Shape Optimization of Planar Inductors for RF Circuits using a Metaheuristic Technique based on Evolutionary Approach

A R T I C L E I N F O A B S T R A C T Article history: Received: 14 July, 2020 Accepted: 08 September, 2020 Online: 21 September, 2020 In this article, we concentrate on the use of a metaheuristic technique based on an Evolutionary Algorithm (EA) for determining the optimal geometrical parameters of spiral inductors for RF circuits. For this purpose, we have opted for an optimization procedure through an enhanced Differential Evolution (DE) algorithm. The proposed tool allows the design of optimized integrated inductors not only with a maximum quality factor(Q), but also with a maximum self-resonant frequency (SRF), and a minimum surface area, in addition to being adapted to any model of any technology. This paper presents also a comparison between performances of the optimized inductors (inductor square shape and inductor circular shape), in terms of the quality factor, SRF, and circuit size. For the purpose of mitigating the impact of parasitic effects, design basics have been taken into consideration. Then, in order to investigate the efficacy of evaluated results, an (EM) simulator has been employed.


Introduction
Integrated Inductors are of paramount importance elements, layout-optimization for spiral inductors has been the focus issue of several studies for the last few years, as for application, the four main characteristics that are required for the design of spiral inductors are: high inductance, high current capability, energy density, and low losses, with the inductors properties being identified by its geometrical and technological parameters [1].
For the sizing of spiral inductors, the designer should consider three main parameters [2], [3], the inductance value which is one of the most sensitive parameters, then, the quality factor (Q), and finally the self-resonant frequency (SRF).
Many works have been conducted for the sake of modeling and optimizing of spiral inductors. Formulation, modeling, and implementation remain the main steps for designing an integrated inductor [4], [5]. However, to ameliorate the optimization, the operation could be repeated many times till an acceptable solution is found.
Metaheuristic's techniques are especially applied to the optimal sizing of analog circuits [6], such techniques have proven to be efficient in solving difficult problems because they necessitate less time to converge and yield better solutions.
Nevertheless, for the sake of achieving the optimal sizing of the (RF) spiral inductors, the Differential Evolution (DE) is to be the focus technique in this paper since it has been widely used in circuit design in the last decade.
In order to design circular and square spiral inductors for operating frequencies around 2.5 GHz, the inductor π-model has been embedded in the improvement device.
The next sections of the paper layout introduce as follows: Section 2 is devoted to the descriptions of the inductor π-model used, afterward, section 3 provides the synopsis of the DE   ASTESJ   ISSN: 2415-6698  algorithm, while the optimal values of DE parameters have been  determined by a proposed technique. Then, section 4 highlights the  inductor sizing-optimization method, the technological  parameters, and the design constraints as well, besides, the  optimization results are presented, where analytical results  obtained with DE are investigated by ADS momentum simulation  software. Last and not least, the conclusion is offered in section 5.

Planar Spiral Inductors
All the shapes of spiral inductor known by four main geometrical parameters, the spacing between lines (s), the number of turns (n), the line width (w), and the outer length of a side (dout), while the inner length of a side (din) defined by: din = (dout -2.(n .(s + w) -s)).
There are other important geometry parameters such the inductor length, while: L = 4.n.davg for the square shape, and L = 4.n.davg for the circular shape, then, the inductor area: A=dout 2 , and finally, the average diameter: davg = 0.5.(dout + din).
Layouts of the circular and the square inductor have been showing respectively in Figure 1 and Figure 2

The electrical Model of Integrated Inductors
It is important, thus, to present the expressions of the electrical model components for the inductor π-model, Figure 3 presented the electrical circuit for this type, while, Cs, Csi, Cox, Rs, and Rsi are respectively the substrate capacitance, the series capacitance between the spiral and the metal underpass, the substrate-oxide capacitance, the series resistance, and the substrate resistance, these parameters are determined by equations (1 2,3,4,5,6,7,8, 9): where (t) is the turn thickness, (tox) is the oxide thickness between the spiral and the substrate, (σm) is the conductivity of the metal, (ω) is the frequency, (tox, M1-M2) is the oxide thickness between the spiral and the under-pass, (εox) is the oxide permittivity, (Gsub) is the substrate conductance per unit area, (Csub) is the substrate capacitance per unit area, (hsub) is the substrate height, (σsub) is the substrate conductivity, (δ) is the skin depth, (µ) is the magnetic permeability of free space, and finally, (Rsh) is the sheet resistance. A similar inductor model has been shown in Figure 4, the quality factor (Q) was calculated by equations (10) and (11), where (Cp) is the shunt capacitance, and (Rp) is the shunt resistance.

Inductance Ls
The model of the inductance Ls for the square inductor is expressed [19], [20] in equation (12): The expression of the inductance Ls for the circular inductor is given in equation (13) [21]: The coefficients ci, β, and αi are not depending on the technology but on the structure of the inductor. With ϼ is the fill ratio, inductances in nH, and dimensions in μm.
The expression of the inductance for a given frequency (f) for two ports [20] defined as follow:

The Quality factor (Q)
The quality factor is presented as follows: The Q-Factor can be formed as: An ideal inductor has an infinite Quality factor [19].
When the peak magnetic energy is the same as the electric energy, the Q-Factor is equal to zero, this phenomenon is defined as the self-resonant frequency phenomena.
The energy stockpiled in the inductor is attached to the imaginary part of the input admittance (Yin), whereas the real part of (Yin) is proportional to the energy dissipated in resistances, with this approach is abridged to [20]:

The Differential Evolution Algorithm
It is possible to say that the DE algorithm, as is the genetic algorithm, is a population-based using identical operators' mutation, crossover, and selection. However; what makes the genetic algorithms yield a better solution is the fact that it builds on the crossover operation while the DE builds on the mutation one [8].
At the beginning of the DE process, the population of the n-pop solution vectors is randomly selected. This population is then ameliorated by stratifying mutation, crossover, and selection operators. First, the algorithm uses the mutation process as its search mechanism. Then, the DE uses crossover (recombination) operators, and the child vector that takes parameters from one parent more than the other. Afterward, a selection process is carried out in order to change the parent vectors if their fitness is less than of the newly generated child vectors. This three-stage process is repeated until a better solution is found [22].
The principal steps of the DE algorithm are defined mathematically as follows:

Crossover
The trial vector is produced by the mixture of the parent vector with the mutated vector: , +1 = , where (Pc) is the crossover probability parameter.

Selection
The comparison between a parent and its identical offspring called the selection and can be expressed as: , +1 = , ℎ where g(x) is the objective function value of the trial vector. The DE algorithm can be declared in 1:

The DE Algorithm Parametrization
To determine the optimal values of DE parameters, the Ackley function presented in equation (21) was investigated for 100 population and 1000 number of iterations.
The function evaluated on xj ∈ [ − 32, 32] for all (j = 1 , … , 32) . From Figure 5, the values of DE parameters that gave the best convergence are presented in Table 1.

Inductors Sizing
In the following section, we aim to maximize the Q-Factor for a specific value of the inductance for two structures, square and circular, by combining the inductor π-model and the DE optimization procedure. Afterward, simulations with ADSEM are adopted.

Constraints of the study
To minify the parasitic phenomena [20], [23], the liaison between geometry parameters in (22) is well respected as a sort of included design-rules [20], [23].  The condition for a minimum self-resonant frequency which SRF ≥ SRFmin can be formed as [20]:

Optimization Procedure
The goal of this optimization is to find the optimum geometrical parameters of the spiral inductor to get a higher value of Q-Factor, the problem can be formulated as follows: where (si) is the penalty coefficient, and P(x) is the sum of constraints.
Constraints g8(x), g9(x), g10(x), and g11(x) are boundary constraints, as result, they can be examined, while the DE was not allowed to generate a candidate vector farther these limitations.

Results and Discussions
In the following, we will be adopting a sizing of square and circular inductors, with distinct values of the inductance Lsreq in the field beyond 2.5 GHz, as shown in Table 3 the technological and physical parameters have been well presented, while Table 4 represents the geometry parameter boundaries.
The details of the optimization have been presented in Table  5 and Table 6. On aim to verify our procedure, Figure 7 gives the cost function versus the number of iterations for square inductors, in this case, the constraint for minimum self-resonant frequency is added as SRFmin=22 GHz. The optimization results of the maximum Q-Factor and area (A) for both circular and square inductors versus the inductance obtained using the DE algorithm are presented in Figure 8.
The Q-Factor versus frequency for each value of the inductance has been shown in Figure 9 and Figure 10. The simulation using momentum software has also been shown in Figure 11, Figure 12, Figure 13, and Figure 14.
The comparison between optimization results and simulations is presented in Table 7 and Table 8.         What the results show is that when the inductance value increases, the quality factor decreases, and the self-resonant frequency decreases as well.
However, the results are very good in terms of the circuit's size, and the constraints are very robust.
The DE algorithm provides better results concerning the circuit's size and has a faster convergence as shown in the results.
We can notice that the simulation of square inductors is very accurate, with an error below 8.50% for the inductance value, and 5.21% for the quality factor for Ls less than 11 nH. It is possible to explain the increase of the error when the value of inductance is greater than 11 nH, in that we have taken similar limits of geometrical parameters for all values of the inductance, when the value of Ls has augmented, the number of turns became higher and Dout increased with a small percentage, in such circumstances, the quality factor decreases owing to the parasitic phenomena effects, this problem can be solved by increasing the parameters of the allowable range in the proportion of the outer diameter.
As for the circular inductor, generally, the error is below than 5.66% for inductance value, and 21.56% for the quality factor, Although, this type has the shortest perimeter, and with a circular configuration, a higher quality factor (Q) is obtained. Yet, this type shows a response to the parasitic phenomena effects.
We notice through the simulation that the circular inductor is not significantly affected by parasitic phenomena in terms of the self-resonant frequency. From Figures 10 and 13, we observe that the inductor of Ls equal to 11 nH reaches its maximum of Q-Factor when fmax ~ 2 GHz, the area on the left of fmax, is an area where the Q-Factor is fundamentally affected by the magnetic induced losses, skin and proximity effects, and the DC resistance [24], [25]. On the opposite side of fmax, in addition to the preceding effects, the Q-Factor is also affected by the substrate noise coupling [23]. The evaluated SRF equal to 10.1 GHz, and the SRF obtained via simulation equal to 8.5 GHz, at this time, the Q-Factor is equal to 0, starting from this point, the peak magnetic energy is less than the electric energy, due to the perturbation of this last because of the parasitic phenomena.
The layout constraints for circular inductors required extensive research, in order to mitigate the parasitic phenomena effects.
Moreover, the degradation of the Q-Factor can be seen more clearly for square inductors, from Figures 9 and 11, for Ls equal to 11 nH, the Q-Factor equal to 0 when the evaluated SRF equal to 15.59 GHz and the SRF obtained via simulation equal to 7 GHz, we conclude that this type is extremely influenced by the parasitic phenomena.

Conclusion
For dealing with the optimal sizing of spiral inductors for (RF) circuits, we proposed on this paper an application of the Differential Evolution (DE) algorithm. Two inductor structures have been optimized i.e. shape square and shape circular, with a maximum Q-Factor, a maximum self-resonant frequency (SRF), and a minimum surface area. The performances of optimized inductors showed good results in terms of the Q-Factor, with the square inductor presenting a higher SRF and a smaller area (A) than the circular one. The π-model does not allow for the assimilation of noises parasitic effects in a good way, leading to a lower SRF value, that is why we are focusing on using the double π-model, instead, for the integrated inductors optimal sizing.