constant gm biasing question

[Julian18]

positive feedback in constant gm bias circuit

Hi, there.
I am kinda confused with the analysis of constant gm circuit . I am referring Tom Lee's book "The Design of CMOS Radio-Frequency Integrated Circuits", 1st edition and the derivation of constant gm circuit is as follows(see also the attachment):
Cutting the point at V1 and applying a test incremental voltage at the gate of M8, this voltage gets amplified at the diode connected MOS M9, and as Tom Lee says, "A self-consistent solution is possible only if that voltage equals the original gate drive." Thus:
gm9=gm8/(1+gm8xRs)
But, were the above equation to hold, the positive feedback will force the circuit to oscillate, which is not very good for a biasing circuit, furthermore, as I simulated the circuit, I found the above equation does not hold(I considered all the nonidealities as possible as I can). So if there is something wrong with this derivation or I have done something dumb?

BTW, in the 2ed Version of this book, the derivation has changed, and a more reasonable relation is given, which seems accurate for long channel devices.

Thanks.

 

[Julian18]

hi, anybody wants to say something about this circuit?

 

[transbrother]

Hello Julian,
If you calculate the loop gain of the circuit, it is ~ 1/(1+gm8RS) if you assume (first order) that currents thru M9 and M8 are equal; current thru M2 and M3 are equal. This also does not take into account body effect.

If this is the case then although there is a positive feedback it will die because the loop gain < 1. So, there shouldnt be any oscillations.

 

[Julian18]

Hi,
but In Tom Lee's book(1st edition), it says that for self-consistent reason, this gain must be equal to 1(if I am not wrong), which is the condition for oscillation.

 

[transbrother]

Yes. oscillation results if loop gain is 1 when there is a total system phase shift of 360 degrees. However, if the magnitude of loop gain is always greater than 1, oscillations will not die out. And, as I have said, your oscillations in the ckt will die out if the loop gain is <1, which is the case in this bias circuit.

 

[Julian18]

Yes. oscillation results if loop gain is 1 when there is a total system phase shift of 360 degrees. However, if the magnitude of loop gain is always greater than 1, oscillations will not die out. And, as I have said, your oscillations in the ckt will die out if the loop gain is <1, which is the case in this bias circuit.

HI:
But based on the derivation of the Book, the biasing circuit only works when loop gain is equal to 1.

 

[transbrother]

I think you are referring to the startup phoenomenon. Yes, there needs to a positive feedback loop in order to startup this loop. during startup, when the system is slightly perturbed and when you have a small current flowing thru the transistors, gm is also very low. At this point, you'll have a loop gain of close to 1 (according to the equation, if gm ~ 0, then loop gain ~ 1) In such a case, the bias circuit is regenerative and node voltages will rise because of the positive feedback. However, when the nodes voltages have started to rise, the loop will become less and less regenerative because gm8 is rising causing the oscillations to die down.

I misunderstood your question at first. I thought you said you were seeing oscillation after startup. And that is why I told you that it could not occur if devices are sized according to the equation because the loop gain is less than 1. However, during startup this is not the case.

 

[Julian18]

I think you are referring to the startup phoenomenon. Yes, there needs to a positive feedback loop in order to startup this loop. during startup, when the system is slightly perturbed and when you have a small current flowing thru the transistors, gm is also very low. At this point, you'll have a loop gain of close to 1 (according to the equation, if gm ~ 0, then loop gain ~ 1) In such a case, the bias circuit is regenerative and node voltages will rise because of the positive feedback. However, when the nodes voltages have started to rise, the loop will become less and less regenerative because gm8 is rising causing the oscillations to die down.

I misunderstood your question at first. I thought you said you were seeing oscillation after startup. And that is why I told you that it could not occur if devices are sized according to the equation because the loop gain is less than 1. However, during startup this is not the case.

Thanks tran:
In fact, now i am thinking maybe this is an error in the derivation of gm-constant circuit in that Book, after all, the author changed this piece of derivation in the 2nd edition.

BTW, do you have a copy of that book(Tom Lee's RF CMOS, 1st edition), maybe you can check it out, and find if I am wrong.

B.R.

 

[transbrother]

Hey Julian,
I dont have a copy of the book. However, if anything I think the book is probably a little confusing from what you say.

How is the explanation in the second edition? Is it more consitent with what we are discussing?

 

[Julian18]

Hi tran:
in fact two different design equations are given in two editions.
In the first edition, the derivation is based on small siganl analysis, and gives the result:
gm9=gm8/(1+gm8xRs)
while in the second edition, a large signal derivation is given and the result is
gm9=2(1-1/sqrt(m))/Rs
where m = (width of M8)/(width of M9)

B.R.

 

[yantom]

The derivative of the first edtion is wrong, the second edition is right.
When i first saw the first edtion, i also found the problem, and i asked my boss, he said Thomas Lee made a lot of mistakes in the first edition and he changed in the second edition.

 

[fendy]

Hi,
Please check the attached file, hope this can help you.

 

[jszair]

The derivative of the first edtion is wrong, the second edition is right.
When i first saw the first edtion, i also found the problem, and i asked my boss, he said Thomas Lee made a lot of mistakes in the first edition and he changed in the second edition.

I'm also studying this ckt.

Why is first edition equation wrong? Break the loop at V1 The loop gain from gate of M8 to gate of M9 is just gm8/gm9/(1+gm8Rs)

But I don't know how making M8 larger than M9 renders this loop gain less than unity for stability... Any hints?