Frequency-Scaled Semisymbolic Analysis

1. Introduction

The semisymbolic analysis is utilized in many various areas of the circuit design [1,2,3,4,5,6,7,8,9,10,11]. A general idea how to use the semisymbolic analysis for computing the poles and zeros of a transfer function was initially described in [12], where a very important method how to utilize the sparsity of matrices is also suggested. This procedure systematically transforms the general eigenvalue problem to the standard one that is solved by the QR or QZ [13] algorithms afterwards. However, during many years, several numerical problems of this procedure have been recognized with miscellaneous suggestions how to overcome them:

• More works have been devoted to solving (very) large-scale problems, [14,15,16,17] e.g., both from efficiency and accuracy point of views. Furthermore, using a parallelism could also be a natural answer to this problem – especially contemporary versions of Fortran (2018 or 2023) are very suitable for programming these tasks [18,19].
• Solving the well known (and almost annoying) problem of the standard eigenvalue task: accuracy of the computation of multiple or nearly multiple eigenvalues [20,21,22,23].
• Many works have also been generally focused on the frequently insufficient precision of the semisymbolic analyses, which is caused by utilized numerical processes [24,25,26].

Frequency scaling techniques were successfully used in many areas of microwave electronics, antenna constructions, neural networks, and numerical methods in physical design [27,28,29,30,31], and it is not surprising that these procedures often concern of the microwaves, because there are huge differences in the values of individual circuit elements or technological parameters in the area of very high frequencies. However, we did not find any use of the frequency scaling in the semisymbolic analysis, i.e., in the reduction of the general eigenvalue problem to the standard one solved by the QR of QZ algorithms afterwards, which provides the poles and zeros of a transfer function of a linear or linearized circuit. For this reason, a utilization of the frequency-scaled semisymbolic analysis for more accurate computing the poles and zeros of the circuit transfer function is designated as the fundamental goal of the article.

As this theme is unusual, we also demonstrate a simple method how to formulate the modified system of the equations for the frequency-scaled semisymbolic analysis, and how to determine the real poles and zeros of the transfer function corresponding to original (unscaled) system. Moreover, the way of transforming the general eigenvalue problem to the standard one is demonstrated on an analytically solved (extraordinary) so-called dynamically degenerate circuit. Finally, increasing the precision by means of frequency-scaled semisymbolic analysis is demonstrated on a wide class of electrical circuits, especially on the microwave ones where the accuracy enhancement is required most of all.

The article is organized as follows. At the beginning, there is a brief overview of the semisymbolic analysis utilized for computing the poles and zeros of linear circuits or circuits that are linearized at an operating point. (Full details are not included because they can be found in our previous conference paper [32] and book chapter [17].) The method is then demonstrated on an unusual example – on a dynamically degenerate circuit [33], where a nonstandard step of the reduction procedure (in the semisymbolic analysis) must be used. (This step has already been shown for a digital filter in [17], however, here, its necessary use also for an analog circuit is demonstrated, which is quite strange in this area.) Afterwards, there is a section in which a simple way of formulating equations for the frequency-scaled semisymbolic analysis is explained, and how the real (non-transformed) poles and zeros (and constant of transfer function) are obtained after its end. Finally, the accuracy and reliability of the algorithm are verified by determining the poles and zeros of transfer functions of the following four electronic circuits that are known (and tested) for considerable numerical demands in the semisymbolic analysis:

• Low-noise antenna amplifier for a multi-constellation radio receiver for all the (five) satellite navigation systems [26] – details of methods for creating the low-noise devices as well as some corresponding computer-aided design tools are described in [34,35,36,37].
• An AB-class power amplifier [38] linearized at an operating point – more of negative feedbacks as well as tiny capacitors of some transistors cause huge differences among the magnitudes of poles and (especially) zeros of a transfer function.
• A testbench circuit with the MDA272 operational amplifier [32,39] – as this circuit is (much more) complicated than the previous one, it constitutes more demanding test as well.
• A distributed microwave oscillator [40] represents the most demanding test – the LRCG models of microstrip lines as well as the models of pHEMTs contain some (indeed) extremely small values, therefore, the frequency-scaled semisymbolic analysis is thoroughly verified here.

And because of course there is another logical possibility of improving the numerical stability of the calculations by combinations of the frequency scaling and 128-bit arithmetic, the benefits of this approach are demonstrated on one of the examples above. As expected, there is a further increase in the numeric robustness of the whole process in terms of choosing the accuracy parameter. However, because this article is primarily focused on the frequency-scaled semisymbolic analysis (in which all the above tasks could be managed in the standard 64-bit arithmetic), we do not expand on this matter so much in detail in this article. (In addition, greater numerical calculation stability can be naturally expected in the 128-bit arithmetic.)

The purpose of the work and its significance are clear:

• First of all, the paper shows that the formulation of the system of equations for the frequency-scaled semisymbolic analysis is very simple, only a slightly more complicated in comparison with the standard (unscaled) semisymbolic analysis. Moreover, recalculation of the results of the frequency-scaled semisymbolic analysis to the actual (untransformed) values of the poles and zeros of the circuit is also very simple.
• Although the above operations (both before and after the semisymbolic analysis) are very simple to implement, they lead to a substantial accuracy improvement, which is clearly demonstrated in the four selected examples. And this uncomplicated adjustment of the algorithm, leading to much more accurate results, is the main purpose of the article.

2. Brief Characteristic of Semisymbolic Analysis

In this section we will focus on the brief characteristic of the semisymbolic analysis only with the necessary definitions that will be needed in the following sections. Much more details about this very useful type of analysis can be found in [1,12,13,14,15,16,17,21,22,24,32,33,38]. However, everything needed to explain the principle of the frequency-scaled semisymbolic analysis this section contains.

2.1. Reduction of Generalized Problem of Eigenvalues to Standard Problem of Eigenvalues

The system of the circuit equations of a linear circuit or a nonlinear circuit linearized at an operating point is defined by the matrix equation (see [12], e.g., and many new others):

$$\left(s\mathit{A}+\mathit{B}\right)\mathit{v}=\mathit{e},$$

(1)

where s stands for Laplace operator, the $\mathit{A}$ and $\mathit{B}$ matrices are composed of circuit reactances and resistances depending on the way of formulating equations, and $\mathit{v}$ and $\mathit{e}$ are the Laplace images of circuit variables and input sources, respectively [14].

Poles (of all transfer functions) and zeros (of a specific transfer function) are solutions of generalized problems of eigenvalues:

$$\begin{array}{ccc}\hfill & det\left(s\mathit{A}+\mathit{B}\right)=0\hfill & \hfill \mathrm{for}\mathrm{poles},\end{array}$$

(2a)

$$\begin{array}{ccc}\hfill & det\left(s{\mathit{A}}_k\left(\mathbf{0}\right)+{\mathit{B}}_k\left({\mathit{e}}_l\right)\right)=0\hfill & \hfill \mathrm{for}\mathrm{zeros},\end{array}$$

(2b)

where the matrix ${\mathit{A}}_k\left(\mathbf{0}\right)$ is formed by replacing the $k^{\mathrm{th}}$ column of $\mathit{A}$ by the column of zeros, and the matrix ${\mathit{B}}_k\left({\mathit{e}}_l\right)$ is created from $\mathit{B}$ by replacing its $k^{\mathrm{th}}$ column by the column $\mathit{e}$ with all its elements zeroed with the exception of one representing the $l^{\mathrm{th}}$ source of input signal.

It goes without saying that the solution of the generalized problem of eigenvalues is more complicated than solving the standard problem of eigenvalues $det(s\mathbf{I}-\mathit{X})=0$ (which is carefully processed in the literature and for which there are numerous software libraries). Therefore, a systematic reduction of the generalized eigenvalue problem is performed to the standard eigenvalue problem (so-called deflation to the standard form) by a sequence of operations that do not change the value of the determinant except for its sign.

Using this systematically performed sequence (the system arrangement is described in detail, for example, in [12,33,38], etc.), the matrices in (Section 2.1) are converted to the shape shown in Figure 1. Although the procedure resembles the Gaussian elimination method, it is a more complex process that also contains specific operations such as differentiation of a row described in Subsect. Section 2.2.

For the shape of the submatrices in Figure 1, we can simplify the calculation of the determinant in (2a) (the artificial multiplication by the unit matrix ${\mathit{A}}_{\mathbf{11}}{\mathit{A}}_{\mathbf{11}}^{-1}(=\mathbf{I})$ from the left enables transforming the general eigenvalue problem to the standard one):

$$det\left(s\mathit{A}+\mathit{B}\right)={\left(-1\right)}^{n_{\leftrightarrow \updownarrow }}det\left[\underset{{\displaystyle =\mathbf{I}}}{\underbrace{{\mathit{A}}_{\mathbf{11}}{\mathit{A}}_{\mathbf{11}}^{-1}}}\left(s{\mathit{A}}_{\mathbf{11}}+{\mathit{B}}_{\mathbf{11}}\right)\right]\prod _{i=n_1+1}^{n_2}{\left({\mathit{B}}_{\mathbf{22}}\right)}_{ii}={\left(-1\right)}^{n_{\leftrightarrow \updownarrow }}\prod _{i=1}^{n_1}{\left({\mathit{A}}_{\mathbf{11}}\right)}_{ii}\prod _{i=n_1+1}^{n_2}{\left({\mathit{B}}_{\mathbf{22}}\right)}_{ii}det\left(s\mathbf{I}+{{\mathit{A}}_{\mathbf{11}}}^{-1}{\mathit{B}}_{\mathbf{11}}\right),$$

(3)

where $n_{\leftrightarrow \updownarrow }$ is the total number of column and row interchanges during the transformation.

Thus, the standard eigenvalue problem has been created, and the poles of (all) the transfer functions can be calculated as eigenvalues of the matrix $\mathit{X}=-{{\mathit{A}}_{\mathbf{11}}}^{-1}{\mathit{B}}_{\mathbf{11}}$. (And the same method is applied to the general eigenvalue problem () for the calculation of zeros.)

2.2. Extraordinary Step for Reduction of “Irreducible” Non-diagonal Elements

The reduction process that leads to the final form on the right side of (3) seems to be a modification of the Gaussian elimination method. In certain cases, however, there is a non-diagonal element in the matrix ${\mathit{A}}_{\mathbf{11}}$ that is not reducible by any other diagonal element of this matrix. In such a case, a suitable row of the lower part of the matrix ${\mathit{B}}_{\mathbf{22}}$ is multiplied by the operator s to move this row to the left. (In the time area, this operation corresponds to differentiation.) And now the originally irreducible element in the matrix ${\mathit{A}}_{\mathbf{11}}$ can be easily reduced using some element of this moved row. In [17], we have already demonstrated the necessity of this step on an irregular digital filter described by $\mathcal{Z}$ transform. In this article, we will show a more illustrative example based on a so-called dynamically degenerate analog circuit [33]. However, compared to this book, we will introduce significantly easier formulation of circuit equations.

2.3. Pivoting

Obviously, the accuracy of the process of the reduction of the generalized eigenvalue problem to the standard form is significantly influenced by the selection of pivot elements. The best results are achieved by full pivoting when the pivot elements are selected from the whole remaining submatrix. However, the full pivoting is unsuitable for solving large tasks, and therefore in this article we only deal with the partial pivoting, where the pivot elements are selected only from the respective subcolumns of the matrix. (And, moreover, the partial pivoting is well compatible with the procedures exploiting the sparsity of the matrices.) The pivoting strategies are defined in a very detailed way in [38]. Here, we only describe checking the pivot element by the $ϵ$ parameter, which is also used in the examples.

The $n^{\mathrm{th}}$ pivot element is selected from the rest of the $n^{\mathrm{th}}$ column of the reduced matrix:

$$A_{nn}\underset{n\leqq i\leqq n_2}{argmax}\left|A_{in}\right|,n=1,\cdots ,n_2-1,$$

(4a)

but this pivot element is considered non-zero only if it is not too small in comparison with the largest element in the remainder of the $n^{\mathrm{th}}$ row:

$$\mathtt{if}\left|A_{nn}\right|\leqq ϵ\underset{n<j\leqq n_2}{max}\left|A_{nj}\right|\mathtt{then}{A}_{nn}0.$$

(4b)

2.4. Final Form of Transfer Function

After applying procedures described in the previous subsections for the calculations of both poles and zeros, we obtain the circuit transfer function in the form

$$c{\displaystyle \frac{{\displaystyle \prod _{i=1}^{n_{1,z}}(s-z_i)}}{{\displaystyle \prod _{i=1}^{n_1}(s-p_i)}}}.$$

(5)

(Of course, the number of zeros $n_{1,z}$ may be different from the number of poles $n_1$.)

3. Analytically Solved Example of Reduction Algorithm on Dynamically Degenerate Circuit

Simple but unusual circuit in Figure 2 was used in [33] to demonstration of multiple methods. For example, it was used to demonstrate the formulation and reduction of the state equations, during which some unusual operations had to be performed due to character of this circuit. However, the original formulation of the state equations in [33] had five equations – for example, the current of the voltage source E was also a variable. In this section, a simpler way of formulation of the system of equations containing only three variables is used. The purpose is to easily demonstrate that even for simple analog circuits, it may be necessary to use the non-standard step of the reduction consisting in differentiation of one of the equations.

The matrix equation corresponding to the circuit in Figure 2 is the following:

$$\left[s\left(\begin{array}{ccc}{C}_1& -C_1& \\ C_2& & \\ & & \phantom{0}\end{array}\right)+\left(\begin{array}{ccc}G& & 1\\ & & \beta -1\\ & 1& \end{array}\right)\right]\left(\begin{array}{c}V\hfill \\ V_{\mathrm{inp}}\hfill \\ I\hfill \end{array}\right)=\left(\begin{array}{c}\\ \\ E\hfill \end{array}\right).$$

(6)

For the calculation of pole(s), we will start modifying the $\mathit{A}$ and $\mathit{B}$ matrices to obtain the shapes shown in Figure 1. Initially, the first column will be added to the second column:

$$s\left(\begin{array}{ccc}{C}_1& & \\ C_2& C_2& \\ & & \phantom{0}\end{array}\right)+\left(\begin{array}{ccc}G& G& 1\\ & & \beta -1\\ & 1& \end{array}\right);$$

(7)

The second column will be subtracted from the first column:

$$s\left(\begin{array}{ccc}{C}_1& & \\ & C_2& \\ & & \phantom{0}\end{array}\right)+\left(\begin{array}{ccc}& G& 1\\ & & \beta -1\\ -1& 1& \end{array}\right);$$

(8)

${\mathit{A}}_{\mathbf{11}}$ is now diagonalized, however, ${\mathit{B}}_{\mathbf{22}}=\mathbf{0}$. Therefore, the third row must be differentiated, i.e., moved to the left, which is the step absent in the standard Gaussian elimination:

$$s\left(\begin{array}{ccc}{C}_1& & \\ & C_2& \\ -1& 1& \phantom{0}\end{array}\right)+\left(\begin{array}{ccc}& G& 1\\ & & \beta -1\\ \phantom{0}& & \end{array}\right);$$

(9)

$(-C_2)$-multiple of the third line will be added to the second line:

$$s\left(\begin{array}{ccc}{C}_1& & \\ C_2& & \\ -1& 1& \phantom{0}\end{array}\right)+\left(\begin{array}{ccc}& G& 1\\ & & \beta -1\\ \phantom{0}& & \end{array}\right);$$

(10)

$\left(-{\displaystyle \frac{C_2}{C_1}}\right)$-multiple of the first line will be added to the second line, and the third line is also de-differentiated (integrated), i.e., moved to the right (to the original position) as was in (8):

$$s\left(\begin{array}{ccc}{C}_1& & \\ & \phantom{0}& \\ & & \phantom{0}\end{array}\right)+\left(\begin{array}{ccc}& G& 1\\ & -G{\displaystyle \frac{C_2}{C_1}}& -{\displaystyle \frac{C_2}{C_1}}+\beta -1\\ -1& 1& \end{array}\right);$$

(11)

${\mathit{A}}_{\mathbf{11}}$ is now diagonalized again. As the circuit is degenerate, it only consists of one element. The second and third rows will be exchanged for a future diagonalization of ${\mathit{B}}_{\mathbf{22}}$:

$$s\left(\begin{array}{ccc}{C}_1& & \\ & \phantom{0}& \\ & & \phantom{0}\end{array}\right)+\left(\begin{array}{ccc}& G& 1\\ -1& 1& \\ & -G{\displaystyle \frac{C_2}{C_1}}& -{\displaystyle \frac{C_2}{C_1}}+\beta -1\end{array}\right);$$

(12)

$\left(G{\displaystyle \frac{C_2}{C_1}}\right)$-multiple of the second row will be added to the third row:

$$s\left(\begin{array}{ccc}{C}_1& & \\ & \phantom{0}& \\ & & \phantom{0}\end{array}\right)+\left(\begin{array}{ccc}& G& 1\\ -1& 1& \\ -G{\displaystyle \frac{C_2}{C_1}}& & -{\displaystyle \frac{C_2}{C_1}}+\beta -1\end{array}\right);$$

(13)

${\mathit{B}}_{\mathbf{22}}$ is now diagonalized, and we will start zeroing the ${\mathit{B}}_{\mathbf{12}}$ matrix as shown in Figure 1. $(-G)$-multiple of the second row will be added to the first row:

$$s\left(\begin{array}{ccc}{C}_1& & \\ & \phantom{0}& \\ & & \phantom{0}\end{array}\right)+\left(\begin{array}{ccc}G& & 1\\ -1& 1& \\ -G{\displaystyle \frac{C_2}{C_1}}& & -{\displaystyle \frac{C_2}{C_1}}+\beta -1\end{array}\right);$$

(14)

$\left(-{\displaystyle \frac{1}{-\frac{C_2}{C_1}+\beta -1}}\right)$-multiple of the third row will be added to the first row:

$$s\left(\begin{array}{ccc}{C}_1& & \\ & \phantom{0}& \\ & & \phantom{0}\end{array}\right)+\left(\begin{array}{ccc}G+{\displaystyle \frac{G\frac{C_2}{C_1}}{-\frac{C_2}{C_1}+\beta -1}}& & \\ -1& 1& \\ -G{\displaystyle \frac{C_2}{C_1}}& & -{\displaystyle \frac{C_2}{C_1}}+\beta -1\end{array}\right);$$

(15)

As ${\mathit{B}}_{\mathbf{12}}$ is now zeroed, the only pole of (all) transfer functions can be easily evaluated using the last part of (3):

$$\mathbf{I}=1,{\mathit{A}}_{\mathbf{11}}^{-1}={\displaystyle \frac{1}{C_1}},{\mathit{B}}_{\mathbf{11}}=G+{\displaystyle \frac{G\frac{C_2}{C_1}}{-\frac{C_2}{C_1}+\beta -1}},$$

(16)

therefore, the pole is the solution of the equation

$$det\left(s+{\displaystyle \frac{1}{C_1}}\left(G+{\displaystyle \frac{G\frac{C_2}{C_1}}{-\frac{C_2}{C_1}+\beta -1}}\right)\right)=0$$

(17)

and thus

$$\begin{array}{cc}\hfill p_1& =-{\displaystyle \frac{1}{C_1}}\left(G+{\displaystyle \frac{G\frac{C_2}{C_1}}{-\frac{C_2}{C_1}+\beta -1}}\right)=-{\displaystyle \frac{G}{C_1}}\left(1-{\displaystyle \frac{C_2}{C_2+C_1(1-\beta )}}\right)\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & =-{\displaystyle \frac{G}{C_1}}{\displaystyle \frac{C_1(1-\beta )}{C_2+C_1(1-\beta )}}=-{\displaystyle \frac{G(1-\beta )}{C_2+C_1(1-\beta )}},\hfill \end{array}$$

(19)

which is in complete agreement with [33]. However, the procedure defined in [33] is conceived more generally and therefore uses (repeatedly in more sections) matrices $5\times 5$. Although the exceptional row differentiation is also shown in [33], we are of the opinion that our version shown in (9) is more illustrative, and the general reduction strategy is clearer overall here due to the usage of the smaller matrices $3\times 3$.

4. Modifying Equations for Frequency-Scaled Semisymbolic Analysis

4.1. Formulating Modified System

In the frequency-scaled semisymbolic analysis, the formulation of the modification of (1) is controlled by a user-selected factor $\varpi$. One option is to directly modify the elements of the matrix A. However, locating these elements in memory is a bit complicated due to efficient utilization of sparsity of this matrix. Therefore, for simple and immediate testing the proposed algorithm, we have used modifying the capacitive and inductive elements during the formulation of the circuit equations.

Regarding the passive elements, the values of capacitors and inductors used for the formulation arise from original ones by multiplying by $\varpi$:

$$C_{\varpi }=C\times \varpi ,L_{\varpi }=L\times \varpi .$$

(20)

Regarding the active elements, the following modified model parameters are created and used in the formulation instead of the original ones:

$$\begin{array}{cc}\hfill {\tau }_{F,\varpi }& ={\tau }_F\times \varpi (\mathrm{ideal}\mathrm{forward}\mathrm{transit}\mathrm{time}),\hfill \end{array}$$

(21a)

$$\begin{array}{cc}\hfill {\tau }_{R,\varpi }& ={\tau }_R\times \varpi (\mathrm{ideal}\mathrm{reverse}\mathrm{transit}\mathrm{time}),\hfill \end{array}$$

(21b)

$$\begin{array}{cc}\hfill C_{JE,\varpi }& =C_{JE}\times \varpi \text{(zero-bias base-emitter depletion capacitance),}\hfill \end{array}$$

(21c)

$$\begin{array}{cc}\hfill C_{JC,\varpi }& =C_{JC}\times \varpi \text{(zero-bias base-collector depletion capacitance), and}\hfill \end{array}$$

(21d)

$$\begin{array}{cc}\hfill C_{JS,\varpi }& =C_{JS}\times \varpi \text{(zero-bias collector-substrate depletion capacitance)}\hfill \end{array}$$

(21e)

for bipolar junction transistors (a definition of the entire BJT model can be found in [41], but the quasi-saturation part of the model was not included because at the operating point – where the nonlinear models are linearized – transistors are far from the quasi-saturation),

$$\begin{array}{cc}\hfill {\tau }_{D,\varpi }& ={\tau }_D\times \varpi (\mathrm{transit}\mathrm{time}),\mathrm{and}\hfill \end{array}$$

(22a)

$$\begin{array}{cc}\hfill C_{J0,\varpi }& =C_{J0}\times \varpi (\mathrm{zero-bias}\mathrm{junction}\mathrm{capacitance})\hfill \end{array}$$

(22b)

for PN-junction diodes (the whole diode model is defined in [41] as well), and

$$\begin{array}{cc}\hfill {\left(\epsilon W\right)}_{\varpi }& =\left(\epsilon W\right)\times \varpi (\mathrm{permittivity}\mathrm{width}\mathrm{product}),\hfill \end{array}$$

(23a)

$$\begin{array}{cc}\hfill C_{JS,\varpi }& =C_{JS}\times \varpi (\mathrm{zero-bias}\mathrm{gate-source}\mathrm{junction}\mathrm{capacitance}),\hfill \end{array}$$

(23b)

$$\begin{array}{cc}\hfill C_{JD,\varpi }& =C_{JD}\times \varpi (\mathrm{zero-bias}\mathrm{gate-drain}\mathrm{junction}\mathrm{capacitance}),\mathrm{and}\hfill \end{array}$$

(23c)

$$\begin{array}{cc}\hfill C_{DS,\varpi }& =C_{DS}\times \varpi (\mathrm{drain}-\mathrm{source}\mathrm{capacitance})\hfill \end{array}$$

(23d)

for pHEMTs. (The original GaAs FET model is entirely defined in [42], however, for testing, we have used our original modification [43], which has been verified that it attains precision of standard models, e.g., see comparisons [44] for SiC MESFETs or [45] for GaN HEMTs.)

Let’s note some of the details of the models listed here are used to illustrate the possible way of the formulation, the accuracy of the models is not, of course, the topic of the article.

4.2. Determining Actual Poles, Zeros, and Constant of Transfer Function

Adjusting the formulation described in Subsect. Section 4.1 will certainly change the original transfer function (5), its poles, zeros, and constant of the transfer function, i.e., we will obtain modified values $p_{i,\varpi }$, $i=1,\cdots ,n_1$, $z_{i,\varpi }$, $i=1,\cdots ,n_{1,z}$, and $c_{\varpi }$. The actual poles, zeros, and constant of the transfer function (5) are simply determined in the following way:

$$\begin{array}{cc}\hfill p_i& =p_{i,\varpi }\times \varpi ,i=1,\cdots ,n_1,\hfill \end{array}$$

(24a)

$$\begin{array}{cc}\hfill z_i& =z_{i,\varpi }\times \varpi ,i=1,\cdots ,n_{1,z},\hfill \end{array}$$

(24b)

$$\begin{array}{cc}\hfill c& =c_{\varpi }\times {\varpi }^{n_1-n_{1,z}}.\hfill \end{array}$$

(24c)

4.3. Note About Controlling Factor

As the semisymbolic analysis – deflation of the generalized task of characteristic values to standard form – and subsequent determining characteristic numbers of matrix represent a very complex numerical process, an exact determination of the factor $\varpi$ is not possible. For the vast majority of tasks, however, a rather accurate result is achieved for a fairly wide class of values of this factor, mostly even for more orders of $\varpi$. (This circumstance was also utilized to deal with relatively difficult test tasks in this article.) However, the semisymbolic analysis is a relatively rapid numerical process (unlike, for example, optimization tasks), and hence there can always be performed multiple tests with different values of $\varpi$. And what is positive and very important: the inappropriate choice of $\varpi$ leads to calculating the wrong (at first glance) poles and zeros (with nonsensical orders, etc.) and can be so easily recognized. (In all our tests, it was thus possible to identify incorrect results immediately.)

5. Sample Examples of Different Levels of Complexity

5.1. Antenna Low-Noise Preamplifier for Multi-Constellation Receiver of Satellite Navigation

The circuit diagram of the antenna low-noise preamplifier (for the multi-constellation receiver of satellite navigation) is shown in Figure 3. (Note that the accurate values of the passive elements originated from a sophisticated design by multi-objective optimization.)

In this case, the frequency-scaled semisymbolic analysis was able to provide accurate results even with the standard 64-bit arithmetic. Therefore, the poles of (all) the transfer functions $-0.005093217\pm 0.5618716\mathrm{j}$, $-0.7139237$, $-0.2145403\pm 0.7367593\mathrm{j}$, $-0.5549448\pm 1.486727\mathrm{j}$, $-3.117465$, $-8.955381$, $-1.750791\pm 14.44991\mathrm{j}$, $-8.49903\pm 17.95136\mathrm{j}$, and $-20.28827$ as well as the zeros of the ${\displaystyle \frac{\mathrm{load}\mathrm{voltage}}{\mathrm{input}\mathrm{voltage}}}$ transfer function 0 (six-time zero), $18.70084$, and $-21.63001\pm 19.72322\mathrm{j}$ (poles/zeros written in gigahertz – GHz – units) were determined identically (!) by both the frequency-scaled semisymbolic analysis (64-bit compilation) and a controlling semisymbolic analysis with variable-length arithmetic (2048-bit compilation). Hence, a comparison table is not necessary in this case.

In this first and simplest example, we can briefly demonstrate that the algorithm is not extremely sensitive to the suggested new factor $\varpi$. For example, for the $\varpi$ values ${10}^8$, ${10}^9$, ${10}^{10}$, ${10}^{11}$, and ${10}^{12}$, we always get exactly the same poles and zeros listed in the previous paragraph. However, for the $\varpi$ value ${10}^7$ and several other lesser ones, we only get five-time zero instead of the correct six-times zero. Therefore, for example, the $\varpi$ value ${10}^{10}$ should be safe for analyses of this circuit.

For this “medium” $\varpi$ value (${10}^{10}$), we have also tried to examine the effect of the $ϵ$ parameter defined in (4b) on the accuracy of the solution. For the $ϵ$ values ${10}^{-12}$, ${10}^{-11}$, and ${10}^{-10}$, we again received exactly the same poles and zeros listed above, which were also confirmed by the extremely accurate controlling analysis using 2048-bit compilation. The $ϵ$ values ${10}^{-9}$, ${10}^{-8}$, and ${10}^{-7}$ were also usable, all the poles and zeros were again the same with the exception of one pole, which differed only in the seventh significant digit. However, the $ϵ$ values ${10}^{-14}$ and ${10}^{-13}$ led only to a four-time zero (instead of the correct six-time zero), which implies that the $ϵ$ parameter in (4b) cannot be too small in some tasks. Nevertheless, it should be emphasized that a suitable $ϵ$ parameter selection must be made in every semisymbolic analysis, the only new parameter in the suggested frequency-scaled semisymbolic analysis is $\varpi$, and hence the new task here is only to determine this $\varpi$ factor.

5.2. Discrete Operational Power Amplifier Working in AB Class Mode

The circuit diagram of the power operational amplifier (with a brief description of the basic parts of the circuit) is shown in Figure 4.

All the differing poles and zeros of the transfer function (the output is of course on the ungrounded contact of $R_{10}$) determined by both suggested frequency-scaled semisymbolic analysis and another extremely accurate but VERY slow analysis are shown in Table 1. It clearly confirms that the 64-bit implementation is sufficient if the frequency scaling is used.

5.3. MDA 272 Integrated Operational Amplifier

The circuit diagram of integrated operational amplifier of 272 class is shown in Figure 5.

The diagram also contains a description of the basic parts of this integrated circuit, and a test-bench circuit is shown as well. Certainly, the transfer function of this circuit is much more complicated than that for the previous one (in total, both 108 poles and zeros).

In Table 2, only the differing poles and zeros are shown. This comparison again clearly demonstrates that the 64-bit implementation of the frequency-scaled semisymbolic analysis is able to provide sufficient accuracy, because the differences between its results and the results of the extremely accurate 2048-bit implementation are negligible – for each pole and zero, less than one thousandth.

This example emphasizes how the use of the frequency scaling suggested in this article is very important – in [17], we have shown that the 64-bit implementation of the traditional (i.e., frequency-unscaled) semisymbolic analysis practically crashed in the case of using the sparse-matrix procedures. (And this article – as already mentioned above – is primarily focused on algorithms utilizing the sparsity of matrices that are promising for analyzes of huge circuits.) Another interesting matter is here that the $ϵ$ value ${10}^{-100}$ was also used for the 64-bit case, which implies that an application of (4b) probably even was not necessary.

5.4. Distributed Tunable Microwave Oscillator

The circuit diagram of the distributed (tunable) microwave oscillator is shown in Figure 6, together with the schemes of the pHEMT, filter and microstrip.

This is the most complicated and difficult example due to especially the microstrips that generate a really large number of poles and zeros. Although the circuit itself is an oscillator, it is possible to calculate a virtual operating point for it (with inductors replaced by a short circuit and capacitors omitted). And the circuit can be linearized at this operating point and the semisymbolic analyses can be performed with the results shown in Table 3.

Table 3 contains only selected poles (especially the interesting ones with the smallest and the biggest absolute values) and a “zero at zero” that is also a good test of accuracy. (Please see the footnotes below the table for more details.) Moreover, Table 3 also contains an important couple of poles 23,24, where the imaginary part can be used very well to estimate the oscillation frequency. This further confirms the meaningfulness of the semisymbolic analysis for weakly nonlinear circuits (and thus for weakly nonlinear oscillators as well)! Finally, let’s emphasize the really remarkable result shown in Table 3: all the poles included in the table were the same (to six significant digits) for both 2048-bit and 64-bit precisions. This is obviously a clear evidence, without the use of the suggested frequency scaling in the 64-bit implementation (here, $\varpi ={10}^{13}$ chosen), such an accordance would not be possible!

6. Combination of Frequency Scaling and More Accurate Arithmetic

In the previous sections, we have shown that the frequency-scaled semisymbolic analysis provides sufficiently accurate results even using standard 64-bit arithmetic. This fact confirms the basic idea of the frequency scaling itself and is also very important because a number of mathematical libraries are provided in the 64-bit version. (E.g., the LAPACK DLL libraries are available in 32- and 64-bit arithmetics, but not in the 80- or 128-bit version.) It is logical that the use of more accurate arithmetic in the frequency-scaled semisymbolic analysis will further increase the numerical stability of the whole process.

This can be clearly illustrated in the results for the low-noise antenna preamplifier described in the last paragraph of Subsect. Section 5.1. When using 64-bit arithmetic and factor $\varpi$${10}^{10}$, the analysis provided accurate zeros of the transfer function for the values of $ϵ$${10}^{-10}$, ${10}^{-11}$, and ${10}^{-12}$, and inaccurate zeros for ${10}^{-13}$ and lesser. However, when using 128-bit arithmetic (and the same factor $\varpi$${10}^{10}$), the analysis will provide accurate zeros of the transfer function for the values of $ϵ$${10}^{-10}$, ${10}^{-11}$, ..., ${10}^{-30}$, and ${10}^{-31}$, and inaccurate zeros for ${10}^{-32}$ and lesser. In other words, possible range of $ϵ$ in (4b) is far greater for the 128-bit arithmetic!

It can be noted that the response of the analyses to the $ϵ$ parameter described in the previous paragraph quite corresponds to expectations due to the used arithmetics. The 64-bit arithmetic provides accuracy up to 17 significant digits, and the 128-bit arithmetic provides accuracy up to 34 significant digits. Therefore, the minimum applicable values of $ϵ$${10}^{-12}$ and ${10}^{-31}$ correspond to these accuracies quite well. (The value of the $ϵ$ parameter suitable for the 64-bit arithmetic is also confirmed by the results in Table 1 and Table 3, where ${10}^{-16}$ was primarily used, although there exist tasks in which no tiny element of the matrix had to be declared zero by (4b), and hence ${10}^{-100}$ could be used as seen in Table 2.)

7. Another Minor, but Important Improvement

In connection with the overall increase of the robustness of all subroutines for the semisymbolic analysis, we have introduced an important adjustment in both new and existing procedures concerning the products ${\prod }_{i=1}^{n_1}{\left({\mathit{A}}_{\mathbf{11}}\right)}_{ii}{\prod }_{i=n_1-2mu+1}^{n_2}{\left({\mathit{B}}_{\mathbf{22}}\right)}_{ii}$ in (3). For (very) large circuits, there is a considerable risk of overflow (or underflow) in these multiplications. Therefore, we only store the logarithms of absolute values and signs during the gradual execution of these products, which turned out to be an absolutely safe solution in all the alternatives of the semisymbolic analysis.

8. Discussion

Although problems with the accuracy of semisymbolic analysis have been known for a long time, most of the tasks can be solved by procedures programmed in the usual 64-bit arithmetic. However, we have found several difficult cases in the library of our circuits analyzed in the previous period, which could not be solved precisely even by the latest semisymbolic analysis procedures. (And poles and zeros of a transfer function for some of them were determined completely incorrectly when using standard 64-bit arithmetic.)

Certainly, the first way for solving this problem is to use a more precise arithmetic, e.g., 80-bit or 128-bit implementations of the algorithms. We first presented using this so-called brute force at the conference [32], and an extended version of this paper was subsequently selected for publication in the book [17]. Both publications [17,32] documented the ability of the 128-bit arithmetic to improve the accuracy of the calculation of poles and zeros of a transfer function. In addition, both publications contain very illustratively solved tasks of the reduction of the general eigenvalue problem to the standard eigenvalue problem. In [32], the reduction for of an unproblematic analog circuit was shown; however, in [17], the reduction for a much more complicated digital circuit was demonstrated in a detailed way.

In this article, a novel method based on the frequency scaling has been suggested. This new method represents a much more sophisticated way of solving the accuracy problem compared to simple enlargement of the lengths of numbers in memory described in our previous works [17,32]. In addition, the implementation of the new method into existing algorithms of semisymbolic analysis is very easy (one simple way is also shown in the article), so it is a very cheap yet very effective solution.

In addition, in the article we naturally mentioned probably the most powerful form of solution to the problem of accuracy: a combination of the newly suggested frequency scaling and a more accurate (e.g., 128-bit) arithmetic. Of course, this solution is probably the most robust, but the main objective of this article was to check the proposed frequency scaling in the standard 64-bit arithmetic. (There are not many compilers with 128-bit arithmetic; however, for example, Fortran/C compilers from Intel or GNU have this arithmetic implemented.)

Another minor but valuable improvement in the implementation of the semisymbolic analysis concerning the prevention of possible overflow is shown as well.

This article also contains the illustrative example of the reduction of the general eigenvalue problem to the standard one for a simple but extraordinary dynamically degenerate analog circuit, where the special step consisting in differentiating a row had to be used.

9. Conclusions

Briefly, the article summarizes these new achievements:

• A new method based on the frequency scaling is suggested for the semisymbolic analysis that significantly improves the accuracy of poles and zeros of transfer functions.
• The proposed procedure is particularly important for modern microwave circuits, for which the semisymbolic analysis leads to a huge difference in the magnitude of the matrix elements and hence to a numeric instability.
• In this way, even very difficult tasks can be operated even by using the standard 64-bit implementations of algorithms for the semisymbolic analysis.
• Implementation of the frequency scaling into existing subroutines for semisymbolic analysis is very easy, and one of the possible ways is shown in the article.
• Four difficult tasks have been demonstrated, for which it is not possible to achieve accurate poles and/or zeros by a 64-bit algorithm implementation. These tasks can only be solved with the newly proposed frequency scaling in the 64-bit implementation (or by less common 128-bit implementation, e.g., or by a combination of both).
• Possible combination of the frequency scaling and a more accurate (especially 128-bit) arithmetic is also considered. (Although the article is primarily focused on the scaling.)
• The article also contains an illustrative analytically solved example of an unusual dynamically degenerate circuit in which an extraordinary step of the reduction must be used.