Trying to understand how poles (vertical asymptotes) and zeros apply to an amplifier.

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edbarx

edbarx

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Joined 2018
#1
Searching in the hope I understand more about poles and zeros, I found the following document, but it is very terse, and there are no specific examples using amplifiers. The transfer function is used as a representative of an abstruct input/output system with no reference to a real amplifier.
Mathematically, it is easy to understand what poles and zeros are by studying the transfer function H. Zeros are solutions for the numerator and poles are solutions for the denominator.

Now the fun: according to the document, plotting all solutions for the transfer function on an Argant Diagram should give a clear idea as to how stable a system is. Poles in the (-, +) quadrant and the (-, -) cause a system to be unstable. Poles in the (+,+) quadrant and the (+,-) quadrant point to a stable system.

Why? There must be a logical reason. And, more importantly, how does the transfer function for a real amplifier look?
 
#2
The poles are a notable increase in some magnitude. The term pole itself make reference to those poles used in circus to sustain the soft big top. This increase in the magnitude (current or voltage) come toguether with a severe change in the phase of this magnitude. Zeros are in certain mode, the inverse. The magnitude is at a minimum but also a big change in phase occurs.
 
#3
Those sudden changes in 0hase convert NFB into positive making the amp to oscillate. Else they can create a null in the transference like a notch filter (not too common). A series capacitor with a shunt capacitor places a pole at the frequency f = 1/(2 pi R C). If R and C are interchanged a zero is inserted. If in place of R and we have L and C they are the same but reversed.

Each single zero or pole has a + or - 3dB voltage or current gain and a phase shift of 45 electrical degrees.
 
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#4
Welcome to the first year of an Electrical Engineering degree! (well it was back when I did it). That document is nice and concise, as you'd expect from MIT. As this type of analysis is applicable to a wide range of dynamic systems (mechanical, electrical, ....), first introducing it as a generalised concept then allows you to understand and apply it wherever you choose. If you can describe a system as a set of transfer functions, you can analyse its dynamic response and hence stability. To apply it to an amplifier, requires this information for whatever level of abstraction you care to use (e.g. passives, active components, macromodels, parasitics etc.).
As to your 'why', the relationship between pole position and stability (i.e. time domain response decays to zero) is nicely demonstrated in Figure 2 and section 1.3, it's worth following through their examples to see this in action.
 
#5
edbarx said:
And, more importantly, how does the transfer function for a real amplifier look?
Click to expand...
In practice, amplifiers are designed to have a dominant pole whose purpose is to reduce the gain around the loop to much less than unity at the frequency of the amplifier's second pole.

The designer chooses the loop gain and frequency of the dominant pole to ensure this.

The response has a -6dB/octave slope above the frequency of the dominant pole. The slope increases to -12dB/octave above the frequency of the second pole.

This is the simplest form of frequency compensation. It works well.
Ed
 
#6
Roughly, poles are points (frequencies) at which frequency response starts to decline with a rate of 6dB/octave. Conversely, frequency responce begins to rise at zeros with 6dB/octave rate.

Cumulative effect of two poles is FR decline rate of 12dB/octave. Two poles colocated at the same frequency form a double pole. A double pole have an additional property compared to a single pole - a Q factor. It's an indicator of the FR shape at the pole's frequency. For Q above sqrt(2)/2 there is a peak in the FR. The greater Q results in higher peak. Zeroes behave completely analogous to this

Phono preamps should have a special FR standized by RIAA. It is defined by pole at 50Hz, zero at 500hz and yet another pole at 2122Hz.

Amplifiers with global feedback usually use so called Dominant Pole Compensation in order to insure amplifier's stability. This is intentionally introduced low frequency pole (e.g. 10Hz is not uncommon).
 
#7
edbarx said:
And, more importantly, how does the transfer function for a real amplifier look?
Click to expand...
Again roughly, a flat FR and a pole at the frequency fd * Aol / Acl

Where:
fd - frequency of the dominant pole
Aol - open loop gain
Acl - closed loop gain

From the low frequency side FR is usually defined by two factors. Input coupling capacitor and a large (usually electrolitic) capacitor in the feed back loop. Each of them introduce a zero at 0Hz and a pole at certain freq f. One pole is determined by the input capacitor and input resistance (usually 10-47k resistor to gnd). The other pole by mentioned electrolytic cap and the resistor immediatly connected to that cap.

This is just a good approximation.
 
#8
Some may find the math instructive. Mostly, I do not. The object is to prevent positive feedback that will regenerate and oscillate. Too much phase shift converts negative feedback into positive feedback. Every stage in an amplifier adds at least 90 degrees of phase lag above some frequency, aka "pole". Note that the phase lag of a single pole approaches 90 degrees while the amplitude continues to fall at 1/f, so a loop with only one pole could never oscillate. To get to 180 degrees, you need a two or more poles. For stable feedback, the loop gain must be gone, less than 0dB before you get 180 degrees of phase shift. A single dominant pole will eventually take you down to 0dB loop gain with only 90 degrees of phase shift. But of course, amplifiers have more stages so there will be more poles. Cascading multiple poles gives you lots of phase shift with little amplitude/gain loss. To be stable, those additional poles must be at a much higher frequency than the dominant pole, so the first pole has reduced the gain to less than unity (0dB) before the total phase lag gets to 180 degrees.
Using SPICE, we place a stimulus in the negative feedback and graph how the amplifier cancels the stimulus over the frequency range. This shows us how much "phase margin", ie difference from 180 degrees we have at zero loop gain (unity), and/or "gain margin" , ie loop loss at 180 degrees phase shift. We have to remember that a capacitive load, for example, may make the situation worse, so we want a healthy phase margin.
 
#10
richb said:
Welcome to the first year of an Electrical Engineering degree!
Click to expand...
Like you, I finished my degree some decades ago. I happened to be browsing some books I own about electronics to reread about poles and zeros, and found a book written by F.A. Wilson, and since I read a whole book series about electronics during my adolescence by the same author, I searched for the topic and found it. Searching online, I found the linked document. At the university it was custom that theories and models were first applied generally and then to more specific situations/problems.

I wish I were still that young, but years have the bad habit of passing so quickly, I am now approaching retirement age. What I am doing is a mental refreshment because of an incessant passion to understand more deeply.
 
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#13
I never heard of an Argant diagram, but each pole p in the s plane corresponds to a term a exp(pt) in the impulse response. When Re(p) > 0, the magnitude tends to infinity for increasing t.

(For completeness: physical systems normally don't predict the future, so the response to an impulse at t = 0 is 0 for t < 0.)
 
#15
Yes, that's what it is called. They used to be impopular because conditionally stable valve amplifiers tended to oscillate when the valves were not fully heated up yet, but they are used all over the place nowadays in things like sigma-delta modulators and class D amplifiers.
 
#16
edbarx said:
Like you, I finished my degree some decades ago. I happened to be browsing some books I own about electronics to reread about poles and zeros, and found a book written by F.A. Wilson, and since I read a whole book series about electronics during my adolescence by the same author, I searched for the topic and found it. Searching online, I found the linked document. At the university it was custom that theories and models were first applied generally and then to more specific situations/problems.

I wish I were still that young, but years have the bad habit of passing so quickly, I am now approaching retirement age. What I am doing is a mental refreshment because of an incessant passion to understand more deeply.
Click to expand...
Ed do you have a link to that book? Or a title?
Wilson appears to be a prolific writer.

Jan
 
#19
edbarx said:
I made a typographic mistake. It should read Argand Diagram which is the Cartesian Two dimensional coordinate system with Y replaced by imaginary numbers, that is, multiples of i {with i = sqrt(-1)}.
Click to expand...
Then I do know what it is, I just never heard that name for it. Re(p) > 0 corresponds to exponentially growing terms in the impulse response. (I'm now referring to the poles of the closed-loop transfer, not of the loop gain. Some other posts are about the poles of the loop gain.)
 
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