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Question on Hajimiri-Lee model (Read 627 times)
iVenky
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Question on Hajimiri-Lee model
Feb 03rd, 2019, 11:47pm
 
I am reading the Hajimiri-Lee phase noise model, and got a question on that. If you have an LC tank circuit that is free-running and I inject a current i(t) (dirac current) at instants either t1 or t2 (shown in the figure), depending on when you inject the phase of the output changes (as shown). If I inject it at t1, then phase of the output changes while if I inject it at t2 only the amplitude changes but not the phase. What I don't understand here is when I inject the current i(t) at instant t1, shouldn't the amplitude also change just by superposition of oscillation waveform and the input i(t) energy? This is the part that's confusing to me. And what happens if I inject a really large i(t) that the delta voltage step at t1 exceeds the initial peak voltage to begin with, would the amplitude still change?

In the dV/dt and V plot, t1 corresponds to 'b' and t2 corresponds to 'a'.
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Ken Kundert
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Re: Question on Hajimiri-Lee model
Reply #1 - Feb 4th, 2019, 1:10am
 
You are correct. The Hajimiri-Lee model is not to be taken too literally. It is more a conceptual/intuitive explanation. The impulses are assumed to be small.

-Ken
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Geoffrey_Coram
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Re: Question on Hajimiri-Lee model
Reply #2 - Feb 6th, 2019, 6:51am
 
It's intuitive, but not always right:
https://ieeexplore.ieee.org/abstract/document/933331/
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iVenky
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Re: Question on Hajimiri-Lee model
Reply #3 - Feb 6th, 2019, 8:09am
 
Thanks for the replies.

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iVenky
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Re: Question on Hajimiri-Lee model
Reply #4 - Feb 21st, 2019, 10:41pm
 
1) While reading about the model again, the author mentions that the ISF is a dimensionless quantity. If you consider this expression (attached), I feel that ISF function Γ(ωτ) has a unit of radians otherwise, I don't see the equation balance on both sides. Is this right?
2) Also, does the ISF change with the Q factor of LC tank? It's interesting that the expression that the author derives is only for an ideal lossless tank.
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Ali Hajimiri
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Re: Question on Hajimiri-Lee model
Reply #5 - Mar 2nd, 2019, 3:50pm
 
This is a good question and deserves a more nuanced answer. The impulse as you have drawn it at t1 does indeed produce both phase and amplitude, changes, but will mostly produce phase and a little bit of amplitude shift. However, there exist an impulse near the transition point for the ideal LC that would produce only phase shift with no net amplitude change (as long as the instantaneous voltage change is not greater than twice the amplitude of the oscillation). The exact location of such an impulse depends on its size. This can be seen from the time-domain waveform shown and the corresponding state-space diagram, shown in the attachment, and can be verified both analytically or using simulations. Now, the smaller this impulse is relative to the oscillation amplitude, the closer the point with zero net amplitude change will be to the zero crossing point. In the limit of an infinitesimal impulse it will be exactly at the zero-crossing point, making that point have zero phase sensitivity from a perturbation perspective. This is not an issue in practice since the model was developed for noise, which in almost all cases can be viewed as a small perturbation.

You also raised another good question about the energetics of this impulse response and how it can induce no amplitude change (namely no change in the net energy of the tank). There is no contradiction here. The answer is that the impulse has a fixed charge (its net area) not necessarily a constant energy. More precisely, the instantaneous power delivered to the tank is the product of the instantaneous tank voltage and the injected current. For the injection shown in the attachment, the voltage is negative for half of the infinitesimal cycle and positive for the other half so the integral (the total power delivered) is actually zero. If you have trouble seeing this, assume that impulse is a short pulse with a non-zero width. So there is no inconsistency here. No net energy is delivered to the tank and thus no net change in the tank amplitude will be observed.

One of the replies to your question on this forum that cites another paper requires some additional clarification. The original paper describing the ISF-based model published in the February 1998 issue of JSSC, offers a model based on the ISF. The model is accurate for phase noise calculation as long as the ISF in determined accurately. The accuracy of this model and its mathematical equivalence have been shown to hold independently by others as well. In addition to the full discussion of the model and its implications for the design, the original paper also offers three methods to determine the ISF in the order of accuracy in an Appendix. The first method (method A) of the appendix is based on direct evaluation of the ISF based on injection of small impulses and measurement of the resultant phase shift. The next two methods are approximations based on some simplified analysis. The paper clearly states that method A "is the most accurate of the three methods presented." The approximate methods B and C are just that, approximations to calculate the ISF. Now the validity of the primary approach of the paper is not affected by the method used to calculated the ISF, it is just a question of accuracy of the method used to calculate the ISF. If method A is used to calculate the ISF, the results are correct, accurate, and intuitive. Even the cited paper by the commenter clearly states that "The first method, a direct calculation in SPICE, with an impulse swept through offsets from a reference point in the oscillator waveform, is correct." Again the point of the other paper is related to the accuracy of the method used to calculate the exact ISF, not the general result of the model. Methods B and C are quick simplifications that may or may not be useful in determining a useful estimate of the ISF, and as simplifications they are by definition "wrong". This is somewhat similar to stating that pi is 3.14 and somebody responding: "Since it is proven that pi is irrational and 3.14=157/50, i.e., a rational number, this is wrong." It does not make 3.14 as useless as -56.8. That is the nature of approximation, which people who design generally appreciate. But regardless, even that inaccuracy is in some of the possible methods of calculating the ISF, not the model itself.

I guess the key take away is that as long as you calculate the ISF correctly (for example using method A), the qualitative and quantitative results obtained from the proposed approach are accurate. Also the insight obtained from the general properties of the oscillator and how its phase noise behave in response to design changes are not affect by the choice of the method used to calculate the ISF.

In a follow-on post, you ask two additional questions, which I try to answer quickly while I am here:
1. ISF is unitless, because radian is unites. It is defined as the ratio of two lengths (the length of the arc facing an angle in a circle divided by the radius.). So as a ratio it is unitless; No inconsistency.
2. The ISF can be determined for any kind of tank. For an ideal (as you put it lossless) tank, it can be determined analytically as a closed-form solution for small perturbation (e.g., noise). The choice of tank Q, will affect the shape of the ISF through oscillation waveform change, and will affect the in^2/Df in the form of the noise it introduces.

I hope you find this helpful.
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