# transfer function

Started by September 4, 2007
```I am having difficulty calculating the inverse Laplace transform of
the following transfer function:

H(s) = (Gdc * w^2) / (s^2 + 2*eta*w*s + w^2)

where:
eta = damping ratio
Gdc = DC gain
w = undamped natural frequency

What steps do you use? I am having difficulty getting the transfer
function into a form that will allow me to take advantage of Laplace
Transform tables for easy inversion.

J

```
```On Sep 4, 7:48 pm, Jennifer Williams <jjenniferwwilli...@gmail.com>
wrote:
> I am having difficulty calculating the inverse Laplace transform of
> the following transfer function:
>
> H(s) = (Gdc * w^2) / (s^2 + 2*eta*w*s + w^2)
>
> where:
>   eta = damping ratio
>   Gdc = DC gain
>   w = undamped natural frequency
>
> What steps do you use? I am having difficulty getting the transfer
> function into a form that will allow me to take advantage of Laplace
> Transform tables for easy inversion.
>
> J

A method that will work, but is tedious, is good old partial fraction
expansion. Factor the denominator to find your poles, then do a
partial fraction expansion to find the coefficients that correspond to
the time-domain term associated with each pole. I think this method is
usually taught when Laplace transforms are examined.

Jason

```
```On Sep 4, 8:11 pm, cincy...@gmail.com wrote:
> On Sep 4, 7:48 pm, Jennifer Williams <jjenniferwwilli...@gmail.com>
> wrote:
>
> > I am having difficulty calculating the inverse Laplace transform of
> > the following transfer function:
>
> > H(s) = (Gdc * w^2) / (s^2 + 2*eta*w*s + w^2)
>
> > where:
> >   eta = damping ratio
> >   Gdc = DC gain
> >   w = undamped natural frequency
>
> > What steps do you use? I am having difficulty getting the transfer
> > function into a form that will allow me to take advantage of Laplace
> > Transform tables for easy inversion.
>
> > J
>
> A method that will work, but is tedious, is good old partial fraction
> expansion. Factor the denominator to find your poles, then do a
> partial fraction expansion to find the coefficients that correspond to
> the time-domain term associated with each pole. I think this method is
> usually taught when Laplace transforms are examined.
>
> Jason

Yes. For some worked out examples, see Wikipedia:

http://en.wikipedia.org/wiki/Laplace_transform#Table_of_selected_Laplace_transforms

John

```
```On Sep 5, 11:48 am, Jennifer Williams <jjenniferwwilli...@gmail.com>
wrote:
> I am having difficulty calculating the inverse Laplace transform of
> the following transfer function:
>
> H(s) = (Gdc * w^2) / (s^2 + 2*eta*w*s + w^2)
>
> where:
>   eta = damping ratio
>   Gdc = DC gain
>   w = undamped natural frequency
>
> What steps do you use? I am having difficulty getting the transfer
> function into a form that will allow me to take advantage of Laplace
> Transform tables for easy inversion.
>
> J

A pic would be nice Jenny. We don't get many females on this NG of
nerd do wells.

```
```On Tue, 04 Sep 2007 23:48:43 +0000, Jennifer Williams wrote:

> I am having difficulty calculating the inverse Laplace transform of
> the following transfer function:
>
> H(s) = (Gdc * w^2) / (s^2 + 2*eta*w*s + w^2)
>
> where:
>   eta = damping ratio
>   Gdc = DC gain
>   w = undamped natural frequency
>
> What steps do you use? I am having difficulty getting the transfer
> function into a form that will allow me to take advantage of Laplace
> Transform tables for easy inversion.
>
> J

If eta < 1 then that's the Laplace transform of a damped sinusoid.  Look
in your tables for the two transforms of damped sinusoids (there'll be one
for e^(this * t) * cos (that * t), and another one for e^(this * t) *
sin(that * t)).  Find the values of this and that to make it match your
expression, then find the linear combination of the damped sin and the
damped cosine.

You'll spend some time with a machete hacking through the underbrush in
mathemagic land, but you'll find an answer if you're diligent.  For
hacking through _real_ underbrush with a real machete, you should keep a
whetstone in your pocket.  For this sort of hacking I suggest beer...

--
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html
```
```gay.highlander@yahoo.co.uk wrote:
> A pic would be nice Jenny. We don't get many females on this NG of
> nerd do wells.
>

...and now you know why.

--
Jim Thomas            Principal Applications Engineer  Bittware, Inc
jthomas@bittware.com  http://www.bittware.com    (603) 226-0404 x536
People think it must be fun to be a super genius, but they don't realize
how hard it is to put up with all the idiots in the world - Calvin
```
```On Sep 4, 9:49 pm, Tim Wescott <t...@seemywebsite.com> wrote:
> On Tue, 04 Sep 2007 23:48:43 +0000, Jennifer Williams wrote:
> > I am having difficulty calculating the inverse Laplace transform of
> > the followingtransferfunction:
>
> > H(s) = (Gdc * w^2) / (s^2 + 2*eta*w*s + w^2)
>
> > where:
> >   eta = damping ratio
> >   Gdc = DC gain
> >   w = undamped natural frequency
>
> > What steps do you use? I am having difficulty getting thetransfer
> >functioninto a form that will allow me to take advantage of Laplace
> > Transform tables for easy inversion.
>
> > J
>
> If eta < 1 then that's the Laplace transform of a damped sinusoid.  Look
> in your tables for the two transforms of damped sinusoids (there'll be one
> for e^(this * t) * cos (that * t), and another one for e^(this * t) *
> sin(that * t)).  Find the values of this and that to make it match your
> expression, then find the linear combination of the damped sin and the
> damped cosine.
Thanks for the responses. I now understand.

As for a pic...

<img src="http://www.swarthmore.edu/NatSci/echeeve1/Class/e12/Lecture/

>
> You'll spend some time with a machete hacking through the underbrush in
> mathemagic land, but you'll find an answer if you're diligent.  For
> hacking through _real_ underbrush with a real machete, you should keep a
> whetstone in your pocket.  For this sort of hacking I suggest beer...
>
> --
> Tim Wescott
> Control systems and communications consultinghttp://www.wescottdesign.com
>
> Need to learn how to apply control theory in your embedded system?
> "Applied Control Theory for Embedded Systems" by Tim Wescott
> Elsevier/Newnes,http://www.wescottdesign.com/actfes/actfes.html

```
```Jennifer Williams wrote:

...