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

2. 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

3. 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,

4. 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

5. 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,

6. 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

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:
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())

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 :