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The 1storder lag element (PT_{1} element)
The 1storder 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 1storder lag element

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

(4.31) 
For the general notation of a
element one obtains the
differential equation

(4.32) 
If the initial condition
is set to zero, on taking the
Laplace transform the
transfer function is

(4.33) 
and it follows that with
the frequency response is

(4.34) 
With the breakpoint
frequency
one obtains

(4.35) 
The amplitude response is

(4.36) 
and the phase response is

(4.37) 
The magnitude characteristic derived from Eq. (4.36) is

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

by
( initial asymptote) 

with
 b)

by
(final asymptote) 

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
and 

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
.
Next: The proportional plus derivative
Up: Some important transfer function
Previous: The derivative element (D
Contents
Christian Schmid 20050509