The nature of the system is reflected by impulse response, and therein lies its importance. For impulse input, the use of Laplace transforms tends to be easier. Impulse response is different from the response to other inputs, since only the particular solution needs to be obtained. Since the impulse response reflects the nature of the system, the assumed particular solution is the same as the assumed homogeneous solution. However, this is not the preferred technique for obtaining the impulse response. Since the impulse input is the derivative of the unit step input, the impulse response of a linear system is the derivative of its unit step response. . Hence in order to get the impulse response, first obtain the unit step response, and then differentiate the unit step response with respect to time. This procedure is easier.
A first-order system, with impulse input, is shown in equation (5.1). Unlike other excitation signals, the response of a first order and second order system to an impulse input does not contain an impulse function as its part. The impulse response is similar to the zero input or natural response, and therein lies the importance of the impulse function. Impulse response reveals the nature of the system. As we shall see later, impulse response is an integral part of the convolution integral, that validates the application of Laplace transforms to analysis of physical systems. We obtain the unit-step response first and then we obtain the impulse response, as shown below.
Equation (5.2) presents the differential equation of a first order system with an unit-step input. Equation (5.3) presents the unit step response of a first order system. Since the impulse function is the derivative of the unit step function, the impulse response of a linear system is the derivative of the unit step response, as stated by equation (5.4). Hence we obtain the impulse response from the unit step response, as shown by equation (5.5). Hence in order to get the impulse response, first obtain the unit step response, and then differentiate the unit step response with respect to time. This procedure is easier, and this is the only procedure followed in this page. There are other techniques available, but they are more complicated.
An example is presented now. Equation (5.6) presents the differential equation of a first-order system with an impulse input. Equation (5.7) shows the equation of the same system with an unit-step input. Equation (5.8) presents the unit-step response and the impulse response is obtained as shown by equation (5.9).
Equation (5.10) presents the differential equation of a second order over-damped system with an impulse input. Equation (5.11) presents the unit-step response of this system. The page on the unit step response has explained how equation (5.11) can be obtained. Since the impulse function is the derivative of unit step function, the impulse response of the over damped system is the derivative of its unit step response. Equation (5.12) presents the impulse response, which has been derived from the unit step response. Hence in order to get the impulse response, first obtain the unit step response, and then differentiate the unit step response with respect to time.
A numerical example is presented now. Equation (5.13) presents the differential equation of a second order over-damped system with impulse input. Equation (5.14) presents the unit step response of this system. Since the impulse function is the derivative of unit step function, the impulse response of the over damped system in this example is the derivative of its unit step response. Equation (5.15) presents the impulse response, derived from the unit step response.
Equation (5.16) presents the differential equation of a second order critically-damped system with impulse input. Equation (5.17) presents the unit step response of this system. The page on the unit step response has explained how equation (5.17) can be obtained. Since the impulse function is the derivative of unit step function, the impulse response of the critically damped system is the derivative of its unit step response. Equation (5.18) presents the impulse response, derived from the unit step response. Hence in order to get the impulse response, first obtain the unit step response, and then differentiate the unit step response with respect to time.
A numerical example is presented now. Equation (5.19) presents the differential equation of a second order critically-damped system with impulse input. Equation (5.20) presents the unit step response of this system. Since the impulse function is the derivative of unit step function, the impulse response of the critically damped system in this example is the derivative of its unit step response. Equation (5.21) presents the impulse response, derived from the unit step response.
Equation (5.22) presents the differential equation of a second order under-damped system with an impulse input. Equation (5.23) presents the unit step response of this system. The page on the unit step response has explained how equation (5.23) can be obtained. Since the impulse function is the derivative of unit step function, the impulse response of the under damped system is the derivative of its unit step response. Equation (5.24) presents the impulse response, derived from the unit step response. Hence in order to get the impulse response, first obtain the unit step response, and then differentiate the unit step response with respect to time.
A numerical example is presented now. Equation (5.25) presents the differential equation of a second order under damped system with an impulse input. Equation (5.26) presents the unit step response of this system. Since the impulse function is the derivative of unit step function, the impulse response of the under damped system in this example is the derivative of its unit step response. Equation (5.27) presents the impulse response, derived from the unit step response.
Equation (5.28) presents the differential equation of a second order undamped system with an impulse input. Equation (5.29) presents the unit step response of this system. The page on the unit step response has explained how equation (5.30) can be obtained. Since the impulse function is the derivative of unit step function, the impulse response of the undamped system is the derivative of its unit step response. Equation (5.30) presents the impulse response, derived from the unit step response.
The differential equation of a second-order over-damped system with an impulse input is presented by equation (5.31). Equation (5.32) presents the unit-step response of this system. Equation (5.33) presents the impulse response, derived from the unit step response.
The differential equation of a second-order critically-damped system with an impulse input is presented by equation (5.34). Equation (5.35) presents the unit-step response of this system. Equation (5.36) presents the impulse response, derived from the unit step response.
The differential equation of a second-order under-damped system with an impulse input is presented by equation (5.37). Equation (5.38) presents the unit-step response of this system. Equation (5.39) presents the impulse response, derived from the unit step response.
This page has described how the impulse response of a first-order and a second-order system can be obtained. The next page describes how the ramp response of a first-order and a second-order system can be obtained.