Rolling Element Research Articles

Improved Calculation of Dynamic Load Capacity for Cylindrical Roller Thrust Bearings: Numerical Update of the Lifetime Reduction Factor for Bearings with Small and Medium Spin-to-Roll Ratios

The standard procedures for calculating the lifetime of rolling bearings, defined by DIN ISO 281 and ISO/TS 16281, have been revisited in this work with a specific focus on redefining the η factor for cylindrical roller thrust bearings (CRTBs). The new η factor proposed in this study accounts for the additional spinning motion of the rolling elements on the raceway, which affects the lifetime of thrust roller bearings. By considering different spin-to-roll ratios (SRRs), the revised η factor results in a smaller lifetime reduction, improving from a 42% reduction with η = 0.85 to a 27% reduction with η = 0.91. This modification opens industrial opportunities for bearings that can handle higher loads or feature fewer or smaller rolling elements while maintaining the same lifespan target as bearings sized with the original η factor. An analytical and numerical methodology was developed to calculate the η factor for various bearing configurations. Two bearing geometries were selected to assess the influence of the SRR on bearing life. The methodology integrates calculations of the total friction coefficient, 2D and 3D stress distributions, and lifetime predictions based on subsurface-initiated fatigue failure modes. The numerical results demonstrate the impact of contact stresses and bearing kinematics on η. Although this study was based on numerical simulations, it sets the groundwork for experimental validation. Future work includes experimental testing to validate these findings, with a focus on subscale CRTBs subjected to varying γ values. Accelerated testing strategies, including higher rotational speeds and optimized lubrication, are proposed to enhance the accuracy of the results. These experiments would provide further insights into the life expectancy differences between various configurations, contributing to more precise lifetime calculations for CRTBs.

Bearing fault feature extraction method for aperiodic stationarity caused by strong noise background

1. Rolling element bearings are the critical parts of every rotating machinery and their failure is one of the main reasons of the machine downtime and even breakdown. The significance of early failure detection cannot be overstated, as it plays a crucial role in maintaining the proper functioning of equipment, enhancing production efficiency, and ensuring safety. Among them, envelope analysis is the most effective and widely used approach, working according to the principle of linear filtering process of signals to remove undesirable components. However, the characteristic frequencies can be no longer evident or even overwhelmed due to the weak early failure signal. We propose an early fault feature extraction method that combines correlation entropy with an improved 2.5-dimensional square envelope spectrum. This approach is designed to overcome the relatively weak impact of early failures, which can be easily obscured by external background noise. Specifically, the correlation entropy matrix is simplified into a series of correlation entropies, and the formula for the higher-order spectrum is enhanced to achieve superior noise removal performance. Furthermore, to enhance the detection of the fault characteristic frequency (FCF), the Fourier transform in the improved 2.5-dimensional spectrum has been refined into a square envelope spectrum transform. Simulation results demonstrate that our proposed method excels at extracting early bearing fault characteristics and detecting inner and outer ring FCFs, thereby offering practical value.

An Investigation of the Effect of the Fault Degree on the Dynamic Characteristics and Crack Propagation of Rolling Bearings

The incipient faults of rolling bearings are dynamically propagating in the service status. However, the bearing material and the size of the fault will affect its expansion trend and direction. Furthermore, bearings manufactured from different materials behave differently when they fail. Therefore, the influence of the fault degree on the dynamic characteristics and crack propagation of rolling bearings is investigated in this paper. First, a dynamic model of the bearing, both under fault-free conditions and with varying fault sizes on the outer ring, is established by considering the actual working conditions of rolling bearings. Then, the reliability of the dynamic model is verified theoretically and experimentally. Finally, the study examined the slip behavior of rolling elements, the variation trends in the maximum shear stress and principal stresses on the outer ring, and the direction of crack propagation under different fault severities. The results indicate that (1) the severity of roller slip becomes more pronounced with the expansion of the fault size; (2) material differences will affect the timing of macro-slip during faults; (3) crack propagation tends to initiate at the edge of the fault exit, with the propagation rate increasing as fault severity escalates; and (4) tensile stress was observed in the first principal stress, which accelerates crack bifurcation at the faulted edge, while both the second principal stress and third principal stress exhibit compressive stress, playing a suppressive role in crack bifurcation at the faulted edge. These findings provide a theoretical basis for further research on the evolution of faults in rolling bearings.

Investigating the Effect of Vibration Signal Length on Bearing Fault Classification Using Wavelet Scattering Transform

Bearing condition monitoring and prognosis are crucial tasks for ensuring the proper operation of rotating machinery and mechanical systems. Vibration signal analysis is one of the most effective techniques for bearing condition monitoring and prognosis. In this study, the wavelet scattering transform, derived from wavelet transforms and incorporating concepts from scattering transform and convolutional network architectures, was utilized to extract quantitative features from vibration signals. The number of wavelet scattering coefficients obtained from vibration signals of different lengths varied due to the use of predefined wavelet and scaling filters in the wavelet scattering network. Additionally, these wavelet scattering coefficients are associated with different scattering paths within the corresponding wavelet scattering networks. Eight different lengths of vibration signals, associated with fifteen classes of rolling element bearing faults and conditions, were investigated using wavelet scattering feature extraction. The classes of rolling element bearing faults and conditions included normal bearings as well as ball and inner race faults with various fault diameters ranging from 0.007 inches to 0.028 inches. For the specific dataset validated, the computational results indicated that excellent bearing fault classification was achievable using wavelet scattering feature vectors derived from vibration signals with lengths of at least 6000 samples.

Feature extraction of rolling bearing based on adaptive variational multi-harmonic mode extraction

Variable multi-harmonic mode extraction (VMHME) not only has the advantages of high computational efficiency and extraction accuracy similar to variational mode extraction (VME), but also could extract the multi-harmonic components of periodic narrowband impulse signals in frequency band as wide as possible, making it very suitable for feature extraction in the event of rolling bearing failure. VMHME needs to accurately estimate the fault characteristic frequency of rolling bearing as its prior parameter, and small errors in estimating the fault characteristic frequency will cause significant deviations in the target extraction components. At present, the theoretical fault characteristic frequency of rolling bearings is commonly used as the estimated fault characteristic frequency. However, due to the installation deformation of rolling bearings and the random sliding between the rolling elements and the raceway during operation, it can cause a deviation between the actual fault characteristic frequency and the theoretical fault characteristic frequency. The most scientific and effective method is to enable VMHME to adaptively obtain the fault characteristic frequency based on the characteristics of the analyzed signal itself. Therefore, this paper introduces the envelope harmonic product spectrum (EHPS) theory into VMHME and proposes an adaptive VMHME (AVMHME) method to effectively extract the multi harmonic components of the periodic narrowband impulse signal when rolling bearings fail. Feasibility of the proposed method is verified through simulation and rolling bearing’ early weak fault experiment, and its superiority is also verified through comparative analysis.

A coupled dynamics model for studying the outer ring fault characteristics and mechanisms of axle-box bearings in urban rail vehicles

With the rapid and large-scale development of urban rail transit in China, the operational safety of urban rail vehicles is highly concerned. The axle-box bearing, as the core component of vehicle bogies, directly determines the operational safety and quality of vehicles. To ensure the service performance of axle-box bearings, their fault characteristics should be monitored and diagnosed in time. A more realistic and in-depth understanding of fault characteristics is a prerequisite for fault diagnosis. Thus, a double-rows tapered roller bearing (DTRB)-vehicle-track coupled dynamic model was built via the co-simulation in this paper, which comprehensively considers a DTRB model with the localised outer ring raceway fault, as well as the flexibilities of the wheelset, axle-box, and bogie frame. Its correctness and rationality were verified by the theoretical comparison and the field test. The results indicate that the distribution of contact load is consistent with the theoretical value, the fault features of the outer ring raceway can be found in the periodic impulse of the roller-raceway contact load. The slip rates of rolling elements are less than 4.3 %. Compared with the field test, the fault order harmonics in the order spectrum of the simulation are closer to the theoretical fault characteristic order under variable and constant speed conditions, and the maximum error is only 0.37 %. Therefore, the proposed coupled dynamic is correct and can be employed to investigate the fault characteristics of axle-box bearings in urban rail vehicles.

Dynamic modeling and analysis of wind turbine gear-bearing coupling system considering tooth root crack and slicing coupling effect

Tooth root cracks would change the meshing stiffness excitation of the gears, increasing the vibration and noise of wind turbine gearbox. Currently, there is relatively little literature that focuses on the slicing coupling effect of cracked gears, leading to incomplete accuracy in solving the time-varying mesh stiffness (TVMS) of gears under cracked conditions, making it difficult to accurately reflect the impact of root crack excitations on the dynamic behavior of wind turbine gear-bearing coupled systems. In this work, The improved analytical calculation method is proposed for the TVMS and system vibration coupling solution of the cracked gear pair, considering the slicing coupling effect. The dynamic model of the wind turbine high-speed stage gear set is developed using the lumped parameter approach, and the rolling bearing is modeled using the Hertz contact theory. Then, the excitations of the gear-bearing coupling system are developed using an improved slicing method. Lastly, the dynamic model of the wind turbine gear-bearing coupling system that considers tooth root cracks and the slicing coupling effect is established and validated. The results indicate that The TVMS modification algorithm considering the slicing coupling effect can better depict the dynamic change of the gear meshing stiffness under the crack failure. Cracks could reduce the TVMS of the gear pair, neglecting the coupling effect between the sliced gear tooth gains the potential risks of overestimating the influence of the gear crack depth on the TVMS. In addition, Gear cracks excitation may alter the dynamic contact load amplitude of the bearing rolling elements, resulting in a tendency for the load-bearing area to decrease. Moreover, When the cracked teeth are involved in meshing, The decrease in mesh stiffness will result in a larger amplitude periodic shock response to the system vibration displacement and a sideband phenomenon near the mesh frequency, neglecting the coupling effect between the sliced gear tooth or simplifying the bearing to a linear support stiffness matrix gain the potential risks of underestimating the influence of the gear crack depth on the vibration displacement of the gear-bearing coupling system.

Mathematical Mechanism of Gini Index Used for Multiple-Impulse Phenomenon Characterization

The Gini index (GI) is widely used for measuring the sparsity of signals and has been proven to be effective in the extraction of fault features. A fault-induced vibration, which involves the obvious phenomenon of multiple impulses, is a kind of sparse signal and the GI has been widely used in the diagnosis of rotating machine faults. However, why the GI can be used to evaluate the sparsity or impulsiveness of a signal has not been revealed directly. In this study, the mathematical mechanism of the GI, used for the representation of the multiple-impulse phenomenon, is deeply researched based on the theoretical deviation of the GI with regard to several typical signals. The theoretical results show that the GI increases with the increment in the number of impulses in the signal when the signal is interrupted by relatively low degrees of white noise. The bigger the difference between the amplitude of the impulse and the variance in the noise, the bigger the value of the GI. Namely, the signal-to-noise ratio has a great influence on the value of the GI. However, the GI is still a powerful tool for the characterization of the impulsive intensity of the multiple-impulse phenomenon. Both simulation and experimental data analysis are introduced to show the application of the GI in practice. It is shown that the fault diagnosis method based on the maximization of the GI is more powerful than that of kurtosis in terms of the extraction of fault features of rolling element bearings (REBs).

Finite Element Analysis of Layered Hollow Tapered Roller Bearings: Identifying Optimal Hollowness for Stress Reduction

Abstract Tapered roller bearings with hollow rollers offer advantages such as improved roller stiffness, enhanced speeds, and lower running temperatures due to better lubrication. Designing precise tapered roller bearings with hollow rollers is crucial. This study introduces a novel design of a layered hollow roller element aimed at minimizing contact pressure and stresses within the bearing, using a finite element analysis (FEA) approach. The study successfully employs elastic finite element analysis to determine the optimal hollowness level for the proposed design. The investigation covers hollowness levels from 30% to 80% for the rollers. The minimum contact pressure and Von-Mises stress were found to be 492.63 N/mm2 and 710.63 N/mm2, respectively. Results indicate that the optimal hollowness, yielding minimum stresses and contact pressure, is at 36%. These design modifications have potential applications in manufacturing industries where bearings are extensively used.

Study on the dynamic characteristics of double row self-aligning roller bearings with surface faults

Purpose The aim of this study is to establish the nonlinear dynamics equations of roller bearings with surface faults on outer raceway, inner raceway and rolling elements to analyze the dynamic characteristics of the double row self-aligning roller bearings, and provided theoretical basis for bearing fault diagnosis and life prediction. Design/methodology/approach First, based on the momentum theorem, the formulas for quantitative calculation of impact load were established, when roller was in contact with the fault of the inner or outer raceway. Then, the fault position piecewise functions and the load-carrying zone piecewise functions were established. Based on these, the nonlinear dynamic equations of double row self-aligning roller bearings are established, and Matlab is used to simulate the faulty bearings at different positions, sizes and rotational speeds. Finally, the vibration test of the fault bearings are completed, and the correctness of the nonlinear dynamic equations of the rolling bearing are verified. Findings The simulation and test results show that: the impact load increased with the increasing rotate speed and fault size, and the larger the fault size, the longer the impact load existed and the shorter vice versa. Originality/value The nonlinear dynamic equation of double row self-aligning roller bearings is established, which provides a theoretical basis for bearing faults diagnosis and fatigue life prediction.

FIR noise reduction using GWO and Fast Kurtogram for extracting fault characteristic frequencies in rolling bearings

Abstract Fast Kurtogram (FK) is a widely recognized method for detecting characteristic fault frequency bands. Nevertheless, its sensitivity to noise often causes diagnostic errors. This research seeks to enhance FK’s identification performance by implementing an adaptive FIR filter that mitigates noise. To validate this approach, single-point fault signals were obtained from the outer race, inner race, and rolling elements of bearings operating at four distinct motor speeds. These raw signals were then subjected to statistical analysis after normalization. To design the adaptive FIR filter, the study employs an enhanced Grey Wolf algorithm along with the Parks-McClellan algorithm to determine the optimal filter order and impulse response coefficients. Subsequently, band-pass filtering is applied to the fault characteristic frequency bands identified through FK. The fault characteristic frequencies are then extracted using envelope spectrum analysis. The bandwidth and fault characteristics derived from the processed signals are compared against those obtained from the unfiltered original signals. The findings indicate that optimal normalization preprocessing is crucial before filter design. The determined optimal order and impulse response coefficients enable the adaptive design of FIR filters. As a result, the FIR-filtered signal exhibits significantly fewer noise-induced anomalous peaks, thereby improving the signal-to-noise ratio and enhancing FK’s robustness against noise interference. Implementing an adaptive FIR filter for noise reduction significantly aids in more precise analysis and diagnosis of bearing conditions. Looking ahead, this study proposes integrating machine learning algorithms to further enhance the accuracy in identifying characteristic frequency bands associated with single-point faults in rolling elements.

Defect quantification evaluation of a rolling element bearing based on physical modelling and instantaneous vibration energy investigation

The estimation of defect size of rolling element bearing (REB) is expected to remain a significant and challenging task in vibration-based condition monitoring of rotating machinery. The conventional approaches focus on extracting the time spacing of the double impulse signal to estimate the defect size of the REB which is subsequently found to be inaccurate and arbitrary. In this paper, a novel model is developed to estimate the defect size of the REB based on the theories of physics and kinematics, in which the multi-event excitations, i.e. the entry, destressing and collision generated by the rolling element-defect interaction, are considered and investigated. The kinematic and geometric contact mechanism, which is mapped onto the rolling element-defect interaction as the rolling element passes over the defect zone, are analyzed and modelled as a function of the time information. In view of the fact that the variation of the vibration energy contained in the multi-impact vibration response is deeply mapped to the time information, a novel idea is proposed to extract the time information by analyzing the instantaneous energy variation of the defect-induced impulse using the scheme of the autoregressive model and the smooth pseudo Wigner-Ville distribution. The proposed model and idea are validated by experiments under different defect size and rotational speed scenarios. The results calculated by the proposed model show good agreement with the real defect size. Experimental and modelling comparisons show that the proposed model has reliable effectiveness and good performance and confidence in quantifying the size of the defect area localized on the outer raceway of the REB, and can provide some theoretical guidance for damage detection and remaining useful life prediction of the REB.

Predicting Rolling Element Bearings’ Deterioration Vibration Trend based on Limited Historical Data for a Desired Confidence Level using Machine Learning Algorithms

Predicting the deterioration trend of bearings and determining the optimal replacement time is crucial for maintenance teams to effectively manage consumable parts, enhance equipment reliability, and prevent potential breakdowns. Given the common practice of periodic condition monitoring based on vibrations in large industries, utilizing this data to forecast future vibration trends is a key focus of this research. Considering the limited availability of data in industrial settings, it is essential for the model used to be capable of working with minimal data. To address these challenges, a run-to-failure test has been conducted on an industrial bearing sample, with its vibration data recorded. Acknowledging the impact of operational variations and uncertainties in vibration behavior even among identical bearings, leveraging the bearing’s vibration history for predictive modeling has been deemed critical. This study proceeds in three distinct phases: selecting the key characteristic for predicting its deterioration trend, identifying the feature to pinpoint the failure onset, and determining the most suitable model for forecasting future bearing vibration states. RMS has emerged as the optimal characteristic for trend prediction, while Peak and Kurtosis have been identified as effective indicators for failure onset detection. The selection of the deterioration estimation model has involved comparing the outcomes of SVR + Bootstrapping and RVR models under various limited data scenarios. Ultimately, the RVR method has proved superior, requiring just three data points for estimation and delivering results with a confidence level leading to an accuracy of approximately 94% for the analyzed bearing.

Examination of condition of rolling bearings

Bearings operating properties impact the function of the whole machine therefore their investigation is an important topic. Early detection of failure of bearings allows for the replacement of them during planned maintenance, thus avoiding sudden downtime or unexpected accidents. There are several methods for monitoring the operating conditions of bearings, which help determine the residual lifetime of the bearings. This paper focuses on a test procedure which can be used to determine the condition of rolling element bearings.

Enhancement of cyclic spectral coherence map by statistical testing approach—application to bearing faults diagnosis in electric motors

Abstract Efficiency of fault detection in rolling element bearings is heavily influenced by the quality of data. In controlled environments, such as test rigs designed for bearing diagnostics, data quality is relatively good. Similarly, diagnosing bearings that support shafts in industrial machinery is relatively straightforward. However, diagnosing bearings in electric motors presents greater complexity due to the influence of additional cyclic components on vibration signals. These extra components, originating from mechanical or electrical sources, complicate frequency-based analysis. This paper proposes a novel approach for diagnosing bearings in electric motors, utilizing statistical analysis within the bi-frequency domain through a cyclostationary framework. The method involves applying a statistical testing procedure to individual pixels on the cyclic spectral coherence (CSC) map. The statistical significance of these pixels is assessed based on quantiles of CSC maps obtained from a dataset representing a healthy bearing. This process results in an enhanced or cleaned CSC map, facilitating the identification of fault-related components. Consequently, this approach enables the detection of defects in electric motor bearings, even when additional signal components unrelated to the defect, but characteristic of a healthy bearing, are present.

An improved approach on ball bearing characteristic frequencies

Bearing Characteristic Frequencies (BCF) are crucial for diagnosing faults in rotary systems, as they depend on the number of balls, bearing geometry, and shaft speed. However, divergences often arise between theoretical and actual BCF values due to additional influencing factors. This paper addresses these divergences by incorporating two key factors into the BCF equation. First, the speed variation among the shaft, inner race, outer race, and rolling elements is considered, as fault conditions can alter these speeds. Second, the fit factor, representing the impact of bearing mounting and component fits, is introduced to account for variations in BCF. The initial section of the paper reviews existing BCF equations proposed by various researchers, while the latter part presents modified equations that integrate these additional factors, offering a more accurate range of BCF values.

Application of the AVMD-AIMCKD method to the enhancement of rolling bearing fault features

Abstract Rolling bearings are prone to failure due to harsh working environments, which can affect production efficiency. Given the background environmental noise interference, it is difficult to extract the fault characteristics of rolling bearings in the early stages of weak fault signals. Based on adaptive VMD combined with adaptive IMCKD (AVMD-AIMCKD), a rolling bearing fault diagnosis method is proposed in this paper. The vibration signal from the rolling element bearing is processed by adaptive variational modal decomposition (AVMD) to extract and select the time-frequency domain signal characteristics. The Adaptive Improved maximum correlated kurtosis deconvolution (AIMCKD) algorithm is used for noise reduction optimization to obtain the best extraction results. Because the traditional method requires input parameter values for VMD and IMCKD, it is not adaptive. The particle swarm optimization algorithm (PSO) is utilized in this study to optimize two variables: the penalty factor and decomposition mode number of variable mode decomposition (VMD). The filter length and shift order of the improved maximum correlated kurtosis deconvolution (IMCKD) are optimized to make it adaptive. The viability of the method is substantiated through simulations, and the efficacy and precision are validated by comparative studies. The experimental results show that the AVMD-AIMCKD method has high accuracy and robustness in rolling bearing fault diagnosis and provides an accurate reference for rolling bearing health monitoring in engineering practice. The AVMD-AIMCKD method effectively extracts the early fault features through adaptive optimization, overcomes the restrictions of traditional methods, and provides a more accurate and reliable tool for bearing fault signal diagnosis.

Graph-based feature engineering for enhanced machine learning in rolling element bearing fault diagnosis

Abstract Fault diagnosis in rolling element bearings is critical for ensuring machinery reliability. This study improves machine learning techniques for predictive fault detection using the benchmark CWRU bearing dataset. Vibration signal data is preprocessed via balancing and graph-based feature engineering is performed to enable effective model training. Diverse classifiers including Random Forests, Support Vector Machines and Neural Networks are systematically evaluated through 10-fold cross-validation. Most of the models demonstrate exceptional performance, with top accuracies and AUC scores of 1.00. The research highlights the potential of hidden features that consider the implicit relations between the entities to improve predictive maintenance through data-driven bearing fault diagnosis.

Vibration properties of full ceramic bearing under elastohydrodynamic fluid lubrication based on the energy approach

As an important part of rotating machinery, heat generation due to friction is inevitable during the rotating process of bearings. A large amount of heat causes the temperature of lubricating oil to increase, viscosity of lubricating oil to decrease, and lubricating performance decreases. Addressing the issue of inadequate lubrication between the rolling element and raceway due to increased lubricating oil temperature, this study focuses on full-ceramic ball bearings. It examines the impact of temperature fluctuations on the bearing's lubrication condition and investigates the vibration characteristics of the full-ceramic bearing. The variation in the viscosity of lubricating oil with respect to temperature is measured via a rotary viscometer. A dynamic model of the silicon nitride full-ceramic ball bearing, which considers the effects of elastohydrodynamic lubrication, is established. The impacts of changes in lubricating oil viscosity due to different temperatures on oil film pressure, oil film thickness, and bearing vibration are analyzed and then compared with experimental data. It is observed that under conditions of constant load and speed, the film thickness of the lubricating oil decreases as the viscosity decreases. Additionally, the second peak value of oil film pressure is reduced with the declining viscosity of the lubricating oil. As the viscosity decreases, the vibration of the bearing increases. The amplitude discrepancy between the simulated vibration signal and experimental vibration signal of the full-ceramic bearing is found to be less than 10 %, verifying the validity of the dynamic model.

Search for optimal parameters of active structural suspension elements by determining their integral stiffness and damping characteristics

The paper presents an approach to the modelling of the “Rotor in active magnetic bearings” system and the subsequent search for optimal support parameters to ensure non-resonant operation of the machine in the entire range of operating rotational speeds. The object of the study is active magnetic bearings for the rotor of an industrial fan for local ventilation of the mine, which was previously installed in the rolling element bearings and due to the lack of the necessary load on them, have large frictional efficiency losses. To solve the problems with the loss of efficiency, it is proposed to use active magnetic bearings with a control system based on a proportional-integral-derivative controller with control principles of single input-single output, which is a cheaper and simpler system to implement compared to multiple input-multiple output systems, however, can be an effective tool for solving the described problems in the operation of the machine. The subject of the study is the dynamics of the rotor of an industrial fan in active magnetic bearings, which are modeled using spring-damper elements with stiffness and damping coefficients which are determined using the approach developed in the work. It is based on the representation of active magnetic bearings as an electromechanical system with components the dynamics of which are described by transfer functions in the frequency domain. The formation of the transfer function of the active magnetic bearing as a whole and its subsequent inclusion in the system of rotor dynamics equations allows obtaining relatively simple dependences of the stiffness and damping coefficients of the bearings on the rotor rotational speed. These dependencies contain the physical parameters of active magnetic bearing components and the parameters of their control systems which allows not only simulating the dynamics of the rotor in such bearings, but also searching for their optimal parameters, in particular, the value of the bias current in the electromagnets, which is necessary to ensure the absence of resonances of the rotor with synchronous excitation loads in the entire range of operating rotational speeds, which is confirmed by plotting the dependences of the rotor critical speeds on the bias current, as well as the Campbell diagram for the optimal value of the bias current.