Models
- Transfer Functions
What
is a Transfer Function?
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What
is a Transfer Function?
Given a linear system, then the transfer function, G(s), of the system
is the ratio of the transform of the output to the transform of the input.
Example
1
Examine the diagram below.
In this system, we
have the following:
-
The system is linear.
(Actually, it is a first order system. More later.)
-
The input is u(t), and
the transform of the input is U(s).
-
The output is y(t), and
the transform of the output is Y(s).
Then, the transfer function
is given by:
How
Do You Compute a Transfer Function?
If you have a differential equation relating input and output, you can
easily compute the transfer function by transforming both sides of the
differential equation and isolating the input transform and the output
transform. Let's look at an example.
Example 1
Let's assume that we have a system described by a first order linear differential
equation. Here is the differential equation.
In this differential equation,
we have:
-
y(t) = Response
of the System,
-
u(t) = Input
to the System,
-
t
= The System Time Constant,
-
Gdc
= The DC Gain of the System.
Now, if we transform both
sides of this differential equation, we get:
Rearrange terms to isolate the input and output transforms
and we have;
-
(ts
+ 1)Y(s) = GdcU(s)
-
Y(s) = GdcU(s)/(ts
+ 1)
-
G(s) = Y(s)/U(s) = Gdc/(ts
+ 1)
By the way, the result
is the generic form for a transfer function of a first order, linear system.
That's
all there is to it. Transform both sides of the differential equation
- even if it is much higher order. Then, solve for the transfer function.
Problems
Links
To Related Lessons
Lessons on Getting Models
From Data
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