A Geometry-Based Design Methodology for Fair Topology Comparison of Rotary Traveling Wave Oscillators

Abstract

This article presents a geometric analysis of the influence of topology on the performance of Rotary Traveling Wave Oscillators (RTWOs), enabling a fair comparison between RTWO topologies by enforcing a constant total resonator length regardless of dimensions or topology. Based on this approach, a comparison is carried out to evaluate the impact of different topologies on the performance of the oscillator. Results show that geometric modifications can increase the oscillation frequency by up to $530\mathrm{M}\mathrm{H}\mathrm{z}$ while slightly reducing power consumption. Validation is performed through electromagnetic simulations in ADS Momentum for RLCG parameter extraction and electrical simulations in HSPICE to obtain oscillation frequency and power consumption. Under identical geometric conditions, the resonator topology significantly modifies oscillator behavior, particularly in structures with fewer abrupt geometric discontinuities. Finally, figures of merit are used to evaluate RTWO performance, highlighting the trade-offs between oscillation frequency, power consumption and integration area.

1. Introduction

Periodic signal generation systems enable process timing and information transmission. Synchronization within these systems allows different functional blocks to operate in a coordinated manner through a common signal, enabling simultaneous operations within precise and well-defined time intervals. In this way, synchronous systems constitute the foundation of modern technology and are present in a wide range of applications, including consumer electronics, military systems, communication applications and healthcare technologies.

In this context, oscillators play a fundamental role as they generate the reference periodic signal. This work focuses on Rotary Traveling Wave Oscillators (RTWOs), which have demonstrated the ability to achieve high oscillation frequencies with moderate power consumption and strong phase noise robustness. RTWOs exploit the distributed inductance L and capacitance C of metal interconnects in VLSI technologies to generate and confine an electromagnetic wave within a closed structure, provided that losses associated with resistance R and conductance G are properly compensated. The structure of an RTWO consists of three main components: the resonator, the Möbius crossover and the compensation stage.

Since the first RTWO implementations, it has been observed that the oscillation frequency is strongly related to both the physical dimensions and the resonator topology, highlighting the inverse relationship between the total length of the structure and the oscillation frequency. In other words, more compact RTWOs tend to achieve higher operating frequencies. However, in practice, the oscillator geometry is not defined solely by the total length l, but also depends on the conductor width $W$, the spacing between rings $S$, the Möbius crossover angle ${\theta }_M$ and the resonator topology $N_c$ in the case of polygonal structures.

Previous works on RTWOs have mainly focused on circuit-level optimization, phase noise reduction and implementation strategies in CMOS technologies. While these studies have significantly improved performance, limited attention has been given to the role of resonator geometry and topology as primary design variables. Existing approaches do not provide a consistent framework for comparing different RTWO topologies under equivalent geometric conditions, which may lead to biased or inconclusive results.

In a previous work [1], a geometric analysis was conducted and a model was derived to estimate the oscillation frequency as a function of the structural dimensions of an RTWO. This model improved the understanding of the relationship between frequency and geometry. However, its validation was limited to a single resonator topology and did not address the problem of comparing different geometrical configurations under controlled conditions.

Recent studies on resonant electromagnetic structures have demonstrated that geometrical modifications can significantly alter not only the effective electrical length, but also the electromagnetic field distribution, coupling mechanisms and quality factor of the resonator [2]. Geometry-driven tuning approaches have shown that variations in structural configuration can directly impact resonance frequency, electromagnetic confinement and propagation behavior within distributed resonant systems.

In the context of integrated oscillators, previous works have also highlighted the importance of simultaneously considering active and passive elements during the design process, since the oscillator performance cannot be fully explained by inductance or resonator length alone [3]. Instead, distributed parasitic effects, geometrical configuration and electromagnetic interactions jointly determine the final oscillation behavior.

Additionally, recent studies on oscillator optimization have emphasized that the interaction between resonator geometry, distributed parasitics and layout-dependent effects plays a significant role in determining oscillation behavior and overall performance [4].

This work extends the previous analysis by introducing a geometry-based design methodology that incorporates both the physical dimensions and the resonator topology [1]. By enforcing a constant total resonator length across different structures, the proposed approach enables a fair comparison between topologies. A comparative study of oscillation frequency, power consumption and integration area efficiency is carried out for different configurations, including square, beveled and polygonal resonators.

The proposed method is validated through electromagnetic simulations using ADS Momentum for the extraction of the electrical parameters of the structures, as well as circuit-level simulations in HSPICE to evaluate the oscillation frequency and the oscillator behavior under different configurations.

2. Geometry-Based Design Framework

2.1. State of the Art and Problem Statement

Throughout the development of Rotary Traveling Wave Oscillators, significant emphasis has been placed on optimizing the active part of the device. Since their introduction in 2001 [5], it has been identified that PMOS transistors in distributed compensation stages limit the oscillation frequency due to their low carrier mobility, which has led to the development of new compensation stages that minimize the use of these devices [6].

Additionally, systems for controlling the direction of the traveling wave within the structure have been proposed [7], as well as methods to mitigate random phase variations. In [8], a physical model is introduced that approximates the RTWO as a superposition of multiple oscillators, enabling the estimation and validation of phase noise, while in [9], techniques based on the nonuniform distribution of capacitors along the structure are proposed to correct phase errors caused by asymmetries.

However, a recurring issue is that the passive part of the oscillator is either neglected or treated as secondary. In [10], the optimization of oscillator performance as a function of the structural dimensions $W$ and $S$ is proposed using neural networks, demonstrating the dependence between oscillation frequency, power consumption and phase noise.

More recent works [11] have explored alternative resonator structures and layout strategies to improve RTWO performance. For instance, the use of eight-shaped transformer-based resonators has been proposed to suppress magnetic coupling noise. These approaches highlight the strong influence of resonator topology. To better position the contribution of this work,

Table 1. Comparison of representative RTWO design approaches reported in the literature.

Refs.	Main Objective	Design Approach	Topology Treatment	Geometry Consideration	Fair Comparison Capability
[2]	Geometrical tuning of resonant structures	Electromagnetic and geometrical modeling	Explicit	Explicit	Not addressed
[5]	Introduction of RTWO concept	Distributed LC-based oscillator	Implicit	Limited	Not addressed
[6]	CMOS implementation of RTWO	Circuit-level design	Fixed topology	Limited	Not addressed
[7]	Directional control and stability	Circuit-level enhancement	Limited	Limited	Not addressed
[8,9]	Phase noise analysis and reduction	Analytical and circuit-level modeling	Not explicitly	Not considered	Not addressed
[10,12]	Performance optimization	Optimization-based design	Limited	Indirect	Not addressed
[11]	Advanced resonator layout for noise reduction	Transformer-based resonator design	Specific topology	Explicit	Not addressed
This Work	Fair topology Comparison methodology for RTWOs	Geometry-based parametrization and distributed modeling	Explicit and systematic	Explicit	Addressed

provides a structured comparison of representative RTWO design approaches reported in the literature.

As shown in Table 1, most existing approaches focus on circuit-level optimization, while the impact of resonator topology is either not explicitly addressed or treated as a secondary factor. Consequently, no consistent framework exists for comparing different topologies under equivalent geometric conditions.

RTWOs can adopt virtually any geometry allowed by the design rules of the integration technology, and in [12], it is shown that both the dimensions and the topology directly influence RTWO performance. This line of research is further developed in [1], where it is established that abrupt changes in carrier flow degrade oscillator performance. Both works agree that avoiding certain design configurations improves oscillation frequency and reduces power consumption. For example:

• Avoid excessive use of thin metal layers;
• Avoid overlapping structures across different metal layers;
• Minimize abrupt geometric discontinuities;
• Optimize via arrangement and stacking in the Möbius crossover depending on the technology.

The main limitation in most of the literature on RTWOs is the lack of a systematic design methodology. These oscillators are highly sensitive to their geometric dimensions, particularly to the total length. Any undesired variation in $l$ results in a deviation in the oscillation frequency, which is especially critical in high-frequency applications.

The structure of an RTWO, composed of concentric rings with conductor width $W$, separated by a distance $S$ and connected through a Möbius crossover, introduces a strong dependency among its geometric parameters. Consequently, variations in $W$, $S$, or in the resonator topology indirectly affect the total length of the structure, making it difficult to establish a direct correlation and a consistent comparison between different configurations.

In [1], a geometric analysis and a parametrization method are presented for polygonal resonators and for a square RTWO with 45° beveled corners. However, before introducing these models, it is necessary to describe the general structure of the RTWO.

2.2. Structure of the Rotary Traveling Wave Oscillator

The structure of an RTWO consists of three fundamental elements that together enable the generation and sustained propagation of an electromagnetic wave along a closed path. The interaction between these elements defines the behavior of the oscillator as well as its main electrical and geometric characteristics.

Resonator: The main contributor of the LC parameters exploited by the oscillator. It is typically located on the top metal layer farthest from the substrate and can adopt virtually any geometric shape, either regular or irregular. Under the analysis presented in this work, it is composed of horizontal interconnects forming an inner ring and an outer ring.

Möbius crossover: A type of connection that links the inner and outer rings, closing the loop and creating a continuous conductive path along which the wave propagates. It employs vertical metal interconnects known as vias, which enable connections between different metal layers. Additionally, via stacking allows connections between nonadjacent metal layers.

Compensation stage: A set of active devices uniformly distributed along the structure, responsible for compensating resistive and dielectric losses. Additionally, these compensators generate a 180° phase shift between the inner and outer rings.

Figure 1 shows different views of the RTWO structure.

Figure 1a shows a general view of the RTWO structure and the different metal layers that compose it. Figure 1b presents a top view of the structure, where the length $L_t$ is indicated, representing a design variable adjusted as a function of the dimensions $W$, $S$ and $l$. The angle ${\theta }_M$ corresponds to the inclination of the diagonal lines in the Möbius crossover, and its value is defined by the integration technology. These angles are restricted to discrete values; however, 45° is valid for most modern technologies.

2.3. Geometric Analysis of RTWOs with Polygonal Resonators

The simplest way to analyze the structure of an RTWO [1] is to apply a simplification in which the Möbius crossover is removed, and only the two concentric rings are considered. In this way, the total length can be approximated as:

$$l\approx l_{in}+l_{out}$$

(1)

where $l_{in}$ and $l_{out}$ represent the lengths of the inner and outer rings, respectively. An important parameter in RTWOs with polygonal resonators is the variable $N_c$, which represents the number of sides or corners of the outer ring. In other words, it defines the resonator topology. For example, for $N_c=6$, a hexagonal resonator is obtained. Figure 2 shows examples of this simplification.

Figure 2a,b show that the parameter $L_t$ adapts to the resonator topology to maintain a constant total length. The geometric analysis is based on projecting lines from the center point toward the vertices and toward the midpoints of the polygon edges, dividing the structure into $2N_c$ equal parts. Therefore, the angle between each section is constant and equal to $\pi /N_c$. With this consideration, the lengths of the inner and outer rings are derived by performing the analysis along the conductor midline. The resulting expressions are given as Equations (2) and (3):

$$l_{in}=N_c\left(L_t+Wtan\left({\displaystyle \frac{\pi }{N_c}}\right)\right)$$

(2)

$$l_{out}=N_c\left(L_t+\left(3W+2S\right)tan\left({\displaystyle \frac{\pi }{N_c}}\right)\right)$$

(3)

Therefore, by substituting the expressions for the inner and outer ring lengths given in Equations (2) and (3) into the approximation in Equation (1), the following expression is obtained:

$$l=N_c\left(2L_t+\left(4W+2S\right)tan\left({\displaystyle \frac{\pi }{N_c}}\right)\right)$$

(4)

The parameter $L_t$ in Equation (4) can be adjusted as a function of the dimensions $W$, $S$ and $l$ and it also depends on the polygonal topology defined by $N_c$. The limitation of Equation (4) lies in the fact that it is based on a simplified representation of the RTWO structure, where the diagonal lines of the Möbius crossover are neglected. Möbius crossover diagonals incorporate a length $\Delta l$; this additional length is negligible at low frequencies, where $l\gg \Delta l$.

Including $\Delta l$ provides a more complete geometrical representation of the structure compared to formulations that neglect the Möbius crossover. This effect becomes more relevant for smaller values of the total length, where the relative contribution of $\Delta l$ increases.

$$\Delta l=2\left(\sqrt{2}-1\right)\left(W+S\right)$$

(5)

Equation (5) accounts for the length increment introduced by the Möbius crossover, which employs diagonal lines at 45°. Therefore, by incorporating this increment into the total length, Equation (6) is obtained.

$$L_t={\displaystyle \frac{1}{{2N}_c}}\left[l-\left(4N_{c}tan\left({\displaystyle \frac{\pi }{N_c}}\right)+2\sqrt{2}-2\right)W-\left(2N_{c}tan\left({\displaystyle \frac{\pi }{N_c}}\right)+2\sqrt{2}-2\right)S\right]$$

(6)

Equation (6) constitutes a design tool that enables the proper dimensioning of RTWOs independently of the conductor width, the spacing between rings, the total length and the polygonal topology employed. It also enables a direct comparison between different structures, allowing the identification of an optimal configuration. In this way, a design criterion is established that facilitates the evaluation and selection of different topologies under equivalent conditions.

It should be noted that the present study focuses on isolating the effect of topology by fixing the main geometrical parameters. This approach enables a fair and controlled comparison between different resonator configurations by eliminating the influence of multiple interacting variables.

While a broader analysis including variations of $W$, $S$ and topology could provide additional insight into multi-dimensional optimization, such an exploration is beyond the scope of this work and is considered a direction for future research.

2.3.1. Integration Area of RTWOs with Polygonal Resonators

To calculate the integration area of RTWOs with polygonal resonators, a geometric concept described in Equation (7) is used, which relates the resonator geometry to the total area occupied by the structure. This analysis is relevant for evaluating efficiency in terms of integration area and its impact on oscillator performance.

$$A={\displaystyle \frac{P{d}_a}{2}}$$

(7)

Equation (7) states that the area of any regular polygon is directly related to the product of its perimeter $P$ and the segment that connects the center of the polygon perpendicularly to the midpoint of any of its sides, known as the apothem $d_a$. The perimeter of a regular polygon can be determined as:

$$P=N_{c}L$$

(8)

and the apothem is given by:

$$d_a={\displaystyle \frac{L}{2tan\left(\frac{\pi }{N_c}\right)}}$$

(9)

where $L$ represents the length of one side of the polygon, that is, the segment between two vertices (Figure 1). In addition, the RTWO structure itself provides the necessary information to obtain the apothem, as shown in Equation (9). For the perimeter calculation, the outer edge of the outer ring is considered. Therefore, the length of one side of the polygon is determined as:

$$L=L_t+\left(4W+2S\right)tan\left({\displaystyle \frac{\pi }{N_c}}\right)$$

(10)

By resolving the chain of dependencies, where each geometric variable is expressed in terms of the preceding ones, a general expression for the integration area is obtained solely as a function of $W$, $S$, $l$ and the topology $N_c$.

$$A={\displaystyle \frac{1}{16N_{c}tan\left(\frac{\pi }{N_c}\right)}}{\left[l+\left(4N_{c}tan\left({\displaystyle \frac{\pi }{N_c}}\right)-2\sqrt{2}+2\right)W+\left(2N_{c}tan\left({\displaystyle \frac{\pi }{N_c}}\right)-2\sqrt{2}+2\right)S\right]}^2$$

(11)

Equation (11) describes the behavior of the integration area of RTWOs with polygonal resonators. It provides a direct analytical formulation that facilitates the comparison between different structures. In this way, it establishes a tool that enables the evaluation of the impact of geometric parameters on the integration area and guides the design toward more efficient configurations.

2.3.2. Geometric Analysis of an RTWO with a Square Resonator with 45° Beveled Corners

In [12], it is stated that avoiding abrupt geometric discontinuities reduces traveling wave losses, thereby improving oscillator performance. Beveling is a technique used to smooth right angles by introducing inclined edges, depending on the application.

In integrated circuit technologies, conductor beveling is mainly limited by the availability of discrete angle values. For this reason, similar to the Möbius crossover, 45° bevels are commonly used. Figure 3 shows the RTWO with a beveled resonator.

A key issue arises when designing this topology, as beveling the corners reduces the total length of the structure, directly affecting the oscillation frequency. Therefore, the lost length must be compensated through the parameter $L_t$.

As in polygonal topologies, the beveled RTWO is based on the simplification given by Equation (1), while also considering the length increment introduced by the diagonal lines of the Möbius crossover as described in Equation (5). Therefore, $l$ can be expressed as:

$$l=l_{in}+l_{out}+\Delta l$$

(12)

Advantageously, the beveled corners form squares of dimensions W × W for the inner bevels and (W + S) × (W + S) for the outer ones. As in the previous geometric analysis, the calculation is performed along the conductor midline. Therefore, the following expressions are obtained:

$$l_{in}=4L_t+2\sqrt{2}W$$

(13)

$$l_{out}=4L_t+\left(8+2\sqrt{2}\right)W+4\sqrt{2}S$$

(14)

Considering the increase in the total length due to the Möbius crossover, the following expression is finally obtained:

$$L_t={\displaystyle \frac{1}{8}}\left[l-6\left(1+\sqrt{2}\right)W-2\left(3\sqrt{2}-1\right)S\right]$$

(15)

Equation (15) allows adjusting the parameter $L_t$ as a function of the structural dimensions of an RTWO with a square resonator with 45° beveled corners. This topology is slightly larger than a conventional square RTWO, which is reflected in the required integration area. A practical and efficient way to calculate the integration area is to use Equation (10) and consider $tan(\pi /4)=1$. Therefore, the integration area of the beveled oscillator can be approximated by modeling the structure as a square and subtracting the corner sections. Hence:

$$A={\left({\displaystyle \frac{1}{8}}\left[l-6\left(1+\sqrt{2}\right)W-2\left(3\sqrt{2}-1\right)S\right]+4W+2S\right)}^2-2{\left(W+S\right)}^2$$

(16)

Equation (16) provides an approximation of the area once the corners are removed. However, this does not represent a limitation, as the neglected region is minimal and can be disregarded. In this way, a sufficiently accurate estimation of the integration area is obtained for design purposes and for comparison between different topologies.

3. Comparative Analysis of RTWO Topologies

3.1. Operating Principles

RTWOs exploit the RLCG parameters of metal interconnects, which are nonuniformly distributed along the structure and depend on the material properties, connection configuration, size and geometry. The generation of the traveling wave relies on the resonance condition of the structure, which occurs when the system is perturbed such that the inductive reactance $X_L$ and the capacitive reactance $X_C$ become equal $X_L=X_C$. Under this condition, energy is stored in the electric field during the first half of the cycle and transferred to the magnetic field during the second half.

Ideally, the energy transfer between fields occurs without losses, exhibiting adiabatic behavior. In practice, this is not the case, as the elements R and G introduce losses due to Joule heating and dielectric leakage, causing the traveling wave to attenuate over time. In [5], an active circuit is proposed to compensate for these losses and to enforce a phase shift between the inner and outer rings, enabling self-oscillation. These compensators introduce both passive and active elements that mainly contribute capacitance. Therefore, the simplest way to estimate the oscillation frequency is given by:

$$f_{osc}={\displaystyle \frac{1}{2\pi l\sqrt{L_0{C}_0}}}$$

(17)

where $C_0$ and $L_0$ represent the capacitance and inductance per unit length. However, as shown in [1], this model presents an error margin of up to ±30%, which decreases when the structure has a large total length (≥1400 μm). This behavior is explained by the dependence of $C_0(N_c,W,S,l)$ and $L_0(N_c,W,S,l)$ on both dimensions and topology. As a result, two oscillators with the same topology and total length but different values of $W$ and $S$ can exhibit frequency differences of up to ±20%.

A comparison is carried out between five RTWOs under identical geometric conditions but with different topologies. The objective is to identify the structure that achieves higher oscillation frequency, lower power consumption and reduced integration area.

3.2. Geometric Comparison of RTWOs

The comparison is conducted with the objective of identifying the topology that minimizes the required integration area under identical geometric conditions, while evaluating the impact of different resonator topologies.

Equations (6) and (15) are used to adjust the parameter $L_t$ for each topology, ensuring that all structures maintain a constant total resonator length. The analyzed topologies include square, 45° beveled square, hexagonal, octagonal and decagonal geometries. In all cases, the geometric parameters are fixed as $W=10$ μm, $S=15$ μm, $l=1400$ μm and ${\theta }_M={45}^{\circ }$. All topologies are implemented using only angles compatible with the design rules of the 65 nm UMC CMOS technology.

The proposed methodology is primarily based on geometric parameters, such as total length, conductor width, spacing and resonator topology. As such, it can be extended to different CMOS technology nodes.

However, electrical parameters, including resistance, capacitance and associated losses, are technology dependent. Consequently, while the comparative behavior between different topologies is preserved, the absolute values of oscillation frequency and power consumption may vary with technology scaling.

The compared structures are shown in Figure 4.

Figure 4 shows that all structures maintain a similar size. RTWO arrays consist of multiple oscillators that enable signal distribution over longer distances while preserving their advantages. A higher density of RTWOs within a circuit allows for more efficient signal distribution. For this purpose, the concept of filling factor $FF$ is introduced, which describes the ratio of the effective area occupied by the structure to the available area. The filling factor is defined as:

$$FF={\displaystyle \frac{A}{A_{box}}}$$

(18)

where $A_{box}$ is defined as the minimum square that encloses the structure. Square and hexagonal resonators exhibit a higher filling factor compared to octagonal and decagonal resonators, which introduce unused regions within the integration area. This characteristic is advantageous for the formation of RTWO arrays. Finally,

Table 2. Comparison of the geometric parameters of the analyzed RTWOs.

Topology	${\mathit{N}}_{\mathit{c}}$	${\mathit{L}}_{\mathit{t}}$ [μm]	$\mathit{L}$ [μm]	$\mathit{A}$ [mm2]	$\mathit{F}\mathit{F}$ [%]
Square	4	137	207	0.0428	100
Beveled	4	145	215 1	0.0449	97.30
Hexagonal	6	95	135.4	0.0476	75.08
Octagonal	8	72	101	0.0492	82.72
Decagonal	10	58	80.7	0.0501	77.20

¹ The value of $L$ for the beveled topology corresponds to the equivalent side assuming a perfectly square structure.

presents a comparison of the different geometric parameters.

Table 2 clearly shows how the parameter $L_t$ decreases to compensate for the increase in the number of sides $N_c$ while maintaining a constant total length. It is also observed that the integration area increases with $N_c$.

By keeping the total length of the structure constant across all topologies, a limited amount of metallic material is distributed along the resonator. Under this condition, the perimeter of the structures remains approximately constant among the different configurations, as described by Equation (8). In this context, the geometry that maximizes the enclosed area for a given perimeter is the circular one; therefore, those topologies that tend to approach this shape exhibit a better area-to-perimeter ratio.

The RTWO with a square resonator requires the smallest integration area while also providing a favorable filling factor. From a geometric perspective, square RTWOs present several advantages over other structures. However, from an electrical standpoint, they may exhibit drawbacks due to the abrupt geometric discontinuities they introduce.

3.2.1. Electrical Comparison of RTWOs

The structure of the oscillators is directly related to their electrical performance. Similar behavior has been reported in other resonant electromagnetic structures, where geometrical modifications alter the field distribution and resonance characteristics [2]. Variations in geometry modify the propagation conditions of the traveling wave along the resonator. These changes affect the distribution of electrical parameters and may increase the energy losses associated with resistive and coupling effects. Consequently, a topology that exhibits favorable geometric characteristics does not necessarily guarantee better electrical performance.

The extraction method presented in [12] and used in [1,13] was employed. In ADS, the structure is designed and full-wave simulations are performed to extract the total RLCG parameters, the propagation constant γ, the characteristic impedance $Z_c$ and the variables of the electrical model [8]. Subsequently, circuit-level simulations are carried out in HSPICE to obtain the oscillation frequency and the total power consumption, including both active and passive contributions.

This simulation framework is widely adopted in the analysis of RTWOs, as it enables accurate modeling of distributed interconnect effects that strongly influence oscillation behavior. The combination of full-wave electromagnetic extraction and circuit-level simulation provides a consistent representation of the resonator and compensation stages.

A total of eight compensators are uniformly distributed along the resonator, each associated with a transmission line section. These compensators are implemented using two inverters connected in antiparallel. This distributed configuration enables effective compensation of losses along the structure and supports stable oscillation. The NMOS transistor channel width is $W_n=10\mathsf{\mu }\mathrm{m}$ and the symmetry factor ${\beta }_0$ is calculated as:

$${\displaystyle \frac{1}{{\beta }_0}}={\displaystyle \frac{{\mu }_p}{{\mu }_n}}$$

(19)

where ${\mu }_p$ and ${\mu }_n$ represent the carrier mobility of PMOS and NMOS transistors, respectively. The symmetry factor has an approximate value of 2.9.

The parameters considered in the comparison are the total resistance $R_{total}$, total inductance $L_{total}$, total capacitance $C_{total}$, oscillation frequency $f_{osc}$ and power consumption including passive $P_{pas}$, active $P_{act}$ and total $P_{total}$ components.

As reported in [8] and observed in [14], the total power consumption is mainly composed of two contributions—the energy dissipated by the structure itself and the consumption of the active devices, the latter being dominant and accounting for approximately 90% of the total power consumption.

Table 3. Comparison of the electrical parameters of the RTWOs.

Topology	${\mathit{R}}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$ [Ω]	${\mathit{L}}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$ [pH]	${\mathit{C}}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$ [fF]	${\mathit{f}}_{\mathit{o}\mathit{s}\mathit{c}}$ [GHZ]	${\mathit{P}}_{\mathit{p}\mathit{a}\mathit{s}}$ [mW]	${\mathit{P}}_{\mathit{a}\mathit{c}\mathit{t}}$ [mW]	${\mathit{P}}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$ [mW]
Square	8.15	432.02	159.13	23.07	1.90	25.36	27.26
Beveled	7.23	416.81	154.82	23.60	1.83	24.75	26.58
Hexagonal	7.57	430.13	158.88	23.15	1.81	24.56	26.37
Octagonal	7.12	417.04	157.53	23.50	1.82	24.83	26.65
Decagonal	6.42	424.96	158.24	23.42	1.73	24.06	25.79

presents the electrical comparison among the different RTWOs.

Table 3 shows that the total resistance exhibits a clear trend with respect to the number of polygon sides. The resistance decreases in polygonal resonators as $N_c$ increases, due to the reduction in abrupt geometric discontinuities, which leads to a more uniform current flow along the conductor contour.

As reported in [1], increasing the conductor width $W$ enables higher oscillation frequencies, since a larger $W$ increases the lateral area of the conductor and facilitates carrier transport. However, increasing W also enlarges the contact area between the conductor and the dielectric, resulting in higher leakage. Therefore, it is expected that an optimal width $W_{opt}$ exists that maximizes frequency while minimizing losses. Unfortunately, this optimal width exceeds the maximum width allowed by the technology.

The total inductance and capacitance do not show a clear trend or strong dependence on $N_c$. This behavior can be explained by the fact that the geometric model defines the resonator length along the conductor midline, while the actual current propagation is influenced by distributed electromagnetic effects and tends to follow the path of minimum impedance. Consequently, each topology exhibits a slightly different effective propagation length $l_{opt}$, along which the traveling wave propagates more efficiently.

Nevertheless, the proposed geometrical approximation provides a sufficiently consistent reference for comparative analysis among different RTWO topologies. Therefore, although small deviations between the geometrical and effective propagation lengths are expected due to topology-dependent electromagnetic effects, the overall trends of $L_{total}$ and $C_{total}$ remain comparable across all analyzed structures.

All structures exhibit very similar values of $L_{total}$ and $C_{total}$, suggesting that the optimal resonator length is nearly the same in all cases. Interestingly, the beveled and octagonal structures show the highest values of the product $L_{total}{C}_{total}$, which can be explained by the fact that a square resonator with 45° beveled corners behaves as an irregular octagon.

The behavior of the oscillation frequency can be explained using the same concept. The traveling wave is confined within the structure and propagates along the optimal length. Therefore, structures with smaller $l_{opt}$ achieve higher $f_{osc}$. The square resonator exhibits the lowest frequency, while the beveled structure achieves the highest. Through geometric modifications, the oscillation frequency can be increased by approximately $530\mathrm{M}\mathrm{H}\mathrm{z}$.

Finally, the active and passive power consumptions are analyzed. RTWOs exhibit a quasi-adiabatic behavior in which part of the energy is dissipated in each cycle and must be compensated by the active circuitry. Therefore, structures with higher losses require greater power compensation, as observed in the square and decagonal topologies. The square resonator shows the highest total power consumption due to abrupt changes in the propagation direction of the traveling wave, while the decagonal topology exhibits lower total power consumption, associated with a more uniform geometry that reduces losses. This behavior is reflected in the total resistance of the oscillator.

3.2.2. Performance Comparison of RTWOs

Although oscillation frequency is the main parameter of interest in an RTWO, it is not sufficient on its own to establish a complete comparison between different topologies. A structure may achieve a higher $f_{osc}$ while requiring higher power consumption or larger integration area. Therefore, figures of merit $FoM$ are required to relate frequency with area and power costs. These figures of merit are defined in Equations (20)–(22).

$${FoM}_1={\displaystyle \frac{f_{osc}}{P_{total}}}$$

(20)

The first figure of merit, energy efficiency, relates oscillation frequency to power consumption. It represents the number of cycles per second delivered by the oscillator per unit of consumed power, so $Fo{M}_1$ should be as high as possible.

$${FoM}_2={\displaystyle \frac{f_{osc}}{A}}$$

(21)

Similarly, $Fo{M}_2$, referred to as area efficiency, compares structures in terms of oscillation frequency and occupied area. An important characteristic of RTWOs is that oscillation frequency increases as the total length decreases. In the same way, RTWOs with smaller $l$ require less integration area, enabling higher oscillation frequencies within reduced areas.

$${FoM}_3={\displaystyle \frac{f_{osc}}{P_{total}\ast A}}$$

(22)

$Fo{M}_3$, defined as global efficiency, integrates both performance variables into a single expression, allowing the identification of a balanced structure. It can be interpreted as the oscillation frequency benefit relative to the cost in power consumption and integration area. A high $Fo{M}_3$ indicates that the RTWO achieves high frequencies with low power consumption and reduced area.

Table 4. Comparison of the performance figures of merit of RTWOs.

Topology	$\mathit{F}\mathit{o}{\mathit{M}}_1$ $\left[\mathbf{G}\mathbf{H}\mathbf{z}/\mathbf{m}\mathbf{W}\right]$	$\mathit{F}\mathit{o}{\mathit{M}}_2$ $\left[\mathbf{G}\mathbf{H}\mathbf{z}/\mathbf{m}{\mathbf{m}}^2\right]$	$\mathit{F}\mathit{o}{\mathit{M}}_3$ $\left[\mathbf{G}\mathbf{H}\mathbf{z}/(\mathbf{m}\mathbf{W}\mathbf{\ast }\mathbf{m}{\mathbf{m}}^2)\right]$
Square	0.846	539.02	19.77
Beveled	0.887	525.61	19.77
Hexagonal	0.877	486.61	18.44
Octagonal	0.881	477.64	17.92
Decagonal	0.908	467.47	18.12

presents the comparison of the figures of merit.

Table 4 shows that the decagonal topology exhibits the highest energy efficiency $Fo{M}_1$. This is due to its lower power consumption combined with an intermediate oscillation frequency. The beveled, hexagonal and octagonal topologies present very similar energy costs for signal generation, while the square resonator exhibits the lowest performance in terms of energy efficiency.

For $Fo{M}_2$, an opposite trend is observed. Topologies with fewer sides achieve better area efficiency, as they require smaller integration area. In addition, square structures exhibit a higher filling factor, enabling greater oscillator density.

Finally, $Fo{M}_3$ reveals a near tie between two topologies. RTWOs with a square resonator and those with 45° beveled corners exhibit the same global efficiency. This behavior can be explained as a balance between competing effects. The beveled topology improves oscillation frequency and reduces power consumption, but requires a larger integration area due to the geometric compensation through $L_t$. In contrast, the square topology exhibits lower frequency and higher power consumption, but benefits from a more compact layout.

As a result, these opposing effects compensate for each other, leading to similar values of the global figure of merit. This demonstrates that different topologies can achieve comparable overall efficiency through distinct physical trade-offs.

4. Discussion

The proposed geometric analysis provides design tools for the proper dimensioning of RTWOs, as it prevents undesired increases in the total length and minimizes deviations in the oscillation frequency. However, the analysis presents a limitation in that the physical length defined by the designer does not coincide with the effective length followed by the traveling wave. As a result, some topologies exhibit performance that deviates from expected trends.

The comparative analysis reveals several geometric electrical trade-offs, highlighting the flexibility of RTWO designs. For instance, when energy efficiency is the main objective, aiming for low power consumption and high oscillation frequency without area constraints, topologies with a higher number of sides are recommended, provided that the geometry is compatible with the integration technology.

For other applications, such as RTWO arrays where higher oscillator density is required to distribute synchronization signals over longer distances with improved quality, square topologies are preferred. These structures optimize the available chip area more effectively and their simplicity facilitates interconnection between oscillators.

The RTWO with a square resonator with 45° beveled corners exhibits outstanding performance. Under identical geometric conditions, the figures of merit show that this topology ranks third in energy efficiency, second in area efficiency and ties for first place in $Fo{M}_3$. A simple structural modification of a square resonator by trimming the corners to reduce abrupt changes in the propagation direction of the traveling wave leads to a significant improvement in performance.

Phase noise was not included in the comparison, as the main objective of this work is to study the influence of topology on oscillation frequency, power consumption and integration area. Although phase noise is primarily determined by the active circuitry and noise sources, the geometry of the resonator and its distributed properties can indirectly influence its behavior.

Distributed losses, geometrical discontinuities and the uniformity of wave propagation affect the effective quality factor of the system. In this sense, topologies with smoother transitions and fewer discontinuities tend to promote a more uniform energy distribution, which may translate into improved phase noise performance. Although this effect is not explicitly analyzed in this work, the proposed geometrical framework provides qualitative insight into these trends.

5. Conclusions

The geometric, electrical and performance comparison reveals the existence of fundamental trade-offs between structural design and dimensioning, energy efficiency and integration area optimization in Rotary Traveling Wave Oscillators.

From a geometrical perspective, simpler structures enable better utilization of the available area, resulting in higher filling factors. However, this advantage does not necessarily translate into improved electrical performance.

The electrical analysis shows that the presence of geometric discontinuities directly affects the propagation of the traveling wave, altering the distribution of electrical parameters and increasing energy losses. In this context, topologies that minimize abrupt geometric transitions promote more uniform wave propagation, leading to improved energy efficiency.

The overall performance comparison highlights the behavior of square-based topologies. The conventional square resonator achieves the highest efficiency in terms of integration area usage and exhibits a maximum filling factor; however, its electrical performance is limited. In contrast, the square resonator with 45° beveled corners provides a balanced trade-off between area utilization and electrical performance, reducing losses without significantly increasing the required integration area.

It is demonstrated that a simple geometric modification applied to the most basic RTWO structure enables simultaneous improvement in oscillation frequency and reduction in power consumption, while requiring only a minimal increase in integration area. The square resonator with 45° beveled corners stands out as an efficient solution, combining a simple geometric design with outstanding electrical performance.

Finally, it should be noted that the validation presented in this work is based on simulation results. Although electromagnetic and circuit-level simulations provide valuable insight into RTWO behavior, experimental measurements are required to fully assess the impact of process-dependent parasitics. Prototype validation is currently in progress and will be addressed in future work.