Poles an Zeros

R

Rob

Guest
Hello,

Could someone explain to me Poles and Zeros.
What does a Pole do and what does a zero do and how they interact
together.
How does phase, gain, and delays relate to them?

If a book or a link could be recommended along with a what math needs
to be used. (maybe a course outline to work from)

I am right now working on systems which I have an good idea on how
they work but I don't seem to have the underlying understanding or how
to calculate.

Some fully worked out examples I can look at would also be helpful. I
just seem to be missing parts. I have found information. But I am
having some issues putting it all together. A number of fields all
converge and I am not able to find a path and seem to be jumping
around and not getting anywhere.
The area I am interested in are digitally controlled power converters
(DC/AC and DC/DC power supplies).

Thank you,
Rob
 
Rob wrote:
Hello,

Could someone explain to me Poles and Zeros.
What does a Pole do and what does a zero do and how they interact
together.
How does phase, gain, and delays relate to them?
Click to expand...
First, you have to wrap your mind around a plane that
represents all possible frequencies in two dimensions,
called the S plane.

The two dimensions of frequency are real and imaginary or
time constant and sinusoidal. Time constants (decaying or
growing) can be described as e (2.7 something, the base of
natural logarithms) raised to a real power. Positive powers
represent exponentially growing signals and negative
exponents represent exponentially decreasing signals. The
amount that any signal is growing or shrinking can be
represented as a line across the S plane that is positioned
along the real axis. If you make the exponent imaginary
(has a square root of -1 factor in it) the result is a sine
wave. Positive and negative imaginary exponentials are just
sine waves going forward or backward in time (and they look
exactly the same, since a sine wave is the same for all
time). Imaginary frequencies are represented as a line
across the S plane at right angles to the real exponential
signals. Low frequencies are represented as being near the
origin and higher frequencies are represented as being
further from the origin. A zero frequency, representing DC,
is the origin. So any exponentially growing or decaying
sine wave of any frequency and decay or growth rate can be
represented as a point on the S plane. Whew!

The amplitude of the signal of any frequency is represented
as a distance at right angles to the S plane, like a rod
sticking up out of it, for big values, or holes down into it
for very small values. Actual linear systems frequency
response to all frequencies can be represented as a surface
or rubber sheet that is lifted up of held down at frequency
points that represent important (resonant or characteristic)
frequencies for the system. If the system produces some
combination of sine wave and decay or growth rate if
energized and left otherwise un-driven, (zero continuous
input energy resulting in a finite result energy), that
natural response can be said to be an infinite response
(because it happens with zero input energy). Such points on
the S plane are represented as having the rubber response
surface held up at that point by an infinitely tall "pole".
There may also be particular combinations of sinusoid and
decay rate or growth that cannot be produced, regardless of
how hard you drive the system. Those are "zero response
points on th S plane rubber surface, and they can be thought
of points tacked down to infinitely deep holes at those
points on the rubber response surface. If the "rubber"
surface is given the right kind of elasticity,
mathematically, once a system is described at its "poles"
and "zeros", the rubber surface predicts its response to all
combinations of wave and decay/growth frequency by the
height of its surface.

Is that enough abstraction for you to work on visualizing
for a while?

If a book or a link could be recommended along with a what math needs
to be used. (maybe a course outline to work from)
Click to expand...
The S plane frequency representation is based completely on
representing the signals and responses as complex
exponentials (e raised to an exponent with real and
imaginary parts, representing the two dimensions of the S
plane. If you Google "S plane" and "exponential frequency"
you should have more hits than you need.

I am right now working on systems which I have an good idea on how
they work but I don't seem to have the underlying understanding or how
to calculate.

Some fully worked out examples I can look at would also be helpful. I
just seem to be missing parts. I have found information. But I am
having some issues putting it all together. A number of fields all
converge and I am not able to find a path and seem to be jumping
around and not getting anywhere.
The area I am interested in are digitally controlled power converters
(DC/AC and DC/DC power supplies).
Click to expand...
The S plane is an extremely useful visualization tool for
capturing every possible response of a linear system to
excitation by almost any signal.
--
Regards,

John Popelish
 
Rob wrote:
Hello,

Could someone explain to me Poles and Zeros.
What does a Pole do and what does a zero do and how they interact
together.
How does phase, gain, and delays relate to them?

If a book or a link could be recommended along with a what math needs
to be used. (maybe a course outline to work from)

I am right now working on systems which I have an good idea on how
they work but I don't seem to have the underlying understanding or how
to calculate.

Some fully worked out examples I can look at would also be helpful. I
just seem to be missing parts. I have found information. But I am
having some issues putting it all together. A number of fields all
converge and I am not able to find a path and seem to be jumping
around and not getting anywhere.
The area I am interested in are digitally controlled power converters
(DC/AC and DC/DC power supplies).

Thank you,
Rob
Click to expand...
This is too big a subject for a news group reply. Read what John said,
and read this:
http://www.wescottdesign.com/articles/zTransform/z-transforms.html.

If you're really interested in digital control, check out my book:
http://www.wescottdesign.com/actfes/actfes.html. You _will_ have to
wrap your brain around some mathematics, but I tried to make it as
accessible as I could, and keep the discussion rooted in the real world
at all times.

Basically, you can express the behavior of a linear, continuous-time,
time-invariant system with a mathematical construct called a "transfer
function". The transfer function is a consequence of analyzing a system
using the Laplace transform (do a web search, or check Wikipedia).

Most such transfer functions (at least those that you don't run away
from, screaming) are rational ratios of polynomials in s, the Laplace
domain 'frequency' variable. So, I may have a PID controller whose
transfer function is

(s + 250)(s + 1000)
H(s) = -------------------
s(s + 5000)

This guy has zeros at s = -250 and s = -1000, and poles at s = 0 and s =
-5000.

It turns out to be very easy to calculate the response of such a system
to a continuous sine wave at a particular frequency -- in this case the
poles of a system will tend to make the system gain go down as the
frequency is increased, and the zeros will make the gain go up.

You can also use the transfer function concept in the digital domain.
Here the system has to be linear and shift-invariant, the analysis
method is called the 'z transform', and the 'frequency domain' variable
is z (instead of s).

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" gives you just what it says.
See details at http://www.wescottdesign.com/actfes/actfes.html
 
"Rob
Could someone explain to me Poles and Zeros.
What does a Pole do and what does a zero do and how they interact
together.
Click to expand...

** I have spent countless hours reading about & watching films and
documentary stories about of the air war that dominated WW2. However, I
have never heard of any confrontation occurring between the Polish air force
( or indeed any Polish WW2 flyer) and a Japanese Zero.

Probably, had a "one on one" showdown occurred somewhere over South East
Asia, the Zero would have defeated the Pole.

The Pole would then become attenuated at a huge number of dB per thousand
feet - until the poor fellow reached the stop band. Forever.



...... Phil
 
On Thu, 28 Feb 2008 19:14:25 -0800 (PST), Rob <robd@ody.ca> wrote:

Hello,

Could someone explain to me Poles and Zeros.
What does a Pole do and what does a zero do and how they interact
together.
How does phase, gain, and delays relate to them?

If a book or a link could be recommended along with a what math needs
to be used. (maybe a course outline to work from)

I am right now working on systems which I have an good idea on how
they work but I don't seem to have the underlying understanding or how
to calculate.

Some fully worked out examples I can look at would also be helpful. I
just seem to be missing parts. I have found information. But I am
having some issues putting it all together. A number of fields all
converge and I am not able to find a path and seem to be jumping
around and not getting anywhere.
The area I am interested in are digitally controlled power converters
(DC/AC and DC/DC power supplies).

Thank you,
Rob
Click to expand...

OK, quick simplified version:

If you have a box with an input and an output, with linear response,
there exists a mathematical expression, a transfer function, that
describes how it behaves. If you know the transfer function, then for
any input signal you can predict the output.

There are many ways to express the transfer function, most of them
mathematically messy. Engineers prefer the Laplace Transform, which
expresses the transfer function as a polynomial, using the complex
varible "S". You'll have to read up on the theory to understand what S
really is.

But if you express a transfer function as a polynomial on S, and
factor it out nicely, and sweep S, the polynomial has "zeroes" where
the numerator hits zero, and "poles" where the denominator hits zero.
With a little practice, one can eyeball the equation, spot the poles
and zeroes, and guess the frequency response.

Take this circuit:


input---------------R--------+--------output
|
|
|
C
|
|
|
ground

which is a simple "single-pole" resistor-capacitor lowpass filter. If
R = 1 ohm and C = 1 farad, it has a pole at radian frequency w = 1,
namely at 0.16 Hertz.

The transfer function is

1 / (S+1)

and

Output = Input * 1 / (S+1)

so has a pole at S = -1, sort of. (S is actually a complex variable...
dig into the theory for gory details.)

You estimate the frequency response by pretending that S is the input
frequency, in radians/second. Ignore the sign!

For very small S, very low frequencies, 1/(S+1) = 1, so frequency
response is flat, unity gain. At high frequencies, S is big, so
1/(S+1) is almost the same as 1/S, so the output is dropping off
inverse with frequency.

If you graph the frequency response, it will look close to...



|
|
|
1 | _______________________
| \
| \
G | \
A | \
I | \
N | \
| \
| \
| \
| etc forever
|
|
|
|
|
|
|
|
|
__________________________________________________________
radian freq 1


where the gain is 1.00 at low frequencies up until 0.16 Hz, where it
starts to roll off as 1/f, namely at -6 dB per octave. Engineers
casually say that this frequency plot "has a pole at omega = 1"
because the Laplace polynomial really does.

(Because S is actually a complex number, the rolloff region has a 90
degree phase shift. A 90 degree phase lag is associated with a 6
dB/octave rolloff in simple networks like this one.)


Now if you add a small resistor in series with that cap, say 0.1 ohms,
it adds a zero. That looks like...



|
|
|
1 | _______________________
| \
| \
| \
| \
| \
| \
| \
| \
| \
0.1| ----------------------------
|
|
|
|
|
|
|
|
|
__________________________________________________________
1 10


where the second break is at w=10, namely 1.6 Hz. Phase is 0 in the
flats and 90 lag on the slopey part.

I find it helpful to start with a rough, practical explanation of
stuff like this before I hit the books for the formal stuff.


John
 
On Sat, 01 Mar 2008 01:27:17 +1100, Phil Allison wrote:

"Rob

Could someone explain to me Poles and Zeros. What does a Pole do and
what does a zero do and how they interact together.


** I have spent countless hours reading about & watching films and
documentary stories about of the air war that dominated WW2. However,
I have never heard of any confrontation occurring between the Polish air
force ( or indeed any Polish WW2 flyer) and a Japanese Zero.

Probably, had a "one on one" showdown occurred somewhere over South
East Asia, the Zero would have defeated the Pole.

The Pole would then become attenuated at a huge number of dB per
thousand feet - until the poor fellow reached the stop band.
Forever.
Click to expand...
Actually, it may have happened. When Poland fell many poles emigrated to
free countries to fight for the Allies. I imagine that most of them
stayed close to the continent, but there may have been some polish
nationals in the American, British, or Australian airplanes over the
Pacific.

But a polish pilot in a PLZ "fighter", going against a zero, would have
looked an awful lot like a small bird going against a falcon -- it would
be a short exchange.

--
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html
 
How about that! Phil has a sense of humor! Who could have imagined?
Lets all go to the pub and hoist a pint!
 
On Fri, 29 Feb 2008 10:34:01 -0600, Tim Wescott <tim@seemywebsite.com>
wrote:

On Sat, 01 Mar 2008 01:27:17 +1100, Phil Allison wrote:

"Rob

Could someone explain to me Poles and Zeros. What does a Pole do and
what does a zero do and how they interact together.


** I have spent countless hours reading about & watching films and
documentary stories about of the air war that dominated WW2. However,
I have never heard of any confrontation occurring between the Polish air
force ( or indeed any Polish WW2 flyer) and a Japanese Zero.

Probably, had a "one on one" showdown occurred somewhere over South
East Asia, the Zero would have defeated the Pole.

The Pole would then become attenuated at a huge number of dB per
thousand feet - until the poor fellow reached the stop band.
Forever.

Actually, it may have happened. When Poland fell many poles emigrated to
free countries to fight for the Allies. I imagine that most of them
stayed close to the continent, but there may have been some polish
nationals in the American, British, or Australian airplanes over the
Pacific.
Click to expand...
There were certainly Poles in the Royal Air Force, including during
the Battle of Britain- see

http://www.lynneolson.com/questionofhonor/olson_cloud_QA.htm

among others.
--
Only three people have ever understood the Schleswig-Holstein problem
One's dead, one's gone mad, and I've forgotten.
 
On Feb 29, 8:27 am, John Larkin
<jjlar...@highNOTlandTHIStechnologyPART.com> wrote:
On Thu, 28 Feb 2008 19:14:25 -0800 (PST), Rob <r...@ody.ca> wrote:


Could someone explain to me Poles and Zeros.

If you have a box with an input and an output, with linear response,
there exists a mathematical expression, a transfer function, that
describes how it behaves. If you know the transfer function, then for
any input signal you can predict the output.

There are many ways to express the transfer function, most of them
mathematically messy. Engineers prefer the Laplace Transform, which
expresses the transfer function as a polynomial, using the complex
varible "S". You'll have to read up on the theory to understand what S
really is.

But if you express a transfer function as a polynomial on S, and
factor it out nicely, and sweep S, the polynomial has "zeroes" where
the numerator hits zero, and "poles" where the denominator hits zero.
Click to expand...
Just to fill in another detail: if one sets up the known conditions
for a
network of electrical components, the result is a set of simple linear
equations. Solving the simultaneous equations will (at worst) result
in
an expression which can be simplified to a ratio of two polynomials in
S,
and the factoring of those polynomials is in general going to result
in the numerator having zeroes (the zeroes of the ratio) and the
denominator
having zeroes (the poles of the ratio). The poles and zeroes
express ALL of the resulting expression except for a single scaling
constant.

The handling of a messy expression then is simplified to the
values of some key numbers (the poles and zeroes).
 
"Tim Wescott"

But a polish pilot in a PLZ "fighter", going against a zero, would have
looked an awful lot like a small bird going against a falcon -- it would
be a short exchange.
Click to expand...

** You mean the Polish PZL 11C - right?

http://imansolas.freeservers.com/Aces/PZL-11c.JPG




........ Phil
 
On Sat, 01 Mar 2008 09:31:44 +1100, Phil Allison wrote:

"Tim Wescott"

But a polish pilot in a PLZ "fighter", going against a zero, would have
looked an awful lot like a small bird going against a falcon -- it
would be a short exchange.


** You mean the Polish PZL 11C - right?

http://imansolas.freeservers.com/Aces/PZL-11c.JPG
Click to expand...
Yse. Uh, Yes.

Apparently it was a very good fighter for 1932, or even 1935 -- the poles
just didn't get the newer version in production soon enough for 1939.
They had seen the writing on the wall, but much too late to save
themselves.

--
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html
 
Thank you all.

I guess I need to step back a bit before going into poles and zeros.
I guess it all starts with the transfer function.
I don't know what anybody else thinks but if there would be some
interest would some of the more knowledgeable people be able to do a
little tutorial of a practical example? I have Tim's book and a few
others. I like Tim's book and I need to start going through it
again. The first time I started reading it things didn't start coming
together. I am a practical and graphical person and need to touch,
see and smell to put everything together. I am now starting to put
things together but need to understand a lot more. Would it be
possible to do a tutorial using MathCAD, Mathlab, or other tools to do
a simple regulator design. For example a small micro design which
closes the loop (PIC or AVR)? Maybe a design like an old classic
LM7805? I can do the electronics and someone else can help with the
analysis, someone else could help with the firmware, someone else
could help with corrections, etc. I am not proposing something which
could be used as a product but something which mimics an IC already on
the market. Something which is practical so it can be built and tested
but also one can see what it takes as a design start. Maybe we could
also look at different ways to close the loop and then continue on a
servo design (but only mimicking devices on the market already, this
is only for education)
What does everyone think? If it sounds good then maybe I could setup a
website so files can be exchanged?

Thoughts, suggestions?
Rob
 
On Thu, 20 Mar 2008 07:49:58 -0700 (PDT), Rob <robd@ody.ca> wrote:

Thank you all.

I guess I need to step back a bit before going into poles and zeros.
I guess it all starts with the transfer function.
I don't know what anybody else thinks but if there would be some
interest would some of the more knowledgeable people be able to do a
little tutorial of a practical example? I have Tim's book and a few
others. I like Tim's book and I need to start going through it
again. The first time I started reading it things didn't start coming
together. I am a practical and graphical person and need to touch,
see and smell to put everything together. I am now starting to put
things together but need to understand a lot more. Would it be
possible to do a tutorial using MathCAD, Mathlab, or other tools to do
a simple regulator design. For example a small micro design which
closes the loop (PIC or AVR)? Maybe a design like an old classic
LM7805? I can do the electronics and someone else can help with the
analysis, someone else could help with the firmware, someone else
could help with corrections, etc. I am not proposing something which
could be used as a product but something which mimics an IC already on
the market. Something which is practical so it can be built and tested
but also one can see what it takes as a design start. Maybe we could
also look at different ways to close the loop and then continue on a
servo design (but only mimicking devices on the market already, this
is only for education)
What does everyone think? If it sounds good then maybe I could setup a
website so files can be exchanged?

Thoughts, suggestions?
Rob
Click to expand...

You can do most everyday closed-loop design for analog circuits, PLLs,
and servo loops, using just Bode plots, which are graphical and
intuitive, and not bother with the LaPlace pole/zero polynomial math.
I haven't done that since college.

I actually prefer the graphical approach as opposed to the "big math"
all-your-eggs-in-one-equation thing. The problem with the math is that
it's easy to make one mistake somewhere in a big equation, and come up
with truly silly results, and either miss the silliness or see it but
not know why.

Bode based loop design is also better for estimating safety margins.

John
 
On Mar 20, 2:32 pm, John Larkin
<jjlar...@highNOTlandTHIStechnologyPART.com> wrote:
On Thu, 20 Mar 2008 07:49:58 -0700 (PDT), Rob <r...@ody.ca> wrote:
Thank you all.

I guess I need to step back a bit before going into poles and zeros.
I guess it all starts with the transfer function.
I don't know what anybody else thinks but if there would be some
interest would some of the more knowledgeable people be able to do a
little tutorial of a practical example? I have Tim's book and a few
others. I like Tim's book and I need to start going through it
again. The first time I started reading it things didn't start coming
together. I am a practical and graphical person and need to touch,
see and smell to put everything together. I am now starting to put
things together but need to understand a lot more. Would it be
possible to do a tutorial using MathCAD, Mathlab, or other tools to do
a simple regulator design. For example a small micro design which
closes the loop (PIC or AVR)? Maybe a design like an old classic
LM7805? I can do the electronics and someone else can help with the
analysis, someone else could help with the firmware, someone else
could help with corrections, etc. I am not proposing something which
could be used as a product but something which mimics an IC already on
the market. Something which is practical so it can be built and tested
but also one can see what it takes as a design start. Maybe we could
also look at different ways to close the loop and then continue on a
servo design (but only mimicking devices on the market already, this
is only for education)
What does everyone think? If it sounds good then maybe I could setup a
website so files can be exchanged?

Thoughts, suggestions?
Rob

You can do most everyday closed-loop design for analog circuits, PLLs,
and servo loops, using just Bode plots, which are graphical and
intuitive, and not bother with the LaPlace pole/zero polynomial math.
I haven't done that since college.

I actually prefer the graphical approach as opposed to the "big math"
all-your-eggs-in-one-equation thing. The problem with the math is that
it's easy to make one mistake somewhere in a big equation, and come up
with truly silly results, and either miss the silliness or see it but
not know why.

Bode based loop design is also better for estimating safety margins.

John
Click to expand...
Hi John,

Thanks, I am not a big math person. I would like to be but I have a
hard time making sense of all the nomenclature. It also seems to
change depending on how one was taught.
Even with doing it graphically a transfer function would still be
required correct? So some big math would be required to put the
transfer function in a form which can then be graphed.
Or am I making a to big of deal out of this? How do you do it?

Thanks,
Rob
 
On Fri, 21 Mar 2008 07:30:08 -0700 (PDT), Rob <robd@ody.ca> wrote:

On Mar 20, 2:32 pm, John Larkin
jjlar...@highNOTlandTHIStechnologyPART.com> wrote:
On Thu, 20 Mar 2008 07:49:58 -0700 (PDT), Rob <r...@ody.ca> wrote:
Thank you all.

I guess I need to step back a bit before going into poles and zeros.
I guess it all starts with the transfer function.
I don't know what anybody else thinks but if there would be some
interest would some of the more knowledgeable people be able to do a
little tutorial of a practical example? I have Tim's book and a few
others. I like Tim's book and I need to start going through it
again. The first time I started reading it things didn't start coming
together. I am a practical and graphical person and need to touch,
see and smell to put everything together. I am now starting to put
things together but need to understand a lot more. Would it be
possible to do a tutorial using MathCAD, Mathlab, or other tools to do
a simple regulator design. For example a small micro design which
closes the loop (PIC or AVR)? Maybe a design like an old classic
LM7805? I can do the electronics and someone else can help with the
analysis, someone else could help with the firmware, someone else
could help with corrections, etc. I am not proposing something which
could be used as a product but something which mimics an IC already on
the market. Something which is practical so it can be built and tested
but also one can see what it takes as a design start. Maybe we could
also look at different ways to close the loop and then continue on a
servo design (but only mimicking devices on the market already, this
is only for education)
What does everyone think? If it sounds good then maybe I could setup a
website so files can be exchanged?

Thoughts, suggestions?
Rob

You can do most everyday closed-loop design for analog circuits, PLLs,
and servo loops, using just Bode plots, which are graphical and
intuitive, and not bother with the LaPlace pole/zero polynomial math.
I haven't done that since college.

I actually prefer the graphical approach as opposed to the "big math"
all-your-eggs-in-one-equation thing. The problem with the math is that
it's easy to make one mistake somewhere in a big equation, and come up
with truly silly results, and either miss the silliness or see it but
not know why.

Bode based loop design is also better for estimating safety margins.

John

Hi John,

Thanks, I am not a big math person. I would like to be but I have a
hard time making sense of all the nomenclature. It also seems to
change depending on how one was taught.
Even with doing it graphically a transfer function would still be
required correct? So some big math would be required to put the
transfer function in a form which can then be graphed.
Or am I making a to big of deal out of this? How do you do it?

Thanks,
Rob
Click to expand...

Get a piece of semilog paper. Vertical (linear) axis will be gain in
dB. Horizontal (log) axis is frequency. Actually, after a bit of
practice, you can do it freehand, on plain paper or a whiteboard. We
do this all the time.

Pick each little part of a circuit and sketch its frequency response.

An RC lowpass is a horizontal line of gain=1, that breaks at some
corner frequency (where Xc = R) and then drops at -6 dB per octave, a
straight 45 degree down slope on the graph.



0 dB -----------
\
\ -6 dB/octave
\
\
\
etc down forever






An opamp integrator is just a down slope, with a gain of 1 at the
frequency where Xc = R.



\
\ -6 dB/octave
\ everywhere (except at
\ extremes, for the purists)
\
\
\




Simple RC networks can usually be graphed by inspection. Like...


in---------R1---------+--------------out
|
|
R2
|
|
C
|
|
gnd


usually plots like...



-----------
\
\
\
\
\
-----------------





So, for a closed-loop system, cascade all the gain curves around the
loop; that's simple addition of the curves when you're working in
dB's.

Recall that for simple networks like this, a 6 dB per octave down
slope corresponds to 90 degrees phase lag.

The frequency/slope at which the overall closed loop crosses unity
gain (0 dB) determines loop stability. If the slope is -6, phase
margin is 90 degrees and the loop is heavily damped and bog stable. A
little steeper, -9 db or around 45 degrees phase margin, is close to
critically damped, snappier response. Steeper slopes, less phase
margin, will make the loop start to ring. -12 dB or more is 180
degrees or more, unstable.

If you slide the whole curve up or down, you're simulating gain
changes anywhere in the loop. Examine how that affects the slope of
the zero crossing, and that tells you how much gain margin you have
for stable operation.

With a little practice, you can do a lot of loop stabilization math in
your head, and sketch the approximate Bode plot freehand, and get nice
stable loops.

I could teach almost anybody how to do this in 20 minutes, if I had a
whiteboard handy.

John
 
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