TABLE OF CONTENTS The natural response discussed in the previous section was obtained for the zero input condition, since the input force was removed after the time t > = 0. Now we consider the zero state condition where an input force is applied after the time t > = 0. We begin with the discussion of the step and impulse response and then derive the response to an arbitrary input using the convolution process, as we did with the first order differential equations.
A constant current I of a unit magnitude applied to an RLC network as shown in Figure 4.13 may be considered a step input u(t). The response to such an excitation should be considered independently for the three conditions, namely roots complex ( ? 2 < ? 2 0) the under-damped condition, roots real ( ? 2 > ? 2 0) the over-damped, and roots equal ( ? 2 = ? 2 0) the critically damped condition. The following analysis uses the parallel RLC network as shown in Figure 4.13.
Applying Kirchoff's current law to the circuit of Figure 4.13 for the unit step function u(t), we get Equation 4.36.
The particular solution is obtained from the new input condition of the constant current, and we get the solution shown in Equation 4.37.
The general solution is obtained for the three damped conditions of roots real, roots complex, and roots equal by...
