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Poles and Zeros: book recomendation

Hello everybody. Due to my current job, over the last months I have been
finding myself working more and more often with -relatively simple-
electronic circuits, like amplifiers or filters. While I have a reasonable
understanding of those circuits in practical terms (capacitor here,
resistor there, we have a low-pass, the 3dB point is here...), I get
completely lost when people starts talking in terms of transfer functions
(that is, poles, zeros... the laplace transform domain). I have a
scientific formation, so I have a basic familiarity with the laplace
transform (or at least I had it over ten years ago, in college, so
refreshing my memory should be easy). However, I was never introduced to
its applications in electronics or any other practical field, for that
matter.

What I would like is a few recommendations of books that could give me a
basic understanding of these topics when applied to electronics, so that
when somebody says "we stabilize this circuit by adding a pole here" I
understand what he is saying. I have the mathematical background to
understand things in terms of differential equations, complex variables...
I just need a good source to learn these things from!

Thanks.
 
K

Kevin Aylward

Jan 1, 1970
0
Hello everybody. Due to my current job, over the last months I have
been finding myself working more and more often with -relatively
simple- electronic circuits, like amplifiers or filters. While I have
a reasonable understanding of those circuits in practical terms
(capacitor here, resistor there, we have a low-pass, the 3dB point is
here...), I get completely lost when people starts talking in terms
of transfer functions (that is, poles, zeros... the laplace transform
domain). I have a scientific formation, so I have a basic familiarity
with the laplace transform (or at least I had it over ten years ago,
in college, so refreshing my memory should be easy). However, I was
never introduced to its applications in electronics or any other
practical field, for that matter.

What I would like is a few recommendations of books that could give
me a basic understanding of these topics when applied to electronics,
so that when somebody says "we stabilize this circuit by adding a
pole here" I understand what he is saying. I have the mathematical
background to understand things in terms of differential equations,
complex variables... I just need a good source to learn these things
from!

Don't have any book recommendations, but the basics are this:

I have two tutorials/description http://www.anasoft.co.uk/EE/index.html,
the "Laplace" link and the "Feedback stability part1"

99.99% of all real circuits have a laplace transfer function that is a
ratio of polynomials. All polynomials can be written as a product of
(1+a.x) like terms, so all transfer functions can be writen as a ratio
of products of (1+a.x) terms. i.e.

V0/Vi = (1+a1.x)(1+a2.x)(1+a3.x).../(1+b1.x)(1+b2.x)(1+b3.x)...

The fundamental characteristics are the roots of the polynomials, i.e.
(1+an.x) = 0 or (1+bm.x) = 0. The roots on the numerator are called
zeros, because it makes the function zero, the roots in the denominator
are called poles, they make the transfer function go to infinity. Why
they are called poles is anyone's guess.

Each zero, represents an attempt at an increase in gain and phase as
frequency increases. each pole represents an attempt at a decrease in
gain and phase as frequency increases.

Its far easier to say "add a zero" then "add a gain and phase as
frequency increases"

Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.
 
L

Le Chaud Lapin

Jan 1, 1970
0
Kevin Aylward said:
The fundamental characteristics are the roots of the polynomials, i.e.
(1+an.x) = 0 or (1+bm.x) = 0. The roots on the numerator are called
zeros, because it makes the function zero, the roots in the denominator
are called poles, they make the transfer function go to infinity. Why
they are called poles is anyone's guess.

Your guess is correct.

Imagine indicating |H(s)| as a height over the s-plane. At zeros,
|H(s)| will be zero. At poles, |H(s)| will be infinite. If you cover
the entire s-plane with a latex membrane, and use something to push up
the membrane to indicate |H(s)|, there will be "nothing" pushing up
the membrane at the zeros, and it will appear that a pole, in the
literal sense, is pushing up the membrane at the the poles. Since
H(s) will be ratio of polynomials, and therefore continuously
differentiable (analytic) except at pole singularities, the membrane
will have nice, curved slopes from the poles to the zeros, much like a
circus tent, except the peaks will be very tall indeed.

Once this membrane is visualized, it then becomes very easy to
visualize what effect H(s) will have on the magnitude of the input
picking out the s of e^st and examining where it falls on the
membrane. Naturally, choices of s close to, but not equal to, a zero,
will yield an output signal close to zero, while choices of s close
to, but not equal to, a pole, will yield outputs close to "infinity".

Now pick your arbitrary vector (jw), and grab it at its tip and move
it around in s-plane. As you do, you can not only see what happens to
magnitude of H(s), but also the phase. Visualizing the phase requires
little more intuition, but the process is essentially the same. Keep
a mental sum of the total phase of H(s) as you move, by taking
difference in phase between your jw, and that of the poles and zeros
of H(s). For poles, add phase difference to your sum, and for zeros,
subtract.

Practice imagining this membrane, and after a while, you will see how
to quickly pick poles and zeros to get a desired magnitude and phase
of H(s) for values of (jw) of interest.

As far as books go, I cannot remember anything specific, but you
should definitely look for illustrations of this membrane.

-Chaud Lapin-
 
F

Fred Bloggs

Jan 1, 1970
0
Kevin said:
Why they are called poles is anyone's guess.
Clearly, by the fundamental theorem of algebra, first proved by a 13
year old German boy, KG, the denominator factors into terms (s-Pk)^n so
that the magnitude will have radial symmetry about Pk in much the same
way as a pole pushing on canvas does locally.
 
K

Kevin Aylward

Jan 1, 1970
0
Le said:
Your guess is correct.

Imagine indicating |H(s)| as a height over the s-plane. At zeros,
the entire s-plane with a latex membrane, and use something to push up
the membrane to indicate |H(s)|, there will be "nothing" pushing up
the membrane at the zeros, and it will appear that a pole, in the
literal sense, is pushing up the membrane at the the poles. Since
H(s) will be ratio of polynomials, and therefore continuously
differentiable (analytic) except at pole singularities, the membrane
will have nice, curved slopes from the poles to the zeros, much like a
circus tent, except the peaks will be very tall indeed.

Well, what do you know. You learn something every day.


Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.
 
K

Kevin Aylward

Jan 1, 1970
0
Fred said:
Clearly, by the fundamental theorem of algebra,

Actually, I consider this theorem a cheat. It relies on one *defining* a
multiple root as two or more roots.
first proved by a 13
year old German boy, KG, the denominator factors into terms (s-Pk)^n
so that the magnitude will have radial symmetry about Pk in much the

Yes.yes yes.
same way as a pole pushing on canvas does locally.

Never imagined this in my wildest dreams. Guess I'm just not visually
minded.

Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.
 
F

Fred Bloggs

Jan 1, 1970
0
Kevin said:
Actually, I consider this theorem a cheat. It relies on one *defining* a
multiple root as two or more roots.




Yes.yes yes.




Never imagined this in my wildest dreams. Guess I'm just not visually
minded.

The true story of mathematical discovery is intuition followed by
formalism- and the custom of theorem statement and proof is completely
reverse from the natural thought process- just a gimmick to exclude the
immature-)
 
A

Active8

Jan 1, 1970
0
Your guess is correct.

i was going to say that it's probably because the function
asymptotically approaches infinity at the pole which looks kinda like a
pole - at least you could draw a pole in there. i forgot the membrane
analogy, duh.

talk about a busy group. go away for a few and there's a slew of
responses to a post that had just 1 reply.

brs,
mike[snip]
 
J

Jim Thompson

Jan 1, 1970
0
Clearly, by the fundamental theorem of algebra, first proved by a 13
year old German boy, KG, the denominator factors into terms (s-Pk)^n so
that the magnitude will have radial symmetry about Pk in much the same
way as a pole pushing on canvas does locally.

I remember it described as a "rubber" sheet with "poles" (or tacks
pinning it down to make zeroes).

...Jim Thompson
 
A

Active8

Jan 1, 1970
0
Jim-T@golana- said:
I remember it described as a "rubber" sheet with "poles" (or tacks
pinning it down to make zeroes).

...Jim Thompson
yup. i forgot about the tacks, too.

mike
 
T

Tim Stinchcombe

Jan 1, 1970
0
Having a similar-sounding background, I suffer the same frustrations. The
best I have found so far is

'Design of Analog Filters', Rolf Schaumann & Mac E. van Valkenburg, OUP 2001

It's full of friendly diagrams showing how poles and zeroes affect the
frequency response of filters, and thankfully doesn't shy away from the
maths involved. The first four or five chapters are quite a straightforward
read, and include diagrams relating the movement of poles and zeroes around
the complex plane to the response of the filter, and to the transfer
functon. However, I don't think it will necessarily equate to the stability
situation in general, so you would probably need to couple it with something
else - Sedra & Smiths 'Microelectronic Circuits' comes to mind (OUP, 4th ed,
1997), which has nice sections on stability etc., but does less specifically
on poles/zeroes/filters. Probably best to check copies out first and not to
buy blind just on my recommendation!

Tim
 
F

Frank Miles

Jan 1, 1970
0
[snip]
What I would like is a few recommendations of books that could give me a
basic understanding of these topics when applied to electronics, so that
when somebody says "we stabilize this circuit by adding a pole here" I
understand what he is saying. I have the mathematical background to
understand things in terms of differential equations, complex variables...
I just need a good source to learn these things from!

While this may seem counter-intuitive, I'd suggest looking at some
introductory-level books on control systems (analog/continuous-time, not
digital: e.g Dorf, "Modern Control Systems"). Alternatively, you could
read some books on filter theory (e.g. Budak, "Passive and Active Network
Analysis and Synthesis"). They might help you make the bridge between the
math and the circuit concepts.

-frank
--
 
P

Phil Hobbs

Jan 1, 1970
0
What I would like is a few recommendations of books that could give me a
basic understanding of these topics when applied to electronics, so that
when somebody says "we stabilize this circuit by adding a pole here" I
understand what he is saying. I have the mathematical background to
understand things in terms of differential equations, complex variables...
I just need a good source to learn these things from!

First of all, the reason that the terminology is so confusing is that when a circuit
designer shows you an RC lowpass circuit that "has a pole at 1 kHz", you, knowing
complex variables, naturally look for an infinite singularity on the real frequency
axis, and are confused when you see a limp-looking smooth rolloff instead.

What you don't know (and it took me awhile to realize myself, long ago) is that sheer
laziness has prevented the designer from saying that the pole is really at i(1 kHz),
on the imaginary axis [or -i(1 kHz) depending on your Fourier transform sign
convention].*

Other than that, it really is just like complex variables in school, except that the
rigour is replaced with some rules of thumb.

If you want the connection made between feedback networks and complex variables, and
don't mind working a bit, I recommend going to the source: Network Analysis and
Feedback Amplifier Design, by H. W. Bode, Van Nostrand, New York, 1945. You can find
a used copy on Advanced Book Exchange for probably ten bucks or so--that's where I
got mine. This is the same Bode that the plot is named after. Books written by the
guy who invented the idea usually have the combination of conceptual simplicity and
technical rigour that will give you a really solid grounding.

You don't have to go through the whole thing to get the basic idea.

Cheers,

Phil Hobbs
(Physicist masquerading as an engineer all this time)


* The laziness may have been on the part of his professor, of course.
 
G

gwhite

Jan 1, 1970
0
Kevin said:
Well, what do you know. You learn something every day.


May I offer the kind suggestion of a remedial course in Comm Systems?
 
G

gwhite

Jan 1, 1970
0
Hello everybody. Due to my current job, over the last months I have been
finding myself working more and more often with -relatively simple-
electronic circuits, like amplifiers or filters. While I have a reasonable
understanding of those circuits in practical terms (capacitor here,
resistor there, we have a low-pass, the 3dB point is here...), I get
completely lost when people starts talking in terms of transfer functions
(that is, poles, zeros... the laplace transform domain). I have a
scientific formation, so I have a basic familiarity with the laplace
transform (or at least I had it over ten years ago, in college, so
refreshing my memory should be easy). However, I was never introduced to
its applications in electronics or any other practical field, for that
matter.

What I would like is a few recommendations of books that could give me a
basic understanding of these topics when applied to electronics, so that
when somebody says "we stabilize this circuit by adding a pole here" I
understand what he is saying. I have the mathematical background to
understand things in terms of differential equations, complex variables...
I just need a good source to learn these things from!


If the question is stability, and you sort of imply that, then a book
on Control Systems will deal specifically with that aspect. I have
the Kuo book -- I suppose it is pretty good. Kuo has the odd habit
of including the "Complementary Root Loci" (CRL) concept along with
"Root Loci" (RL) concept. Since these are mirrors of each other, no
new information is given by including CRL. I haven't seen another text
include CRL. Other than that, I thought it was good. I'd look before
I'd buy though.

The pole-zero concept is also used by filter designers. Frank Miles
suggested the Budak book. I have that one too and think it is one of
the better filter synthesis books I've seen. Steve Winder's book is more
cookbook like (not a thing wrong with that) with a bit of concise theory,
and he does include a pole-zero discussion. Maybe look at Winder first.
 
C

Chuck Simmons

Jan 1, 1970
0
Kevin said:
Actually, I consider this theorem a cheat. It relies on one *defining* a
multiple root as two or more roots.

Correctly, the Fundamental Theorem of Algebra merely states that any
polynomial can be factored uniquely into linear and quadratic factors.
It makes no statement about roots at all although some modern books may
present the theorem in less general form (mumbling about roots) than my
statement above which is close to the original statement. The reason for
allowing quadratic factors is to avoid the need for the factors having
coefficients outside the real numbers.

Chuck
 
K

Kevin Aylward

Jan 1, 1970
0
gwhite said:
May I offer the kind suggestion of a remedial course in Comm Systems?

You may, but it will be ignored as as not required in the slightest. I
am very well acquainted with communications theory, thank you. Indeed,
my Degree is in "Communications and Electronic Engineering", having done
probably 10 times more formal work in communications theory than
analogue design. In addition I am not aware of any demonstrably lack in
my knowledge in this subject matter. However, I am aware of an
individual who presented reams of trivial observations cursorily related
to communications, but was in fact 101 linear analysis mathematics, who
made some related assertions, but that they were not relevant to the
matter actually being discussed. If you would be so kind as to actually
present some examples of where I lacked such knowledge, rather then
simply not presenting knowledge already held, and ignored as not
relevant to the subject in concern, you suggestion might have some
value. However, this would seem doubtful, in light of the fact that it
has been shown that you make grandiose claims that have been proved
fundamentally flawed at their heart.

Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.
 
P

Peter O. Brackett

Jan 1, 1970
0
Thread:

Re: pole and zero terminology for complex functons...

I always thought the name "pole" arose from the fact that the surface of
the [complex function] "membrane" around and near a "pole" or denominator
singularity seems to be stretched and pokes upward and so looks almost like
that part of a canvas tent around where the supporting "tent pole" holds it
up.

--
Peter
Consultant
Indialantic By-the-Sea, FL.


Active8 said:
Your guess is correct.

i was going to say that it's probably because the function
asymptotically approaches infinity at the pole which looks kinda like a
pole - at least you could draw a pole in there. i forgot the membrane
analogy, duh.

talk about a busy group. go away for a few and there's a slew of
responses to a post that had just 1 reply.

brs,
mike[snip]
 
G

gwhite

Jan 1, 1970
0
Kevin said:
You may, but it will be ignored as as not required in the slightest. I
am very well acquainted with communications theory, thank you. Indeed,
my Degree is in "Communications and Electronic Engineering", having done
probably 10 times more formal work in communications theory than
analogue design. In addition I am not aware of any demonstrably lack in
my knowledge in this subject matter. However, I am aware of an
individual who presented reams of trivial observations cursorily related
to communications, but was in fact 101 linear analysis mathematics, who
made some related assertions, but that they were not relevant to the
matter actually being discussed. If you would be so kind as to actually
present some examples of where I lacked such knowledge, rather then
simply not presenting knowledge already held, and ignored as not
relevant to the subject in concern, you suggestion might have some
value. However, this would seem doubtful, in light of the fact that it
has been shown that you make grandiose claims that have been proved
fundamentally flawed at their heart.


That was beautiful -- just what I was expecting and looking for.
The style... The phrasing... Almost like music... Thanks.
 
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