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
You need to graph out the required frequency-impedance slope then approximate the required roll off rates using a segmented breakpoint scheme consisting of a number of CR series sections in parallel. Essentially it's a straight line approximation to the required Z-F curve. The CR's adding zeroes as the frequency goes up.
Estimating the individual time constants can be irksome as each has effect outside it's area of interest. Use a 'least-squares approximation' to obtain a best curve fit for the number of sections involved.
It's an interesting subject but I've come across nothing out there that's of use.
In other words YES. You use combinations of resistors and capacitors or inductors. Understanding the concept of "poles" and "Zeroes" is one way which allows the synthesis of such circuits. Another is the concept of "corner Frequency".
This does exactly what you want, in the beginning the slope is less than
3dB/oct. and at 10kHz it goes to 6dB/oct. This is how to produce a pink noise that rolls off faster at the end of range, or to make some weighted filters (dBA) etc.
This can be done in a number of ways employing active components (I've just seen the drift of the thread take on this complexity).
The Bi-Quad filter comes to mind and has been around implemented with Op-Amps for quite a while.
Another is the cascaded, bucket-brigade chip from reticon (haven't played with this for 20 years tho').
You could also build an MCU interfacing ADC and DAC chips or simply step up to DSP chips for filters that are impossible to implement in any combination of passive L-C-R combinations.
However, the Bi-Quad offers simultaneous Low Pass, Band Pass, Band Reject, and High Pass outputs from one circuit configuration. For playing around with, that flexibility is hard to beat.
"Diego Stutzer" in news: snipped-for-privacy@posting.google.com...
What you are asking about is a form of what's traditionally called the network synthesis problem (creating a network of components to realize a prescribed signal response) and specifically the synthesis of a one-port, or impedance.
At one time (when phone companies ruled the earth and computers had conquered few signals and DSP was reserved for BIG things like the US Perimeter Acquisition Radar at Concrete, North Dakota -- affectionately the "PAR"), this was a popular subject in engineering schools at the advanced-undergrad or graduate level. It is still extremely important sometimes, especially with the sophisticated signal processing used today on continuous-time signals in consumer products. A host of applied-mathematical techniques (Foster and Cauer synthesis, Brune's impedance-synthesis lemma, etc.) apply even to one-ports. Some of them are highly counterintuitive. Not, in other words, a subject perfectly matched to the contraints of brief advice on newsgroups. (Note also that Butterworth and Chebyshev approximants are mathematical methods to approach one group of curves out of things that naturally give you a different type of characteristic -- "Butterworth and Chebyshev" have nothing to do with specific circuit topologies or components). If you want to pursue it further I could suggest investigating "network synthesis." Temes and LaPatra had a reasonable modern (1970s) book about it. Karl Willy Wagner started it all in 1915 by inventing filters.
Richard Clark suggested also investigating the small op-amp "biquad" networks for designable frequency response (actually you can turn them into one-ports, the so-called shunt-filter class, but again a bit of a subject for a brief response). Note that technically a "bi-quad" is any network giving a biquadratic transfer function (2nd-order numerator and denominator) though in RC-active filters it's often applied to the closely related Åkerberg-Mossberg and Tow-Thomas configurations. For practical info see van Valkenberg's excellent general introductory book on filters from the 1980s.
For an accessible modern example of these small op-amp-based "biquad" networks, look up the LTC1562 from Linear Technology, a commercial chip with four trimmed "biquad" networks, programmable by outboard components for applications from a few kHz to a few hundred kHz.
You cannot do *exactly* what you propose, but you can get arbitrarily close to it.
The "closeness" being a function of the cost you are prepared to pay.
The closer you want to get to the desired function [curve] of impedance versus frequency, the more the cost [cost = total number of R-L-C elements in the design].
Basically what you are trying to doe 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 researched, 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 in great depth.
I'll post a few such references here below for your reference.
Before you can actually perform the DPI synthesis you will first have to find an appropriate rational polynomial function, to form the basis for your synthesis, which approximates the impedance function [curve] you desire to match. To obtain such a rational polynomial you will have to solve an appropriate approximation problem.
Approximation theory and the techniques for doing this with rational polynomial 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, such as Remez second method, which you may have to write yourself!
Unless you can find consultant to help you... :-)
Check out the following classic texts on network synthesis for a complete run down on what you need to do to accomplish your objective:
1.) Ernst A. Guillemin, "Synthesis of Passive Networks", John Wiley & Sons, NY, 1957. [LC# 57-8886. On technical library shelves at LCShelf Call # TK3226.G84. See Chapters 3, 4, 9, 10 which cover the 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, in appropriate frequency ranges, they can often be synthesized with active RC networks [R, C and Op-Amps] by appropriate transformations of the passive synthesis results.
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 myself 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 mentioned above but under a suitable *warping* of the real frequency axis.
Hope that all helps... and good luck
:-)
-- Peter K1PO Consultant - Signal Processing and Analog Electronics Indialantic By-the-Sea, FL
Gawwwwd! You list makes me feel really old. Not only do I recognize all your references, I knew "Ernie" personally... but I had Harry B. Lee as my instructor, since I was in VI-B.
...Jim Thompson
--
| James E.Thompson, P.E. | mens |
| Analog Innovations, Inc. | et |
Think War of Independence, this guy was a descendent of "Light-Horse" Harry Lee ;-)
Harry B. Lee was, IMNSHO, better than Ernie. Everything Ernie did was
1H, 1F and 1 ohm... Harry was a realist, and I still have my notes... I don't think anyone could have taught nodal and loop analysis better; which is why I attack IC design in that fashion.
...Jim Thompson
--
| James E.Thompson, P.E. | mens |
| Analog Innovations, Inc. | et |
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