First Order Circuit Capacitors and inductors RC and RL circuits

RC and RL circuits (first order circuits) Circuits containing no independent sources ‘source-free’ circuits Excitation from stored energy Natural response Circuits containing independent sources DC source (voltage or current source) Sources are modeled by step functions Step response Forced response Complete response = Natural response + forced response

RC circuit – step response + vc  Vs R Objective of analysis: to find expression for vc(t) for t >0 , i.e. to get the voltage response of the circuit to a step change in voltage source OR simply to get a step response Taking KCL,    For vc(0) = Vo, , where  = RC = time constant For vc(0) = 0,

RC circuit – step response vc(t) Vs 0.632Vs t  2 3 4 5 Vs -- is the final value i.e. the capacitor voltage as t   In practice vc(t) considered to reach final value after 5 When t = , the voltage will reach 63.2% of its final value

RL circuit – step response + vL  Vs R iL(t) Objective of analysis: to find expression for iL(t) for t >0 , i.e. to get the current response of the circuit to a step change in voltage source OR simply to get a step response  Taking KCL,   For iL(0) = Io, , where  = L/R = time constant For iL(0) = 0,

RL circuit – step response iL(t) Vs/R 0.632(Vs/R) t  2 3 4 5 (Vs/ R) -- is the final value i.e. the inductor current as t   In practice iL(t) considered to reach final value after 5 When t = , the current will reach 63.2% of its final value

The complete response The combination of natural and step (or forced) responses For RC circuit, the complete response is: Natural response: Forced response: Response due to initial energy stored in capacitor Vo is the initial value, i.e. vc(0) Response due to the present of the source Vs is the final value i.e vc() Note: this is what we obtained when we solved the step response with initial energy (or initial voltage) at t =0

The complete response Complete response is also can be written as the combination of steady state and transient responses: Steady state response: Transient response: Response that exist long after the excitation is applied Response that eventually decays to zero as t   For DC excitation, this is the term in the complete response that does not change with time For DC excitation, this is the term in the complete response that changes with time This is the final value, (i.e. vc()) Vo is the initial value (i.e. vc(0)) and Vs is the final value (i.e. vc())

The complete response Complete response of an RL circuit can be written as: (natural response) + (forced response) (steady state response) + (transient response)

The General Solution In general, the response to all variables (voltage or current) in RC or RL circuit can be written as: x(t) can be v(t) or i(t) for any branch of the RC or RL circuit x() – final value of x(t) (long after to) x(to) – initial value of x(t) – for continuous variables, x(to+) = x(to-) For to = 0, the equation becomes :