Applied the Laplace Transformation step function on the Tank Filling System .
Input Functions
The formula of step function is:
The formula of time delay is:
Derivative
Example
The equation means f(t) has value of 0 when t < 3 and 1 when t > 3.
f(t) = u(t-3)
2- Pulse function (Rectangular Pulse)
The formula of rectangular pulse is:
Assume the constants a, b, with a < b and A are positive.
We write the function using the rectangular pulse formula.
Example:
3- Impulse function
4- Ramp function
The formula of Ramp function is:
f(t) = At
The formula of Ramp function with Ramp function with time delay is:
f(t) = At.(t-a)
Example:
f(t) = sin t . u(t-2?)
The sin(t) starts at t = 2?, because we have multiplied sin(t) by u(t ? 2?).
Step response for first-order system
The basis for the definition of ? given above is the simplest case with one linear differential equation (first-order system). Here, we study this system in more detail. A first-order system can be written in the following standard form:
where
• u is the independent variable (input)
• y is the dependent variable (output)
• ? is the time constant
• k is the gain
The time constant ? characterizes the system’s dominant “inertia” against changes. It is depends on the system.
EX: for level tank:
It is reach to 63% of the steady state of a system output.
when t= ?
A fraction 1 – e ?1 = 1 ? 0.3679 ? 0.63
As proven below, the solution (“step response”) can then be written as:
when t = ? ; we have:
when t ? ? we have e?t/? ? 0 and the system approaches a new steady state where
?y(t) = ?y(?). Note that the exponential term (1 ? e?t/?) describes how fast the system approaches its new steady state, and as a function of the non-dimensional time t/? we have:
The time response is plotted in above Figure. We note that at time t = ? (the time constant), we have reached 63% of the total change, and after four time constants, we have reached 98% of the change (and we have for all practical purposes arrived at the new steady state).
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .
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