Notice andy_c...
....that in equations 4 an 5 respectively, as the magnitude of 'T' tends to zero, Tv tends to Z2/Z1, while Ti asymptotes to Z1/Z2....
This will occur irrespective of the behaviour of 'T' with frequency...
Thus this method is inherently flawed beyond unity loop-gain frequency, where 'T' is necessarily small...
This is also evident in the final expression, where the product of Tv and Ti in the numerator tends to unity, as T falls with frequency....giving rise to a spurious zero....
....that in equations 4 an 5 respectively, as the magnitude of 'T' tends to zero, Tv tends to Z2/Z1, while Ti asymptotes to Z1/Z2....
This will occur irrespective of the behaviour of 'T' with frequency...
Thus this method is inherently flawed beyond unity loop-gain frequency, where 'T' is necessarily small...
This is also evident in the final expression, where the product of Tv and Ti in the numerator tends to unity, as T falls with frequency....giving rise to a spurious zero....
Hi Mike,
Thanks for posting that. I went through your analysis, and I believe you're missing a minus sign on the Tv calculation. Your diagram was confusing me because the current source is unlabeled. If it is indeed gm*Vx, and you have the same sign convention with respect to voltage arrow tip and tail as a vector uses, and gm is positive, then you've got a positive feedback situation.
I decided instead to attempt to verify what you're saying by using the diagrams and formulas shown here: http://www.spectrum-soft.com/news/spring97/loopgain.shtm. I modeled an op-amp with open-loop differential input impedance Zi, open-loop output impedance Zo, and a VCVS of A(s)*Vd, where Vd is the difference-mode input voltage. To simplify, I used a unity-feedback voltage follower configuration. I grounded the non-inverting input of the op-amp. This gives a similar mathematical form to what you are using. I found Gi and Gv (referring to the Spectrum Soft notation). These turned out to be:
Gi = Zi / Zo * (1 + A(s))
Gv = -(A(s) + Zo / Zi)
Note the minus sign on Gv. Next I assumed A(s)->0 as s->infinity. Taking this limit and plugging into the formula:
G = (Gi * Gv - 1) / (Gi + Gv + 2)
it's seen that the numerator does not go to zero, but rather to -2. I ended up with:
G = (-2 * Zi * Zo) / ((Zi - Zo)^2)
Thanks for posting that. I went through your analysis, and I believe you're missing a minus sign on the Tv calculation. Your diagram was confusing me because the current source is unlabeled. If it is indeed gm*Vx, and you have the same sign convention with respect to voltage arrow tip and tail as a vector uses, and gm is positive, then you've got a positive feedback situation.
I decided instead to attempt to verify what you're saying by using the diagrams and formulas shown here: http://www.spectrum-soft.com/news/spring97/loopgain.shtm. I modeled an op-amp with open-loop differential input impedance Zi, open-loop output impedance Zo, and a VCVS of A(s)*Vd, where Vd is the difference-mode input voltage. To simplify, I used a unity-feedback voltage follower configuration. I grounded the non-inverting input of the op-amp. This gives a similar mathematical form to what you are using. I found Gi and Gv (referring to the Spectrum Soft notation). These turned out to be:
Gi = Zi / Zo * (1 + A(s))
Gv = -(A(s) + Zo / Zi)
Note the minus sign on Gv. Next I assumed A(s)->0 as s->infinity. Taking this limit and plugging into the formula:
G = (Gi * Gv - 1) / (Gi + Gv + 2)
it's seen that the numerator does not go to zero, but rather to -2. I ended up with:
G = (-2 * Zi * Zo) / ((Zi - Zo)^2)
Hi Mike,
Thanks for your reply. How are you authoring the PDF with all the equations by the way?
Anyway, look carefully at your second and third equations after Figure 2 in the PDF. Clearly
i3 = -vin / Z2
But you've already got a minus sign before i3, so those two minus signs become a plus. So we must have
iy = gm * vin + vin / Z2
Thanks for your reply. How are you authoring the PDF with all the equations by the way?
Anyway, look carefully at your second and third equations after Figure 2 in the PDF. Clearly
i3 = -vin / Z2
But you've already got a minus sign before i3, so those two minus signs become a plus. So we must have
iy = gm * vin + vin / Z2
Hi again Mike,
After following through with the calculations having the sign change per my above post, I ended up with:
T = (1 + TiTv) / (Tv - Ti - 2)
referenced to the sign conventions of figures 1 and 2, which is the same as your equation 13 except that Ti is negated. As frequency becomes large, Tv and Ti approach negative reciprocals of each other (Tv being negative and Ti being positive) and indeed the numerator of the expression for T does go to zero as you originally stated. So it looks like somewhere in the spectrum-soft equations there's a sign flipped. I'll have to go through the algebra with their notation to figure that one out. Of course, it could be my screwup too.
Do you agree with the sign issue I brought up in my previous post? And if so, after making changes do you come up with the formula above for the sign convention of figures 1 and 2?
After following through with the calculations having the sign change per my above post, I ended up with:
T = (1 + TiTv) / (Tv - Ti - 2)
referenced to the sign conventions of figures 1 and 2, which is the same as your equation 13 except that Ti is negated. As frequency becomes large, Tv and Ti approach negative reciprocals of each other (Tv being negative and Ti being positive) and indeed the numerator of the expression for T does go to zero as you originally stated. So it looks like somewhere in the spectrum-soft equations there's a sign flipped. I'll have to go through the algebra with their notation to figure that one out. Of course, it could be my screwup too.
Do you agree with the sign issue I brought up in my previous post? And if so, after making changes do you come up with the formula above for the sign convention of figures 1 and 2?
Huge Errors in previous attachements....
....looks i got myself tangled up in conventions.....
...Obviously, (with hindsight...
), one must obtain the same answer regardless of conventions used....
...thanks for the corrections Andy......
apologies for any inconvenience caused.......
O.K....so here we go..:
....looks i got myself tangled up in conventions.....
...Obviously, (with hindsight...
), one must obtain the same answer regardless of conventions used.... ...thanks for the corrections Andy......
apologies for any inconvenience caused.......
O.K....so here we go..:
Please Note...
...that the observations made here are still valid:
http://diyaudio.com/forums/showthread.php?postid=464668#post464668
http://diyaudio.com/forums/showthread.php?postid=463340#post463340
http://diyaudio.com/forums/showthread.php?postid=463349#post463349
http://diyaudio.com/forums/showthread.php?postid=463989#post463989
Cheers!
...that the observations made here are still valid:
http://diyaudio.com/forums/showthread.php?postid=464668#post464668
http://diyaudio.com/forums/showthread.php?postid=463340#post463340
http://diyaudio.com/forums/showthread.php?postid=463349#post463349
http://diyaudio.com/forums/showthread.php?postid=463989#post463989
Cheers!
Re: Huge Errors in previous attachements....
No inconvenience whatever. In fact, it was a great exercise going through all this, as the algebra of the approach you used was much simpler than Thompson's. It gets even easier if the Z's are replaced with Y's too. Also, the Hurst articles were quite good and very thought-provoking. I'm now going to repeat this exercise using the notation and sign conventions of the spectrum-soft site to see if it was an error in my algebra or their formulas (possible wrong formula for a given sign convention) that led me to an incorrect conclusion.
I now agree with you that the computed loop gain going to zero at high frequencies is due to subtraction from one of a number close to one in the numerator of the loop gain expression. However, your assertion that the data obtained is essentially useless above the unity loop gain frequency seems, frankly, a bit bold to me. I understand your reasoning in a heuristic sense, but when it comes down to brass tacks it beomes a question of how much error is really encountered as a function of the magnitude of the loop gain. This might be expressed, say, as the error in degrees of the phase margin and the error in dB of the gain margin. I'll mess around with some simulations and maybe some math to try to get some idea of this. In fact, I think one could dream up some simple circuits where the gain and phase margins could be computed exactly by hand. Then the results could be compared with simulation to get an idea how much error is really being introduced here.
Anyway, thanks for all your efforts and the references too. It's been an interesting and illuminating discussion.
mikeks said:
No inconvenience whatever. In fact, it was a great exercise going through all this, as the algebra of the approach you used was much simpler than Thompson's. It gets even easier if the Z's are replaced with Y's too. Also, the Hurst articles were quite good and very thought-provoking. I'm now going to repeat this exercise using the notation and sign conventions of the spectrum-soft site to see if it was an error in my algebra or their formulas (possible wrong formula for a given sign convention) that led me to an incorrect conclusion.
I now agree with you that the computed loop gain going to zero at high frequencies is due to subtraction from one of a number close to one in the numerator of the loop gain expression. However, your assertion that the data obtained is essentially useless above the unity loop gain frequency seems, frankly, a bit bold to me. I understand your reasoning in a heuristic sense, but when it comes down to brass tacks it beomes a question of how much error is really encountered as a function of the magnitude of the loop gain. This might be expressed, say, as the error in degrees of the phase margin and the error in dB of the gain margin. I'll mess around with some simulations and maybe some math to try to get some idea of this. In fact, I think one could dream up some simple circuits where the gain and phase margins could be computed exactly by hand. Then the results could be compared with simulation to get an idea how much error is really being introduced here.
Anyway, thanks for all your efforts and the references too. It's been an interesting and illuminating discussion.
Well, I went through the diagrams and equations at the spectrum-soft site at http://www.spectrum-soft.com/news/spring97/loopgain.shtm. Turns out they use the same convention for the signs of voltages and directions of currents used in your PDF file in post #66, which happens to also be the configuration I worked out in detail. There's an overall sign flip of what they call G vs what we've been calling T, but that's explained by the difference between return ratio and loop gain, which is a non-issue. But even after flipping that sign, there's still an error in the sign of Gv, which is what we've been calling Tv. In other words, their formula for G is in error given the sign convention for voltages and currents that they state in their diagrams. That's where the error in my calculations came from, where it looked like the numerator of the loop gain expression wasn't going to zero as frequency became large. So, another mystery solved. Ain't science wonderful?
Hi Andy...
By the way Andy,
I think Hurst is incorrect...when he suggests that the foward path through the feedback network ensures that return ratio is not equal to loop gain....
http://www.ece.ucdavis.edu/~hurst/papers/ExactSimFB,CAS91.pdf
http://www.ece.ucdavis.edu/~hurst/papers/Compare2FbApproaches,EDU.pdf
Infact (my opinion anyway), the foward path through the feedback network only affects the 'open loop' gain and the closed loop gain, as defined for the inverting configuration in section 6.4, pg. 96 here:
http://focus.ti.com/lit/an/slod006b/slod006b.pdf
...and does not alter the fact that the modulus of return ratio (with independent sources disabled) and loop gain are always identical.....
what do you think..?
andy_c said:
By the way Andy,
I think Hurst is incorrect...when he suggests that the foward path through the feedback network ensures that return ratio is not equal to loop gain....
http://www.ece.ucdavis.edu/~hurst/papers/ExactSimFB,CAS91.pdf
http://www.ece.ucdavis.edu/~hurst/papers/Compare2FbApproaches,EDU.pdf
Infact (my opinion anyway), the foward path through the feedback network only affects the 'open loop' gain and the closed loop gain, as defined for the inverting configuration in section 6.4, pg. 96 here:
http://focus.ti.com/lit/an/slod006b/slod006b.pdf
...and does not alter the fact that the modulus of return ratio (with independent sources disabled) and loop gain are always identical.....
what do you think..?
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