Complex Frequency Research Articles

Equilibrium optimiser tuned frequency-shifted internal model control proportional-derivative decoupled dual-loop design for industrial plants followed by experimental validation

Integrating and unstable processes require rugged control demand by virtue of their non-self-regulation character. The occurrence of system poles located at the origin and on the right half of the complex frequency plane causes instability in an open-loop configuration. When these processes proceed with dead time, it demands more sophisticated control requirements. Therefore, binal-loop control schemes are recommended over single-loop controller schemes. For plants with prevalent unstable and integrating dynamics and dead time, a novel modified inferred internal model control-proportional derivative decoupled binal-loop control system is suggested. The inner-loop is controlled by the stabilising proportional derivative controller with Routh–Hurwitz stability constraints. The outer-loop contains an inferred internal model controller for reference-point tracking. The equilibrium optimiser is then used to minimise the integral squared error by tuning the inner and outer-loop controller settings in the confined search space. The suggested strategy delivers appreciable enhancement in the quantitative performance measures as compared to some of the contemporary control techniques. The effect of time-varying disturbance and robust stability analysis is also presented. Finally, utilising a magnetic levitation laboratory setup, the suggested method is experimentally verified using the hardware-in-loop method.

Charge superradiance on charged BTZ black holes

We study superradiant scattering for a charged scalar field subject to Robin (mixed) boundary conditions on a charged BTZ black hole background. Scalar field modes having a real frequency do not exhibit superradiant scattering, independent of the boundary conditions applied. For scalar field modes with a complex frequency, no superradiant scattering occurs if the black hole is static. After exploring some regions of the parameter space, we provide evidence for the existence of superradiantly scattered modes with complex frequencies for a charged and rotating BTZ black hole. Most of the superradiantly scattered modes we find satisfy Robin (mixed) boundary conditions, but there are also superradiantly scattered modes with complex frequencies satisfying Dirichlet and Neumann boundary conditions. We explore the effect of the black hole and scalar field charge on the outgoing energy flux of these superradiantly scattered modes, and also investigate their stability.

Effects of Operational and Environmental Conditions on Estimated Dynamic Characteristics of a Large In-service Wind Turbine

Abstract Purpose The reliable and continuous operation of wind turbines is of utmost importance, making it necessary to thoroughly understand their dynamic behavior under various operational and environmental conditions. Methods To achieve this, a data acquisition system distributed throughout the tower height is designed. The system records data such as the acceleration, temperature, and relative humidity from the sensors, along with the rotor speed, wind speed, temperature, pitch angle, nacelle direction, and wind direction from the data acquisition system of the turbine. The acquired data is synchronized and processed by Autonomous and Continuous System Identification system based on the poly-reference Least Squares Complex Frequency method. The extensive dataset, gathered over a 7-month period, allows for the estimation of modal parameters of the wind turbine. The modal parameters are then correlated with the operational and environmental conditions that were recorded. The relationships between these conditions are thoroughly analyzed and explained. Additionally, the operational principles of the wind turbine are elucidated in detail. The correlations between the modal parameters and operational or environmental factors are presented and interpreted, shedding light on the complex interplay between wind turbine dynamics and external conditions. Conclusion It can be said that changes in operational and environmental conditions affect the modal parameters of the wind turbine differently across various structural modes. Without considering these effects, structural health monitoring systems may produce false alarms. Failure to consider these effects in the development of structural health monitoring systems may lead to incorrect damage alarms.

Magnetic loss versus temperature and role of doping in Mn-Zn ferrites

We investigate the effect of different doping schemes on the broadband magnetic losses and their temperature dependence in Mn-Zn ferrites. CaO, Nb2O5, ZrO2, and SiO2 are added with increasing proportions to TiO2-doped prefired powders and, after sintering at either 1275 °C or 1300 °C, the obtained ring samples are tested versus frequency f (DC-1 GHz) and peak polarization Jp (2 mT – 200 mT) up to T = 160 °C. Appropriately enhanced impurity contents are shown to induce further decrease of the energy loss in materials already prepared for best performance at high temperatures (140 – 160 °C). This behavior can be hardly ascribed to the impurity-related increase of the electrical resistivity brough about by extra-doping, being it rather connected to a corresponding monotonical decrease of the effective magnetic anisotropy < Keff > with T. The decreasing anisotropy makes the balance between the contributions of domain wall (dw) displacements and reversible rotations to the magnetization process evolving in favor of the latter. The energy loss correspondingly develops with frequency and peak polarization in a complex fashion, according to the specific dissipative mechanisms sustained by the spins precessing either inside the moving walls or in the bulk. A dividing line in the (Jp − f) plane is identified, which separates dominant dw- and rotation-generated losses. It moves downward (i.e. lower f) with increasing temperature, the higher T the lower the frequency at which the rotations, theoretically assessed via the Landau-Lifshitz equation, supersede the domain wall contribution. Once accomplished, however, the transition to rotations can lead, according to the theoretical model, to higher losses when moving to higher temperatures. Following the experimental trend of the complex resistivity versus frequency at different T values, the calculations and the experiments show that eddy currents start to contribute to the energy loss, in the 5 mm thick ring samples, around a few MHz, accounting for about 50 % of measured loss beyond some 50 MHz. The chief dissipative process at applicative frequencies and induction values is therefore identified with spin damping, to which the generalized loss decomposition method can be applied.

Unified framework of forced magnetic reconnection and Alfvén resonance

A unified linear theory that includes forced reconnection as a particular case of Alfvén resonance is presented. We consider a generalized Taylor problem in which a sheared magnetic field is subject to a time-dependent boundary perturbation oscillating at frequency ω0. By analyzing the asymptotic time response of the system, the theory demonstrates that the Alfvén resonance is due to the residues at the resonant poles, in the complex frequency plane, introduced by the boundary perturbation. Alfvén resonance transitions towards forced reconnection, described by the constant-psi regime for (normalized) times t≫S1/3, when the forcing frequency of the boundary perturbation is ω0≪S−1/3, allowing the coupling of the Alfvén resonances across the neutral line with the reconnecting mode, as originally suggested in Uberoi and Zweibel, (1999). Additionally, it is shown that even if forced reconnection develops for finite, albeit small, frequencies, the reconnection rate and reconnected flux are strongly reduced for frequencies ω0≫S−3/5.

Design, Simulation and Analysis of Circular and Elliptical Antennas for 5G Applications

This research examines the design and simulation of both circular and elliptical microstrip antennas tailored for use in 5G technology, utilizing Ansoft HFSS software. The study assesses these antennas on parameters such as bandwidth, gain, and radiation patterns, aiming to meet the complex frequency demands of 5G networks. The findings suggest that circular antennas provide consistent omnidirectional coverage, whereas elliptical antennas excel in targeted coverage, essential for effective 5G communications. Moreover, elliptical antennas demonstrate significant benefits in point-to-point communications through adjustable radiation patterns. The paper offers insights into refining these antenna designs to cater to diverse 5G networking needs and proposes specific design changes to enhance their performance.

Vehicle detection in varied weather conditions using enhanced deep YOLO with complex wavelet

Traffic congestion is prevalent in many major and medium-sized cities throughout different countries in contemporary society. In traffic images, various multi-sized vehicles are tightly clustered together and obstructed from one another. Identifying vehicles in such instances is crucial for urban traffic surveillance, safety monitoring, and legal concerns but it also presents major challenges. The remarkable detection accuracy and efficiency of deep learning-based systems have led to their recent and extensive use in vehicle identification. There are significant advanced YOLO models with different backbone architectures and frameworks developed for vehicle detection. Yet, the performance of YOLO variants are facing the challenges of handling false detection against occluded and densely sophisticated scenarios. The proposed model is developed to address such types of limitations, for example; dynamic illumination, noisy images, and scale sensitivity to improve the vehicle detection rate in different traffic scenarios and varying weather conditions. The proposed study employs an improved YOLOv4 to identify moving vehicles in different lighting conditions including daylight, cloudy, rainy, and night. For hybridization, three techniques are utilized such as the Multiscale Retinex, Dual tree complex wavelet transform (DTCWT), and Pulse Coupled Neural Networks (PCNN). The DTCWT is employed for multiscale decomposition and to denoise the complex high frequency subband information, then the denoised subbands are reconstructed into a denoised image. The Multiscale retinex is utilized to reduce the halo artifacts on high-contrast edges and maintain the balance with dynamic range compression and color reproduction. The synchronizing pulse burst property of PCNN is used to detect the isolated noisy pixels and modify the detected noisy pixels. From the results it is worth noting that the developed model surpasses state-of-the-art methods in sunny, night, cloudy, and rainy modes. The proposed method using the DTCWT technique can detect the vehicles with mAP of 91.09% and 35FPS.

Chromosomal Abnormalities in Chromosome 5 and Chromosome 8 in Iraqi Patients with Acute Myeloid Leukemia

Acute myeloid leukemia (AML) is the mostly diagnosed leukemia in adults, resulting from accumulate immature blasts in the bone marrow that are replaced instead of normal blasts that cannot function like normal blood cells. AML resulting from genetic abnormalities, including multiple gene mutations and chromosomal aberrations, are found in approximately 80% of children with AML. These shown correlations between specific recurrent chromosomal abnormalities and clinical-biological characteristics and outcomes, also, novel therapies are being developed that target some of the identified genetic defects. The aim of this study was to investigate and record the role of chromosome 5 and 8 aberrations in the development of newly diagnosed and relapsed cases of AML in Iraqi patients. Chromosomal changes were studied in peripheral blood samples from 30 Iraqi patient samples diagnosed with acute myeloid leukemia and divided into 9 newly diagnosed and 13 chemotherapy-treated subjects who were incomplete remission and 8 relapsed subjects. The results of the chromosomal analysis of this study revealed numerical and structural abnormalities of both chromosomes: 5 and 8 in 16 (53.33%) and 12 (40%), respectively, within a complex karyotype and high frequencies in numerical abnormalities of chromosomes for chromosome 5 and structural abnormalities for chromosome 8. Chromosomal aberrations involving chromosomes 5 and 8 are linked to AML development and prognosis, and their presence in a complex karyotype can lead to disease progression and the worst prognosis, especially for abnormalities on chromosome 5, which were detected the most frequently. Future molecular genetic tests and complete karyotype analysis should enhance AML diagnosis and treatment.

Modal Identification of a Steel Platform Specimen with a Subspace Implementation of the Poly-reference Complex Frequency Method Formulated in Modal Model (pCF-MM)

Modal Identification of a Steel Platform Specimen with a Subspace Implementation of the Poly-reference Complex Frequency Method Formulated in Modal Model (pCF-MM)

Dietary sodium intake and atrial myopathy

Abstract Background Diagnostic markers associated with atrial cardiomyopathy, a precursor state to atrial fibrillation (AF), include premature atrial complex (PACs) frequency, left atrial volume index (LAVi) and p-wave indices. Life-style factors associated with atrial myopathy markers are insufficiently studied. Sodium intake is a modifiable risk factor for hypertension, cardiovascular disease and AF. Purpose We aimed to study the association between sodium intake and atrial myopathy markers in the population-based multicenter Swedish CArdioPulmonary bioImage Study. Methods We included 5,309 randomly invited 50–65-year-olds at the Malmö (n=1,125,) and Uppsala sites (n=4,184) without AF who had contributed a 24hECG and a fasting morning urine sample, from which daily sodium intake was estimated using the Kawasaki formula. At the Malmö site resting ECGs from which P-wave indices could be measured were available in 1,066 individuals, and LAVi was available in a random subsample (n=357). The association between sodium intake and atrial myopathy markers was modeled adjusted for age, sex, body mass index, smoking, systolic blood pressure, antihypertensive treatment, coronary artery disease, and heart failure using negative binomial regression for PAC frequency, linear regression for P-wave duration and LAVi and logistic regression for P-wave terminal force in V1, all using sodium intake as a continuous variable and strata at &amp;lt;2, 2.0-2.99g, 3.0-3.99g, 4.0-4.99g and ≥5 g/day, with 3.0-3.99 as the reference category. We also modeled the association between sodium intake and PACs using restricted cubic splines. Results The mean age was 58.1 (±4.1) years, and 52.7% were female. The mean estimated sodium intake was 3.3 (±1.4) g/day. Mean 24h PAC count was 27, 4.7% had ≥500 PACs. High sodium intakes were associated with increased PAC frequency, of 14% (95% confidence interval (CI 1-29% p=0.03)) for 4.0-4.99g/day and 25% (95%CI 8-45%, p=0.002) for intakes ≥5.0g/day, compared to sodium intakes of 3.0-3.99 g/day. There was no significant association between low sodium intakes (&amp;lt;3.0 g/day) and PACs, Figure 1. Sodium intakes of 4.0-4.99 g/day were associated with increased P-wave duration, ß-coefficient 2.4 (95%CI 0.3-4.1, p=0.03), compared to 3.0-3.99g/day, but there was no association between sodium intakes and LAVi, or P-wave terminal force in V1, either as a continuous variable or in strata. Conclusion High dietary sodium intake is associated with PAC frequency, a known prediction of AF and stroke. Whether reduced sodium diets could reduce PAC frequency, and whether that would impact AF and stroke risk needs further study.Sodium intake and PAC frequency

A Spread-Spectrum Modulation Scheme for a 3 × 6 Indirect Matrix Converter Based on a Current Ripple Model

Focused on addressing harmonic suppression in multiphase indirect matrix converters (IMCs), this study explores spread-spectrum modulation technology through ripple analysis calculations. We introduce a current ripple spread-spectrum modulation (CR-SSM) method tailored for multiphase IMC systems. In this approach, a 3 × 6-phase IMC is modeled as a two-port network, and a small-signal model of the output side is established. The transfer function is utilized to analyze the two-port network in the complex frequency domain (s-plane). The time-domain expression of the output current ripple is derived in vector form. Subsequently, the distribution of the output ripple and locus are determined based on specified constraints. The carrier frequency is dynamically adjusted online according to the specified ripple locus. Compared to classical periodic PWM methods, this approach offers a broader range of frequency variations and achieves a more uniform output spectrum. Furthermore, CR-SSM optimizes system efficiency and enhances spread-spectrum modulation. Experimental results demonstrate that this method effectively enhances the quality of input and output waveforms in multiphase IMC systems.

Hydrodynamics and instabilities of relativistic superfluids at finite superflow

We study the linear response of relativistic superfluids with a non-zero superfluid velocity. For sufficiently large superflow, an instability develops via the crossing of a pole of the retarded Green’s functions to the upper half complex frequency plane. We show that this is caused by a local thermodynamic instability, i.e. when an eigenvalue of the static susceptibility matrix (the second derivatives of the free energy) diverges and changes sign. The onset of the instability occurs when ∂ζ(nsζ) = 0, with ζ the norm of the superfluid velocity and ns the superfluid density. The Landau instability for non-relativistic superfluids such as Helium 4 also coincides with the non-relativistic version of this criterion. We then turn to gauge/gravity duality and show that this thermodynamic instability criterion applies equally well to strongly-coupled superfluids. In passing, we compute holographically a number of transport coefficients parametrizing deviations out-of-equilibrium in the hydrodynamic regime and demonstrate that the gapless quasinormal modes of the dual planar black hole match those predicted by superfluid hydrodynamics.

Penalty Sponge Layers (PSL) for hyperbolic systems. General formulation, well-posedness and stability

In this work we design and analyze an alternative to the Absorbing Boundary Conditions (ABC) or Perfectly Matched Layers (PML) technique to solve symmetric first order hyperbolic systems in unbounded domains. We propose a general formulation of absorbing layers that we call Penalty Sponge Layer (PSL). Well-posedness as well as stability properties are quite simply inherited from the original problem. Such a formulation is written in terms of a set of auxiliary variables similar to those encountered in the Complex Frequency Shifted PML (CFS-PML) technique, but this time they have the role of a penalty term, hence the name PSL. This second role of penalty, ensured by stretching an artificial frequency to infinity in the vicinity of the interface, makes the layer asymptotically non-reflecting. Numerical tests are performed to compare them with Standard Sponge Layers (SSL) and PMLs, and to validate their success, even in severe situations where the latter fail (e.g., the Euler system at oblique subsonic/critical or supersonic flow and anisotropic waves).

Measuring source properties and quasinormal mode frequencies of heavy massive black-hole binaries with LISA

The Laser Interferometer Space Antenna (LISA) will be launched in the mid-2030s. It promises to observe the coalescence of massive black-hole (BH) binaries with signal-to-noise ratios (SNRs) reaching thousands. Crucially, it will detect some of these binaries with high SNR in both the inspiral and the merger-ringdown stages. Such signals are ideal for tests of general relativity (GR) using information from the whole waveform. Here, we consider astrophysically motivated binary systems at the high-mass end of the population observable by LISA and simulate their signals using the newly developed multipolar effective-one-body model: pSEOBNRv5HM. The merger-ringdown signal in this model depends on the binary properties (masses and spins) and also on parameters that describe fractional deviations from the GR quasinormal mode (complex) frequencies of the remnant BH. Performing full Bayesian analyses, we assess to which accuracy LISA will be able to constrain deviations from GR in the ringdown signal when using information from the whole signal. We find that these deviations can typically be constrained to within 10% and in the best cases to within 1%. We also show that with this model we can measure the binary masses and spins with great accuracy even for very massive BH systems with low SNR in the inspiral. In particular, individual source-frame masses can typically be constrained to within 10% and as precisely as 1%, and individual spins can typically be constrained to within 0.1 and, in the best cases, to within 0.001. We also probe the accuracy of the SEOBNRv5HM waveform family by performing synthetic injections of GR numerical-relativity waveforms. Using a novel method that we develop here to quantify the impact of systematic errors, we show that, already for sources with SNR O(100), we would measure erroneous deviations from GR due to waveform model inaccuracies. One of the main sources of error is the mismodeling of the relative alignment between harmonics. These results confirm the need for improving waveform models to perform tests of GR with binary BHs observed at high SNR by LISA. Published by the American Physical Society 2024

COMPREHENSIVE ANALYSIS OF ENERGY DISSIPATION IN SOIL-EMBEDDED PILES: AN ANALYTICAL APPROACH

The study delves into a comprehensive examination of the total energy dissipation associated with a pile entrenched in the soil, with the soil being conceptualized as viscoelastic springs. "Dissipation" can be utilized to gauge the decay rate of displacement amplitude over time. Understanding dissipation in piles is pivotal for discerning the pace at which a pile achieves its equilibrium state when exposed to impact or free vibrations. Elevated dissipation levels indicate a swifter attenuation of transient vibrations, a crucial aspect for effective vibration mitigation during significant occurrences such as earthquakes, severe weather conditions, explosions, and impacts. The dispersion relation, reliant on wavenumber and free wave frequency, is derived for the pile-soil system using the Bloch's Floquet theorem. This relation is approached via the free-wave method, where a real wavenumber produces a sequence of complex frequencies. We propose an analytical closed-form expression in this study to forecast the overall dissipation from these complex frequency values. Additionally, we validate the accuracy of the predicted dissipation against a finite element-based numerical model of the soil-pile system. Furthermore, we examine the influence of factors such as material damping, Young's modulus, and shear modulus of the soil on the dissipation characteristics of the pile-soil system.

COMPREHENSIVE ANALYSIS OF ENERGY DISSIPATION IN SOIL-EMBEDDED PILES: AN ANALYTICAL APPROACH

The study delves into a comprehensive examination of the total energy dissipation associated with a pile entrenched in the soil, with the soil being conceptualized as viscoelastic springs. "Dissipation" can be utilized to gauge the decay rate of displacement amplitude over time. Understanding dissipation in piles is pivotal for discerning the pace at which a pile achieves its equilibrium state when exposed to impact or free vibrations. Elevated dissipation levels indicate a swifter attenuation of transient vibrations, a crucial aspect for effective vibration mitigation during significant occurrences such as earthquakes, severe weather conditions, explosions, and impacts. The dispersion relation, reliant on wavenumber and free wave frequency, is derived for the pile-soil system using the Bloch's Floquet theorem. This relation is approached via the free-wave method, where a real wavenumber produces a sequence of complex frequencies. We propose an analytical closed-form expression in this study to forecast the overall dissipation from these complex frequency values. Additionally, we validate the accuracy of the predicted dissipation against a finite element-based numerical model of the soil-pile system. Furthermore, we examine the influence of factors such as material damping, Young's modulus, and shear modulus of the soil on the dissipation characteristics of the pile-soil system.

Analysis of dynamic characteristics of whipping effect on l-shaped main steam pipe considering fluid-structure interaction effect

To investigate the whipping effect on the L-shaped main steam pipe utilized in the marine steam systems, the fluid-structure interaction (FSI) method is utilized to numerically analyze its dynamic characteristics. The numerical results illustrate that the dynamic response of whipping effect with rupture position B on the oblique pipe section is more severe with the significantly larger displacement of L-shaped pipe and anti-whip restraints and more complex frequency characteristics than that of whipping effect with rupture position A on the right-angled elbow. The mode transition phenomenon is also observed in the case with rupture position B. The maximum deformation concentrates on different parts when burst of pipes occurs at different positions and the deformation with rupture position B is several times larger than that with rupture position A. The increase of gap between anti-whip restraints intensifies the whipping effect, but it also reduces the maximum restraint force. The friction acts as the resistance in the slip process and thus contributes to the protection against whipping effect. The anti-whip restraints E closed to rupture position B on the oblique pipe section has better performance than anti-whip restraints D in the protection against horizontal whipping effect.

Hollow Fe3O4/Fe@C nanocubes for broadband microwave absorption spanning low- and high-frequency bands

Rationally constructing hollow structures with heterogeneous interface is usually an effective strategy to regulate the electromagnetic response behavior for lightweight and strong absorption, but achieving low-frequency and broadband absorption is a major challenge. This work adopts a novel Fe(II)/Fe(III) ions regulation strategy to design and synthesize hollow Fe(OH)X (x = 2, 3) nanocubes, which addresses the disadvantages of time-consuming or using etchant. Subsequently, a unique hollow Fe3O4/Fe@C nanoboxes with adjustable magnetic composition and uniformly coated shell have been successfully synthesized by in-situ carbonization with the assistance of phenolic resin and hydrogen thermal reduction. Notably, the different magnetic components induce the magnetic coupling assembly and play a crucial role in determining its electromagnetic parameters, whereas Fe3O4 contributes to more dipole polarization and stronger low-frequency response characteristics. For Fe3O4@C hollow nanoboxes, the minimum reflection loss (RLmin) is as low as –119.9 dB at 2.3 mm, and extremely wide effective absorption bandwidth (EAB) of 11.4 GHz is achieved spanning low-frequency band of 2.8 ∼ 7.2 GHz and high-frequency band of 11.0 ∼ 18 GHz at 5.0 mm. Surprisingly, the Fe3O4@C also has satisfactory EAB20 (RL＜ –20 dB) reaching 7.1 GHz (14.9 ∼ 18 GHz) at a thin thickness of 1.8 mm. The RLmin of other hollow nanoboxes is also below –30 dB, and the EAB is greater than 9 GHz. This study provides a competitive candidate for efficient microwave absorption in complex frequency bands, especially low-frequency bands.

Damping analysis of a transversely isotropic piezothermoelastic nanobeam resonator based on the MGT thermoelasticity

The present work investigates the vibration of a piezothermoelastic nanobeam in the frame of Moore–Gibson–Thompson (MGT) thermoelasticity theory. Closed form analytical expressions for the thermoelastic damping in terms of quality factor and frequency shift for a homogeneous transversely isotropic piezothermoelastic (PTE) beam is derived by using Euler–Bernoulli beam theory and complex frequency approach. Detailed analysis of damping of vibration owing to thermal fluctuations and electric potential in the present context under three sets of boundary conditions is attempted to investigate the influence of modes of vibration, electric potential, and beam length on energy dissipation caused by thermoelastic damping in PTE beam resonators. Analytical results are illustrated with the help of graphical plots on numerical findings in the case of the nanobeam resonator of Lead Zirconate Titanate (PZT-5A) material. The investigation brings out some significant key findings and observations in view of MGT heat conduction model.

Taxonomic implications of normal and abnormal stomatal complexes in Indigofera L. (Indigofereae, Faboideae, Fabaceae)

This study is the first to report the foliar and stem epidermal micro-morphology of 13 taxa of Indigofera L. (Fabaceae) using light (LM) and scanning electron microscopy (SEM). The micro-morphological characteristics studied here are related to the epidermal cell shape, size, frequency, anticlinal wall pattern, and stomatal complex types, size, position, frequency, and index. The study revealed 19 major normal stomatal types with eight subtypes and seven major abnormal stomatal types with 13 subtypes. The stomatal index was lower on the abaxial leaf surface than on the adaxial surface. Notably, the adaxial surface of I. hochstetteri had the highest stomatal index (27.46%), while the abaxial surface of I. oblongifolia had the lowest (9.95%). The adaxial surface of I. hochstetteri also displayed the highest average stomatal frequency (38.67), while the adaxial surface of I. spinosa had the lowest average frequency (9.37). SEM analysis revealed that most leaves had slightly sunken to sunken stomata, while stem stomata were positioned at the same level as epidermal cells in most taxa. Indigofera's foliar and stem epidermal anatomy recommends their application as baseline data coupled with other taxonomic data for the delimitation and differentiation of closely related taxa in the genus. The study provides a comprehensive description, illustrations, images, and micrographs of the stomatal types, as well as a taxonomic key for distinguishing the studied taxa of Indigofera.