### Summary

*Interval Analysis of Analog Circuits with Component Tolerances*

*Intervall-Analyse analoger Schaltungen mit Bauteiletoleranzen*

**Keywords**: interval analysis, interval arithmetic, analog circuits, symbolic preprocessing, tolerance analysis

In this thesis methods for interval analysis of tolerance affected analog circuits are investigated. For an efficient implementation, it is crucial to regard possible dependencies in circuit equations. Part and parcel of of this strategy is the handling of fill-in patterns for those parameters related to uncertain components. These patterns arise from linear circuit analysis, where they are used for efficient equation setup. With a view towards the incorporation of the methods in the industrial design flow, procedures are presented, which are capable of the treatment of a large number of tolerance affected components.

First, a brief description of tolerance-related issues and the application of interval analysis in analog circuit design is given. After a short introduction to interval arithmetic and derived algorithms, the basics of analog circuit analysis are explained. Additionally, symbolic preprocessing procedures are suggested for an improved equation formulation.

Then a convenient mathematical formulation for the special structure of matrix equations arising from linear analog circuits is introduced and extended to nonlinear components like diodes and transistors. On the top of this, algorithms tuned for accurate treatment of linear circuit equations are shown. Alternative variants are suggested, which admit the choice between accuracy or computation time reduction. This also includes routines for calculating tight bounds to the variance of the results of complex-valued linear systems from frequency-domain analysis.

Finally, the DC behavior of nonlinear circuits with uncertain parameters is analyzed using an extended interval-Newton solver. In particular, a methodology for computing a verified estimation of solution set around a given operating point is presented.

The algorithms developed in the context are illustrated by giving detailed examples in the environment of the circuit analysis tool Analog Insydes. The usability of the proposed methods in analog circuit design is demonstrated using practical application to real-world devices.