The derivative element (D element)

The relationship between the input and output of a derivative element is described by

This element differentiates the input signal and therefore is called a derivative element, or in short D element. The associated transfer function is

and with it follows that the frequency response

from which the magnitude

and the phase response

follow. It can easily be seen, that the I and D elements are related by an inversion. Therefore the curves of the magnitude and phase response of the D element can be found - as shown above - by mirroring those of the I element at the 0- line and line, respectively. This is obvious from Eqs. (4.29) and (4.30). Figure 4.8 shows the Bode diagram and the Nyquist plot of the frequency response of the D element. The slope of the line is +20 /decade and the phase response is independent of the frequency.

Figure 4.8: (a) Magnitude and phase response (b) Nyquist plot of the frequency response of a D element

The D element discussed here is - as already mentioned in section 3.3 - an idealisation and therefore not a physically realisable element. For practical applications the D element will be approximated by the element (see section 4.4.6).