The 2nd-order lag element (PT₂ element and PT₂ S element)

A 2nd-order lag element is characterised by two independent energy storages. Depending on the damping properties and the position of the poles of , respectively, one distinguishes between oscillating and aperiodic behaviour. If a 2nd-order lag element has a conjugate complex pair of poles, then it shows an oscillating behaviour ( behaviour). If both poles are on the negative real axis, then the element has a lag behaviour ( behaviour).

The network of Figure 4.16 is an example of such an element. From the equation of the mesh

with

the differential equation becomes

Interactive Questions 4.4   Test using other example

The transfer function is

Figure 4.16: Simple network as an example of a 2nd-order lag element

For the 2nd-order lag element the general notation of the transfer function

is chosen. Introducing terms which characterise the time behaviour, that is the damping ratio

and the natural frequency (frequency of the undamped oscillation)

one obtains from Eq. (4.51)

For the frequency response

results. The amplitude response is

and the phase response

Here the ambivalence of the function has to be observed. For the magnitude characteristic one has from Eq. (4.56)

The progression of can be approximated by the following asymptotes:

a)

For by

with

b)

For by

with

In the Bode diagram the final asymptote is a line with a slope of -40 /decade. The point of intersection of both asymptotes follows from

to be the normalised frequency . The exact value of may deviate considerably from the point of intersection at , because it is

according to Eq. (4.56). For this value is above, for below the asymptotes.

Figure 4.17 shows for and the magnitude and phase responses in a Bode diagram. This graphical representation contains the cases of and behaviour, which will be discussed in section A.3.2. From Figure 4.17 it can be seen that a maximum magnitude exists for some values of the damping ratio . This maximum occurs at the so called resonant peak frequency . A detailed analysis of this resonance can be found in section A.3.1.

Figure 4.17: Bode diagram of a 2nd-order lag element with the transfer function

Figure 4.18 shows the Nyquist plots of 2nd-order lag elements with high and low damping. From Eq. (4.55) it follows that for the real part of is zero. Therefore the locus of the frequency response intersects with the imaginary axis at independent of the values of , as can be seen in Figure 4.18.

Figure 4.18: Nyquist plots of 2nd-order lag elements, (a) element, (b) element

The modes of a dynamical system are determined according to Eq. (3.12) by the roots of the characteristic equation or by the poles of the transfer function, respectively. From the characteristic equation of the 2nd-order lag element

one obtains the poles of the transfer function as

The oscillating behaviour of a 2nd-order lag element is dependent on the position of the poles in the plane. For a graphical analysis the step response is a useful tool. Table 4.2 shows step responses and the corresponding poles for different values of . These five cases will be analysed and discussed in more detail in section A.3.2.

Table 4.2: Pole positions in the plane and step responses for elements with the transfer function