Electronics > Beginners

RC time constant

<< < (7/8) > >>

The Electrician:

--- Quote from: Dave on August 22, 2019, 10:34:32 am ---If you do need an analytic solution, the URL I provided above has everything you need.

--- End quote ---
It doesn't have everything; an analytic solution is provided for the frequency response but not for the time response.

An expression for the time response of the low pass topology shown here: http://sim.okawa-denshi.jp/en/CRCRkeisan.htm
can be derived with the help of Mathematica:



The numerical values shown on this page: http://sim.okawa-denshi.jp/en/CRCRtool.php

can be used with the derived expression to produce a plot of the time response of the circuit shown:



A time response plot with the two sections reversed (R2/C2 first, R1/C1 second) is as shown:

Dave:

--- Quote from: The Electrician on August 26, 2019, 07:30:34 am ---It doesn't have everything; an analytic solution is provided for the frequency response but not for the time response.

--- End quote ---
A boatload of exponents doesn't seem mighty helpful, IMO.
Manually doing inverse Laplace transforms only makes sense when expressions are simple enough to even make sense doing it (learning mathematics, for example). For anything more complex you'd use a computer anyhow.

The Electrician:

--- Quote from: Dave on August 27, 2019, 12:40:15 pm ---
--- Quote from: The Electrician on August 26, 2019, 07:30:34 am ---It doesn't have everything; an analytic solution is provided for the frequency response but not for the time response.

--- End quote ---
A boatload of exponents doesn't seem mighty helpful, IMO.
--- End quote ---

I suppose that if one isn't interested in an analytical solution for the time response, exponents aren't helpful.  But, if an analytical solution is of interest, exponents are inevitable.


--- Quote from: Dave on August 27, 2019, 12:40:15 pm ---Manually doing inverse Laplace transforms only makes sense when expressions are simple enough to even make sense doing it (learning mathematics, for example). For anything more complex you'd use a computer anyhow.

--- End quote ---

I did use a computer to obtain the expression shown.  You don't think I came up with all those exponents by hand, do you?

Wimberleytech:
Pick the node of interest and use Elmore's approximation.

The Electrician:

--- Quote from: Wimberleytech on August 28, 2019, 01:49:20 am ---Pick the node of interest and use Elmore's approximation.

--- End quote ---

Before software packages such as Mathematica or Maple were available, the calculations involved in obtaining an Elmore approximation were less than those for finding the exact time response:

https://ieeexplore.ieee.org/document/545648 (This is a 22 year old paper.  Mathematical software has become much more powerful in the intervening years.)

and

https://en.wikipedia.org/wiki/Elmore_delay  (moments or Pade approxmations are needed)

But now with the availability of software like Mathematica with built-in Inverse Laplace transforms, and fast computers to go along with the software, it's quicker to get the analytical expression for the time response than to calculate the Elmore delay.

Here is the calculation and plot of the time response of the twin tee filter shown here: http://sim.okawa-denshi.jp/en/TwinTCRtool.php

It took Mathematica about 1 second to find the Inverse Laplace transform.  The full transform expression isn't shown because it's several pages long.

Navigation

[0] Message Index

[#] Next page

[*] Previous page

There was an error while thanking
Thanking...
Go to full version
Powered by SMFPacks Advanced Attachments Uploader Mod