The parts of time response of a system can be classified and grouped in different ways. The types of response are listed below.
The responses listed above are now explained.
Example 1:
Let the homogeneous differential equation of a first-order system be defined as follows.
Equation (1.1) is the homogeneous differential equation of first-order. The solution is presented by equation (1.2). The response of the homogeneous differential equation is the homogeneous response. The homogeneous response contains the response due to the pole of the system. For this system, the pole is at - k. In technical terms, we state that the response contains characteristic or the natural mode of the system. In this case, there is only a single pole, as the system is of first-order. A second-order system contains two poles and the homogeneous solution for the second-order system has two factors, one for each pole. The parts containing characteristic or the natural modes of the system combine to form the natural response. For the first-order system defined by equation (1.1), equation (1.2) represents both the homogeneous response and the natural response. It is not necessary that both the homogeneous response and the natural response be equal to each other.
Example 2:
Equation (1.3) presents the differential equation of the first-order system with unit-step input. The solution is the sum of particular solution and complementary solution, as stated by equation (1.4). The particular solution presented by equation (1.5) is the forced response. In this case, equation (1.5) is also the steady-state response, because it does not contain any decaying exponential term. The complementary solution presented by equation (1.6) is the natural response. Equation (1.7) presents the zero-state response. We can call it as the unit-step response too. It is seen that the zero-state response is the sum of the forced response and the natural response. Note that it is not correct to state that zero-state response is the sum of the forced response and the homogeneous response, since the natural response is not always equal to the homogenous response. When the forcing function is of zero value, then the natural response is the same as the homogenous response. Since the initial condition is zero, the zero-state response is also the total response. Hence the total response is the sum of the forced response and the natural response. For any system, the the total response is the sum of the forced response and the natural response. Equation (1.6) represents also the transient response of the system. The transient response contains the component of response that decays with time. It can be seen that the total response is the sum of the transient response and the steady-state response.
Example 3:
Equation (1.8) presents the differential equation of the first-order system with a step input and the initial condition for this system is not zero. The solution is the sum of particular solution and complementary solution, as stated by equation (1.9). The particular solution is presented by equation (1.10) and the complementary solution is presented by equation (1.11). The particular solution is called the forced response.
The total response is expressed by equation (1.12). It is the sum of the particular solution and the complementary solution. Remove the response due to the initial condition from the total response and then we get the zero-state response. The zero-state response is the response obtained due to excitation signal, with zero initial condition. Equation (1.13) presents the zero-state response. Since the excitation signal is a step function, equation (1.13) represents the step response also. The zero-input response,which is the response due to the non-zero initial condition alone, is represented by equation (1.14). The natural response contains the response due to the natural modes of the system and equation (1.15) expresses the natural response. In this case, the the natural response is the same as the transient response, but it is not always the case. When the excitation signal is a step function or a sinusoidal signal, the natural response and the transient response are the same. If the excitation signal contains an exponential function, then the natural response and the transient response are different. Equation (1.16) is the steady-state response. The steady-state response has no decaying component.
Example 4:
Equation (1.17) presents the differential equation of the first-order system with an exponential input and the initial condition for this system is not zero. Equation (1.18) presents the total response of this system. Equation (1.19) presents the homogeneous response. The zero-input response is equal to the homogeneous response and hence equation (1.19) presents the zero-input response. Equation (1.20) presents the natural response. It can be seen that the natural response is not equal to the homogeneous response, since part of the natural response occurs due to the forcing function. The particular solution is presented by equation (1.21) and this equation presents the forced response. The zero-state response, which is the response obtained with zero-initial condition, is presented by equation (1.22). Since the forced response and the zero-input response contain only decaying terms, equation (1.23) presents the transient response. Ultimately the response of the system becomes zero and the steady-state response of this system is zero. It can be seen that the transient response is different from the natural response. The transient response can have terms besides those involving the natural modes or poles of the system. If the excitation signal has an exponential function, then the excitation signal has either pole or poles in the left-half of s-plane and the response contains terms that have poles of the excitation signal. Since this response decays with time, it is also part of the transient response.
In summary, we can state that the total response can be expressed as follows.
When the excitation signal is a periodic non-sinusoidal signal, then the total response contains both the periodic response and the transient response. After the transient response lapses, what remains is the periodic response. When the excitation signal is a dc signal or a sinusoidal signal, the response that remains after the lapse of the transient response is the steady-state response. The steady-state response is a dc value if the excitation signal is a dc signal, and it is a sinusoidal signal if the excitation signal is a sinusoidal signal. The frequency of the excitation signal and the steady-state response is the same. When the excitation signal is a dc signal or a sinusoidal signal, the steady-state response contains no term representing the characteristic or the natural modes of the system. On the other hand, the periodic response contains terms representing the characteristic or the natural modes of the system. Hence you can see that there is difference between steady-state response and periodic response. Examples for periodic response are presented in the pages to follow. The next page is on the homogeneous response of first-order circuits.