# Laplace Function for nth order LC Bandpass Filter

#### Guenter

Joined May 24, 2021
5
Hi Guys,

First - I'm a hobbiest not an engineer. I never studied electronics or had an professional education in electronis.

So for my hamradio Superhet Project, I try to speed up some simulations.
For this request, I found this Laplace Funktion for a 2nd order Bandpass filter:

Laplace=(w/Q * s) / (s**2 + w/Q*s + w**2) * (w/Q * s) / (s**2 + w/Q*s + w**2)

So I need to simulate LSB/USB (SSB) Filters and a 2nd Order is not sharp enough.
Using XTAL-Filters in my case are not flexible with the frequency in my "try and errer" phase.

So, I there some guy who can post some Laplace Funktions for 4th, and 9th order in above style (with w and Q in it).

For myself I tried to understand how can I do it myself. But I can't because I do not understand the mathematics behind.

THX

Guenter

#### MisterBill2

Joined Jan 23, 2018
14,710
Hi Guys,

First - I'm a hobbiest not an engineer. I never studied electronics or had an professional education in electronis.

So for my hamradio Superhet Project, I try to speed up some simulations.
For this request, I found this Laplace Funktion for a 2nd order Bandpass filter:

Laplace=(w/Q * s) / (s**2 + w/Q*s + w**2) * (w/Q * s) / (s**2 + w/Q*s + w**2)

So I need to simulate LSB/USB (SSB) Filters and a 2nd Order is not sharp enough.
Using XTAL-Filters in my case are not flexible with the frequency in my "try and errer" phase.

So, I there some guy who can post some Laplace Funktions for 4th, and 9th order in above style (with w and Q in it).

For myself I tried to understand how can I do it myself. But I can't because I do not understand the mathematics behind.

THX

Guenter
I did go through engineering school at a well respected college, graduated in 1973 and have been an EE ever since.
I do recall studying the Laplace Series functions back in college in 1971, and it seems that we found that they were indeed rather powerful. That was in the upper level engineering math calculus class, I think.
I have not had to use Laplace since then in my career, and so I am out of practice. BUT I can offer a suggestion, which is a textbook in the reference section of your local library, or possibly in a used book store near a college campus. That part of math does not change and so an older textbook will be totally adequate if it contains that subject.

#### LvW

Joined Jun 13, 2013
1,646
Guenther - You speak about "LaPlace functions", but what you need (what you really are interested on) are the transfer functions of filter circuits in the frequency domain. So - when you try to do some search, use the keyword "filter transfer functions" for 2nd or 4th order bandpass filters.
These transfer functions are nothing else than "LaPlace"-transformed differential equations in the time domain.
As far as bandpass filters are concerned, you carefully should discriminate between "sharpness" (depends on the Q-value only for a second order bandpass) and "bandwidth" (usually expressed in 3dB points).
In order to see what you really need, you should specify the midfrequency of the bandpass as well as your damping requirenments (Example: "At least 25 dB 100 Hz below and above the midfrequency").

#### Papabravo

Joined Feb 24, 2006
20,151
I recommend picking up a used version of:
Van Valkenburg, M.E., Analog Filter Design. (1982)
It will go through a development that is essential to your progress: How to write a proper filter specification.
From that specification you can pick the proper filter topology AND order to meet that specification.
AFAIK, there is nothing in the literature that refers to a "Laplace" filter.
All of the following filters have transfer functions that can be expressed with Laplace Transforms, with methods for extracting parameter values from the corner frequency and the 'Q':

Butterworth Filter - Maximally flat in the passband
Chebyshev Filter - Steep attenuation, at the expense of controlled ripple in the passband and the stop band
Bessel Filter - Constant Group Delay in the passband, at the expense of a shallower rolloff.

Those three will keep you off the streets and on your toes for the time being. There are other more exotic types and topologies but those will have to wait for the time being.