The second order lagging element

The general form of a differential equation for this element is

. (3.42)

then the operational equation is

. (3.43)

We can rewrite Eqs. (3.42) and (3.43) as follows

, (3.44)

, (3.45)

where T=T₂,

is the damping ratio.

The characteristic equation is given by

. (3.46)

The roots of the preceding equation are

. (3.47)

Three cases should be considered:

n two complex conjugate roots,

n two imaginary roots,

n two different real roots.

Oscillating element. For the roots are complex conjugate. Solving Eq.(3.45) for the unit step input, x(t)=1(t), when all the initial conditions are zero, we obtained the equation for the transient response to a step unit input as follows

(3.48)

where is the natural frequency of the oscillation, sec^-1,

is the period of the natural oscillation, sec^-1,

In general, the transfer function of the oscillating element is written as

. (3.49)

The transfer function in the frequency domain is represented by

, (3.50)

and its real and imaginary parts are expressed as follows

, .(3.51)

the frequency response of the oscillating element is described by the following equations

, (3.52)

(3.53)

(3.54)

The transient response to a step unit input and the log-magnitude and phase curves for the oscillating element are shown in Fig.3.15(a) and Fig.3.15(b) respectively.

Fig. 3.15. Transient response to a step unit input (a) and log-magnitude and phase diagram (b) for the oscillating element.

The block diagrams for the oscillating element are shown in Fig. 3.16.

Fig.3.16. Block diagram representation of the oscillation element.

Conservative element. For , roots of characteristic equation are imaginary, then Eq. (3.42) is written as follows

. (3.55)

The transfer function for this case is given by

. (3.56)

A pair of complex conjugate roots on the imaginary axis yields a sinusoid with a constant amplitude. Then the time response to a step unit input is written as

, (3.57)

where is the natural frequency of the oscillation, sec^-1,

is the period of oscillation, sec^-1.

The transfer function in the frequency domain is

. (3.58)

The frequency response is described by the following equations

, (3.59)

. (3.60)

(3.61)

The transient response for this case is shown in Fig. 3.17(a). Graphs of log-magnitude and phase versus are shown in Fig. 3.17(b).

Fig. 3.17. Transient response to a step unit input (a) and log-magnitude and phase diagram (b) for the conservative element.

Thesecond order lag element. For roots of the characteristic equation Eq.(3.46) are real. Expanding the characteristic function in a partial fraction gives the transfer function in the form

, (3.62)

where

, if the differential equation is represented by Eq. (3.42)

or

, if the differential equation is represented by Eq.(3.44)

The second order lag element can be represented as a cascade of two first order lag elements. The time response has the form

The frequency response is expressed by following equations

, (3.63)

, (3.64)

, (3.65)

. (3.66)

The transient response for this case is shown in Fig. 3.18(a), where is the time delay, is the increasing time.

The log-magnitude and phase angle diagram are drawn in Fig.3.18(b). The block diagram for the second order lag element is shown in Fig. 3.19.

Fig. 3.18. Transient response to a step unit input (a) and log-magnitude and phase diagram (b) for the second order lag element.

Fig. 3.19. Block diagram of the second order lag element.

Integrating Elements