2 : Homogeneous solution , characteristic equation2 R 1S + S+ =0L LC2 2 R 2 1S +2αS+ω0=0 ⇒ α @ , ω0@2L LCω : undamped resonant frequency (rad/s)0α : damping factor or neper frequencyC.T. Pan 278.4 The Natural Response of a Series/ParallelRLC Circuitinitial conditions⎧i(0)= I0⎨⎩v(0)= V0characteristic roots (natural frequencies)S =-α+ α -ω2 21 0S =-α- α -ω2 22 0St 1 St 2() 1 2∴ i t = Ae + Ae+ d +Need two initial conditions , i.e. , i(0 ) and i(0 ) .diC.T. Pan 28