# Mathematical Circuit Root Simplification Using an Ensemble Heuristic–Metaheuristic Algorithm

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## Abstract

**:**

## 1. Introduction

- Introducing a combined mathematical modeling and optimization technique for the extraction and simplification of symbolic poles and zeros in OTAs.
- Proposing an enhanced root splitting technique, named ERS, to accurately extract the exact pole/zero expressions.
- Applying a combined heuristic–metaheuristic optimization algorithm to solve the proposed symbolic root simplification problem, utilizing the heuristic knowledge available in the circuit model and SA.
- Programming the proposed method in a MATLAB m-file, wherein simplified root equations are automatically generated from the circuit netlist.
- Successfully driving symbolic pole/zero expressions for three OTAs.

## 2. Literature Review

#### 2.1. Symbolic Simplification Techniques

^{34}terms within its voltage transfer function [16]. Therefore, symbolic analysis tools must rely on simplification techniques to tackle the complexity and hardness of real-world circuits. Based on the steps taken in the simplification process, simplification algorithms can be categorized into SAG (simplification-after-generation), SDG (simplification-during-generation), and SBG (simplification-before-generation) [17]. It is worth noting that the proposed PZSA algorithm in this study is an SAG technique. An SAG is performed once the symbolic circuit analysis is done and, as a result, the exact symbolic expressions have been obtained. Therefore, simplified functions can be constructed from some terms of the exact expressions. In the following section, we discuss the details of the SAG technique used in the proposed method.

**,**according to Equation (1) [8], where each polynomial ${\stackrel{\xb4}{f}}_{i}\left(\mathbf{x}\right)$ or ${f}_{i}\left(\mathbf{x}\right)$ is a sum-of-product (SOP) of $\mathbf{x}$. This is expressed as ${h}_{k}\left(\mathbf{x}\right)={h}_{k1}\left(\mathbf{x}\right)+{h}_{k2}\left(\mathbf{x}\right)+\dots +{h}_{kT}\left(\mathbf{x}\right)$, where ${h}_{k}\left(\mathbf{x}\right)$ is the $k$-th polynomial within the circuit transfer function $H\left(s,\mathbf{x}\right)$, comprising $T$ terms.

#### 2.2. Symbolic Pole/Zero Extraction Techniques

## 3. Proposed Method

- The input circuit netlist is loaded as a text file (in .txt format).
- All transistors are replaced via proper small-signal modeling.
- The symbolic circuit is solved via a modified nodal analysis (MNA).
- The exact transfer function (TF) is achieved in the expanded symbolic form.
- The exact expressions of poles and zeroes are derived using ERS.
- The numerical results of the exact symbolic pole/zero expressions are stored.
- A heuristic algorithm is performed to generate a near-optimal solution, utilizing the circuit-based knowledge available in the exact poles and zeroes.
- SA is performed to improve further the quality of the heuristic solution in order to generate the final simplified symbolic pole/zero expressions.
- The numerical results of the obtained simplified symbolic pole/zero expressions are calculated.
- The numerical results of the exact and simplified poles/zeroes are compared against HSPICE and other simplification algorithms.

#### 3.1. Symbolic Pole/Zero Extraction via ERS

Algorithm 1. Symbolic Pole/Zero Extraction using ERS |

Inputs: |

Symbolic exact expanded transfer function |

Numerical values of the circuit parameters in the nominal point |

$\mathrm{Maximum}\text{}\mathrm{allowable}\text{}\mathrm{root}\text{}\mathrm{displacement}\text{}({T}_{ERS}$) |

Output: |

Symbolic expressions of poles and zeros |

Numerical Analysis: |

1. $\text{}\mathrm{Extract}\text{}\mathrm{coefficients}\text{}\mathrm{of}\text{}\mathrm{the}\text{}\mathrm{numerator}\text{}({\stackrel{\xb4}{f}}_{0}$$,\text{}{\stackrel{\xb4}{f}}_{1}$$,\dots ,\text{}{\stackrel{\xb4}{f}}_{\stackrel{\xb4}{n}}$$\left)\text{}\mathrm{and}\text{}\mathrm{the}\text{}\mathrm{denominator}\text{}\right({f}_{0}$$,\text{}{f}_{1}$$,\text{}\dots ,\text{}{f}_{n}$) |

2. Numerically evaluate the exact expanded transfer function in the nominal point |

3. $\mathrm{Numerically}\text{}\mathrm{find}\text{}\mathrm{roots}\text{}\mathrm{of}\text{}\mathrm{the}\text{}\mathrm{numerator}\text{}\mathrm{and}\text{}\mathrm{the}\text{}\mathrm{denominator}\text{}\mathrm{in}\text{}[{f}_{min},{f}_{max}]$ |

4. $\mathrm{Sort}\text{}\mathrm{the}\text{}\mathrm{exact}\text{}\mathrm{numerical}\text{}\mathrm{zeroes}\text{}({z}_{E,j}$$\left)\text{}\mathrm{and}\text{}\mathrm{poles}\text{}\right({p}_{E,i}$) by their magnitude |

Extraction of Symbolic Zeros: |

5. $\mathbf{for}\text{}j\in ZeroSet$ |

6. Numerically estimate the position of zero j as ${z}_{ERS,j}=-{\stackrel{\xb4}{f}}_{j-1}/{\stackrel{\xb4}{f}}_{j}$ |

7. $\mathbf{if}\text{}\left|\left({z}_{ERS,j}-{z}_{E,j}\right)/{z}_{E,j}\right|\le {T}_{ERS}$ |

8. $\mathrm{Extract}\text{}\mathrm{the}\text{}\mathrm{real}\text{}\mathrm{zero}\text{}j\text{}\mathrm{as}\text{}{z}_{j}={\stackrel{\xb4}{f}}_{j-1}/{\stackrel{\xb4}{f}}_{j}$ |

9. else |

10. $\mathrm{Consider}\text{}{z}_{j}$$\text{}\mathrm{and}\text{}{z}_{j+1}$ in one cluster 11. $\mathrm{Calculate}\text{}a=\left({\stackrel{\xb4}{f}}_{j}/{\stackrel{\xb4}{f}}_{j-1}\right)+\left({\stackrel{\xb4}{f}}_{j+1}/{\stackrel{\xb4}{f}}_{j}\right)$$\text{}\mathrm{and}\text{}b={\stackrel{\xb4}{f}}_{j+1}/{\stackrel{\xb4}{f}}_{j-1}$ |

12. $\mathbf{if}\text{}{a}^{2}\ge 4b$ |

13. $\mathrm{Extract}\text{}\mathrm{the}\text{}\mathrm{real}\text{}\mathrm{zero}\text{}j\text{}\mathrm{as}\text{}{z}_{j}=-a+\left(\sqrt{{a}^{2}-4b}/2b\right)$ |

14. $\mathrm{Extract}\text{}\mathrm{the}\text{}\mathrm{real}\text{}\mathrm{zero}\text{}j+1\text{}\mathrm{as}\text{}{z}_{j+1}=-a-\left(\sqrt{{a}^{2}-4b}/2b\right)$ |

15. else |

16. $\mathrm{Extract}\text{}\mathrm{the}\text{}\mathrm{complex}\text{}\mathrm{conjugate}\text{}\mathrm{zeroes}\text{}j\text{}\mathrm{and}\text{}j+1\text{}\mathrm{as}\text{}{z}_{j,j+1}=-a\pm j\left(\sqrt{{a}^{2}-4b}/2b\right)$ |

17. end if18. end if19. end for |

Extraction of Symbolic Poles: |

20. $\mathbf{for}\text{}i\in PoleSet$ |

21. $\mathrm{Numerically}\text{}\mathrm{estimate}\text{}\mathrm{the}\text{}\mathrm{position}\text{}\mathrm{of}\text{}\mathrm{pole}\text{}i\text{}\mathrm{as}\text{}{p}_{ERS,i}=-{f}_{i-1}/{f}_{i}$ |

22. $\mathbf{if}\text{}\left|\left({p}_{ERS,i}-{p}_{E,i}\right)/{p}_{E,i}\right|\le {T}_{ERS}$ |

23. $\mathrm{Extract}\text{}\mathrm{the}\text{}\mathrm{real}\text{}\mathrm{pole}\text{}i\mathrm{as}\text{}{p}_{i}={f}_{i-1}/{f}_{i}$ |

24. else |

25. $\mathrm{Consider}\text{}{p}_{i}$$\text{}\mathrm{and}\text{}{p}_{i+1}$ in one cluster 26. $\mathrm{Calculate}\text{}a=\left({f}_{i}/{f}_{i-1}\right)+\left({f}_{i+1}/{f}_{i}\right)$$\text{}\mathrm{and}\text{}b={f}_{i+1}/{f}_{i-1}$ |

27. $\mathbf{if}\text{}{a}^{2}\ge 4b$ |

28. $\mathrm{Extract}\text{}\mathrm{the}\text{}\mathrm{real}\text{}\mathrm{pole}\text{}i\mathrm{as}\text{}{p}_{i}=-a+\left(\sqrt{{a}^{2}-4b}/2b\right)$ |

29. $\mathrm{Extract}\text{}\mathrm{the}\text{}\mathrm{real}\text{}\mathrm{pole}\text{}i+1\text{}\mathrm{as}\text{}{p}_{i+1}=-a-\left(\sqrt{{a}^{2}-4b}/2b\right)$ |

30. else |

31. $\mathrm{Extract}\text{}\mathrm{the}\text{}\mathrm{complex}\text{}\mathrm{conjugate}\text{}\mathrm{poles}\text{}i\mathrm{and}\text{}i+1\text{}\mathrm{as}\text{}{p}_{i,i+1}=-a\pm j\left(\sqrt{{a}^{2}-4b}/2b\right)$ |

32. end if34. end if35. end for |

#### 3.2. Symbolic Pole/Zero Simplification via PZSA

#### 3.2.1. Solution Encoding/Decoding

#### 3.2.2. Generation of the Initial Solution

#### 3.2.3. Objective Function Evaluation

_{n}) demonstrates the proportion of the selected symbolic terms to the total number of exact terms extracted by the ERS method. Additionally, the second and third sub-objectives (E

_{p}and E

_{z}) express the mean displacements of the simplified poles and zeroes, respectively. According to Equation (35), these three sub-objectives are merged into a single objective function (OF) to be minimized by the PZSA method, where the sub-objectives E

_{n}, E

_{p}

_{,}and E

_{z}are calculated via Equations (36)–(38), respectively.

#### 3.2.4. Generation of a New Solution

#### 3.2.5. Acceptance Rule Checking

## 4. Performance Evaluation

#### 4.1. Results for a Three-Stage Amplifier in the RCg_{m} Model (Circuit 1)

_{m}model. First, the exact expanded TF is derived according to Equation (43) using the MNA technique. Next, by applying the simplification algorithm in [26], the simplified expanded TF is obtained according to Equation (44). By performing PZSA, 3 poles and 2 zeroes can be achieved as Equations (45)–(49).

_{SA}= 20%. Although in some cases the existing methods have achieved less pole/zero displacements, the resulting expressions are not as compact as the proposed method. This occurs due to the selection of a much larger value for ${w}_{n}$ compared to the values ${w}_{p}$ and ${w}_{z}$ in the defined objective function. However, as previously mentioned, these values can be modified by the circuit designer based on the desired application requirements.

#### 4.2. Results for a Two-Stage Miller Compensated Amplifier (Circuit 2)

_{SA}= 20%.

#### 4.3. Results for a Three-Stage Amplifier in Transistor Model (Circuit 3)

#### 4.4. Discussion

_{n}, E

_{p}, and E

_{z}) for the different circuits. As previously mentioned, the total number of terms in the exact poles/zero expressions extracted by the ERS method are 68, 204, and 2061 for circuits 1, 2, and 3, respectively. Subsequently, these expressions are then simplified by the PZSA method resulting in a total of 11, 9, and 19 terms. As a result, the first sub-objective (E

_{n}) equals the proportion of the number of simplified terms to the exact terms for each circuit. Additionally, the obtained results for the sub-objectives E

_{p}and E

_{z}demonstrate that all pole/zero displacements do not exceed the pre-specified threshold of T

_{SA}= 20%.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Encoding of a solution: If ${S}_{i}$ = 1, the $i$-th symbolic term is present in the solution; otherwise, if ${S}_{i}$ = 0, the $i$-th term has been discarded from the solution.

Sets/Parameters | Definition |
---|---|

$i$ | Index of poles, $i=\mathrm{1,2},\dots ,n$ |

$j$ | Index of zeroes, $j=\mathrm{1,2},\dots ,\stackrel{\xb4}{n}$ |

$n$ | Degree of the denominator within the exact expanded TF |

$\stackrel{\xb4}{n}$ | Degree of the numerator within the exact expanded TF |

$k$ | Index of the symbolic terms, $k=\mathrm{1,2},\dots ,L$ |

$L$ | Number of symbolic terms within all pole/zero expressions |

$[{f}_{min},{f}_{max}]$ | Defined frequency bound range for the pole/zero extraction |

${S}_{k}$ | A binary parameter: 1 if the $k$-th term is presented; 0 otherwise |

${E}_{n}$ | Percentage of the selected symbolic terms |

$PoleSet$ | Set of poles in the frequency range of $[{f}_{min},{f}_{max}]$ |

ZeroSet | Set of zeroes in the frequency range of $[{f}_{min},{f}_{max}]$ |

${p}_{E,i}$ | $i$-th pole within the exact expanded TF |

${p}_{ERS,i}$ | $i$-th extracted pole via ERS |

${p}_{SA,i}$ | $i$-th simplified pole via SA |

${E}_{p}$ | Mean pole displacements |

${z}_{E,j}$ | $j$-the zero of the exact expanded TF |

${z}_{ERS,j}$ | $j$-th extracted zero via ERS |

${z}_{SA,j}$ | $j$-th simplified zero via SA |

${E}_{z}$ | Mean zero displacements |

${T}_{ERS}$ | Maximum allowable pole/zero extraction error via ERS |

${T}_{SA}$ | Maximum allowable pole/zero simplification error via SA |

Phase | Parameter | Value/Description |
---|---|---|

Model Parameters | ${f}_{min}$ | 1 Hz |

${f}_{max}$ | 10 × ${f}_{T}$ | |

${T}_{ERS}$ in Equations (32) and (33) | 10% | |

${T}_{SA}$ in Equations (39) and (40) | 20% | |

${w}_{n}$ in Equation (35) | 0.99 | |

${w}_{p}$ in Equation (35) | 0.005 | |

${w}_{z}$ in Equation (35) | 0.005 |

Phase | Parameter | Parameter Levels | Selected Value | ||
---|---|---|---|---|---|

SA Parameters | Maximum iterations | L | 5 × L | 10 × L | 5 × L |

Local search operators | Swap | Exchange | Swap/Exchange | Swap/Exchange | |

${T}_{initial}$ in Equation (42) | 10^{−5} | 10^{−4} | 10^{−3} | 10^{−5} | |

${T}_{final}$ in Equation (42) | 0 | 10^{−8} | 10^{−10} | 0 |

Expression | Expanded TF (Exact) | Ref. [26] | Ref. [29] | Ref. [31] | Proposed Exact (ERS) | Proposed Simplified (SA) |
---|---|---|---|---|---|---|

P1 | - | - | 1 | 10 | 10 | 1 |

P2 | - | - | 4 | 26 | 26 | 3 |

P3 | - | - | 5 | 25 | 25 | 3 |

Z1 | - | - | 2 | 9 | 3 | 2 |

Z2 | - | - | 2 | 9 | 4 | 2 |

Overall TF | 40 | 10 | - | - | - | - |

Parameter | HSPICE | Expanded TF (Exact) | Ref. [26] | Ref. [29] | Ref. [31] | Proposed Exact | Proposed Simplified |
---|---|---|---|---|---|---|---|

P1 (Hz) | −12.8 | −12.8 | −13.3 | −13.2 | −12.8 | −12.8 | −13.2 |

P2 (MHz) | −3.19 | −3.19 | −3.49 | −3.19 | −2.96 | −2.96 | −3.18 |

P3 (MHz) | −40.6 | −40.6 | −36.3 | −43.9 | −43.8 | −43.8 | −39.8 |

Z1 (MHz) | 2.72 | 2.72 | 2.72 | 3.18 | 3.36 | 3.18 | 3.18 |

Z2 (MHz) | −18.6 | −18.6 | −18.6 | −15.9 | −17.5 | −15.9 | −15.9 |

Mean pole displacement (%) | - | - | 7.8 | 3.8 | 5 | 5 | 1.9 |

Max pole displacement (%) | - | - | 10.6 | 8.36 | 7.9 | 7.9 | 3.5 |

Mean zero displacement (%) | - | - | 0.03 | 15.9 | 14.7 | 15.8 | 15.9 |

Max zero displacement (%) | - | - | 0.04 | 17.1 | 23.6 | 17.1 | 17.1 |

Expression | Expanded TF (Exact) | Ref. [26] | Ref. [29] | Ref. [31] | Proposed Exact (ERS) | Proposed Simplified (SA) |
---|---|---|---|---|---|---|

P1 | - | - | 5 | 104 | 104 | 5 |

P2 | - | - | 7 | 82 | 82 | 2 |

Z | - | - | 4 | 18 | 18 | 2 |

Overall TF | 134 | 11 | - | - | - | - |

Parameter | HSPICE | Expanded TF (Exact) | Ref. [26] | Ref. [29] | Ref. [31] | Proposed Exact | Proposed Simplified |
---|---|---|---|---|---|---|---|

P1 (KHz) | −177.1 | −178.5 | −192 | −152.8 | −178.4 | −178.4 | −152.8 |

P2 (MHz) | −377.4 | −435.4 | −409.1 | −341 | −435.6 | −435.6 | −409.3 |

Z (MHz) | 407.2 | 409.3 | 409.3 | 409.3 | 409.3 | 409.3 | 409.3 |

Mean pole displacement (%) | - | - | 6.8 | 18 | 0.04 | 0.04 | 10.2 |

Max pole displacement (%) | - | - | 7.5 | 21.7 | 0.04 | 0.04 | 14.4 |

Zero displacement (%) | - | - | 0.01 | 0.01 | 0 | 0 | 0.01 |

Expression | Expanded TF (Exact) | Ref. [26] | Ref. [29] | Ref. [31] | Proposed Exact (ERS) | Proposed Simplified (SA) |
---|---|---|---|---|---|---|

P1 | - | - | 29 | 714 | 714 | 9 |

P2 | - | - | 21 | 837 | 837 | 3 |

P3 | - | - | 23 | 330 | 330 | 3 |

Z1 | - | - | 15 | 75 | 75 | 2 |

Z2 | - | - | 13 | 105 | 105 | 2 |

Overall TF | 1320 | 18 | - | - | - | - |

Parameter | HSPICE | Expanded TF (Exact) | Ref. [26] | Ref. [29] | Ref. [31] | Proposed Exact | Proposed Simplified |
---|---|---|---|---|---|---|---|

P1 (Hz) | −27.7 | −27.9 | −20.6 | −20.5 | −27.9 | −27.9 | −22.8 |

P2 (MHz) | −1.84 | −1.84 | −2.03 | −2.16 | −1.76 | −1.76 | −2.07 |

P3 (MHz) | −36.6 | −40.2 | −36.3 | −36.1 | −42.1 | −42.1 | −35.7 |

Z1 (MHz) | 1.4 | 1.4 | 1.4 | 2.23 | 1.62 | 1.62 | 1.62 |

Z2 (MHz) | −10.3 | −10.2 | −10.5 | −7.37 | −8.81 | −8.81 | −9.1 |

Mean pole displacement (%) | - | - | 15.6 | 18 | 3 | 3 | 14.2 |

Max pole displacement (%) | - | - | 26.4 | 26.5 | 4.6 | 4.6 | 18.4 |

Mean zero displacement (%) | - | - | 1.7 | 43.5 | 14.8 | 14.8 | 13.5 |

Max zero displacement (%) | - | - | 2.9 | 59.3 | 15.9 | 15.9 | 16.1 |

Circuit/Objective | E_{n} | E_{p} | E_{z} |
---|---|---|---|

Circuit 1 | 0.1617 | 0.019 | 0.159 |

Circuit 2 | 0.0441 | 0.102 | 0.0001 |

Circuit 3 | 0.0092 | 0.142 | 0.135 |

Circuit/Objective | Circuit 1 | Circuit 2 | Circuit 3 |
---|---|---|---|

Run 1 | 0.1619 | 0.0447 | 0.0119 |

Run 2 | 0.162 | 0.0451 | 0.013 |

Run 3 | 0.1618 | 0.0448 | 0.0128 |

Run 4 | 0.1766 | 0.045 | 0.0123 |

Run 5 | 0.162 | 0.0447 | 0.0114 |

Run 6 | 0.1767 | 0.0445 | 0.0115 |

Run 7 | 0.1617 | 0.0446 | 0.012 |

Run 8 | 0.1616 | 0.0449 | 0.0129 |

Run 9 | 0.1473 | 0.0398 | 0.0117 |

Run 10 | 0.1621 | 0.0447 | 0.0121 |

Average | 0.1634 | 0.0443 | 0.0122 |

Standard Deviation | 0.0083 | 0.0016 | 0.0006 |

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## Share and Cite

**MDPI and ACS Style**

Behmanesh-Fard, N.; Yazdanjouei, H.; Shokouhifar, M.; Werner, F.
Mathematical Circuit Root Simplification Using an Ensemble Heuristic–Metaheuristic Algorithm. *Mathematics* **2023**, *11*, 1498.
https://doi.org/10.3390/math11061498

**AMA Style**

Behmanesh-Fard N, Yazdanjouei H, Shokouhifar M, Werner F.
Mathematical Circuit Root Simplification Using an Ensemble Heuristic–Metaheuristic Algorithm. *Mathematics*. 2023; 11(6):1498.
https://doi.org/10.3390/math11061498

**Chicago/Turabian Style**

Behmanesh-Fard, Navid, Hossein Yazdanjouei, Mohammad Shokouhifar, and Frank Werner.
2023. "Mathematical Circuit Root Simplification Using an Ensemble Heuristic–Metaheuristic Algorithm" *Mathematics* 11, no. 6: 1498.
https://doi.org/10.3390/math11061498