# Derivative

```block Derivative "Approximated derivative block"
import Modelica.Blocks.Types.Init;

parameter Real k(unit = "1") = 1 "Gains";
parameter SIunits.Time T(min = Modelica.Constants.small) = 0.01 "Time constants (T>0 required; T=0 is ideal derivative block)";
parameter Modelica.Blocks.Types.Init initType = Modelica.Blocks.Types.Init.NoInit "Type of initialization (1: no init, 2: steady state, 3: initial state, 4: initial output)"
annotation (
Evaluate = true,
Dialog(group = "Initialization"));
parameter Real x_start = 0 "Initial or guess value of state"
annotation (Dialog(group = "Initialization"));
parameter Real y_start = 0 "Initial value of output (= state)"
annotation (Dialog(
enable = initType == Init.InitialOutput,
group = "Initialization"));

extends Interfaces.SISO;

output Real x(start = x_start) "State of block";
protected
parameter Boolean zeroGain = abs(k) < Modelica.Constants.eps;
initial equation
der(x) = 0;
elseif initType == Init.InitialState then
x = x_start;
elseif initType == Init.InitialOutput then
if zeroGain then
x = u;
else
y = y_start;
end if;
end if;
equation
y = if zeroGain then 0 else k / T * (u - x);
der(x) = if zeroGain then 0 else (u - x) / T;

annotation (
Documentation(info = "<html>\n<p>\nThis blocks defines the transfer function between the\ninput u and the output y\nas <em>approximated derivative</em>:\n</p>\n<pre>\n             k * s\n     y = ------------ * u\n            T * s + 1\n</pre>\n<p>\nIf you would like to be able to change easily between different\ntransfer functions (FirstOrder, SecondOrder, ... ) by changing\nparameters, use the general block <strong>TransferFunction</strong> instead\nand model a derivative block with parameters<br>\nb = {k,0}, a = {T, 1}.\n</p>\n\n<p>\nIf k=0, the block reduces to y=0.\n</p>\n</html>"),
Icon(
coordinateSystem(
preserveAspectRatio = true,
extent = {
{-100, -100},
{100, 100}}),
graphics = {
Line(
points = {
{-80, 78},
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color = {192, 192, 192}),
Polygon(
lineColor = {192, 192, 192},
fillColor = {192, 192, 192},
fillPattern = FillPattern.Solid,
points = {
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points = {
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color = {192, 192, 192}),
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lineColor = {192, 192, 192},
fillColor = {192, 192, 192},
fillPattern = FillPattern.Solid,
points = {
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Line(
origin = {-24.667, -27.333},
points = {
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color = {0, 0, 127},
smooth = Smooth.Bezier),
Text(
lineColor = {192, 192, 192},
extent = {
{-30, 14},
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textString = "DT1"),
Text(
extent = {
{-150, -150},
{150, -110}},
textString = "k=%k")}),
Diagram(
coordinateSystem(
preserveAspectRatio = true,
extent = {
{-100, -100},
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graphics = {
Text(
extent = {
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textString = "k s"),
Text(
extent = {
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textString = "T s + 1"),
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Line(
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end Derivative;

```