The 1st-order lag element (PT₁ element)

The 1st-order lag element or in short element is an element with an output signal that for a step input has a certain initial slope and approaches asymptotically the final value. An example of such an element is the simple RC lag circuit shown in Figure 4.9. When at time a voltage is applied, the voltage at the output will approach exponentially with the time constant the final value .

Figure 4.9: Simple RC lag as an example of a 1st-order lag element

The circuit involves a single energy storage element, the capacitor . The differential equation of this RC lag is

For the general notation of a element one obtains the differential equation

If the initial condition is set to zero, on taking the Laplace transform the transfer function is

and it follows that with the frequency response is

With the breakpoint frequency one obtains

The amplitude response is

and the phase response is

The magnitude characteristic derived from Eq. (4.36) is

Eq. (4.38) can be asymptotically approximated by lines for:

a)

by

with

b)

by

with

In the Bode diagram can be consequently approximated by two lines. The progression of the initial asymptote is horizontal, whereas the final asymptote shows a slope of -20 /decade. The intersection of both lines (breakpoint) can be determined from

and provides the frequency

Therefore is called the breakpoint frequency. As can be easily seen from Figure 4.10a,

Figure 4.10: (a) Magnitude and phase response (b) Nyquist plot of the frequency response of a element

the deviation between and the asymptotes has a maximum at the breakpoint for . The exact values are

The deviation of the magnitude characteristic from the asymptotes for is

The deviations at the other frequencies are symmetrical from the breakpoint on the logarithmic scale, as can be seen directly from Table 4.1. This is the reason why the magnitude and phase characteristics can be easily constructed on a Bode diagram. The phase curve is approximately zero up to one tenth of the breakpoint frequency, and beyond 10 time the breakpoint frequency. In between, it is approximately a straight line with slope per decade through at the breakpoint frequency.

Table 4.1: Magnitude, phase response and deviation of the exact magnitude from the asymptotes for a element with

As already shown in section A.2 the locus of the frequency response of a element is a semicircle, which starts for at on the real axis and stops for at the origin, as shown in Figure 4.10b.

The constant in the transfer function and on the frequency response, respectively, is usually called the time constant of the element. It can be determined also by the point of intersection of the line with the initial slope and the horizontal line of the final asymptote, , of the step response as shown in Figure 4.11.

Figure 4.11: Graphical representation of the step response, , of a element

This time constant can also be physically interpreted. It is the time when the step response has reached approx. 63% of the final value, . is - similar to the P element - called the gain of the element. It is defined as the value of the frequency response at .

DYNAST study example 4.1   Simple automobile model

DYNAST study example 4.2   D.C. motor - open loop