FPGA Based Digital World - First Order Circuits

First Order Circuits

A first order circuit is a circuit with first-order differential equation.An example first order RC circuit is shown below.

Fig.1 Example of RC Circuit

When K1 is switched to position 1, the circuit is excited by the energy(voltage or current) stored in the capacitor, and the response is called natural response;

When K1 is switched to position 2, the circuit is excited by the DC voltage source V1(or current source I1), and the response is called forced response.

Natural Response of Source-Free RC Circuit

Suppose the iniatial voltage of C1 is V0, and apply KCL in the top node, we get

Solve the above solution as below

The final result is:

The corresponding curve is

where

is the time constant where the response decays to 1/e (or 0.368) of its initial value.

Natural Response of Source-Free RL Circuit

For below RL circuit,

The response is

Step Response of RC Circuit

The step response is the circuit behavior when excited by a step voltage or current source.

In Fig.1, when K1 is switched to position 2, it's similiar to being excited by a step voltage source, as shown below.

Applying KCL, we get

Solving the above equation by below steps for t>0,

we get

The response can also be divided into natural response and forced response, as below.

When the inital voltage of the capacitor at t=0 is 0, the voltage and current become

Step Response of RL Circuit

For below RL circuit,

The response is


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