Designing Frequency-Dependent Impedances?

[Diego Stutzer]

Hi,
Every one knows, that e.g. a simple RC-parallel circuit has a
frequency-dependent impedance-characteristic (Absolute Value) - the
impedance (Abs) raises as the Frequency approaches zero. As a formula: Zin
= 1/(1/R + i w C) , where i ist the imaginary number and w the frequency.

Now the hard part. How does one create an Impedance, which decreases
"slower", for frequencies close to zero but then decreases "faster" for
higher frequencies, than the simple parallel RC-Circuit?
Is there some kind of procedure like the one for syntesizeing LC-Filters
(Butterworth, Chebychev,..)?

Simply increasing C does not really help, because this equals a factoring of
the frequency.
Increasing R does not help as well, as it seems.


I hope one of you cracks can help me out.
So far, thanks for reading.
Diego Stutzer

 

[Udo Piechottka]

Diego Stutzer schrieb:

Simply increasing C does not really help, because this equals a factoring of
the frequency.
Increasing R does not help as well, as it seems.

It depends on the slope you want to realize. You could use a combination of Rs
and Cs to build a more complex frequency response or an active filter. There are
many ways.
The best would be if you described the task and the characteristics of the
filter you're looking for...

- Udo

 

[Bob Masta]

On 19 Feb 2004 02:29:02 -0800, abcdstutzer@evard.ch (Diego Stutzer)
wrote:

Hi,
Every one knows, that e.g. a simple RC-parallel circuit has a
frequency-dependent impedance-characteristic (Absolute Value) - the
impedance (Abs) raises as the Frequency approaches zero. As a formula: Zin
= 1/(1/R + i w C) , where i ist the imaginary number and w the frequency.

Now the hard part. How does one create an Impedance, which decreases
"slower", for frequencies close to zero but then decreases "faster" for
higher frequencies, than the simple parallel RC-Circuit?
Is there some kind of procedure like the one for syntesizeing LC-Filters
(Butterworth, Chebychev,..)?

Simply increasing C does not really help, because this equals a factoring of
the frequency.
Increasing R does not help as well, as it seems.


I hope one of you cracks can help me out.
So far, thanks for reading.
Diego Stutzer

I've never looked into changing the slope of impedances,
but depending on what you are doing you might be
interested in how to control the slope of filter
transfer functions to get something other than 6*N
dB/octave. A common requirement is to get 3 dB/octave,
which is done by cascading normal low-pass and high-pass
stages in a piecewise-linear approximation. The slope
starts down at -6 dB/oct then it hits a +6 stage which
flattens it out, then another -6 stage, etc. You can get
closer to the desired slope by having more stages with
closer spacing. If you are going to build a one-off version,
you might want to use buffered stages so they don't
interact and you can design your network from the Bode
plots. If you are in mass production, you may want to
skip the buffers and spend a lot more time dealing
with the interactions. Without buffers, you probably won't
be able to get a very wide frequency range, but I have
seen this approach used for generating pink noise
from white noise over a useful portion of the audio
range.

Hope this helps!


Bob Masta
dqatechATdaqartaDOTcom

D A Q A R T A
Data AcQuisition And Real-Time Analysis
www.daqarta.com

 

[Cliff Curry]

Diego: if you want to make an impedance out of regular parts, I'm afraid you
don't have too much to work with. Impedances made of R, L , and C's have
certain rerqirements: for example, the numberof poles and zeros cannot
differ by more than 1.

If you can use active parts, though, you can get all sorts of impedances.
You have to be careful with this, though, because these impedances likely
will oscillate unless you put them in a circuit that dampens out their
activity.

What did you have in mind?

If you want a big impedance in some frequency ranges, then a small impedance
in others, one way to do that is to use a passive filter design. For
example, the input of a bandpass filter looks a lot like the terminating
resistance in the passband, and is much higher or lower out of the passband.
So, the input impedance of a high pass filter made with several L's and C's
will be large at low frequencies, and resistive (and smaller), at high
frequencies.

Cliff

"Diego Stutzer" <abcdstutzer@evard.ch> wrote in message
news:aca3ec4c.0402190229.7977e8c6@posting.google.com...

Now the hard part. How does one create an Impedance, which decreases
"slower", for frequencies close to zero but then decreases "faster" for
higher frequencies, than the simple parallel RC-Circuit?
Is there some kind of procedure like the one for syntesizeing LC-Filters
(Butterworth, Chebychev,..)?

Diego Stutzer

 

[Roy Lewallen]

I think you'll find that the impedance looking into a properly
terminated highpass filter has this characteristic, provided that the
filter is realized with a series C as the input component.

LC filters accomplish their attenuation by means of mismatch to a fixed
source impedance, since they don't have any (intentionally) dissipative
elements. This mismatch can go in either direction, that is, the
impedance can get either higher or lower with frequency, depending on
how you've chosen to realize the filter. One starting with a series C
will have an infinite impedance at DC, and will approach the impedance
of the load at frequencies substantially higher than the transition
("break") frequency. You can get increasingly fast impedance changes
with increasing filter order.

Roy Lewallen, W7EL

Diego Stutzer wrote:

Hi,
Every one knows, that e.g. a simple RC-parallel circuit has a
frequency-dependent impedance-characteristic (Absolute Value) - the
impedance (Abs) raises as the Frequency approaches zero. As a formula: Zin
= 1/(1/R + i w C) , where i ist the imaginary number and w the frequency.

Now the hard part. How does one create an Impedance, which decreases
"slower", for frequencies close to zero but then decreases "faster" for
higher frequencies, than the simple parallel RC-Circuit?
Is there some kind of procedure like the one for syntesizeing LC-Filters
(Butterworth, Chebychev,..)?

Simply increasing C does not really help, because this equals a factoring of
the frequency.
Increasing R does not help as well, as it seems.


I hope one of you cracks can help me out.
So far, thanks for reading.
Diego Stutzer

 

[Peter O. Brackett]

Diego:

You cannot do exactly what you want, but you can get arbitrarily
close to it. The "closeness" being a function of cost. The closer
you want to get to the desired curve of impedance versus
frequency, the more the cost [cost = total number of elements in the
design].

Basically what you need to do is very well known in the network synthesis
literature as driving point impedance [DPI] synthesis. [e.g. Darlington's
method and other similar techniques. Darlngton's technique approaches
the problem of DPI as the synthesis of a lossless two port terminated
in an appropriate single resistance.] Network synthesis was widely
studied and taught back in the 1940 - 1970 era but... today it is
seldom seen, used, or taught. There are however lots of older textbooks
which cover this field. I'll post a few such references here below.

Before you can actually perform the DPI synthesis you will first have to
find an appropriate rational polynomial which approximates the impedance
curve you desire to match. To obtain such a rational polynomial you will
first
have to solve an appropriate approximation problem. Approximation
theory and the techniques for doing this are a whole 'nother problem, and
other than a few simple graphical straight line segment tricks, will
usually require the use of a computer with an appropriate algorithm
which you may have to write yourself!

Check out the following classic texts on network synthesis for a complete
run down on what you need to do:

1.) Ernst A. Guillemin, "Synthesis of Passive Networks", John Wiley & Sons,
NY, 1957. [LC# 57-8886. On technical library shelves at LCShelf
Call # K3226.G84. See Chapters 3, 4, 9, 10 which cover DPI synthesis
in detail, and Chapter 14 which covers the approximation problem.]

2.) Norman Balabanian, "Network Syntheis" Prentice-Hall, Englewood
Cliffs, NJ 1958. [LC# 58-11650. On technical library shelves at LCC
Shelf Call # TK3226.B26. See Chapters 2 & 3 for DPI and Chapter
9 for the approximation problem.]

3.) Louis Weinberg, "Network Analysis and Synthesis", McGraw-Hill,
New York, 1962. [LC# 61-16969. On technical library shelves at
LC Shelf Call # TK3226.W395. See Chapter's 9 & 10 for DPI
synthesis and Chapter 11 for the approximation problem]

One does not have to realize such designs with purely passive RLC
networks and they can often be synthesized with active RC networks
[R, C and Op-Amps] by appropriate transformations of the passive
synthesis. See for instance...

4.) Adel S. Sedra and Peter O. Brackett, "Filter Theory and Design:
Active and Passive", Matrix Publishers, Portland, OR, 1978. [LC #
76-39745. On technical library shelves at LCC Shelf Call #
TK7872.F5S42.]

Also, and I have done this a couple of times for special low frequency
applications, one can match the analog driving point impedance through
an appropriate Op-Amp reflectometer circuit to a combination analog
to digital A/D and digital to analog converter D/A and perform/emulate
the DPI synthesis in real time using digital signal procssing [DSP]
techniques.
Basically to use the A/D - D/A digital technique to emulate the desired
DPI you will have to solve the same synthesis and approximation
problems.

Hope that all helps... good luck



--
Peter K1PO
Consultant - Signal Processing and Analog Electronics
Indialantic By-the-Sea, FL




"Diego Stutzer" <abcdstutzer@evard.ch> wrote in message
news:aca3ec4c.0402190229.7977e8c6@posting.google.com...

Hi,
Every one knows, that e.g. a simple RC-parallel circuit has a
frequency-dependent impedance-characteristic (Absolute Value) - the
impedance (Abs) raises as the Frequency approaches zero. As a formula:
Zin
= 1/(1/R + i w C) , where i ist the imaginary number and w the frequency.

Now the hard part. How does one create an Impedance, which decreases
"slower", for frequencies close to zero but then decreases "faster" for
higher frequencies, than the simple parallel RC-Circuit?
Is there some kind of procedure like the one for syntesizeing LC-Filters
(Butterworth, Chebychev,..)?

Simply increasing C does not really help, because this equals a factoring
of
the frequency.
Increasing R does not help as well, as it seems.


I hope one of you cracks can help me out.
So far, thanks for reading.
Diego Stutzer

 

[Kristine Hyvang]

"Diego Stutzer" <abcdstutzer@evard.ch> wrote in message
news:aca3ec4c.0402190229.7977e8c6@posting.google.com...

Hi,
Every one knows, that e.g. a simple RC-parallel circuit has a
frequency-dependent impedance-characteristic (Absolute Value) - the
impedance (Abs) raises as the Frequency approaches zero. As a formula:
Zin
= 1/(1/R + i w C) , where i ist the imaginary number and w the frequency.

Now the hard part. How does one create an Impedance, which decreases
"slower", for frequencies close to zero but then decreases "faster" for
higher frequencies, than the simple parallel RC-Circuit?
Is there some kind of procedure like the one for syntesizeing LC-Filters
(Butterworth, Chebychev,..)?

Simply increasing C does not really help, because this equals a factoring
of
the frequency.
Increasing R does not help as well, as it seems.


I hope one of you cracks can help me out.
So far, thanks for reading.
Diego Stutzer

I'm not sure if I understood your question. You want a filter " which
decreases
"slower", for frequencies close to zero but then decreases "faster" for
higher frequencies.
I take it you want a transmission (starting from DC) a curve that "slowly"
decreases until a top point (the resonant freq.), and then falls quickly !?

A second order (or higher polynomial) Butterworth filter will do that for
you.
It has a transmission of Vout/Vin = k/(s^2 +bs+c)

where b is the ratio of the steepness Q (decrease of impedance) to the
resonant frequency.
c is the reciproc of the squared resonant frequency [1/ (w^2)] w-
radians.
k is a scalar - the DC amplification

Now what is the resonant frequency in your question ?
-It is the point where the transmission "decreases" in both direction (the
top point).

You would find a well suited circuit description in textbooks explaining
about the "Sallen Key" set up. It uses one op-amp, two capcitors (one in the
positive feedback !) and three resistors. It even works !


(^ - means squared)