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Simulators >> Circuit Simulators >> Linearization about a varying operating point
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Message started by Cri Azzolini on Dec 21^st, 2005, 1:25am

Title: Linearization about a varying operating point
Post by Cri Azzolini on Dec 21^st, 2005, 1:25am

Hi all,

regarding PAC analysis in SpecreRF, there is a point I am not sure to understand. Spectre's manual describes PAC as "a small-signal analysis like AC analysis, except the circuit is first linearized
about a periodically varying operating point as opposed to a simple DC perating point."

If it is quite clear how a set of non-linear equation is linearized about a specific point, I can not understand how it can be linearized about a trajectory. As far as I can guess, the same equations may be linearized about all the operating points belonging to the trajectory and then "averaged" in some way. Is it correct? May anybody explain this point?

I realize that this topic is not strictly related to design activities but I am sure that a deep understanding of simulators is of the greatest importance for all IC designers.

Thanks and regards,
Cri

Title: Re: Linearization about a varying operating point
Post by Geoffrey_Coram on Dec 21^st, 2005, 8:11am

There is probably a white paper on PAC available from Cadence. Otherwise, try "Efficient AC and Noise Analysis of Two-Tone RF Circuits" by Telichevesky, Kundert, and White, from DAC 1996.

The "averaging" is a reasonable way to think about it.

Title: Re: Linearization about a varying operating point
Post by vborich on Dec 21^st, 2005, 8:24am

>>As far as I can guess, the same equations may be >>linearized about all the operating points belonging to >>the trajectory and then "averaged" in some way. Is it >>correct? May anybody explain this point?

Yes, they are linearized about all operating points belonging to the trajectory, but no averaging takes place. The resulting derivatives (conductances and capacitances) along the solution trajectory are periodic waveforms admitting a Fourier series representation. The Fourier series coefficients establish the coupling between the sidebands of the small signal solution.

In the case of purely linear circuits, derivative waveforms are constant and no cross-coupling exists between the sidebands. The PAC problem then effectively breaks into a series of AC problems, one for each analysis sideband.

Title: Re: Linearization about a varying operating point
Post by Cri Azzolini on Dec 21^st, 2005, 9:02am

Hi Vborich!

Thanks for your reply!
If I well understand the time varying operating point influences the linearized equations leading to time-dependant coefficients: those coefficients may be transformed using the Fourier's series.
For instance, if I am interested in the PAC solution in the baseband (harmonic=0), the PAC analysis is performed as an AC analysis in which the linearized equations are computed using the 0th-order Fourier coefficients. In this way I was improperly thinking about "averaging".
Is this correct? I am not sure ....
In the meanwhile I am going to look for the paper suggested by Geoffrey_Coram (thanks Geoffrey!).

Thanks,
Cri

Title: Re: Linearization about a varying operating point
Post by vborich on Dec 21^st, 2005, 10:13am

The notion of "averaging" as I know it is absent from PAC. PAC computes the solution around all of the large-signal harmonics simultaneously. Roughly, the size of the PAC system is N*S, where N is number of circuit nodes and S is the number of sidebands.

Your 0th harmonic sideband, i.e. the IF component, has contributions from the sidebands of all harmonics at all circuit nodes. The cross-coupling between the sidebands is established by the non-DC F.S. coefficients. The stronger the nonlinearity, the larger the coefficients and the stronger the sideband coupling. In a purely linear circuit, there is no cross-coupling -- linear circuits do not convert frequencies. But PAC still solves the equations simulatenously, it's just that the solution is mathematically equivalent to a sequence of AC simulations at each of the sideband frequencies.

Another good paper is "Nonlinear-linear analysis of resistive mixers" by S. Egami in IEEE Trans MTT, 1974.