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Transfer Function of the Sumudu, Laplace Transforms and Their Application to Groundwater
Published in Abdon Atangana, Mathematical Analysis of Groundwater Flow Models, 2022
Rendani Vele Makahane, Abdon Atangana
A Bode plot is a graph that shows the system’s frequency response. It was initially considered by Hendrick Wade Bode in the 1930s. It combines two logarithmic plots, one expressing the magnitude, and the other the phase shift of a system with respect to a given input frequency (York, 2009). The x-axis displays frequency, whereas the y axis displays magnitude and phase angle (Summer, 2004). The Bode plot is used to test and analyze the filters of a system. Two types of filters are recognized, a high pass filter allowing passage of signals with frequencies higher than the cut-off frequency, and restricts signals with frequencies lower than the cut-off frequency. The second filter is a low pass filter; it permits signals with a frequency of less than a selected cut-off frequency to pass and restricts signals with frequencies greater than the cut-off frequency. In the next few pages, we shall give the subsequent Bode plots for the Laplace and Sumudu transforms with respect to time and space as well as with Caputo, Caputo–Fabrizio and Atangana–Baleanu derivatives. Analysis and comparison between the Bode plots are also presented. The following plots were produced using MatLab software (Figures 7.1 to 7.6).
Introduction to Feedback Control Systems
Published in Ramin S. Esfandiari, Bei Lu, Modeling and Analysis of Dynamic Systems, 2018
The Bode plot technique is widely used to display a frequency response function. It also gives useful information for analyzing and designing control systems. Stability criteria can be interpreted using the Bode plot and numerous control design techniques are based on the Bode plot. In Section 8.4, we introduced the concept of the Bode plot and presented the Bode plot of the frequency response function for two fundamental systems, first-order and second-order. In this section, we first discuss how to use the Bode plot to display the frequency response function for a general dynamic system. Then, we learn how the Bode plot is utilized to determine stability. Finally, we will see how the Bode plot technique is used to design a proportional feedback controller.
Design of Discrete Time Control Systems
Published in Jitendra R. Raol, Ramakalyan Ayyagari, Control Systems, 2020
Jitendra R. Raol, Ramakalyan Ayyagari
As we have seen in Section 10.1, a Bode plot, often known as a Bode diagram, is a graphical tool for drawing the frequency response of a control system. In fact, it is a diagram represented by two plots: (i) the magnitude of a TF v/s frequency and (ii) the phase of the TF v/s frequency; generally, this frequency is in rad/sec. (or rad/s); the magnitude is in dB; and the frequency is plotted in log scale. One important merit is that, in the s-domain the magnitude curve can be approximated by straight lines, and hence, the magnitude plot can be sketched without exact computation of the TF. However, this feature is not available for the Bode diagram in the z-domain. To have this feature, one uses bilinear transformation to transform unit circle of the z-plane into the imaginary axis of another complex plane, the w-plane: () w=1Tln(z);w=2T(z−1)(z+1)⇒z=1+wT/21−wT/2
A Novel SSA Tuned PI-TDF Control Scheme for Mitigation of Frequency Excursions in Hybrid Power System
Published in Smart Science, 2020
As can be observed from the quantitative and qualitative performance as depicted in Table 4 and Figure 6(a,b), respectively, the frequency excursions are arrested more efficiently and effectively with the proposed SSA optimized PI-TDF control strategy with oscillations being less peaky and the minimum value of the ISE being the least as compared to other techniques. Further, the stability of the system, with the proposed controller, is computed in respect of the eigenvalues which are all negative meaning thereby the system structure is stable. The eigenvalues are listed in Table 5. In addition, the stability is assessed also in terms of Bode plots as shown in Figure 7 wherein the gain and phase margins turn out to be both positive and hence the stability of the system with the control structure is ascertained.
Design of support vector machine controller for hybrid power system automatic generation control
Published in Energy Sources, Part A: Recovery, Utilization, and Environmental Effects, 2022
Padmaja Appikonda, Rama Sudha Kasibhatla
Bode plot is an important tool for stability analysis of control systems. It gives key information regarding the closed loop system’s gain margin (GM), phase margin (PM), and stability. A basic requirement of control system design is a reliable response with appropriate GM and PM. The information obtained from a bode plot may be used to assess the stability of a feedback system. The gain and phase margins are the distances between the points where instability could arise. An increase in both gain and phase margin represents greater stability, and hence a larger margin is preferable. For a stable system, both margins must be positive, or the PM should be greater than the GM. Bode plots can be drawn by calculating the amplitude and phase angle mathematically for the transfer function. The frequency response, of a modified AGC scheme shown in Figure 1, in terms of the deviations in magnitude and phase as a function of frequency, is shown by Bode plots (Stankovic, Tadmor, and Sakharuk 1998). By drawing the bode plots for all scenarios, the stability of the AGC scheme of one, two, and three-area power systems with and without integrated to the hybrid power system utilizing PID controller and SVM controller is studied. For simplicity, the stability analysis of the typical scenarios where the hybrid-power system is integrated with single-area, two-area and three-area using bode plots are explained. For this investigation, the load perturbation () and frequency deviation () are considered as input and output signals respectively.
Corrosion monitoring at the interface using sensors and advanced sensing materials: methods, challenges and opportunities
Published in Corrosion Engineering, Science and Technology, 2023
Vinooth Rajendran, Anil Prathuru, Carlos Fernandez, Nadimul Haque Faisal
EIS is usually measured by applying an alternating current (AC) frequency to an electrochemical cell and measuring impedance through the cell. It is a powerful technique to apply in a wide range of the electrochemical system to determine the interface reaction on the different layers. In three-electrode cell, the working electrode, counter electrode and reference electrode work under potentiostatic or galvanostatic control which analysis the corrosion, moisture and interface conditions. In the two-electrode system, the counter electrode works also as a reference electrode. The counter electrode surface area should be large than the working electrode to maintain reaction equality. The Kramers–Kronig (KK) transform check to be made on the validity of an impedance data set acquired for a linear system over a wide range of frequencies [63,104]. The impedance of the working electrode controls the overall impedance which is significantly higher compared to the counter electrode. AC frequency flows over the working electrode [18]. The external AC electric field applied to the working electrode induces the intrinsic atomic and molecular charges to polarise as per the applied electrical field. Material degradation and absorbed moisture molecules makes an impact on the impedance of the electrode. EIS data can be presented as a bode (simple) plot or a Nyquist (complex) plot (illustrated in Figure 8). The bode plot is the combination of AC frequency, phase and amplitude which explains the materials phase and magnitude changes in each frequency level. The Nyquist plot determines the resistance of material from high frequency to low-frequency range [64,105,106].