This section describes how first-order and second-order systems respond to a complex, sinusoid input signal. A complex sinusoidal signal has an envelope decaying exponentially, and the signal variation is sinusoidal within the envelope. Finding the response to a complex sinusoid signal is not any more difficult than obtaining the response to a sinusoidal signal. 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.
A differential equation of a first-order system with a complex sinusoid input is presented by equation (9.1). As shown by equation (9.2), we can represent the differential equation, with the input now shown as an exponential function, with a complex pole. We can also describe function Q( D), as shown by equation (9.2). Equation (9.3) shows the expression for function Q(- 1 + j3). The result is a complex value. The particular solution due to the exponential function is obtained, as shown next.
The particular solution due to the exponential function with a complex pole is obtained as shown by equation (9.4). Then the particular solution that we need is obtained as shown by equation (9.5). The level of difficulty in obtaining the solution is the same as that encountered in getting the response due to a sinusoidal signal.
The response to the complex sinusoid input function is obtained as the sum of the particular solution and the complementary solution, as shown by equation (9.6). The value of A, the constant forming part of the complementary solution, is determined from the initial condition of the differential equation, as shown by equation (9.7). Then response to the complex sinusoid input function is obtained, and it is expressed by equation (9.8).
A numerical example is presented now. A differential equation of a second-order, over-damped system with a complex, sinusoid input is presented by equation (9.9). Equation (9.10) defines the differential equation with an exponential input function. The pole of the input function is a complex value. Function Q(D) is shown by equation (9.11). It is evaluated by replacing D by ( - 4 + j3). The value of function Q(- 4 + j3) is obtained, and it is shown by equation (9.12).
The particular solution is obtained as shown by equation (9.13). The simplified expression for the particular solution is shown by equation (9.14). The assumed complementary solution for the over damped system is displayed by equation (9.15).
The response to the complex sinusoid input function is obtained as the sum of the particular solution and the complementary solution, as shown by equation (9.15). 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 (9.16). Then response to the complex sinusoid input function is obtained, and it is expressed by equation (9.17).
A numerical example is presented now. A differential equation of a second-order, critically-damped system with a complex, sinusoid input is presented by equation (9.18). Equation (9.19) defines the differential equation with an exponential input function. The pole of the input function is a complex value. Function Q(D) is shown by equation (9.20). It is evaluated by replacing D by the complex value of (- 1 + j). The value of function Q(- 1 + j) is obtained, and it is shown by equation (9.21).
The particular solution is obtained as shown by equation (9.22). The simplified expression for the particular solution is shown by equation (9.23). The assumed complementary solution for the critically damped system is displayed by equation (9.24).
The response to the complex sinusoid input function is obtained as the sum of the particular solution and the complementary solution, as shown by equation (9.52). 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 (9.26). Then response to the complex sinusoid input function is obtained, and it is expressed by equation (9.27).
A numerical example is presented now. A differential equation of a second-order, under damped system with a complex, sinusoid input is presented by equation (9.28). Equation (9.29) defines the differential equation with an exponential input function. The under-damped system has a pair of complex, conjugate poles, and the excitation function also has a complex pole. But the poles are not coincident in this case and the solution is easy. The pole of the input function is a complex value. Function Q(D) is shown by equation (597). It is evaluated by replacing D by ( - 1 + j10). It turns out to be a real value in this case. It may turn out to be a complex value for other examples of this type, but the same technique is followed even then. The value of function Q( - 1 + j10), is obtained, and it is shown by equation (9.31).
The particular solution is obtained as shown by equation (9.32). The simplified expression for the particular solution is shown by equation (9.34). The assumed complementary solution for the under damped system is displayed by equation (9.35). It can be seen that the assumed complementary solution is due to a pair of complex, conjugate poles of the under damped system.
The response to the complex sinusoid input function is obtained as the sum of the particular solution, and the complementary solution, as shown by equation (9.36). 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 (9.37). Then response to the complex sinusoid input function is obtained, and it is expressed by equation (9.38). Next we take up another example, where two roots are at the same location.
A numerical example is presented now. A differential equation of a second-order, under damped system with a complex, sinusoid input is presented by equation (9.39). The under-damped system has a pair of complex, conjugate poles, and the excitation function also has a complex pole. In this case, the pole of the excitation function, and one of the poles of the under damped system are coincident. We have to use a slightly deviant technique to get the solution.
Equation (9.40) shows the two poles of the under damped system, and we can express the two parts, as the product of two functions, and we can express the differential equation as shown by equation (9.41). In such a case, the particular solution due to the exponential function with the complex pole can be obtained as shown by equation (9.42). A similar technique is used in the instance of an undamped system with a sinusoidal excitation function, where the poles of the system and the poles of the excitation function are coincident. The same technique has been used in the case of an over-damped system, excited by an exponential input, with the pole of the input function coincident with one of the poles of the system.
Equation (9.43) shows function Q2(D). We can replace D by the complex value of ( - 1 + j), which is the value of the complex pole of the excitation function. We get Q2(-1 + j) to be an imaginary value, as shown by equation (9.44). In a case like this, where the pole of function, and the pole of the excitation function, are not coincident, the value of Q2 is an imaginary value, when it is evaluated, with D replaced by the pole of the exponential input function. Next we obtain the particular function. The value of function Q1, evaluated with D replaced by the complex value of ( - 1 + j), is equal to zero. Hence D in function Q1 is replaced by (D - 1 + j), as shown by equation (9.45). . This is the technique we follow, whenever the result, due to replacing D in function Q by the pole of the excitation function, is zero. Here function Q1 simplifies to D. Function yT(t) is obtained, as shown by equation (9.46). The imaginary part of yT(t) is the particular function, as shown by equation (9.47). To get the response due to complex sinusoid input, obtain the complementary solution and add it to the particular solution, as illustrated in the previous examples.
We can summarize as shown below.
Equations (9.49) and (9.50) describe the particular solution for the two cases that can occur. When the input is a cosine function, the solution is equally easy.
Another numerical example is presented now. A differential equation of a second-order, undamped system with a complex, sinusoid input is presented by equation (9.51). Equation (9.52) represents the differential equation differently. The value of function Q(-1+j) is evaluated, as shown by equation (9.53). Then the particular solution can be obtained, as shown by equation (617).
This chapter has described how the response of a first and a second order system can be obtained, when the system is excited by the commonly used signals. The next page describes how the response of a first and a second order system can be obtained, using the technique of Laplace Transforms.