Advances in Electrical Engineering

Volume 2015, Article ID 328517, 13 pages

http://dx.doi.org/10.1155/2015/328517

## A Cancellation-Free Symbolic Sensitivity Technique Based on Network Determinant Expansion

Department of Electrical Engineering, Ulyanovsk State Technical University, 32 Severny Venets Street, Ulyanovsk 432027, Russia

Received 28 July 2014; Revised 30 November 2014; Accepted 30 November 2014

Academic Editor: Ramón M. Rodríguez-Dagnino

Copyright © 2015 Vladimir Filaretov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The generalization of Bode’s sensitivity analysis technique for all types of the transfer functions and circuit elements is presented in the paper. The proposed formulae for first- and second-order symbolic sensitivity calculation provide the compact size of obtained expression and have the advantages of cancellation-free sum-of-product terms and matrix-free computation. This is achieved by means of the concept of high order summative cofactors and the generalized parameter extraction method. The proposed technique is implemented in symbolic circuit analysis program Cirsym. Illustrative example on symbolic sensitivity circuit analysis and comparison of the presented technique with the transimpedance method and the method based on the modified Coates flow-graph are given.

#### 1. Introduction

G_{1}The influence of network components on transfer function is expressed by sensitivities [1]. Sensitivity analysis is an important part of analog circuit design process. There are many methods of numerical calculation of circuit sensitivities, but only symbolic analysis of circuit sensitivity provides the way to get analytical sensitivity function of all interested circuit parameters. This is certainly advantageous in such applications as circuit parameters optimization and Monte Carlo simulation.

First researches on symbolic approach to sensitivity analysis of analog circuit were presented in [1–5], but those heuristic algorithms were not efficient enough and had limitations by the circuits scale.

Nowadays several more effective symbolic sensitivity analysis techniques have been developed. Some of them are based on sequence of expression computation such as two-port transimpedance method [6]. But the generated expressions are not compact enough, do not fully exploit sharing of subexpressions, cannot be easily manipulated, and complicate the circuit insight.

The others methods are based on implementation of graph theory for symbolic circuit analysis [7–10]. In particular, the work in [7] shows the technique based on the modified Coates flow-graph for performing sensitivity analysis. However, the proposed concept requires the nodal admittance matrix expansion, which may consist of the equal summands with opposite sign. It leads to appearing of the cancellation sum-of-product terms. In addition, the usage of graph-based methods does not provide the optimal size of obtaining expression.

Another widespread approach is implementation of the modified nodal matrix [10, 11]. But obtained results are not compact and include cancellations so the expressions often present only in semisymbolic form.

In this paper we propose a cancellation-free symbolic sensitivity technique for computation of compact expression. The first- and second-order symbolic sensitivities formulas for all types of the transfer functions and circuit elements are derived by means of the concept of higher order summative cofactors (HOSC) [12–15] and generalized parameter extraction method (GPEM) [16–21] to avoid drawbacks of the previous methods mentioned above.

The rest of this paper is organized as follows. In Section 2 we explain the basic idea of sensitivity analysis by means of GPEM and derive the formulae for symbolic sensitivity of different transfer functions with respect to controlled sources (CS) and two ports. The process of obtaining symbolic expressions can be easily automated. The example of the AC sensitivity analysis is presented in Section 3. In Section 4 the comparison of the presented method with the transimpedance method and the method based on the modified Coates flow-graph is performed. Section 5 concludes this paper.

#### 2. Theoretical Basis of the Proposed Method

##### 2.1. Symbolic Approach for Sensitivity Analysis

The sensitivity computation of the transfer function of with respect to change of the parameter requires finding the corresponding derivative:

The study of network response by means of formula 1 is acceptable only when changes in its parameter are small. When some element’s parameters change significantly, this approximated result may be unsatisfied. Analytical sensitivity analysis would be much easier by using a symbolic approach than by a numerical approach where computation cost is a key concern.

If we consider the transfer function as formula 1 will transform into expression proposed by Hoang [5]:

Calculation of the transfer function derivatives leads to appearance of multiple algebraic cofactors with different signs and thus they may contain many cancellations. The well-known technique proposed by Bode [1] presented the sensitivity function in form of ratio of determinant and cofactors of nodal admittance or loop impedance matrix multiplication. For example, the sensitivity of of the circuit in Figure 1 is expressed as follows: where is a transfer impedance of current controlled voltage source (CCVS) or transfer admittance of voltage controlled current source (VCCS), is a determinant of loop impedance matrix or for nodal admittance matrix, and , , , and are cofactors.