THIS BLOG CONTAIN ABOUT THE Natural Response of First Order RC and RL Circuits
Natural Response of an RL Circuit
If we consider the circuit:
It is assumed that the switch has been closed long enough so that the inductor is fully charged. This means that all voltages and currents have reached constant values. Thus only constant (or d.c.) currents can appear just prior to the switch opening and the inductor appears as a short circuit.
As the inductor appears as a short circuit there can be no current in either R0 or R. Hence all of the source current, I0, appears in the inductive branch and the voltage across this branch is zero.
To find the natural response we need to see what happens when the source is disconnected. Hence we say when t = 0 the switch is opened. This then reduces the above circuit to:
To find
i(t) we use Kirchoff's voltage law to obtain an expression involving i, R, and L. Summing the voltages around the closed loop gives:
This is known as a first order differential equation and can be solved by rearranging and then 'separating the variables'. This gives us:
Then by integrating both the right hand side and left hand side and including a constant of integration i(0) gives:
hence taking inverse logs:
if we use 0- to represent the time just prior to switching and 0+ just after switching. Due to the characteristics of an inductor an instantaneous change of current in an inductor is not possible, therefore the current just after switching is equal to the current just prior to switching, then:
this then gives us the final value for the current of:
This leads us to define the time constant for a RL circuit:
We can then derive the voltage across the resistor from a direct application of Ohm's Law:
Natural Response of an RC Circuit
By following the above steps we can calculate the current and voltage in the circuit show below:
The switch remains to the left until the capacitor is fully charged then at time, t = 0 the switch is changed to the right position, so the capacitor is effectively connected to only the resistor. This gives us the equations:
and:
where time constant, 
LEARNING
Our topic is all about the RL and RC Circuit I learn in this topic are Resistive Circuit => RC Ci t => RC Circuit algebraic equations => differential equations
Same Solution Methods (a) N s (a) Nodal Analysis (b) Mesh Analysis. The natural response is due to the initial condition of the storage component ( C or L). The forced response is resulted from external input ( or force). In this chapter, a constant input (DC input) will be considered and the forced response is called step response. When a dc voltage (current) source is suddenly applied to a circuit , it can be modeled as a step function , and the resulting response is called response .
