Understanding First Order Circuits: Capacitors, Inductors, and Response Analysis

1. First Order Circuit • Capacitors and inductors • RC and RL circuits

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

3. t=0 R Vs + vc  Taking KCL,    RC circuit – step response 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 For vc(0) = Vo, , where  = RC = time constant For vc(0) = 0,

4. vc(t) 0.632Vs  2 3 4 5 t RC circuit – step response Vs Vs -- is the final valuei.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

5. t=0 R Vs + vL  iL(t) Taking KCL,    RL circuit – step response 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 For iL(0) = Io, , where  = L/R = time constant For iL(0) = 0,

6. iL(t)  2 3 4 5 t RL circuit – step response Vs/R 0.632(Vs/R) (Vs/ R) -- is the final valuei.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

7. 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

8. The complete response • Complete response is also can be written as the combination of steady state and transient responses: Transient response: Steady state 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())

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

10. 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 :