finding dc operating points of nonlinear circuits using
Post on 25-Dec-2021
8 Views
Preview:
TRANSCRIPT
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
4H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
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
5
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
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
Carleman Linearization: Simple Example
6
• Two DC OPs at and : numerically calculated by Newton-Raphson method• Proposed: Analysis using Carleman linearization
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
• Linear difference equations:
• Desired solution corresponds to:
Carleman Linearization: Procedure
7
Transform to equivalent difference equations:multiply with and useRestricted to polynomial case
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
Carleman Linearization: Procedure
8
Difference equations correspond to an infinite dimensional linear system:
Infinite dimensional linear system
Restricted to polynomial case
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
Transform to equivalent difference equations:multiply with and use
• Linear difference equations:
• Desired solution corresponds to:
Carleman Linearization: Approximation
9
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)
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
Infinite dimensional linear system: no close form solution
Carleman Linearization: Approximation
10
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.
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
Infinite dimensional linear system: no close form solution
• Valid only in the vicinity of the origin (local approximation)
Self-Consistent Technique
11H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
Self-Consistent Technique
12
Replace and approximate over
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
Predefined interval
Self-Consistent Technique
13
Self-consistent technique
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
Predefined interval
Replace and approximate over
Solutions of the Linear Equation
14
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
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
Example: Tunnel Diode Circuit
15
*
*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
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
Example: Tunnel Diode Circuit
16
Initial interval is defined by the supply voltage
Self-consistent Carleman linearization with maximal dimension
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
Example: Tunnel Diode Circuit
17
Solutions obtained by the self-consistent Carleman linearization
• Solutions do not “converge” with the increasing dimension of the system
• Divide into half
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
Self-consistent Carleman linearization with maximal dimension
Example: Tunnel Diode Circuit
18
Solutions obtained by the self-consistent Carleman linearization
• Solutions in “converge”• Solutions in do not “converge”• Divide further
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
Self-consistent Carleman linearization with maximal dimension
Example: Tunnel Diode Circuit
19
Solutions obtained by the self-consistent Carleman linearization
• Solutions in “converge” outside of the sub-interval
• Solutions in do not “converge”• Divide
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
Self-consistent Carleman linearization with maximal dimension
Example: Tunnel Diode Circuit
20
Solutions obtained by the self-consistent Carleman linearization
• Solutions “converge” within sub-intervals
• Is a DC OP in ?• Expand the sub-interval so that
already identified DC OP is included
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
Self-consistent Carleman linearization with maximal dimension
Example: Tunnel Diode Circuit
21
Solutions obtained by the self-consistent Carleman linearization
• Expanded• Solutions “converge” to the
already found DC OP• All DC OPs in the initial interval
are identified
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
Self-consistent Carleman linearization with maximal dimension
General Procedure
22
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
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
General Procedure
23
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
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
General Procedure
24
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 increases3. Divide interval if criteria is not satisfied4. Repeat procedure for a given number of steps
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
General Procedure
25
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 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
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
Example: CMOS Multivibrator
26
*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
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
Example: CMOS Multivibrator
27
• Solutions “converge” within given intervals
• All possible DC OPs in the initial interval are identified
H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
Self-consistent Carleman linearization with maximal dimension
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
28H. Weber, Lj. Trajković, and W. Mathis: MWSCAS 202111.08.2021
Thank you for your attention!
top related