The derivative lag element (DT₁ element)

This element has a step response which initially contains a step and then decreases exponential to zero with a characteristic time constant as shown in Figure 4.13.

Figure 4.13: Graphical representation of the step response, , of a element

Figure 4.14: Simple high-pass circuit as an example of a element

Figure 4.14 shows an example of such a system; a simple high pass filter. The differential equation of this circuit is

which can be written as

Applying the Laplace transform one obtains the transfer function

The generalised notation of the element is

For constructing the Bode plot one starts with the frequency response

which on substituting gives

From Eq. (4.44) it follows that

and

After some calculations the phase response can be shown to be

Comparing Eq. (4.45) with Eqs. (4.38) and (4.41) shows that the magnitude characteristic of the element can be obtained by adding the corresponding curves of a element and a D element. The same also holds for the phase response . With this information the curves of the frequency response characteristics and of the Nyquist diagram can be simply constructed according to Figure 4.15.

Figure 4.15: (a) Magnitude and phase responses (b) Nyquist plot of the frequency response of a element