Nested Miller Compensation

[promach]

Does anyone know how to derive the transfer function of nested miller compensated amplifier found at page 12 of this thesis https://goo.gl/xJ8NgC ?

I have read several other books, yet no author present a step-by-step derivation usng small-signal model.

[amspire]

Quite often, the easiest way to derive the transfer function is to start out the output and work towards the input. Just alternate between solving for current and voltage.

Going from the output to input gives you an ever increasing equation, but it is very systematic. You do not have to think past the voltage or current you are working on.

If you redraw the circuit, you can make it a bit easier - like bundling the RCs together as one step in the chain of equations.

You end up with an equation of the form V_in = big equation with a lot of V_out terms.

You then have to wrangle the equation into a V_out/V_in form. You can use Matlab or the free Octave to simplify the algebraic manipulation. It can be done by hand, but mistakes are easy.

[Wimberleytech]

Here is the nodal admittance matrix (assuming I did not screw up  ).

[danadak]

Si8gnal Flow Graph will save you a lot of time, and provide visual insight into
circuit behavior.

https://sites.google.com/site/ahmedelmorsy89/projects-1/sfg-signal-flow-graph


Regards, Dana.

[Wimberleytech]

Here is the nodal admittance matrix (assuming I did not screw up  ).

Confirmed this result with Symbolic Spice.

Here is the SSPICE output for Vout/Vin [V4/V1].  I have further confirmed the approximation for the numerator but have not tackled the denominator.

[0  ]  [GMM1            sCM1+sC1+G1     0                        -sCM1                                  ][V1 ]
[0  ]=[0                  -GMM2                 sCM2+sC2+G2     -sCM2                                 ][V2 ]
[0  ]  [0                  -sCM1                 -sCM2+GMM3        +sCM2+sCM1+sC3+G3        ][V3 ]
[1  ]  [1                  0                         0                          0                                       ][V4 ]


*nested miller

Numerator of: v4/v1

TERMS SORTED ACCORDING TO POWERS OF s

s**2 terms:

 + sCM2*sCM1*GMM1 + sCM1*sC2*GMM1

s**1 terms:

 + sCM2*GMM2*GMM1 + sCM1*GMM1*G2

s**0 terms:

 - GMM3*GMM2*GMM1

TERMS SORTED ACCORDING TO POWERS OF S

S**0 terms:

 + sCM2*sCM1*GMM1 + sCM2*GMM2*GMM1 + sCM1*sC2*GMM1
 + sCM1*GMM1*G2 - GMM3*GMM2*GMM1

************************************************

Denominator of: v4/v1

TERMS SORTED ACCORDING TO POWERS OF s

s**3 terms:

 - sCM2*sCM1*sC3 - sCM2*sCM1*sC2 - sCM2*sCM1*sC1
 - sCM2*sC3*sC1 - sCM2*sC2*sC1 - sCM1*sC3*sC2 - sCM1*sC2*sC1
 - sC3*sC2*sC1

s**2 terms:

 - sCM2*sCM1*GMM3 + sCM2*sCM1*GMM2 - sCM2*sCM1*G3
 - sCM2*sCM1*G2 - sCM2*sCM1*G1 - sCM2*sC3*G1 - sCM2*sC2*G1
 - sCM2*sC1*GMM3 - sCM2*sC1*G3 - sCM2*sC1*G2 - sCM1*sC3*G2
 - sCM1*sC2*G3 - sCM1*sC2*G1 - sCM1*sC1*G2 - sC3*sC2*G1
 - sC3*sC1*G2 - sC2*sC1*G3

s**1 terms:

 - sCM2*GMM3*G1 - sCM2*G3*G1 - sCM2*G2*G1 - sCM1*GMM3*GMM2
 - sCM1*G3*G2 - sCM1*G2*G1 - sC3*G2*G1 - sC2*G3*G1
 - sC1*G3*G2

s**0 terms:

 - G3*G2*G1

[promach]

@Wimberleytech

This is not shown for two reasons. First, this is considered to be
trivial and no more than algebraic manipulation of a set of equations
of the block diagram of Figure 2.5. Secondly, the "exact" transfer
function is far more complex than the one given in equation 2.7. The
rules applied to the simplification are given on page 11.

For the PFC amplifier, please read section 2.4.2. Again, the "exact"
transfer function is cumbersome, yet uninteresting to an expert
reader. Still, in this section you have all major steps but without
going into excessive detail. Equation (2.16) give the nodal
description of the PFC amplifier given in Figure 2.8 b).

The thesis has only the major steps, but still in as much detail for
you to duplicated the results given.

The above is reply from the thesis author.

I need some time to go through both the calculations made by you (on NMC amplifier) and the thesis author (on PFC amplifier).

[Wimberleytech]

@Wimberleytech


I need some time to go through both the calculations made by you (on NMC amplifier) and the thesis author (on PFC amplifier).

I believe the matrix and resulting transfer function I provided is accurate.  In order to simplify, you need to make approximations like
S(C1 + C2) ~ C2 when C2 >> C1. 
It is that kind of approximation the author is referring to.
For the numerator, you need to factor the third order into a second order and first order.  I have not dont it yet, but it is that process that will reveal opportunities to eliminate terms by approximation.

[promach]

factor the third order into a second order and first order.

Could you show the above by a very simple example ?

Besides, why do you invert the current direction polarity of -gm2*V1 as in the second KCL equation in the matrix ?

[Wimberleytech]

factor the third order into a second order and first order.

Could you show the above by a very simple example ?

The thesis states that Cm1, Cm2, CL >> Co1-3
Therefore, whenever you see Cm1+any Co, ignore the Co term.  Same for Cm2.
He also ignores any factor containing the output conductance.

Please apply these assumptions and modify the Symbolic spice V4/V1 output accordingly, and I will give you the next step.

[promach]

modify the Symbolic spice V4/V1 output

This piece of software is not free and limited demo version is usable for up to only 6 nodes.  Besides, I am using Linux OS.

Let me do further reading first.

[f5r5e5d]

http://cirlab.det.unifi.it/Sapwin/ may be helpful, outputs a Matlab .m file of the symbolic TF

Octave is highly Matlab compatable, now includes a symbolics package wrapping Python's sympy if you want a free SW chain

[promach]

Besides, why do you invert the current direction polarity of -gm2*V1 as in the second KCL equation in the matrix ?

I still do not understand this. Would you mind shedding some light ?

Is it because for -gm2 (example topology would be common source single-stage NMOS amplifier), impedance looking the drain is so much larger ??

[Wimberleytech]

Besides, why do you invert the current direction polarity of -gm2*V1 as in the second KCL equation in the matrix ?

I still do not understand this. Would you mind shedding some light ?

Is it because for -gm2 (example topology would be common source single-stage NMOS amplifier), impedance looking the drain is so much larger ??

I replaced the triangles in Fig 2.5 with VCCS and did nodal analysis at node 2 by summing all currents OUT of the node.  I ignored c02 because it was not necessary to address your gm2 question

[Wimberleytech]

modify the Symbolic spice V4/V1 output

This piece of software is not free and limited demo version is usable for up to only 6 nodes.  Besides, I am using Linux OS.

Let me do further reading first.

I did not intend for you to use SSPICE.  I meant that you should analyze the output (which I provided) by hand.

At any rate, if you were to use SSPICE, 6 nodes is plenty.  I would not want to deal with anything beyond a 6x6 matrix.