## Nonlinear circuits and systems projects

My research projects in the area of nonlinear systems deal with finding dc operating points, steady state, and transient responses of electronic circuits. These are essential tasks in electrical circuit simulation and involve solving nonlinear differential/algebraic equations. Traditional methods for solving such systems of equations often fail, are difficult to converge, and, often cannot find all the solutions. I investigate the application of homotopy methods to solving nonlinear equations describing circuits consisting of bipolar junction and MOS transistors that traditionally pose simulation difficulties. Our experiments with homotopies led to better understanding of homotopy algorithms and the behavior of nonlinear circuits, and, ultimately, to the development of better circuit simulation tools.

Recent projects:

Improving an electronic circuit simulator based on homotopy methods:(software available)
Sudhir Kumar (2016 MITACS Globalink student) and Dr. Ljiljana Trajkovic

Analog circuits have always been the un-avoidable part of the electronics. Even if most of the systems are now digitalized, still analog circuits depict the real case scenario because before we discretize the system we should have an idea of their transients and their analog variation. Every analog circuits usually consists of voltage or current sources, some resistors, transistors etc. Coming to the transistor, they constitute non-linear part of the circuits. They behave differently with AC and DC sources. The value of voltages and currents determined with DC sources, are known as DC operating points. The DC operating points determination needs the solution of system of non-linear algebraic equation. DC operating points are usually calculated by the Newton-Raphson algorithm. The newton-raphson algorithm fails some times because of the unavailability of initial conditions very close to the solution. Providing a starting point very close to the unknown solution is a difficulty in itself. Therefore alternative methods to solve nonlinear system of equations, such as homotopies, are often applied to find the DC operating points of various circuits. Various homotopies can be implemented. Now modified variable stimulus homotopy has been also implemented and tested. The project is composed of two programs: a parser, developed by Edward Chan improved by Andrea Dyess, and then recently further improved by Joao Eric Melo and Jatin Vikram Singh. It generates a system of equations derived from a circuit description file. A Matlab script written by Heath Hoffman and also new matlab homotopy implements the homotopy method to solve the system of equations. The parser had some problem in nodal analysis of the circuit. The main goal of the project is to identify and correct the problems in parser and test more complicated and simple circuits as well on homotopy and analyse the behaviour of homotopy curves.

Improving an electronic circuit simulator based on homotopy methods:(software available)
Jatin Vikram Singh (2015 MITACS Globalink student) and Dr. Ljiljana Trajkovic

DC operating points, also known as the bias points or the quiescent points, are the values of the voltages and currents established in the circuit with DC sources. Devices, such as transistors, behave differently when biased with DC or AC sources. They are usually combined with resistors that ensure their proper functioning. To find the DC operating points of a transistor circuit, we need to solve a system of nonlinear algebraic equations that describe the DC behavior of the circuit. In this project, we have used a software implementation of homotopy algorithm. Homotopy is an alternate iterative method to find the DC operating points. It is preferred over other iterative techniques because it provides all the possible DC operating points. The software implementation of homotopy is composed of two programs: a parser, developed by Edward Chan improved by Andrea Dyess, and then recently further improved by Joao Eric Melo. It generates a system of equations derived from a circuit description file. A Matlab script written by Heath Hoffman implements the homotopy method to solve the system of equations. The output generated by the parser was incomplete and needed modifications that were in past done manually. The main goal of the project is to identify and correct the problems.

Improving an electronic circuit simulator based on homotopy methods:(software available)
João Erik de Andrade Melo (2014 MITACS Globalink student) and Dr. Ljiljana Trajkovic

Determining voltages and currents in electronic circuits is of great importance to industry and academia. Correct biasing makes the electronic components operate properly and have the desired behavior. These specific voltages and currents are called DC operating points of a circuit. Finding the DC operating points of a transistor circuit involved solving a system of nonlinear algebraic equations that describe the circuit's DC behavior. Newton-Raphson method, which is commonly used to find DC operating points, requires an initial point close enough to the solution that are sometimes difficult to provide. Homotopy methods, an alternative approach to solving nonlinear systems of equations, can be applied to find DC operating points of circuits. In this project, we have used a software implementation of a homotopy method. This implementation is composed for two programs:
- a parser, originally developed by E. Chan and revised by A. Dyess and A. Singhal, that derives the system of equations from a circuit description
- a MATLAB code, written by H. Hoffman, that implements the homotopy method to solve the system of equations.

The main contribution of this project is an improved version of the Parser that generates equations in the correct form required by the MATLAB program.

Homotopy algorithms for solving equations emanating from electronic circuits:(software available)
Archit Singhal (2013 MITACS Globalink student) and Dr. Ljiljana Trajkovic

DC behaviour of electronic circuits is described by systems of non-linear algebraic equations whose solutions are known as circuit¢s DC operating points. DC operating points are usually calculated by using the Newton-Raphson method or its variants such as damped Newton methods. These methods are robust and have quadratic convergence when a starting point sufficiently close to a solution is supplied. The Newton-Raphson algorithms sometimes fail because it is difficult to provide a starting point sufficiently close to an often unknown solution. Computational difficulties in computing the DC operating points of transistor circuits are exacerbated by the exponential nature of the diode-type non-linearities that model semiconductor devices.

We describe here simple software implementation of parameter embedding algorithms for calculating DC operating points of non-linear circuits. The implementation described here, relies on commercially available MATLAB tools. In spite of its simplicity, the implementation is powerful enough to solve benchmark non-linear circuits that possess multiple DC operating points. Parameter embedding methods, also known as continuation methods, are robust and accurate numerical techniques employed to solve non-linear algebraic equations. They are used to solve equations that possess more than one solution. Probability-one homotopy algorithms are a class of embedding algorithms that promise global convergence. Various homotopy algorithms have been introduced for finding multiple solutions of non-linear circuit equations and for finding DC operating points of transistor circuits. They have been successful in finding solutions to highly non-linear circuits that could not be simulated using conventional numerical methods. However, they offer a very attractive alternative for solving difficult non-linear problems where initial solutions are difficult to estimate or where multiple solutions are desired.

Simulation of transistor circuits: effect of model parameters
Paulman Chan (Co-op student) and Dr. Ljiljana Trajkovic (supervisor)

We use PSpice, RSpice, and HSpice to simulate simple one-port circuits composed of two bipolar junction transistors (BJT's) and linear resistors connected in a feedback structure. These simulation tools often yield extremely high voltages between the port terminals, and we are seeking to understand why various versions of Spice yield such high voltages. We hope to show that the choice of BJT parameters, such as Early voltage, play an important role in causing such large voltages and unrealistic simulation results.

Periodic steady-state simulation of oscillators using homotopy methods:
Wanling Ma (M.Sc student, Oregon State University), Dr. Ljiljana Trajkovic, and Dr. Kartikeya Mayaram (supervisor, Oregon State University)

We apply homotopy methods to find periodic steady-state response of autonomous circuits, such as sinusoidal oscillators. We use a formulation suitable for application of homotopy-based algorithms that have been shown to have superior convergence. We have developed a new circuit simulator called SSpiceHom that incorporates the new formulation and employs homotopy algorithms for reliable steady-state simulation of oscillators. The new simulator is based on SSpice and HOMPACK. SSpiceHom is also a tool for exploring the use of globally convergent homotopies for the periodic steady-state analysis of both autonomous and non-autonomous circuits.

• W. Ma, Lj. Trajkovic, and K. Mayaram, ''HomSSPICE: a homotopy-based circuit simulator for periodic steady-state analysis of oscillators,'' Proc. IEEE Int. Symp. Circuits and Systems, Scottsdale, AZ, May 2002, pp. I-645-I-648.
Quantitative analysis of feedback structures for determining operating points in resistive networks:
Lars Koronenberg (Ph.D student, University of Magdeburg, Germany), Dr. Wolfgang Mathis (University of Magdeburg, Germany) and Dr. Ljiljana Trajkovic (co-supervisors)

Common numerical approaches for determining a circuit's dc operating points usually yield no more than one solution. Even applications of homotopy methods have performed with a limited success. We propose a new method: a combination of investigating the circuit's topology and analyzing the circuit numerically. The method, named ''test for positive feedback structures,'' was previously introduced to determine which bipolar transistor, as part of a feedback structure, can cause positive feedback, and if so, what the circuit's operating points will be. We also show that the method can be extended to circuits with field effect transistors (FET's).

• A. Reibiger, W. Mathis, T. Nahring, L. Kronenberg, and Lj. Trajkovic, ''Mathematical foundations of the TC-method for computing multiple dc-operating points,'' (invited paper) Proc. XI. International Symposium on Theoretical Engineering, Linz, Austria, August 2001.
• L. Kronenberg, Lj. Trajkovic, and W. Mathis, ''Finding dc operating points: limitations of topological and determinant criteria,'' Proc. NOLTA 2000, Dresden, Germany, Sept. 2000, pp. 209-212.
• L. Kronenberg, W. Mathis, and Lj. Trajkovic, ''Limitations of criteria for testing transistor circuits for multiple dc operating points,'' (invited paper) Proc. 43rd Midwest Symposium on Circuits and Systems, MWSCAS 2000, Lansing, MI, Aug. 2000, pp. 1156-1159.
• L. Kronenberg, W. Mathis, and Lj. Trajkovic, ''Method PFBS for the analysis of transistor circuits: proof of uniqueness,'' Proc. Mixed Design of Integrated Circuits and Systems, MIXDES 2000, Gdynia, Poland, June 2000, pp. 145-148.
• L. Kronenberg, Lj. Trajkovic, and W. Mathis, ''Analysis of feedback structures in FET circuits,'' X International Symposium on Theoretical Electrical Engineering, Magdeburg, Germany, Sept. 1999, pp. 445-450.
• L. Kronenberg, Lj. Trajkovic, and W. Mathis, ''Analysis of feedback structures and their effect on multiple dc operating points,'' European Circuit Theory and Design Conference, Stresa, Italy, Aug. 1999, pp. 683-686.
• L. Kronenberg, Lj. Trajkovic, and W. Mathis, ''Ein Backtracking-Suchalgorithmus fur topologische Gleichstrom-Arbeitspunkt-Analyse,'' 43rd Int. Scientific Colloquium, Technical University of Ilmenau, Sept. 1998, vol. 3, pp. 131-137.

Analysis and simulation of simple transistor structures exhibiting negative differential resistance:
Dr. F. Shoucair (UC Berkeley) and Dr. Ljiljana Trajkovic

We investigate, analytically and via simulations, one-port circuits consisting of two bipolar junction transistors (BJT's) and a few linear resistors connected in a feedback structure. These circuits possess topologies and parameter values such that their terminal one-port i-v characteristics exhibit a negative differential resistance (NDR) region. These structures have been used in the past to model silicon controlled rectifier (SCR) devices and the latch-up phenomena in CMOS integrated circuits. We show analytically that the large voltages across transistor pn junctions predicted by SPICE are not numerical artifacts of this simulator, but are intrinsic properties of the circuits. Our analysis explicitly accounts for the influence of the Early voltage effects on the calculations of the break-over voltages and currents, and suggests that this sometimes neglected effect may indeed play a dominant role in these circuits' behavior.