Chapter 12 - Phasor Algebra

Phase angle (PhA) value determined by bioelectrical impedance analysis (BIA) is an indicator of cell membrane damage and body cell mass. Recent studies have shown that low PhA value is associated with increased nutritional risk in various group of patients. However, there have been only a few studies performed globally assessing the relationship between nutritional risk and PhA in hospitalized geriatric patients. The aim of the study is to evaluate the predictive value of the PhA for malnutrition risk in hospitalized geriatric patients. One hundred and twenty-two hospitalized geriatric patients were included in this cross-sectional study. Comprehensive geriatric assessment tests and BIA measurements were performed within the first 48h after admission. Nutritional risk state of the patients was determined with NRS-2002. Phase angle values of the patients with malnutrition risk were compared with the patients that did not have the same risk. The independent variables for predicting malnutrition risk were determined. SPSS version 15 was utilized for the statistical analyzes. The patients with malnutrition risk had significantly lower phase angle values than the patients without malnutrition risk (p=0.003). ROC curve analysis suggested that the optimum PhA cut-off point for malnutrition risk was 4.7° with 79.6% sensitivity, 64.6% specificity, 73.9% positive predictive value, and 73.9% negative predictive value. BMI, prealbumin, PhA, and Mini Mental State Examination Test scores were the independent variables for predicting malnutrition risk. PhA can be a useful, independent indicator for predicting malnutrition risk in hospitalized geriatric patients.

A simple, low cost instrument that measures impedance and phase angle was used along with a parallel-plate capacitance system to estimate the moisture content (MC) of six types of wheat. Moisture content of grain is important and is measured at various stages of their processing and storage. A sample of about 150 g of wheat was placed separately between a set of parallel plate electrodes and the impedance and phase angle of the system were measured at frequencies 1 and 5 MHz. A semi-empirical equation was developed using the measured impedance and phase angle values, and the computed capacitance, and the MC values obtained by standard air-oven method. Multi Linear Regression (MLR) method was used for the empirical equation development using statistical software. In the present work, a low-cost impedance analyzer designed and assembled in our laboratory was used to measure the impedance and phase angles. MC values of wheat samples in the moisture range of 9% to 25%, not used in the calibration, were predicted by the equations and compared with their standard air-oven values. For over 97% of the samples tested from the six varieties of wheat, the predicted MC values were within 1% of the air-oven values. This method, being nondestructive and rapid, will have considerable application in the drying and storage processes of wheat and similar field crops.

A simple, low cost instrument called the CI meter, that measures impedance and phase angle was used along with a parallel-plate capacitance system to estimate the moisture content (MC) of six types of wheat. MC measurement of grain is important at various stages of their purchase, processing and storage. Samples of about 150 g of wheat at different MC levels were placed separately between a set of parallel plate electrodes, and the impedance and phase angle of the system were measured at 1 and 5 MHz. The MC values of all these samples were obtained by the standard air-oven method A semi- empirical equation was developed using the measured impedance and phase angle values and the computed capacitance, and the oven MC values obtained. Multi Linear Regression (MLR) method was used for the development of the empirical equation. In the present work, a low-cost impedance analyzer designed by Kandala, and assembled in the National Peanut Laboratory was used to measure the impedance and phase angles. MC values of wheat samples in the moisture range of 9% to 25%, not used in the calibration, were predicted by the equation develop, and compared with the standard air-oven values. For over 97% of the samples tested from the six varieties of wheat, the predicted MC values were within 1% of the air-oven values. This method, being nondestructive and rapid, will have considerable application in the buying, processing, drying and storage processes of wheat and similar field crops.

This study aimed to compare phase angle (PhA) and bioelectrical impedance vector analysis (BIVA) values between adult patients with congenital adrenal hyperplasia caused by 21-hydroxylase deficiency (CAH21OHD) and a control group. A total of 22 patients (15 women, 22.9 ± 3.7 years) were compared with 17 controls (11 women, 27.0 ± 2.5 years). Body composition was determined by dual-energy X-ray absorptiometry. Bioelectrical impedance was used to calculate PhA, and BIVA was performed using specific software. Student’s t-test and analysis of covariance were used to compare groups. Hedges’ G and partial n2 were calculated for the effect estimates. Hotelling’s t2 test was used to compare the mean impedance vectors between the groups. The Mahalanobis test was used to determine the distance between confidence ellipses. Patients with CAH21OHD had a higher fat mass percentage than that of the control group (both sexes). There was no significant difference in PhA values between groups (CAH21OHD vs. control) in females (6.9° vs. 6.3°, p = 0.092) and males (8.2° vs. 8.1°, p = 0.849), after adjusting for covariates (age and height). BIVA analysis showed a significant difference in the mean impedance vectors between the female groups (T2 = 15.9, D = 1.58, p = 0.003) owing to the higher reactance/height (Δ = 8.5; p < 0.001) of the patients. The PhA did not significantly differ between the groups. Female patients had significantly higher reactance values. However, further studies are needed to determine the usefulness of bioimpedance parameters in evaluating the hydration status and cellular integrity of patients with CAH21OHD.

e23106 Background: Androgen deprivation therapy (ADT) is standard treatment for prostate cancer but may cause changes in body composition that lead to decreased physical function. Low vitamin D status is also associated with decreased functionality. The Short Physical Performance Battery (SPPB) is an objective assessment tool for evaluating functionality. The bioelectrical impedance analysis (BIA) is a portable tool for assessing lean mass and phase angle, a measure of nutrition status and frailty in chronic disease. Methods: This was a secondary analysis of a randomized controlled trial to assess the impact of high dose vitamin D in older patients (≥60 yrs) with prostate cancer on ADT. Patients with prostate cancer (N = 59, age 67.6 ± 5.4) with vitamin D insufficiency ( &lt; 32 ng/ml) were randomized to high-dose vitamin D (n = 29, 600 IU/daily plus 50,000 IU/weekly) or low-dose vitamin D (n = 30, 600 IU/daily plus placebo weekly) for 24 weeks. SPPB tests and BIA were assessed at baseline, 12 weeks and 24 weeks. Phase angle values were calculated using atan(reactance/resistance) x (180°/π). A phase angle value &lt; 5.7° is a valid cutoff for frailty in older men. Results: Serum analyses showed compliance with vitamin D intake (25-OH vitamin D change: high vitamin D = +32.0 ng/ml vs low dose RDA vitamin D = +4.3 ng/ml; p &lt; .01). There were no difference at all three assessments for SPPB or lean mass between arms. The high-dose vitamin D group had wider phase angle values at 12 weeks (5.81º vs. 5.32º; p = .018) and 24 weeks (5.89º vs. 5.40º; p = .030). The low-dose group had phase angle values &lt; 5.7º over the course of the study. At baseline, over 18% of patients had phase angle values below the 10th percentile of the general population matched on age and BMI. As vitamin D levels increased throughout the study, fewer patients fell below the 10th percentile. Conclusions: The high-dose vitamin D group maintained wider phase angle values over 24 weeks, while phase angle for the low-dose group decreased from baseline. High-dose vitamin D supplementation may impact factors related to phase angle and frailty in patients receiving ADT. Funding: NCI UGCA189961, R21CA175793, K07CA168911, and GR501293.

Performance Grade (PG) Bitumen is bitumen that is graded based on its performance at different temperatures. In the Superpave grading system, asphalt binders are classified according to their performance in extreme hot and cold temperatures and are referred to as performance grade (PG) asphalt. The main objective of assessing and selecting asphalt binder using the PG system is to ensure that the binder has properties suitable for the environmental conditions in the field. Cariphalte is a premium grade polymer modified bitumen used in road and airport construction and other similar applications. Cariphalte is a Styrene Butadine Styrene (SBS) modified bitumen. The SBS in Cariphalte ensures the formation of a 3-dimensional network in the bitumen thereby reducing the temperature susceptibility and increasing the stiffness modulus at high temperatures. Cariphalte is also resistant to oxidation, reducing deformation and slowing secondary solidification. To get an idea of the improvement in rheological properties due to the addition of gilsonite, tests were carried out on a mixture of cariphalte and gilsonite asphalt with 6 variations of gilsonite content. The tests carried out include basic rheological properties testing and mechanistic rheological properties testing. From this research, the test results showed that at the same temperature, asphalt with the addition of 0% gilsonite had the smallest phase angle (δ) compared to asphalt using the addition of gilsonite. This explains that the addition of gilsonite can increase the phase angle (δ) value of PG-76 asphalt, which makes the asphalt material have thicker characteristics.

In this paper, we present results dealing with the non-uniform pollution carried out under 50 Hz applied voltage on a plane model simulating the 1512 L outdoor insulator largely used by the Algerian Company of Gas and Electric Power (SONELGAZ). Many configurations in non-uniform pollution are studied in the ENP's (Ecole Nationale Polytechnique d'Alger) High Voltage Laboratory in order to analyze the impact of polluted layer distribution on the insulator dielectric performances. The polluted solution has a conductivity of 1.2 mS/cm obtained with distilled water and NaCl. Our investigations are particularly focused on the on line monitoring of both position and width of the contaminated layer. The flashover voltage and the leakage current magnitude have been investigated in order to study the flashover process on this insulating surface. A video apparatus is used to reflect the parallel discharges behavior, appearing when the polluted layer reaches a critical width. Phase angle values between applied voltage and leakage current (LC) signals at the fundamental frequency (50 Hz) are calculated using the Fast Fourier Transform (FFT) spectral analysis. Phase angle measurements indicate that the equivalent impedance of the insulator behaves like RC circuit with a high capacitive effect engendered by the pre-established clean band. This effect decreases when electric discharges occur at a particular voltage level. The Discrete Wavelet Transform (DWT) is adopted for the leakage current decomposition in several time-frequency bands. The STD-MRA (Standard deviation-Multi Resolution Analysis) of these frequency bands is calculated and is employed to choose the most interesting details that detect both position and width increasing of the conducting layer. Reported results show that the pollution surface state and the severity of this conducting layer deposited on insulator surface could be identified from the STD-MRA representation of leakage current frequency bands. It was shown that the high frequency band of the leakage current increases before the final flashover when the polluted layer is located in the middle of the plane model. It was established that a good correlation has been found between the insulator state surface and details of the leakage current obtained through the DWT decomposition. In fact, these details provide relevant information on both position and width of the polluted layer nonuniformly distributed on the insulator surface.

Phase angle (PhA) correlates with body composition and could predict the nutrition status of patients and disease prognosis. We aimed to explore the feasibility of predicting PhA-diagnosed malnutrition using facial image information based on deep learning (DL). From August 2021 to April 2022, inpatients were enrolled from surgery, gastroenterology, and oncology departments in a tertiary hospital. Subjective global assessment was used as the gold standard of malnutrition diagnosis. The highest Youden index value was selected as the PhA cutoff point. We developed a multimodal DL framework to automatically analyze the three-dimensional (3D) facial data and accurately determine patients' PhA categories. The framework was trained and validated using a cross-validation approach and tested on an independent dataset. Four hundred eighty-two patients were included in the final dataset, including 176 with malnourishment. In male patients, the PhA value with the highest Youden index was 5.55°, and the area under the receiver operating characteristic curve (AUC) = 0.68; in female patients, the PhA value with the highest Youden index was 4.88°, and AUC = 0.69. Inpatients with low PhA had higher incidence of infectious complications during the hospital stay (P = 0.003). The DL model trained with 4096 points extracted from 3D facial data had the best performance. The algorithm showed fair performance in predicting PhA, with an AUC of 0.77 and an accuracy of 0.74. Predicting the PhA of inpatients from facial images is feasible and can be used for malnutrition assessment and prognostic prediction.

The evaluation of the nutrition status of patients has been the subject of interest of many scientific disciplines. Any deviation from normal values is a serious clinical problem. There are multiple nutrition status evaluation methods used including diet history, scales and questionnaires, physical examination, anthropometric measurements, biochemical measurements, function tests, as well as bioelectric impedance analysis or adipometry. Phase angle, obtained by means of bioelectric impedance analysis, is another parameter that is being more and more frequently applied in nutrition status monitoring. It is proportional to body cell mass. Its direct correlation with the cellular nutrition status has been documented. High phase angle values signify well-being, while low phase angle values indicate poor condition of cells. The purpose of this paper was to review the current state of knowledge about the application of phase angle in evaluation and monitoring of the nutrition status of patients with anorexia nervosa on the basis of available literature. It was proven that the phase angle values in patients with anorexia nervosa are much lower compared to healthy people. Detailed observations showed phase angle value increase in the course of treatment. The relevance of the commonly used body mass index (BMI) has been questioned due to significant degree of generalization in the nutrition status evaluation. Thus, there is a need for new, objective parameters for nutrition status evaluation, which will assist in the treatment and monitoring of patients in a more meaningful and reliable way. The existing independent studies equivocally confirm the usefulness of phase angle in the evaluation of nutrition status of patients with anorexia nervosa and its broader application in clinical practice is only a matter of time. However, these are merely attempts and they have not yet found wider application in clinical practice in the treatment of anorexia nervosa.

Resting energy expenditure (REE) constitutes the largest component of total energy expenditure and undergoes an age-related decline that is unexplained by decreased fat-free mass. Phase angle (PhA) is a cellular health indicator that is possibly associated with REE. We investigated the association of REE and PhA in hospitalized older adults. This single-center, cross-sectional analysis utilized the baseline data from a prospective longitudinal study and included 131 eligible patients aged ≥70 years. The REE was measured using indirect calorimetry, and PhA and body composition were assessed using bioelectrical impedance. The association between REE, PhA, and body composition was examined, and REE was compared using previously reported PhA cutoff values. In this cohort with a mean (±standard deviation) age of 87.4 (±7.0) years, 34.4% of the participants were men. REE and PhA correlated strongly (r: 0.562, p < 0.001) and significantly after adjusting for age and sex (r: 0.433, p < 0.001). Multivariate analysis showed a significant independent association between REE and PhA and skeletal muscle mass (standardized β [95% CI]; 28.072 [2.188-53.956], p = 0.035) without any significant interaction between PhA and age on REE. The low PhA group had a significantly lower REE (kcal/day; 890 [856-925] vs. 1077 [1033-1122], p < 0.001), and this remained significant after adjusting for age, sex, and skeletal muscle mass index. PhA is associated with REE in older adults. Adjusting REE calculation algorithms based on PhA values and correcting predicted REE according to PhA may aid in determining more accurate energy requirements.

We introduce a novel method for complex number representation. The proposed, redundant complex binary number system (RCBNS) is developed by combining a redundant binary number and a complex number in base (-1+j). Donald (1960) and Walter Penny (1964, 1965) represented complex numbers using base -j and (-1+j) in the classified algorithmic models. A redundant complex binary number system consists of both real and imaginary-radix number systems that form a redundant integer digit set. This system is formed by using complex radix of (-1+j) and a digit set of /spl alpha/=3, where /spl alpha/ assumes a value of -3, -2, -1, 0, 1, 2, 3. The arithmetic operations of complex numbers with this system treat the real and imaginary parts as one unit. The carry-free addition has the advantage of redundancy in number representation in the arithmetic operations. Results of the arithmetic operations are in the RCBNS form. The two methods for conversion from the RCBNS form to the standard binary number form have been presented. The RCBNS reduces the number of steps required to perform complex number arithmetic operations, thus enhancing the speed.

The rapid development of mathematics is largely due to the creation of complex numbers. Although complex numbers are just strange and seemingly meaningless notations for most people, complex numbers play important roles in engineering fields. A better understanding of complex numbers' geometric and algebraic structure enables us to study complex analysis better. Complex numbers convince scientists that our world is magical, full of wonderful insights, and even miraculous. In this paper, I first review several basic properties of complex numbers. The set of complex numbers is a group under the operation of the multiplication and under the operation of addition. Then I visualize complex numbers by building a bijective connection between complex numbers and points on the complex plane. I also give several alternative forms of complex numbers, namely the trigonometric and general forms. By invoking the arithmetic properties of complex numbers, I prove two trigonometric identities.

Phase Angle (PhA) is a ratio of whole body reactance and resistance obtained from Bioelectrical Impedance Analysis (BIA). It indicates cellular health and integrity and is considered as prognostic tool in medical disorders. In spite of prognostic potentials of PhA, it has limited usefulness in clinical practice and in population studies because of non-availability of normal population reference limits for comparison. Moreover, it is influenced by various factors like age, sex, race and body composition (i.e. body fat, muscle mass, visceral fat, body cell mass, total body water, etc). The aim of this study was to evaluate predictors of phase angle which will be useful in formulation of reference values for Indian population. BIA was performed by Tanita Body Composition Analyser on healthy adults aged 17-24 years. The inbuilt software measured the phase angle by the formula: Phase angle (PhA) = Reactance (xc)/Resistance (R)* (180/π). Phase angle values were compared across categories of age, sex, weight, height, Body Mass Index (BMI), total fat, visceral fat and muscle mass. Mean value of phase angle was found to be 5.65. Phase angle was significantly (p< 0.001) higher in male than in female. Phase angle was significantly predicted from height (p< 0.001), weight (p< 0.002), muscle mass (p< 0.002) and visceral fat (p< 0.02) in multiple regression models. Phase angle differs across anthropometric and body composition categories. Thus height, weight and muscle mass should also be taken into consideration while deriving population specific reference limits of phase angle.

Purpose The phase angle (PhA) measured by bioelectrical impedance analysis (BIA) is associated with nutritional status and cellular health, and it is a clinically important parameter used to assess the risk of various diseases. It remains unclear whether PhA is associated with nonalcoholic fatty liver. The purpose of this study is to investigate the relationship between the BIA parameter PhA and nonalcoholic fatty liver disease (NAFLD). Design/methodology/approach This cross-sectional study was conducted with 300 adults aged 20–64 years (NAFLD: 196, normal: 104). Some biochemical findings of the participants were collected, and whole-body and segmental PhAs were measured using Tanita (MC-780) at 50 kHz. Findings The PhA values of the whole body (p = 0.003), trunk (p &lt; 0.001), right and left legs (p &lt; 0.001 for both) were found to significantly differ according to the degrees of fatty liver. It was observed that the highest PhA values were in normal individuals. Logistic regression analysis showed that the reduction in PhAs of the whole body (p = 0.038), right (p = 0.019) and left legs (p = 0.049) and trunk (p = 0.001) after adjusting for all confounding factors increased the risk of NAFLD. Additionally, whole body PhA was significantly associated with age (year, p = 0.02), BMI (kg/m2; p &lt; 0.001), fat mass (kg; p = 0.001), fat mass (%; p &lt; 0.001), albumin (g/L; p &lt; 0.001) and CRP (mg/dL; p = 0.001). Originality/value The results of this study showed that PhA can be used in the management of NAFLD. To identify potential mechanisms in the relationship between the angle of the liver and NAFLD, large-scale prospective studies are needed.

&lt;div class="section abstract"&gt;&lt;div class="htmlview paragraph"&gt;The customer perception of ride comfort with vehicle performance is the most important aspect in a vehicle design. The ride comfort and vehicle performance are influenced by driveline components i.e. propeller shaft phase angle, inclination angle and critical frequency of the driveline system. The optimization of the driveline system is essential to ensure the efficient and smooth power transfer. Propeller shaft is one of the critical components in the driveline to influence the vehicle performance.&lt;/div&gt;&lt;div class="htmlview paragraph"&gt;Propeller shaft characteristics influenced by several factors like vehicle max torque, propeller shaft joint type, materials properties, UJ phase and inclination angle and shaft unbalance value. The optimization of the above parameter within the tolerance limit enables to meet the required performance standard. Various methodologies are available to optimize these parameters to enhance the vehicle performance and comfort leads to customer satisfactions. This study focuses on the analytical optimization of the propeller shaft’s universal joint (UJ) phase and inclination angle and validated at vehicle level.&lt;/div&gt;&lt;div class="htmlview paragraph"&gt;An analytical model was employed to evaluate the velocity fluctuations across the full rotation of the propeller shaft by considering the different UJ phase angles (0&lt;sup&gt;°&lt;/sup&gt; − 360&lt;sup&gt;°&lt;/sup&gt;) and inclination angles (1&lt;sup&gt;°&lt;/sup&gt; − 5&lt;sup&gt;°&lt;/sup&gt;). It was observed from the study that optimization of these parameter improves the vibration performance and decreases the velocity fluctuations. The results shows that the well optimized propeller shaft will enhance smoothness in the driveline systems, reduction in NVH levels, vehicle performance and ride comfort. This study suggests the importance of the precise geometry alignment in driveline design and provide the further refinement methodology.&lt;/div&gt;&lt;/div&gt;