This page describes how the first-order and the second-order systems respond to sinusoidal input signal. The nature of a sinusoidal signal has been described earlier. We deal with the first-order system first, and the description of how to obtain the response of second-order system is taken up next.
Given a differential equation with an excitation signal, the steps involved in obtaining the solution are the same. The first step is to obtain the particular solution. In the case of a sinusoidal input signal, it can be determined directly. The second step is to get the assumed complementary solution. The third step is to write the total solution as the sum of the particular solution and the complementary solution. The fourth step is to determine the unknown constants in the complementary solution from the initial conditions. In the case of a sinusoidal input, the particular solution is also called the steady-state solution or response , since it has no decaying component.
A differential equation of a first-order system with a sinusoidal input is presented by equation (8.1). Using Euler’s identity shown by equation (8.2), we can represent the differential equation with a sinusoidal input in a different way. Equation (8.3) displays the differential equation of a first-order system, with the input now shown as an exponential function. We can also describe function Q(D), as shown by equation (8.3). Equation (8.3) also shows the expression for function Q(j.w). Since cos(w.t) is the real part of the exponential function, we get the particular solution, as shown by equation (8.4). As long as we are not dealing with the differential equation of an undamped system, there is no possibility of function Q(j.w) being equal to zero. Why is it so ? It is because a sinusoidal function has its poles on the jw axis, and a system, which is not undamped, does not have its poles on the jw axis.
If the input to the system is a sine function as shown by equation (8.5), the procedure to obtain the particular solution is the same, and we take the imaginary part as the particular solution, as shown by equation (8.7).
A differential equation of a first-order system with sinusoidal input is presented by equation (8.8). We call the particular solution due to the exponential function as yT(t). This particular solution is evaluated, as shown by equation (8.9). It is simplified by multiplying both the numerator and the denominator by the conjugate of the denominator, as shown by second line of equation (8.9). Then the denominator becomes only a real value. Imaginary part of the solution, shown by the second line of equation (8.9), is the solution we are seeking, and the particular solution due to sine input function is displayed by equation (8.10). The solution in equation appears long, but it is not that complicated when real values replace the symbols.
The sinusoidal response is obtained as the sum of the particular and the complementary solution, as shown by equation (8.11). The value of A, the constant forming part of the complementary solution, is determined from the initial value of the sinusoidal response, as shown by equation (8.12). Then the sinusoidal response obtained, is expressed by equation (8.13).
A differential equation of a first-order system with cosine input is presented by equation (8.14). To obtain the particular solution, we follow the same procedure. We call the particular solution due to the exponential function, as yT(t). This particular solution is evaluated, as shown by equation (8.15). It is simplified by multiplying both the numerator and the denominator, by the conjugate of the denominator, as shown by second line of equation (8.15). Then the denominator becomes only a real value. The real part of the solution, shown by the second line of equation (8.15), is the solution we are seeking, and the particular solution due to cosine input function is displayed by equation (8.16).
Alternatively, the reciprocal of Q(j.w) can be set to be a complex value, as shown by equation (8.17a). Then, if the input is a sine function, the particular solution is obtained, as shown by equation (8.17b). If the input is a cosine function, the particular solution is obtained, as shown by equation (8.17c). It is good to remember this, as it can be of help in obtaining the solution.
A numerical example is presented now. A differential equation of a first-order system with cosine input is presented by equation (8.18). We call the particular solution, due to the exponential function, as yT(t). This particular solution is evaluated, as shown by equation (8.19a). First, we get the complex value of function Q(j.w), as shown by the first line of equation (8.19a). In the second line of equation (8.19a), we multiply both the numerator and the denominator, by the conjugate of the denominator.. Then the denominator becomes only a real value. The real part of the solution, shown by the second line of equation (8.19a), is the solution we are seeking, and the particular solution due to cosine input function is displayed by equation (8.19b). This expression is not long. This technique is quite simple. It is much simpler than the technique traditionally followed to get the solution.
The sinusoidal response is obtained as the sum of the particular and the complementary solution, as shown by equation (8.20). The value of A, the constant forming part of the complementary solution, is determined from the initial value of the sinusoidal response, as shown by equation (8.21). As shown by equation (8.21), the initial value of the response is set equal to zero. Then the sinusoidal response obtained, is expressed by equation (8.22).
The differential equation of a second order system with a sinusoidal input is shown by equation (8.23). This equation can represent an over-damped, a critically-damped, an under-damped, or an undamped system, based on the values of b and c. The particular solution is obtained in the same fashion, as outlined for the first order system.
Using Euler’s identity, we can represent the differential equation with the input as an exponential function, as shown by equation (8.24). Equation (8.24) shows the expression for function Q(j.w) . Since cos(w t ), is the real part of the exponential function, we get the particular solution, as shown by equation (8.24). If the sinusoidal input function is a sine function, as shown by equation (8.25), the particular solution is obtained, as shown by equation (8.26). As long as we are not dealing with the differential equation of an undamped system, there is no possibility of function Q(j.w) being equal to zero. It is because a sinusoidal function has its poles on the jw axis, and a system, which is not undamped, does not have its poles on the jw axis.
When the pole due to the excitation function is distinct from the poles of the second order system, then the solution is straight forward. If the pole of the excitation function is at the same location as the pole of the system, then the technique used is different.
A numerical example is presented now. A differential equation of a second-order, over damped system with cosine input is presented by equation (8.27). Equation (8.28) defines function Q(D). The value of function Q(j4), is obtained as shown by equation (8.28). The particular solution is obtained, as shown by equation (8.29). If you are familiar with complex numbers and Euler’s identity, obtaining the particular solution is not complicated. This technique is quite simple. It is much simpler than the technique traditionally followed to get the solution. Equation (8.30) shows the expression for the particular solution, and the assumed complementary solution is shown by equation (8.31).
The sinusoidal response is obtained as the sum of the particular solution and the complementary solution, as shown by equation (8.32). The values of A and B, the constants forming part of the complementary solution, are determined from the initial conditions of the differential equation, as shown by equation (8.33). Equation (8.34) expresses the values of these constants. Then the sinusoidal response obtained, is expressed by equation (8.35).
A numerical example is presented now. A differential equation of a second-order, critically damped system with cosine input is presented by equation (8.36). Equation (8.37) defines function Q(D). The value of function Q(j3), is obtained as shown by equation (8.37). The particular solution is obtained, as shown by equation (8.38). Equation (8.39) shows the expression for the particular solution, and the assumed complementary solution is shown by equation (8.40).
The sinusoidal response is obtained as the sum of the particular solution and the complementary solution, as shown by equation (8.41). The values of A and B, the constants forming part of the complementary solution, are determined from the initial conditions of the differential equation, as shown by equation (8.42). Equation (8.43) expresses the values of these constants. Then the sinusoidal response obtained, is expressed by equation (8.44).
A numerical example is presented now. A differential equation of a second-order, under damped system with cosine input is presented by equation (8.45). Equation (8.46) defines function Q(D). The value of function Q(j2) is obtained as shown by equation (8.46). The particular solution is obtained, as shown by equation (8.47). Equation (8.48) shows the expression for the particular solution, and the assumed complementary solution is shown by equation (8.49). When the system is under damped, it has a pair of complex, conjugate poles, and the assumed solution is as expressed by equation (8.49).
The sinusoidal response is obtained as the sum of the particular solution and the complementary solution, as shown by equation (8.50). The values of A and B, the constants forming part of the complementary solution, are determined from the initial conditions of the differential equation, as shown by equation (8.51). Equation (8.52) expresses the values of these constants. Then the sinusoidal response obtained, is expressed by equation (8.53).
A numerical example is presented now. A differential equation of a second-order, undamped system with sine input is presented by equation (8.55). An undamped system has a pair of conjugate poles on the imaginary axis, and the sinusoidal function is also characterized by a pair of conjugate poles on the imaginary axis. As long as these poles are not coincident, the solution is easily obtained. Equation (8.55) shows the value of function Q(j3). The particular solution is obtained, as shown by equation (8.56). The sinusoidal response is expressed as the sum of the particular solution and the assumed complementary solution, as shown by equation (8.57). The values of A and B, the constants forming part of the complementary solution, are determined from the initial conditions of the differential equation, as shown by equation (8.58). To get the sinusoidal response can be obtained, substitute the values of A and B in equation (8.57).
A numerical example is presented now. A differential equation of a second-order, undamped system with sine input is presented by equation (8.59). Here one of the poles of the system and the pole due to excitation signal, are coincident. The solution is somewhat complicated. Equation (8.59) can be presented as shown by equation (8.60), with the use of two factors. Equation (8.61) shows the next step, where we simplify the solution due to one of the factors, that does not contain coincident poles. Equation (8.62) shows the simplified equation. It is seen that function Q1( jw) is zero, as shown by equation (8.63).
Hence we use function Q1( D + jw) , in place of function Q1( jw) . Function Q1( D + jw) is equal to D. We know that 1/D represents integration, and the integral of 1 is time t , as shown by equation (8.64). The particular solution for the sine input is shown by equation (8.64). If the input is a cosine function, then the particular solution obtained is expressed by equation (8.65). This example is somewhat complicated. If you find it too difficult, skip it. It is a rare occurrence, but it is presented here for the sake of completeness. It is necessary to show how the response of such a system can be obtained.
This page has described how the response of a first and a second order system can be obtained, when the system has a sinusoidal input signal. The next page describes how the response of a first and a second order system can be obtained, if the input signal is a complex sinusoid.