The method of circuit determinants is a symbolic method for analyzing electrical circuits, in which the circuit of a circuit with arbitrary linear elements is used directly to calculate the required currents and voltages, bypassing the compilation of equilibrium equations. The method is designed to obtain optimal in terms of complexity of symbolic expressions of circuit functions, responses, conversion errors and tolerances of elements, as well as parameters of macromodels of subschemes and parameters of unknown elements in linear electric circuits.
Parameter Allocation Formulas
The method of circuit determinants is based on the Foisner formulas for extracting the parameters of bipolar elements [1] [2] , which can be represented in a circuit algebraic form [3] :
In general, an arbitrary parameter can be selected using the following expression:
where χ є (R, g, K, G, H, B); Δ (χ → ∞) is the determinant of the first derivative of the circuit obtained from the initial circuit as a result of assigning a value to the parameter χ tending to infinity (the resistance is removed, the conductivity is replaced with an ideal conductor on the circuit (tightened), the controlled sources are replaced with nullors) [4] ; Δ (χ = 0) is the determinant of the second derivative scheme, which is formed as a result of the neutralization of the selected element, that is, the adoption of χ = 0 (the resistance is tightened, conductivity is removed, controlled sources are neutralized) As determinants, we will consider symbolic determinants, that is, analytical expressions in which all the parameters of a circuit are represented by symbols, not numbers [5] [6] . Nullor is the circuit model of an ideal Tellegen amplifier [7] , that is, a controlled source, the parameter of which tends to infinity. Nullor is an anomalous controlled source, since the current of the norator (the controlled branch of the nullor) is not defined, and the current and the voltage of the nullator (the controlling branch of the nullor) are zero. When a managed source is replaced, its controlled and controlling branch is replaced with a norator and a nullator, respectively. During neutralization, the controlled voltage branch and the control current branch are tightened, and the controlled current branch and the control voltage branch are removed. An ideal conductor and an open branch are special cases of the inclusion of nullor. An ideal conductor is equivalent to a unidirectional parallel connection of a norator and a nullator, and an open branch is equivalent to their opposite series connection. When the direction of the norator or nullator changes, the sign of the identifier of the circuit containing these elements is reversed. If the capacitors are set in the operator form by the capacitance conductivities of the PC, and the inductance by the inductive resistances pL, then the result of the decomposition of the symbolic identifier of the circuit using formulas (1) - (3) becomes an expression that does not contain fractions, which makes it simple and convenient to consider. The circuit elements according to the formula (3) are allocated recursively until the simplest scheme is obtained, the determinant of which is derived from Ohm's law (for example, open resistance or conductivity (Fig. 1, a and b), closed on itself or conductivity ( Fig. 1, c and d), two unconnected nodes (Fig. 1, e), a single node (Fig. 1, e), a contour with a nullor (Fig. 1, g), an open branch with a norator and a nullator (Fig. 1, h), contour with MD (Fig. 1, i-m)).
To the described basis of the simplest schemes, it is also advisable to add the schemes in fig. 1, n and fig. 1, o, consisting of two circuits with an INUN or ITUT, respectively, since the neutralization of one of the MD leads to a node scheme. The generalization of these schemes has a similar property, which consist of m contours with MD (m> 2) and have determinants Δ = K 1 • K 2 • ... • K m +1 and Δ = B 1 • B 2 • ... • B m + 1 respectively.
Degeneration of schemes
In the system identifier (matrix) of the circuit, the appearance of rows is possible, which consist of elements equal to zero. The scheme corresponding to this determinant is called degenerate. Thus, the determinant of the degenerate scheme is identically zero. From the physical point of view, it is assumed that a circuit in which infinitely large currents and voltages develop, or the values of currents and voltages turn out to be undetermined, is degenerate [8] . Thus, the internal resistances of the controlled voltage branch and the control current branch are zero, so an infinitely large current is created in a circuit containing only the voltage controlled branches and control current branches. On the other hand, the internal conductivities of the controlled current branch and the control voltage branch are zero, therefore, infinitely large values of voltages appear on cross section elements formed only by the controlled current branches and control voltage branches. The method of circuit determinants makes it possible to establish the degeneracy of a circuit directly by its structure and composition of elements in order to avoid unnecessary calculations [7] [8] . Below are the conditions for the degeneration of the scheme and the neutralization of elements in the closure and opening of the branches (Table 1) and in contours and sections (Table 2).
| Schema element | A loop | Open branch |
|---|---|---|
| Resistance | Allotment | Neutralization |
| Conductivity | Neutralization | Allotment |
| Controlled voltage branch | Degeneration | Neutralization |
| Control current branch | Degeneration | Neutralization |
| Controlled current branch | Neutralization | Degeneration |
| Control voltage branch | Neutralization | Degeneration |
| Norator | Degeneration | Degeneration |
| Nullator | Degeneration | Degeneration |
| Schema element | Element incident rate | |||
| contour | cross section | |||
| from the controlled voltage branch or norator | from the control current or nullator branch | from the controlled current branch or norator | from the control voltage branch or nullator | |
| Resistance | - | Tightening | ||
| Conductivity | Deletion | - | ||
| Controlled voltage branch | Degeneration | - | Tightening | - |
| Control current branch | - | Degeneration | - | Tightening |
| Controlled current branch | Deletion | - | Degeneration | - |
| Control voltage branch | - | Deletion | - | Degeneration |
| Norator | - | Degeneration | - | Degeneration |
| Nullator | Degeneration | - | Degeneration | - |
Schematic-algebraic formulas
Any circuit function of an electrical circuit can be considered as the ratio N / D [9] . The numerator N here is the determinant of the circuit, in which the independent source and the branch of the desired response are replaced by nullor, and the denominator D is the determinant of the circuit with neutralized input and output. In fig. 2 these rules are illustrated with schematic-algebraic formulas for six well-known circuit functions: voltage transfer coefficient (Fig. 2, a), transfer resistance (Figure 2, b), transfer conductivity (Figure 2, c), current transfer ratio (Fig. 2, d), input conductances (Fig. 2, e) and resistance (Fig. 2, e), respectively [10] .
If there are several independent sources in the circuit for applying the device of circuit determinants, use the overlay method [6] .
The rule for changing characters in schemes with NUI
In schemes that contain more than one directional nullor, they must be numbered in such a way that norator and nullator belonging to the same nullor have the same numbers:
When formulating this rule, the orientation of the norator and nullator does not change (that is, they are directed upwards).
Schema determiner method applications
The method of circuit determinants is used to solve various problems of circuit theory:
- symbolic analysis of linear [5] [6] [11] [12] and nonlinear schemes [13] ;
- diakoptiki [14] [15] ;
- diagnostics [15] ;
- structural synthesis [3] [16] [17] [18] ;
- tolerance analysis [10] [19] ;
- analytical solution of linear algebraic equations [20] .
See also
- Feusner, Friedrich Wilhelm
- Symbolic Circuit Analysis Diagnostics and Synthesis (SCADS)
Notes
- ↑ Feussner W. Ueber Stromverzweigung in netzformigen Leitern // Annalen der Physik. - 1902. - Bd 9, N 13. - S. 1304-1329
- ↑ Feussner W. Zur Berechnung der Stromstarke in netzformigen Leitern // Annalen der Physik. - 1904. - Bd 15, N 12. - S. 385—394
- ↑ 1 2 Gorshkov K.S., Filaretov V.V. Synthesis of electrical circuits based on the circuit approach. - LAP LAMBERT Academic Publishing GmbH & Co. KG, 2011. - 242 c
- ↑ . Hashemian R. Symbolic representation of the network transfer functions using norator-nullator pairs // Electronic circuits and systems.- 1977.- Vol. 1, No. 6 (November) .- P. 193-197
- ↑ 1 2 Filaretov V. V. Topological analysis of electronic circuits by selecting parameters // Electricity.- 1998.- № 5.- P. 43-52
- ↑ 1 2 3 V. V. Filaretov. Topological analysis of electrical circuits based on the circuit approach: Dis. ... Dr. tech. Sciences 05.09.05 (Theoretical Electrical Engineering) / Ulyanovsk State. tech. University, St. Petersburg State. tech. un-t - Ulyanovsk-St. Petersburg, 2002. - 265 seconds
- 2 1 2 Tellegen BDH On nullators and norators // IEEE Transactions on circuit theory .- 1966.- CT-13.- N 4.- P. 466-469
- ↑ 1 2 Kurganov S. А., Filaretov V. V. Schematic-algebraic analysis, diakoptika and diagnostics of linear electric circuits: Tutorial. - Ulyanovsk: UlSTU, 2005. - 320 seconds
- Un Braun J. Topological analysis of networks containing nullators and norators // Electronics letters.- 1966.- Vol. 2, No. 11.- P. 427-428
- ↑ 1 2 Gorshkov K.S., Filaretov V.V. Generalization of the method of symbol analysis of Middlebrook to calculate the tolerances of electric circuits // Electronics and communication: Thematic issue "Electronics and Nanotechnologies". - Kiev, 2010.- № 5. - p. 60-64
- Are Filaretov VV, Korotkov, AS Generalized parameterization of the network method of symbolization analysis // Proceedings of the European Conference on the theory and design (ECCTD-2003) .- Kraków, Poland, 2003.- Vol. 2.- P. 406-409
- Are Filaretov VV, Korotkov AS Generalized parameter extraction method for multiple excitation // Proceedings of the 8th international workshop on Circuit Design.-Wroclaw (September 23-24) .- 2004.-P. 8-11
- ↑ Korotkov A.S., Kurganov S.A., Filaretov V.V. Symbolic analysis of discrete-analog circuits with switchable capacitors // Electricity .- 2009.-№ 4.- P. 37-46
- ↑ Filaretov V.V. The method of binary vectors for the topological analysis of electronic circuits in parts //Elect .-2001.-№ 8.-С.33-42
- ↑ 1 2 Kurganov S. A. Symbolic analysis and diakoptika electrical circuits: Dis. ... Dr. tech. Sciences 05.09.05 (Theoretical Electrical Engineering) / Ulyanovsk State. tech. University, St. Petersburg State. tech. un-t - Ulyanovsk-Saint-Petersburg, 2006. - 328 with
- ↑ Gorshkov, KS. Structural synthesis and symbolic tolerance analysis of electrical circuits by the method of circuit determinants: Author's abstract. dis. ... Cand. tech. Sciences / MEI (TU), 2010
- ↑ Filaretov V., Gorshkov K. Transconductance Realization of Block Diagrams of Electronic Networks // Proc. of International Conference on Signals and Electronic Systems (ICSES`08). - Krakow, Poland. - 2008. - R. 261-264
- ↑ Filaretov V., Gorshkov K., Mikheenko A. A, Proc. of International Conference on Synthesis, Modeling, Analysis and Simulation Methods and Applications to Circuit Design (SMACD) .- Seville, Spain .- Sept. 2012.- P. 293-296.
- Are Filaretov V., Gorshkov K. The Theorem for the Symbolic Circuit Tolerance Analysis // The Journal of Electrical and Computer Engineering. Vol. 2011.- Article ID 652706.- 5 p
- ↑ Filaretov V.V. Schematic mapping of a matrix for symbolic solution of systems of linear algebraic equations // Logic-algebraic methods, models, applied applications: Tr. international conf. KLIN-2001.- Ulyanovsk: UlSTU, 2001.-V.3.-P.13-15