# The Generalization of the Extra Element Theorem for Symbolic

Authors: Vladimir  Filaretov; Konstantin Gorshkov
Journal: Journal of Electrical and Computer Engineering
 (2011)

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## Abstract

A new method of the symbolic circuit tolerance analysis has been proposed. The approach is based on Middlebrook's extra element theorem and the generalized parameter extraction method. It does not need the matrix network description or presentation as algebraic sets or topological graphs. The proposed techniques have been realized in the computer program (Toleralize). To verify the theoretical analysis, computer simulation results are included.


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## Body

## 1. Introduction

The transfer function of the linear time-invariant network is generally presented as a rational fraction. But in many cases, especially in network synthesis, it is more appropriate to write the transfer function as follows [1]:(1)S=S0(1+γ),
where S0 is a nominal transfer coefficient, and γ is a multiplicative relative error. Both the variables S0 and γ can be presented as a rational fraction.The transfer function in the form (1) can be obtained by the usage of the Middlebrook’s extra element theorem [1–3] (2)S=S0(1+Zn1/χ1+Zd1/χ),S=S0′(1+χ/Zn11+χ/Zd1),
where χ is a parameter of an arbitrary circuit element (impedance, conductance, and controlled source), S0 and S0′ are nominal transfer functions of the initial circuit in which χ→∞ or χ→0 correspondingly, Zn1 is a driving-point impedance of the network from the port where χ-element is connected, with the input port open, and Zd1 is the driving-point impedance of the network at the port where χ-element is connected, with the input port short.In case ofχ→∞, the selected element must be omitted from the circuit if χ is an impedance and replaced by a short or by a nullor [4–6] if χ is a conductance or a controlled source correspondingly. In case of χ→0, the selected element must be replaced by a short if χ is an impedance, and deleted if χ is a conductance or a controlled source.So, the usage of formulae (2) required computation of three different circuit functions Zn1,Zd1 and S0 or S0′. This requirement makes problems for the symbolic circuit analysis of complicated network.In Section2 of this paper, the technique for improvement of the estimation efficiency of the extra element theorem is proposed. The obtained results provide the development of new symbolic circuit tolerance analysis method presented in Section 3. A practical example is discussed in Section 4. The conclusion is finally drawn in Section 5.


## 2. Extra Element Theorem and Generalized Parameter Extraction Method

In accordance with [7], the transfer function of electronic network can be expressed as (3)S=ND,
where N is the determinant of the network, in which the independent source  and arbitrary response are replaced by nullor, and D is the determinant of the network, in which the independent excitation and the arbitrary response are zero.The calculation procedure of the network determinants is based on the recursive usage of the parameter extraction formula which generalizes the known Feussner's equations [8, 9](4)Δ=χΔ(χ⟶∞)+Δ(χ=0),
where χ is a parameter of arbitrary circuit element, Δ(χ→∞) and Δ(χ→0) correspond to determinants of the circuit matrix in which the parameter of extracted elements χ→∞ or χ→0, respectively.The described techniques develop the singular elements approach discussed by Braun [7], Parten and Seacat [10], and Hashemian [11] and have been realized in the generalized parameter extraction method, an effective tool for network symbolic analysis [12–14] and synthesis [15]. The method can be used for the analysis of the circuit with all types of controlled sources. It requires neither matrix, nor ordinary graph description of the circuit. The main advantage of the method is that there are no cancellations among the generated terms.The generalized parameter extraction method has been realized in the program CIRSYM as a part of the software tool SCAD (http://intersyn.narod.ru/scad.htm).So, it is possible to present the formulae (2) with the help of network determinants as (5)S=NχDχ(χ+Nχ/Nχχ+Dχ/Dχ),(6)S=NχDχ(1+χ(Nχ/Nχ)1+χ(Dχ/Dχ)),
where Nχ and Dχ are determinants of networks, in which the extracted parameter χ→∞; Nχ and Dχ are determinants of networks, in which the extracted parameter χ→0.As evident from the expressions (5) and (6), we have two pairs of identical network determinants. So, for estimation of the transfer function in the form (1), we need to obtain only four network determinants: Nχ, Dχ, Nχ, and Dχ. It is simpler than a separate computation of circuit functions from (2): S0=Nχ/Dχ  or S0′=Nχ/Dχ, Zn1=Nχ/Nχ, and Zd1=Dχ/Dχ. So, the proposed technique provides more than 30% increase of the computation efficiency.In the case of a multiple extraction, that is, a simultaneous extraction of several parametersχ1,χ2,…,χn, it is possible to modify the formulae (5) and (6) as follows:(7)S=S0χ1(Nχ2,…,nχ1/Nχ1,2,…,n)+⋯+χn(Nχ1,2…,n-1χn/Nχ1,2,…,n)+χ1χ2(Nχ3,…,nχ1,2/Nχ1,2,…,n)+⋯+χ1χ2⋯χn+Nχ(1,2,…,n)χ(1,2,…,n)/Nχ1,2,…,nχ1(Dχ2,…,nχ1/Dχ1,2,…,n)+⋯+χn(Dχ1,2,…,n-1χn/Dχ1,2,…,n)+χ1χ2(Dχ2,…,nχ1,2/Dχ1,2,…,n)+⋯+χ1χ2⋯χn+Dχ(1,2,…,n)χ(1,2,…,n)/Dχ1,2,…,n,S=S0′1+χ1(Nχ2,…,nχ1/Nχ1,2,…,n)+⋯+χn(Nχ1,2,…,n-1χn/Nχ1,2,…,n)+χ1χ2(Nχ3,…,nχ1,2/Nχ1,2,…,n)+⋯+χ1χ2⋯χn(Nχ(1,2,…,n)χ(1,2,…,n)/Nχ1,2,…,n)1+χ1(Dχ2,…,nχ1/Dχ1,2,…,n)+⋯+χn(Dχ1,2,…,n-1χn/Dχ1,2,…,n)+χ1χ2(Dχ2,…,nχ1,2/Dχ1,2,…,n)+⋯+χ1χ2⋯χn(Dχ(1,2,…,n)χ(1,2,…,n)/Dχ1,2,…,n),where S0=Nχ1,2,…,n/Dχ1,2,…,n and S0′=Nχ1,2,…,n/Dχ1,2,…,n.


## 3. Symbolic Circuit Tolerance Analysis

The network multiplicative relative error from (1) has a direct connection with the parameters spread of the circuit elements. In accordance with [16], the influence of the elements tolerances to the network characteristics can be simulated by including additional elements into the network. If we consider the additional elements as “extra” elements, it is possible to use the extra element theorem for the network tolerance analysis.The toleranceδ(χi) of the two-port circuit element χi can be simulated by series resistances or by shunt admittances as shown in Figures 1(a) and 1(b) correspondingly. The circuit simulation of the controlled sources tolerances is presented in Figures 1(c)–1(f).A circuit simulation of the elements tolerances.
(a)(b)(c)(d)(e)(f)The symbolic expression of the network multiplicative relative error corresponding to the notations of circuit functions (1) and (4) can be defined as(8)γ=δ(χ)(Nδ(χ)Dδ(χ)-Nδ(χ)Dδ(χ))δ(χ)Nδ(χ)Dδ(χ)+Nδ(χ)Dδ(χ).The fractional formulae for obtaining the tolerance of the arbitrary circuit elementsδ(χi) can be expressed in the same way(9)δ(χ)=γNδ(χ)Dδ(χ)Nδ(χ)Dδ(χ)-Nδ(χ)Dδ(χ)(γ+1).It is possible to generalize the expression (8) in case when multiple tolerance-simulated elements are added into a circuit simultaneously,(10)γ(1,2,…,n)=[δ(χ1)(Nδ(χ2,…,n)δ(χ1)Dδ(χ1,2,…,n)-Nδ(χ1,2,…,n)Dδ(χ2,…,n)δ(χ1))+⋯+δ(χn)(Nδ(χ1,2,…,n-1)δ(χn)Dδ(χ1,2,…,n-1)-Nδ(χ1,2,…,n)Dδ(χ1,2,…,n-1)δ(χ1))+δ(χ1χ2)(Nδ(χ3,…,n)δ(χ1,2)Dδ(χ1,2,…,n)-Nδ(χ1,2,…,n)Dδ(χ3,…,n)δ(χ1,2))+⋯+δ(χ1χ2⋯χn)(Nδ(χ(1,2,…,n))δ(χ(1,2,…,n))Dδ(χ1,2,…,n)-Nδ(χ1,2,…,n)Dδ(χ(1,2,…,n))δ(χ(1,2,…,n)))]/[Nδ(χ1,2,…,n)(δ(χ1)Dδ(χ2,…,n)δ(χ1)+⋯+δ(χn)Dδ(χ1,2,…,n-1)δ(χn))+δ(χ1χ2)Dδ(χ3,…,n)δ(χ1,2)+⋯+δ(χ1χ2⋯χn)Dδ(χ(1,2,…,n))δ(χ(1,2,…,n))+Dδ(χ1,2,…,n)]Note that the value of network determinantsNχ and Dχ in formulae (8) and (9) as well as determinants Nχ1,2,…,nиDχ1,2,…,n in (10) is invariable and can be found in the stage of nominal transfer function estimation.The proposed techniques of tolerance analysis have been realized in the computer program Toleralize as a part of the software tool SCAD. Toleralize provides the estimation of symbolic expressions of a network transfer function in forms (5) or (6), a network multiplicative relative error in forms (8) or (10), and an element tolerance in form (9), as well as their numerical values.


## 4. Example

As an example, let us consider the procedure of the tolerance analysis of the transistor amplifier shown in Figure2 [16]. The elements parameters are presented in Table 1 (first column).Table 1
The elements parameters of the transistor amplifier shown in Figure2 and the results of tolerance analysis.
χiSymbolic expressions of network determinantsNχ and Dχγ%δ(χi)%δ(χi)=+10%δ(χi)=-10%γ=+15%γ=-15%R1 10 kΩNχ=R6B2R3B1=10000000000;
Dχ=(B1R2+R2+R5)(R3+R4)=222000008−8.319−18R2 0.1 kΩNχ=R6((R3+R4)(B1+1)+(-B2R3)(-B1))   =10202000000;Dχ=(R3+R4)(B1R1+R1+R5)+((R3+R4)R6)(B1+1)  +(-R3R6B2)(-B1)=12224000000−8.910.8−13.418R3 1 kΩNχ=R6((R2(B1+1))+(-B2)(-B1(R1+R2)))   =101010100000;   Dχ=(B1R1R2+R1(R2+R5)+R2R5)  +(R6)(B1R2+R2+R5)+(-R6B2)(-R2B1)   =11222000000.89−1.1−403−64R4 1 kΩNχ=R6R2(B1+1)=10100000; Dχ=B1(R1+R6)R2+(R1+R6)(R2+R5)+R2R5   =122200000−0.970.99−133180R5 1 kΩNχ=0; Dχ=(R1+R2+R6)((R3+R4))=22200000−0.170.17−731989R6 1 kΩNχ=(R3+R4)(R2(B1+1))+(-B2R3)(-B1(R1+R2))   =101020200000; Dχ=(R3+R4)(B1R2+R2+R5)+(-B2R3)(-R2B1)   =10222000001.7−1.9271−49.7B1 100 SNχ=R6(B2R3(R1+R2)+R2(R3+R4))   =1010200000000; Dχ=R2(R6B2R3+(R1+R6)(R3+R4))   =122000000000.17−0.21−117−90B2 100 SNχ=R6R3B1(R1+R2)=1010000000000;Dχ=R6R2B1R3=100000000001.8−2.1198−47Figure 2
The linearized equivalent network of a transistor amplifier.It is possible to use the original formulae (2) for the tolerance analysis, but the technique on the base of network determinants approach considered in Section 3 makes the calculation process much simpler and faster.First, we compute the nominal transfer function of the network by means of a generalized parameter extraction method(11)Nδ(χi)=R6(((R3+R4))(R2(B1+1))+(-B2R3)(-B1(R1+R2)))=101020200000000,(12)Dδ(χi)=(R3+R4)(-B1R1R2+R1(R2+R5)+R2R5)+((R3+R4)R6)(-B1R2+R2+R5)+(R3R6B2)(R2B1)=1244400000000.So, the nominal transfer function of the amplifier isS0=81,18.Then, we use formula (8) for the estimation of network multiplicative relative errors provided that the elements parameters spread is δ(χi)=±10%. Note that we already know the value of network determinants Nδ(χi) (11) and Dδ(χi) (12), so we need to obtain only two network determinants Nδ(χi) and Dδ(χi) for each χi circuit elements by means of a tolerances simulation approach. The results of the computation of the determinants symbolic expressions and their numerical values are presented in Table 1 (second and third columns, correspondingly).Now, we estimate the elements tolerances provided that the multiplicative relative error of the amplifier transfer function isγ=±15%. The numeric values of network determinants included in expression (9) are already obtained, so the computation process becomes really simple.  The calculated results of the parameters spread of circuit elements are presented in Table 1 (fourth column).The results obtained have been verificated by the computer program Toleralize.


## 5. Conclusions

In this paper, a network determinant approach implemented for symbolic circuit analysis by means of the extra element theorem has been considered. The formulae for analytical evaluation of a network multiplicative relative error and the tolerance of the arbitrary circuit elements are presented. The proposed techniques provide more than 30% increase of estimation efficiency as compared to the original Middlebrook’s method. The techniques proposed have been realized in the computer program Toleralize.



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