Parameters of the PT1 Element

Parameters of the PT1 element Classification of the PT 1 element  A time delay delay element of the 1st order is called a PT1 -element. In this context the relationship between the input variable y(t) and the output variable x(t) can be expressed by the differential euation

The parameter !P is referred to as the proportional the proportional coefficient " the parameter T is called the time constant  of  of the PT1 element. The followin# $i#ure shows the step response and the bloc% s ymbol of the PT 1 element.

&ere the final steady-state value of the output variable is assumed to be only asymptotic" i.e. time delayed. The time constant T specifies how fast the output variable tends towards the final value. In mathematical terms the followin# euation expresses the characteristic of the output variable for t ' 

 A PT1 element is thus a system with compensation and time delay  (see  (see the followin# $i#ure).

Determining the time constants on the basis of the step response hereas the proportional coefficient !P of the PT1 element for an input variable step chan#e of the hei#ht 1 can be read directly off the step response (as it corresponds to the final steady-state value of the output variable)" findin# out the time constant T is somewhat more complicated. It can be achieved in two different ways.

Determining the value of T using the tangent method The so-called tan#ent method uses the application of tan#ents on the step response to determine the time constant T. The point where the tan#ent intersects with the final steady-state value of the output variable and then drop a perpendicular line down to the time axis. The resultin# se#ment of the time axis corresponds to the time constant.

$i#. on the left* +eterminin# the time constant T accordin# to the tan#ent method. The tan#ent is drawn as a red line to the step response from the time point t , .

Determining the value of T according to the 63% method The so-called 63% method  is based on the fact that the time correspondin# to the time constant T has elapsed when / of the final value has been reached. This can

be derived directly from the euation #iven above by insertin# the value T for the time variable t. e thus obtain the followin# for the output variable

The followin# #raph illustrates how the time constant can be derived directly from the step response by this method.

$i#. left* +eterminin# the time constant T accordin# to the / method. This method #ives relatively #ood results even when the si#nals are distorted.

Example for a PT1 element PT1 behavior is evident wherever there is a system with precisely one ener#y stora#e element. The $i#ure below shows a mechanical system comprisin# a mass m (ener#y stora#e element) and a shoc% absorber r" whose frictional force is assumed to be proportional to the velocity. $urthermore an external force $ acts on the mass. If you ta%e the sum of the forces" you arrive at the followin# expression for the motion

 As can be seen from a comparison to the differential euation of the #eneral PT1 element shown above" this mechanical system demonstrates PT1 characteristics.

$i#. ri#ht* 0hoc% absorber system for a mass as an example of a mechanical PT 1 element.

hat are the systems proportional coefficient and time constant2 3nter your answer into the followin# answer box.

Experiment In the followin# experiment you shall determine the step response of the two PT1 elements of the P element of the experiment card 4controlled system simulation4 (05671-89). 9se the step response to determine the respective proportional coefficient !P and the time constant T. $irst set up the followin# experiment circuit.

 Activate the step response plotter and confi#ure it as shown in the followin# Table.

Settings nput :hannel A

;eas. ran#e* 1 <

:ouplin#* +:

:hannel =

;eas. ran#e* 1 <

:ouplin#* +:

>an#e* 1

5ffset* 

5ther

Settings !utput 0tep response from ... to ...



8/

+elay time?ms



;easurements

 Settings Diagram

+isplay

:hannel A

x-axis from ... to ...



.7 s

y-axis from ... to ...



1

@ow determine the step response of the left-hand PT1-element and copy the dia#ram into the upper space reserved for the #raph. +etermine the proportional coefficient and time constant in accordance with both the tan#ent and the / method. Then repeat the experiment with the ri#ht-hand PT 1 element" copy the step response into the lower space reserved for #raphs and determine from this the proportional coefficient and time constants. 3nter the numerical values obtained for the parameters in the answer box below. 1

     /

x1*  y 1*

C

.16 x7* .61

B

y 7* 8.1 dx*



.61 dy * 6C.C



dy?dx* 1.1eD

8

6



7

1



 t? s

0tep response of the left-hand PT 1 element

1

     /

x1*  y 1*

C

 x7* .1C6

B

y 7* 6B.C dx*



.1C6 dy * 6B.C



dy?dx* 787

8

6



7

1



 t? s

0tep response of the ri#ht-hand PT 1 element

Proportional coefficients and time constants determined* Feft-hand (!p, 8/ t , .6 s) >i#ht-hand (!p , 6B.C/ t , .1C

@ow repeat the experiment usin# the ri#ht-hand PT 1 element" but for a different amplitude of the input variable step chan#e (alter the step chan#e from  to 78/). +ra# and drop the step response into the space reserved for the #raphic below and use this to also determine proportional coefficient and time constant. +o the parameters chan#e because the hei#ht of the step response chan#es2 3nter the your answer with your reasons into the answer box belowE

1

     /

x1*  y 1*

C

 x7* .1

B

y 7* 76.8 dx*



.1 dy * 76.8



dy?dx* 181

8

6



7

1



 t? s

0tep response of the ri#ht-hand PT 1 element for a chan#e in the hei#ht of the input variable step

 Answer* 6B.C/ - 76.8/ , 76.6/