Finding DC Operating Points of Nonlinear Circuits Using ...

Finding DC Operating Points of Nonlinear Circuits Using Carleman Linearization

Harry WeberTheoretische ElektrotechnikLeibniz Universität Hannover

Ljiljana TrajkovićSchool of Engineering Science

Simon Fraser University

Wolfgang MathisDidaktik der Elektrotechnik und Informatik

Leibniz Universität Hannover

MWSCAS 202111.08.2021

Overview

• Motivation• Analysis of DC operating points

• Carleman linearization• Nonlinear algebraic equations and infinite dimensional linear systems

• Self-consistent technique• Approximation over a predefined interval

• Examples:• Tunnel diode circuit• CMOS multivibrator

• Summary and extensions

2H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021

Motivation: DC Operating Points

3

DC operating points? Nonlinear algebraic equation

• Multiple DC operating points (OPs) possible• DC OPs depend on circuit parameters• Finding DC OPs using numerical methods such

as Newton-Raphson method are difficult due to different attraction domains

• Starting point problem

H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021

Motivation: DC Operating Points


DC operating points? Nonlinear algebraic equation

• Multiple DC operating points (OPs) possible• DC OPs depend on circuit parameters• Finding DC OPs using numerical methods such

as Newton-Raphson method are difficult due to different attraction domains

• Starting point problem

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Embedding

• For a parameter variation additional DC OPs may appear or disappear• System knowledge for devising a successful embedding• Improved global analysis for polynomial models Carleman linearization

x

Two DC OPsappear

Homotopy Methods


Carleman Linearization: Simple Example

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• Two DC OPs at and : numerically calculated by Newton-Raphson method• Proposed: Analysis using Carleman linearization


• Linear difference equations:

• Desired solution corresponds to:

Carleman Linearization: Procedure

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Transform to equivalent difference equations:multiply with and useRestricted to polynomial case


Carleman Linearization: Procedure

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Difference equations correspond to an infinite dimensional linear system:

Infinite dimensional linear system

Restricted to polynomial case


Transform to equivalent difference equations:multiply with and use

• Linear difference equations:

• Desired solution corresponds to:

Carleman Linearization: Approximation

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Approximation by truncation

*H. Weber and W. Mathis, “Analysis and design of nonlinear circuits with a self-consistent Carleman linearization,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 65, no. 12, pp. 4272–4284, Dec. 2018.

• Valid only in the vicinity of the origin (local approximation)


Infinite dimensional linear system: no close form solution

Carleman Linearization: Approximation

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Approximation by truncation Self-consistent technique* with maximal dimension

• Adaption of coefficients over a predefined interval (global approximation)

• Calculation of coefficientsby a least square fit

*H. Weber and W. Mathis, “Analysis and design of nonlinear circuits with a self-consistent Carleman linearization,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 65, no. 12, pp. 4272–4284, Dec. 2018.


Infinite dimensional linear system: no close form solution

• Valid only in the vicinity of the origin (local approximation)

Self-Consistent Technique



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Replace and approximate over


Predefined interval


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Self-consistent technique


Predefined interval

Replace and approximate over

Solutions of the Linear Equation

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Solving

Truncation

DC OP

Self-consistent technique over the interval .

Solving

DC OP

• Calculate a DC OP in a predefined interval• Obtain a starting point for the Newton method• Case: Multiple or no DC OPs located in the

given interval

DC OPs :

• Truncated method provides only an approximation in the vicinity of the origin


Example: Tunnel Diode Circuit

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*

*H.-C. Wu, “Analysis of nonlinear resistive networks having multiple solutions with spline function techniques”, Ph.D. dissertation, Iowa State University, 1977.**A. N. Willson Jr., “The no-gain property for networks containing three-terminal elements,” IEEE Transactions on Circuits and Systems, vol. 22, no. 8, pp 678–687, Aug. 1975.

Self-consistent Carleman linearization over

• Initial interval is given by the sum of all supply voltages if the no-gain property holds**

Small extension in order to include the case



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Initial interval is defined by the supply voltage

Self-consistent Carleman linearization with maximal dimension



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Solutions obtained by the self-consistent Carleman linearization

• Solutions do not “converge” with the increasing dimension of the system

• Divide into half




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• Solutions in “converge”• Solutions in do not “converge”• Divide further




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• Solutions in “converge” outside of the sub-interval

• Solutions in do not “converge”• Divide




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• Solutions “converge” within sub-intervals

• Is a DC OP in ?• Expand the sub-interval so that

already identified DC OP is included




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• Expanded• Solutions “converge” to the

already found DC OP• All DC OPs in the initial interval

are identified



General Procedure

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Proposed procedure for detecting all DC OPs:1. Initial interval is defined by the supply voltage (the no-gain property)

Initial interval defined by the supply voltage


General Procedure

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Proposed procedure for detecting all DC OPs:1. Initial interval is defined by the supply voltage (the no-gain property)2. Check if the procedure “converges” within the given interval as the dimension of

the system increases

Solutions do not “converge”within the interval


General Procedure

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the system increases3. Divide interval if criteria is not satisfied4. Repeat procedure for a given number of steps


General Procedure

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the system increases3. Divide interval if criteria is not satisfied4. Repeat procedure for a given number of steps5. If no DC OP is identified in a sub-interval, expand the interval so that an already

identified DC OP is included and recheck criteria6. If criteria is fulfilled, all DC OPs are found within the initial interval


Example: CMOS Multivibrator

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*A. Buonomo, “A new CMOS astable multivibrator and its nonlinear analysis,” International Journal of Circuit Theory and Applications, vol. 39, no. 2, pp. 91–102, Feb. 2011.

• Initial interval given by supply voltage

• One trivial DC OP at the origin

Polynomial model*:DC operating points:

Decomposition into two sub-networks


Example: CMOS Multivibrator

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• Solutions “converge” within given intervals

• All possible DC OPs in the initial interval are identified



Summary and Extensions

• Considered DC operating points (OPs) of nonlinear circuits described by polynomial models

• Calculated DC OPs over a given interval based on the self-consistent Carleman linearization

• The initial interval was defined by the supply voltage if the no-gain property holds• Successive partition of the initial interval was used to identify all DC OPs• The procedure was illustrated for two nonlinear circuits: tunnel diode and CMOS

multivibrator• The approach is applicable to nonlinear circuits decomposable into sub-networks• Extensions of the procedure for:

• higher dimensional algebraic equations• nonlinear equations with transcendental functions


Thank you for your attention!

Finding DC Operating Points of Nonlinear Circuits Using ...

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