Filters - transfer function

[nForce]

Hello,

I don't know how did we get here the transfer function. If someone can explain.

Thank you.

[KJDS]

How is your understanding of Laplace?

[bson]

The transfer function is Vin/Vout and s=jw.
The RC filter is a voltage divider with the two impedances, Z1=R and Z2=1/jwC
Vout = Vin*(1+Z1/Z2) = Vin*(1+R/(1/jwC)) = Vin * (1 + jwRC)
Vout/Vin = 1 + jwRC
A(s) = Vin/Vout = 1/(1+sRC)

[nForce]

bson, thank you.

Another question, where does here an extra Wc come in a denominator? It's only a passive filter with an addition of op amp which has a gain of 1+(R2/R3). This term is in numerator, which seems logical.

Thanks again.

[bson]

No idea, that's weird and may be specific to this material?  Notationally w_c might refer to the corner frequency of the dominant pole, but that would make no sense as used.   

[T3sl4co1l]

Looks like a typo.

Tim

[CatalinaWOW]

Looks like a typo.

Tim

Looks like a typo.

Tim

Typo being a short and polite way of saying someones fingers were typing faster than their brain was working.  Or maybe someone didn't really understand what they were typing.  The first is unfortunately common in books.  The latter is less common, but unfortunately, not rare.

The denominator should have been either (1+RCs) or (1+s/wc)  with wc=1/RC.  The cutoff frequency is a fairly common convention to make the equations more wieldy, expecially on much larger filters.

[IanB]

I don't know how did we get here the transfer function. If someone can explain.

The transfer function is Vin/Vout and s=jw.
The RC filter is a voltage divider with the two impedances, Z1=R and Z2=1/jwC
Vout = Vin*(1+Z1/Z2) = Vin*(1+R/(1/jwC)) = Vin * (1 + jwRC)
Vout/Vin = 1 + jwRC
A(s) = Vin/Vout = 1/(1+sRC)

Or from first principles, in the time domain:

The differential equation for the filter is:

  [Vin(t)-Vout(t)] / R = C dVout/dt

Multiply through by R:

  Vin(t) - Vout(t) = RC dVout/dt

Now convert to the Laplace domain, noting that dx(t)/dt becomes sx(s):

  Vin(s) - Vout(s) = RC s Vout(s)

Rearranging:

  Vin(s) = (1 + sRC) Vout(s)

The transfer function is Vout/Vin, so:

  A(s) = Vout(s) / Vin(s) = 1 / (1 + sRC)

For the EE's out there, how does the "magic" work that you can replace jw by s?

(OK, I'm sure I can look it up. But it's the first time I've come across it.)

[CatalinaWOW]

s is just shorthand for jw.  If you are writing transfer functions all day (people used to do that in the 1950s through the early 1970s) saving a character is huge.  All of these things are just conventions.  Those who get very involved with Fourier transforms will find three different conventions.  There is a scale factor of 2 pi between the forward and reverse transform.  Some apply that factor on the forward transform, some on the reverse.  And a small minority put the square root of the scale factor on both sides, making the transforms totally symetric.

[coppice]

s is just shorthand for jw.  If you are writing transfer functions all day (people used to do that in the 1950s through the early 1970s) saving a character is huge.

I thought it was just to stop all the arguments between the engineers and mathematicians about whether it should be iw or jw. 

[T3sl4co1l]

s is traditional from Laplace analysis, which is strictly different from Fourier analysis, but for rational, LTI systems, it works out that s == j*w.

And of course, we still can't write a proper omega on this forum, so 'w' has to fill in, in discussion.

Tim

[bson]

For the EE's out there, how does the "magic" work that you can replace jw by s?

(OK, I'm sure I can look it up. But it's the first time I've come across it.)

In the s plane frequency is on the imaginary axis, up/down.  Hence, j*w.  The real axis is sigma, which is the decay factor.  w is converted to a point on the s plane with sigma=0 by simply making it imaginary, jw.

sRC or perhaps more logically RCs, can then be read out as "a point along the frequency axis scaled by RC".

[bson]

s is traditional from Laplace analysis, which is strictly different from Fourier analysis, but for rational, LTI systems, it works out that s == j*w.

And of course, we still can't write a proper omega on this forum, so 'w' has to fill in, in discussion.

Or sigma.

Well, we can, and it will preview correctly, but at some point gets turned into '?'.  Perhaps by a mysql scrubber - mysql historically has had problems with multibyte characters where a text column is set to end with a partial mbc.  It can then 'swallow' the next character and becomes an injection attack vector.

But really shouldn't be so hard to fix.  Just needs a proper scrubber that knows how to parse out UTF-8 rather than just blindly hammer mbc's.