The symbolic analysis of OTAs suffers from NP-hardness [7]. For instance, the μA741 amplifier has about 10³⁴ terms within its voltage transfer function [23]. Therefore, symbolic analysis tools must rely on the simplification techniques to tackle with the complexities and hardnesses of real-world circuits. Based on the step in which a simplification is done, simplification algorithms can be categorized into SAG (simplification-after-generation), SDG (simplification-during-generation), and SBG (simplification-before-generation) [24]. It is worth noting that the PZSA algorithm in this study is a SAG technique. A SAG is performed once the symbolic circuit analysis is done and consequently the exact symbolic expressions have been obtained so that simplified functions can be constructed from some terms of the exact expressions. In the following, we discuss the details of the SAG technique, which is used in the proposed method.

The small-signal transfer function of a linear or linearized circuit can be represented as a function of the frequency $s$ and the circuit parameters $\mathbf{x}$ as follows:

Each polynomial ${\acute{f}}_i\left(\mathbf{x}\right)$ or $f_i\left(\mathbf{x}\right)$ is a sum-of-product (SOP) of $\mathbf{x}$ which 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 $k$-th polynomial within the circuit transfer function $H\left(s,\mathbf{x}\right)$ comprising $T$ terms.

The simplification method in [25] finds the largest term in terms of the magnitude, $h_{km}\left(\mathbf{x}\right)$, for the polynomial $h_k\left(\mathbf{x}\right)$. Then, all other terms within the polynomial $h_k\left(\mathbf{x}\right)$ are taken one by one. The condition for which the term $h_{kl}\left(\mathbf{x}\right)$ can be discarded from the polynomial $h_k\left(\mathbf{x}\right)$, is $\left|h_{kl}\left(\mathbf{x}\right)\right|\le \epsilon \times \left|h_{km}\left(\mathbf{x}\right)\right|$. Here, $\epsilon$ (0 < $\epsilon$ < 1) is a user-specified threshold to limit the maximum error. The main drawback is that the error may be accumulated. To overcome this drawback, the reported criterion in [26] sorts the terms within $h_k\left(\mathbf{x}\right)$ based on their magnitude obtained in the nominal point. Afterwards, $P$ terms with the least accumulated magnitude are discarded from the polynomial, if the error is below $\epsilon$. The condition on the $P$ terms for which they could be discarded, can be expressed as follows:

Although this method achieves more accurate expressions at the nominal point, it may cause significant errors for other values of the parameters. To avoid an elimination of the mutually canceling terms, the method in [27] presented an enhanced condition for which the $P$ terms with the least magnitude can be discarded if:

In the above-mentioned techniques, the maximum error is limited for each polynomial. However, the obtained error in the poles and zeroes is not under consideration. If the same error ${\epsilon }_M$ drives in all polynomials, no pole and/or zero displacement can be observed [28]. To overcome this drawback, an adaptive $\epsilon$ can be used, in which, erm deletion is done step by step while displacements in poles and zeros are monitored at every step, so that the term pruning procedure would be finished if the obtained displacements are beyond a pre-determined threshold [26].

Recently, various swarm and evolutionary metaheuristic algorithms [29,30,31,32,33,34] have been applied for the simplified symbolic analysis of OTAs. In these techniques, different criteria such as the magnitude error, phase error, and pole/zero displacements, have been used to evaluate feasible solutions generated by the metaheuristic algorithm. Although these methods achieve a low mean error rate, the worst cases of the displacements in the poles and zeroes are not accurately under consideration. The common drawback of the existing techniques is that the simplified function is achieved in either expanded or nested form. In other words, the transfer function is not derived in a factorization form which makes it hard to evaluate the contribution of roots.

The symbolic pole/zero analysis also suffers from the NP-hardness, even worse, as some operations between the polynomials have to be performed. Generally, a direct calculation of the roots from the expanded transfer functions yields very complex results for polynomials with degrees larger than two [7]. Since the numerator and denominator of a transfer function in practical OTAs have generally degrees much larger than two, it is rarely possible to mathematically find the exact symbolic pole/zero expressions [10].

In the following, the existing pole/zero extraction methods including root spitting, time-constant analysis, and eigenvalue analysis are discussed. Root splitting [11] is one of the well-known root extraction techniques. It extracts poles assuming them to be reciprocally dominant. By factorization, the denominator of the exact function in Eq. (1) can be re-written as a function of the poles $p_i$ as follows:

It follows from Eq. (1) that

Therefore, assuming $p_1$ to be the dominant pole within the denominator, the first pole can be approximately expressed as:

Consequently, with a similar approach, considering $p_i$ to dominate the other poles, $p_i$ can be given by the negative quotient of the two consecutive coefficients $f_{i-1}$ and $f_i$ [8]. Similar argumentations can be done to calculate simplified zeroes from the numerator of the transfer function. The root splitting method is not appropriate for manual pole/zero calculations (hand-and-paper analysis), as some estimations and simplifications should be developed to allow the circuit designer in extracting approximate dominant poles and zeroes manually. The most popular approach of such techniques is the time-constant method [10,12,13,14], which is expressed in Eq. (7), where ${\tau }_k$ can be achieved by multiplying the resistance $R_k$ to the capacitance $C_k$.

This technique is error-prone, as there is no information about the accuracy of the obtained symbolic results. Moreover, zeroes and higher-order poles cannot be determined by this approach, at all. A more general approach in this context was reported in [13] which has also the ability to extract the higher-order poles. This method is based on open-circuits and short-circuits analysis to calculate the time constants of the circuit.

There are also some pole/zero extraction methods on the basis of the solution derived by the eigenvalue problem. A positive feature of these methods is that the simplification is no longer driven by magnitude and phase errors but by the pole/zero position, allowing an improved error control. For example, a modified Signal-Flow Graph (named MSFG) has been recently developed to represent the equivalences between the system and SPICE outcomes of static nonlinear OTAs [15]. In this method, the circuit is firstly converted into an MSFG, and then, the graph would be simplified in particular polynomials by minimizing the MSFGs. In [19], the implementation of some simplification procedures during the eigenvalue computation via a symbolic LR algorithm was addressed, in which the LR method is applied to compute the reduced matrix corresponding to the eigenvalue cluster. This technique is followed in [20] by an algorithm to reduce the circuit matrix into a row echelon format. After the determination of the symbolic state matrix, the approximated poles and zeroes are achieved using the LR algorithm.

The main drawback of the existing symbolic pole/zero analysis methods is that the simplified expressions of poles and zeroes are not so compact as no SAG is applied on the final expressions. So, in this study, we utilize a combined heuristic-metaheuristic SAG algorithm to ensure obtaining the simplified symbolic pole/zero expressions with the least achievable complexity.

After the netlist pre-processing, the circuit is solved using MNA [7], and consequently, the exact symbolic TF is achieved in expanded form. Then, the obtained exact pole/zero expressions are approximately calculated using the proposed ERS method which is an enhanced version of the traditional RA algorithm. Generally, an expanded TF could be converted into the factorized form according to Eq. (8), where $Z(s,\mathbf{x})$ is a function of $\acute{n}$ (real or complex conjugate) zeroes $z_1$, $z_2$, …, $z_{\acute{n}}$, and $P(s,\mathbf{x})$ is a function of $n$ (real or complex conjugate) poles $p_1$, $p_2$, …, $p_n$.

In the ERS algorithm, only poles and zeroes located in the interval [$f_{min},f_{max}]$ would be extracted, where $f_{min}$ and $f_{max}$ are the minimum and maximum user-specified frequency for the pole/zero extraction, respectively. If the frequency range has not been specified by the user, it is considered as default range of [$\mathrm{0,10}{f}_T]$, where $f_T$ is the frequency of unity-gain of the exact expression in the nominal point. In the following, the pole extraction procedure for $P(s,\mathbf{x})$ is described. By comparing Eqs. (1) and (8), $P(s,\mathbf{x})$ can be calculated as:

Typically, a single real pole is dominant in OTAs. Assuming $p_1$ to be dominant and all other poles to be located at much higher frequencies, i.e., $‖p_1‖<<‖p_2‖,‖p_3‖,\dots ,‖p_n‖$, the first pole can be splitted, and thus, $P(s,\mathbf{x})$ can be approximately written as:

By equating the $s$ coefficients of $P(s,\mathbf{x})$ in Eq. (10) to those in Eq. (9), the dominant real pole $p_1$ can be approximated given as:

Consequently, by assuming the condition in Eq. (12), we can simplify the rightmost expression of Eq. (11) as Eq. (13). By equating the $s$ coefficients of Eq. (13) to the same $s$ coefficients in Eq. (9), the parameters $g_i$ can be calculated as Eq. (14). and thus, the leftmost expression in Eq. (10) can be expressed as:

The circumstances of $s$ coefficients in the original denominator $D(s,\mathbf{x})$, for which Eq. (15) is valid, are:

Equation (15) shows that $P(s,\mathbf{x})$ can be simplified into a product of a first-order polynomial (i.e., first dominant pole) and a high-order polynomial corresponding to other high-frequency poles. In a more general case, assuming the first $m$ poles ($1<m<n$) to be successively dominant in pairs (i.e., $p_1$ dominates $p_2$, $p_2$ dominates $p_3$, and so on, $p_{m-1}$ dominates $p_m$), $P(s,\mathbf{x})$ can be approximated as follows:

By similar approximations as done in the previous case, $P(s,\mathbf{x})$ can be simplified according to:

By equating the $s$ coefficients of Eq. (18) and Eq. (9), the $m$ first poles are derived as Eq. (19). Also, the parameters $g_i$ can be expressed as Eq. (20). So, $P(s,\mathbf{x})$ can be simplified into the multiplication of $m$+1 polynomials: $m$ first-order polynomials (representing the $m$ first poles) and a high-order polynomial, as Eq. (21). The conditions on $s$ coefficients of $D(s,\mathbf{x})$, for which Eq. (21) is valid, can be expressed as Eq. (22).

In the general case, let us extend the above formulations for the case that all the $n$ poles are dominant reciprocally, in which, $P(s,\mathbf{x})$ can be approximated as follows:

Under the assumption that all poles are dominant in pairs (i.e., $p_1$ dominates $p_2$, $p_2$ dominates $p_3$, and so on), the following conditions are satisfied:

Therefore, the rightmost expression in Eq. (24) can be approximated as Eq. (25). By equating $s$ coefficients of Eq. (25) and Eq. (10), $P(s,\mathbf{x})$ in Eq. (25) can be approximately expressed as Eq. (26), where each pole $p_i$ can be calculated according to Eq. (27).

The interesting feature is that all poles are derived from $s$ coefficients of the denominator of the transfer function. The above formulations are under the assumption that all poles are real. In other words, the approach fails for closely spaced or complex conjugate poles. Therefore, the method should be extended for the cases in which two consecutive poles are located in a cluster. Assuming that $p_i$ and $p_{i+1}$ are a pair of poles (real or conjugate), they are remained split off in the expression $P(s,\mathbf{x})$ and can be expressed via a second-order polynomial $(1+as+b{s}^2)$, where $a$ and $b$ can be calculated as follows:

The condition for which the poles $p_i$ and $p_{i+1}$ are real, is $a^2\ge 4b$. If the condition has been satisfied, the real poles $p_i$ and $p_{i+1}$ can be expressed as Eq. (29). Otherwise, these poles can be represented as complex conjugate poles according to Eq. (30).

It should be emphasized that all above formulations could be used for the extraction of simplified zeroes $Z(s,\mathbf{x})$ from the numerator $N(s,\mathbf{x})$ of the expanded TF. In ERS, the pole $p_i$ (or zero $z_j$) can be splitted by means of Eq. (26), if the conditions in Eqs. (31) and (32) are met, where $p_{ERS,i}$ ($z_{ERS,j}$) is the absolute of the numerical value of $i$-th pole ($j$-th zero) extracted via the ERS method, and $p_{E,i}$ ($z_{E,j}$) is the absolute of the $i$-th pole ($j$-th zero) of the exact function of Eq. (1), which are numerically achieved by the calculation of the roots of the transfer function. $PoleSet$ ($ZeroSet$) is the set of poles (zeroes) which are in the range of the interval [$f_{min},f_{max}]$. Also, $T_{ERS}$ is a pre-determined constant to specify the maximum allowable root displacement (in %) for the each ERS-root, compared with the exact one.

The ERS pole/zero extraction method comprises evaluation and extraction steps. In the evaluation step, all poles and zeros within the interval [$f_{min},f_{max}]$ are assumed to be reciprocally dominant, and thus, their ERS values are numerically obtained according to Eq. (26). In the extraction step, the conditions of Eqs. (31) and (32) are checked for all extracted poles and zeroes. Then, each pole (zero) which has satisfied the mentioned condition, can be symbolically extracted according to Eq. (26). On the other hand, the pair of real or complex conjugate poles (zeros) remained split off and the condition $a^2\ge 4b$ is checked for them. If the condition has been satisfied, the poles (zeros) are a pair of real poles (zeroes) and can be calculated by Eq. (29). Otherwise, they are considered as complex conjugate poles (zeroes) which are extracted by means of Eq. (30).

The extracted symbolic pole/zero expressions cannot give an analytical information about the circuit behavior, due to their high complexity. So, a pole/zero simplification based on PZSA is used to simplify the exact pole/zero expressions. SA is a single-solution metaheuristic inspired from the metallurgy annealing, which involves heating and then slowly cooling of the material to reduce its defects [22]. Generally, SA starts its search from a fully random solution, and then, iteratively updates the solution until arriving at the stopping criterion [35]. However, to improve the quality and speed of the search process in SA, we utilize the knowledge from the exact circuit expressions as heuristic information to guide the SA algorithm by starting from a near-optimal solution. After generating the initial solution using the heuristic algorithm, SA is performed for improving further the quality of the solution using local search operators in an iterative procedure. In the following, the main steps of the PZSA algorithm are described.

The block diagram of a three-stage compensation OTA can be shown in Figure 6. The circuit is described by the equivalent RCg_m model. By using MNA, the exact expanded TF has been derived as follows:

The exact TF comprises 40 terms. By performing the simplification algorithm in [34], the simplified expanded TF with 10 terms has been obtained according to Eq. (42). The expanded TF even in the simplified form, cannot give effective insights for the designer to evaluate the positions of poles and zeroes. However, by performing PZSA, three poles and two zeroes can be achieved as Eqs. (43)-(47).

The comparison of the different methods according to the number of simplified terms within the simplified poles/zeroes can be summarized in Table 3. Moreover, the numerical results of the different methods are provided in Table 4, wherein the last four rows illustrate the error of the simplified equations when compared to the exact expressions.

The second circuit is a folded cascode two-stage OTA with the compensation described by MOS transistors, as shown in Figure 7. The exact expanded TF obtained by MNA contains 134 symbolic terms.

By performing the simplification method in [34], the simplified TF has been obtained according to Eq. (48). By utilizing PZSA on Eq. (48), two simplified poles and one zero have been obtained as Eqs. (49)-(51), respectively. Similar to Circuit 1, the comparison of the number of simplified terms and the numerical results are summarized in Table 5 and Table 6, respectively.