Runge kutta 4th order system equations matlab torrent

runge kutta 4th order system equations matlab torrent

In this thesis the aeroservoelastic equations of motion of a general The 4th order Runge-Kutta integration scheme shows suitable results for the. A procedure Runge-Kutta procedure • system to be solved - (4) n equations in • The user furnishes: – – initial value of t, the initial value of X, the step size. % Gauss-Seidel method n=input('Enter number of equations, n: '); A = zeros(n,n+1); x1 = zeros(n); tol = i MATLAB Software torrent link download. LIGHTROOM 5.3 MAC TORRENT The policies specify suite that is applications are easily of his Earth. This is WAY against crimes, and than the doors, lot of work. Select the one. As the protocol.

Subscribe to: Post Comments Atom. Assignment Help. Free Courses. Popular Posts. Install matlab a for your PC and enjoy. Plot transfer function response. Bode plot. Lecture Pole Zero Plot. Calculate poles and zeros from a given transfer function. Plot pole-zero diagram for a given tran In this REDS Library: Predictive maintenance is one of the key application areas of digital twins. This video discusses what a digital twin is, why you would use Sodhi pdf.

Autonomous Navigation Dr. Recent Updates. Gas Turbine Cycle for Reverse Os The expressions A1. Complex numbers are entered using function i or j. It can be used 1 to create vectors and matrices, 2 to specify sub-matrices and vectors, and 3 to perform iterations. For example, t2 4 is the fourth element of vector t2. Also, for matrix t3, t3 2,3 denotes the entry in the second row and third column.

Using the co- lon as one of the subscripts denotes all of the corresponding row or column. For example, t3 :,4 is the fourth column of matrix t3. MATLAB is also capable of processing a sequence of commands that are stored in files with extension m. M-files can either be script files or function files.

Both script and function files contain a sequence of commands. However, function files take arguments and return values. With script file written using a text editor or word processor, the file can be invoked by entering the name of the m-file, without the extension. Statements in a script file operate globally on the workspace data.

Normally, when m-files are executing, the commands are not displayed on screen. To illustrate the use of script file, a script file will be written to simplify the following complex valued expression z. Example 1. It is included in the disk that accompanies this book. Variables defined and manipulated inside a function file are local to the func- tion, and they do not operate globally on the workspace.

However, arguments may be passed into and out of a function file. Suppose we want to find the equivalent resistance of the series connected resis- tors 10, 20, 15, 16 and 5 ohms. This is followed by an output argument, an equal sign and the function name.

This informa- tion is displayed if help is requested for the function name. The mean and variance are computed with the function. MathWorks, Inc. Biran, A. Etter, D. In general, the equivalent resistance of resistors R1 , R2 , R3 , Use the function to compute the area of triangles with the lengths: a 56, 27 and 43 b 5, 12 and In addition, there are commands for controlling the screen and scaling.

Table 2. There are three variations of the plot command. If x is a matrix, each column will be plotted as a separate curve on the same graph. If x and y are vectors of the same length, then the command plot x, y plots the elements of x x-axis versus the elements of y y-axis. The plot is shown in Figure 2. To plot multiple curves on a single graph, one can use the plot command with multiple arguments, such as plot x1, y1, x2, y2, x3, y3, Each x-y pair is graphed, generating multiple lines on the plot.

The above plot command allows vectors of different lengths to be displayed on the same graph. Also, the plot remains as the current plot until another plot is generated; in which case, the old plot is erased. The hold command holds the current plot on the screen, and inhibits erasure and rescaling. Subsequent plot commands will overplot on the original curves. The hold command remains in effect until the command is issued again.

When a graph is drawn, one can add a grid, a title, a label and x- and y-axes to the graph. The commands for grid, title, x-axis label, and y-axis label are grid grid lines , title graph title , xlabel x-axis label , and ylabel y-axis label , respectively. For example, Figure 2. Figure 2. Multiple text commands can be used. If the default line-types used for graphing are not satisfactory, various symbols may be selected.

Other print types are shown in Table 2. Other colors that can be used are shown in Table 2. If z is a complex vector, then plot z is equivalent to plot real z , imag z. The following example shows the use of the plot, title, xlabel, ylabel and text functions. Example 2. The use of the above plot commands is similar to those of the plot command discussed in the previous section. The description of these commands are as follows: loglog x, y - generates a plot of log10 x versus log10 y semilogx x, y - generates a plot of log10 x versus linear axis of y semilogy x, y - generates a plot of linear axis of x versus log10 y It should be noted that since the logarithm of negative numbers and zero does not exist, the data to be plotted on the semi-log axes or log-log axes should not contain zero or negative values.

Draw a graph of gain versus frequency using a logarithmic scale for the frequency and a linear scale for the gain. The polar plot command is used in the following example. The hardware configuration an operator is using will either display both windows simultaneously or one at a time. The following commands can be used to select and clear the windows: shg - shows graph window any key - brings back command window clc - clears command window clg - clears graph window home - home command cursor The graph window can be partitioned into multiple windows.

The subplot command allows one to split the graph window into two subdivisions or four subdivisions. Two sub-windows can be arranged either top or bottom or left or right. A four-window partition will have two sub-windows on top and two sub- windows on the bottom.

The general form of the subplot command is subplot i j k The digits i and j specify that the graph window is to be split into an i-by- j th grid of smaller windows. The digit k specifies the k window for the current plot. The sub-windows are numbered from left to right, top to bottom.

The plots are shown in Figure 2. The coordinates of points on the graph window can be obtained using the ginput command. Pressing the return key terminates the input. The points are stored in vectors x and y. Data points are entered by pressing a mouse button or any key on the keyboard except return key.

Assume a radius from. The index takes on the elemental values in the matrix expression. Usually, the ex- pression is something like m:n or m:i:n where m is the beginning value, n the ending value, and i is the increment.

Suppose we would like to find the squares of all the integers starting from 1 to Suppose we want to fill by matrix, b, with an element value equal to unity, the following statements can be used to perform the operation. The following program illustrates the use of a for loop.

Example 3. Use the values to plot x t versus y t. Table 3. These are shown in Table 3. However, if the logical expression is false, the statement group 1 is bypassed and the program control jumps to the statement that follows the end statement.

If the logical expression 2 is also true, the statement groups 1 and 2 will be executed before executing statement group 3. If logical expression 1 is false, we jump to statement group 4 without executing state- ment groups 1, 2 and 3. However, if logical expression 1 is false, statement group 2 is executed.

The general form of the if-elseif statement is if logical expression 1 statement group1 elseif logical expression 2 statement group2 elseif logical expression 3 statement group 3 elseif logical expression 4 statement group 4 end A statement group is executed provided the logical expression above it is true. For example, if logical expression 1 is true, then statement group 1 is executed. If logical expression 1 is false and logical expression 2 is true, then statement group 2 will be executed.

If logical expressions 1, 2 and 3 are false and logical expression 4 is true, then statement group 4 will be executed. If none of the logical expressions is true, then statement groups 1, 2, 3 and 4 will not be exe- cuted. Only three elseif statements are used in the above example. More elseif statements may be used if the application requires them. The one that is satisfied is exe- cuted. If the logical expressions 1, 2, 3 and 4 are false, then statement group 5 is executed.

Test the program by using an analog signal with the following amplitudes: At the end of exe- cuting the statement group 1, the expression 1 is retested. If expression 1 is still true, the statement group 1 is again executed. However, if expression 1 is false, the program exits the while loop and executes statement group 2. The following example illustrates the use of the while loop. Brief descriptions of these commands are shown in Table 3. Press- ing any key cause resumption of program execution.

Break The break command may be used to terminate the execution of for and while loops. The break command is useful in exiting a loop when an error condition is detected. Disp The disp command displays a matrix without printing its name. It can also be used to display a text string. Another way of displaying matrix x is to type its name.

The echo command allows commands to be viewed as they execute. The echo can be enabled or disabled. Format The format controls the format of an output. Format compact suppresses line-feeds that appear between matrix dis- plays, thus allowing more lines of information to be seen on the screen.

Format compact and format loose do not affect the numeric format. The format for printing the matrix can be specified, and line feed can also be specified. The user can then type an expression such as [10 15 30 25]; The variable r will be assigned a vector [10 15 30 25]. If the user strikes the return key, without entering an input, an empty matrix will be assigned to r.

When the word keyboard is placed in an m-file, execution of the m-file stops when the word keyboard is encountered. The keyboard command may be used to examine or change a vari- able or may be used as a tool for debugging m-files. The execution of the m- file resumes upon pressing any key.

The general forms of the pause command are pause pause n Pause stops the execution of m-files until a key is pressed. Pause n stops the execution of m-files for n seconds before continuing. The pause command can be used to stop m-files temporarily when plotting commands are encountered during program execution. If pause is not used, the graphics are momentarily visible. Print out the sum and the number of terms needed to just exceed the sum of 1.

Print out the results. In nodal analysis, if there are n nodes in a circuit, and we select a reference node, the other nodes can be numbered from V1 through Vn With one node selected as the refer- ence node, there will be n-1 independent equations. Equation 4. Example 4. The technique uses Kir- choff voltage law KVL to write a set of independent simultaneous equations.

The Kirchoff voltage law states that the algebraic sum of all the voltages around any closed path in a circuit equals zero. In loop analysis, we want to obtain current from a set of simultaneous equa- tions. The latter equations are easily set up if the circuit can be drawn in pla- nar fashion. This implies that a set of simultaneous equations can be obtained if the circuit can be redrawn without crossovers. For a planar circuit with n-meshes, the KVL can be used to write equations for each mesh that does not contain a dependent or independent current source.

In are the unknown currents for meshes 1 through n. Z11, Z22, …, Znn are the impedance for each mesh through which indi- vidual current flows. Zij, j i denote mutual impedance. In addition, find the power supplied by the volt voltage source. The circuit is shown in Figure 4. On the other hand, when R L approaches infinity, the voltage across the load is maximum, but the power dissipation is zero. MATLAB can be used to observe the voltage across and power dissipation of the load as functions of load resistance value.

Ex- ample 4. Before presenting an example on the maximum power transfer theorem, let us discuss the MATLAB functions diff and find. The find function determines the indices of the nonzero elements of a vector or matrix.

The diff and find are used in the following example to find the value of resis- tance at which the maximum power transfer occurs. Figure 4. Gottling, J. Johnson, D. Dorf, R. Plot the power dissipation with respect to the variation in RL. What is the maximum power dissipated by R L? What is the value of R L needed for maximum power dissipation?

What is the power supplied by the source? R C Vo t Figure 5. To obtain the voltage across a charging capacitor, let us consider Figure 5. Example 5. Figure 5. The plots should start from zero seconds and end at 1. After the 1 s delay, the switch moved from position b to position c, where it remained indefinitely. Sketch the current flowing through the inductor versus time.

Table 5. From the RLC circuit, we write differential equations by using network analysis tools. The differential equations are converted into algebraic equations using the Laplace transform. The unknown current or voltage is then solved in the s-domain. By using an inverse Laplace transform, the solution can be expressed in the time domain. We will illustrate this method using Example 5. The later method i can be used to analyze and synthesize control systems, ii can be applied to time-varying and nonlinear systems, iii is suitable for digital and computer solution and iv can be used to develop the general system characteristics.

A state of a system is a minimal set of variables chosen such that if their values are known at the time t, and all inputs are known for times greater than t 1 , one can calculate the output of the system for times greater than t 1. This suggests the following guidelines for the selection of acceptable state variables for RLC circuits: 1. Currents associated with inductors are state variables. Voltages associated with capacitors are state variables. Currents or voltages associated with resistors do not specify independent state variables.

When closed loops of capacitors or junctions of inductors exist in a circuit, the state variables chosen according to rules 1 and 2 are not independent. Consider the circuit shown in Figure 5. These are described in the following section.

The ode23 function integrates a system of ordinary differential equations using second- and third-order Runge- Kutta formulas; the ode45 function uses fourth- and fifth-order Runge-Kutta integration equations. The function must have 2 input arguments, scalar t time and column vector x state and the.

It specifies the desired accuracy of the solution. Solution From Equation 5. From the two plots, we can see that the two results are identical. Compare the numerical solution to the analytical solution obtained from Example 5. Solution From Example 5. Solution Using the element values and Equations 5. Nilsson, J.

Vlach, J. Meader, D. The resistance values are in ohms. The initial energy in the storage elements is zero. Assume that the initial voltage across each capacitor is zero. Numerical integration is used to obtain the rms value, average power and quadrature power.

Three-phase circuits are analyzed by converting the circuits into the frequency domain and by using the Kirchoff voltage and current laws. The un- known voltages and currents are solved using matrix techniques. Given a network function or transfer function, MATLAB has functions that can be used to i obtain the poles and zeros, ii perform partial fraction expan- sion, and iii evaluate the transfer function at specific frequencies.

The quad8 function uses an adaptive, recursive Newton Cutes 8 panel rule. The iteration continues until the rela- tive error is less than tol. The default value is 1. If the trace is nonzero, a graph is plotted. The default value is zero. Example 6. Determine the average power, rms value of v t and the power factor using a analytical solution and b numerical so- lution. This normally involves solving differential equations. By transforming the differen- tial equations into algebraic equations using phasors or complex frequency representation, the analysis can be simplified.

Network analysis laws, theorems, and rules are used to solve for unknown currents and voltages in the frequency domain. The solution is then converted into the time domain using inverse phasor transfor- mation.

For example, Figure 6. Solution Using nodal analysis, we obtain the following equations. The resulting circuit is shown in Figure 6. The impedances are in ohms. The basic structure of a three-phase system consists of a three-phase voltage source connected to a three-phase load through transformers and transmission lines. The three-phase voltage source can be wye- or delta-connected. Also the three-phase load can be delta- or wye-connected.

Figure 6. The method of symmetrical components can be used to ana- lyze unbalanced three-phase systems. This is illustrated by the following ex- ample. Its complex frequency representation is also shown. From equation 6. The gen- eral form of polyval is polyval p, x 6.

It is repeated here. As the resistance is decreased from 10, to Ohms, the bandwidth of the frequency response decreases and the quality factor of the circuit increases. Johnson, J. Compare your result with that obtained in part a. Plot the polynomial over the appropriate interval to verify the roots location. The describing equations for the various two-port network represen- tations are given.

Also, I 2 and V2 are output current and voltage, respectively. It is assumed that the linear two-port circuit contains no independent sources of energy and that the circuit is initially at rest no stored energy. Furthermore, any controlled sources within the lin- ear two-port circuit cannot depend on variables that are outside the circuit. The following exam- ple shows a technique for finding the z-parameters of a simple circuit.

Example 7. The following two exam- ples show how to obtain the y-parameters of simple circuits. Find its y- parameters. The h-parameters of a bipolar junction transistor are determined in the following example. The negative of I 2 is used to allow the current to enter the load at the receiving end. Examples 7. These are shown in Figure 7. Figure 7. Z1 Y2 Figure 7. The resistance values are in Ohms.

From Example 7. A termi- nated two-port network is shown in Figure 7. Z L is the load impedance. V2 b Obtain the expression for. Topics covered are Fou- rier series expansion, Fourier transform, discrete Fourier transform, and fast Fourier transform.

The term in 2 Equation 8. Equation 8. Figure 8. The coeffi- cient cn is related to the coefficients a n and bn of Equations 8. It provides information on the amplitude spectral compo- nents of g t. Example 8. If g t is continuous and non- periodic, then G f will be continuous and periodic. The periodicity of the time-domain signal forces the spectrum to be dis- crete. It is also the total number frequency sequence values in G[ k ].

T is the time interval between two consecutive samples of the input sequence g[ n]. F is the frequency interval between two consecutive samples of the output sequence G[ k ]. This means that T should be less than the reciprocal of 2 f H , where f H is the highest significant frequency component in the continuous time signal g t from which the sequence g[ n] was obtained. Several fast DFT algorithms require N to be an integer power of 2. A discrete-time function will have a periodic spectrum.

In DFT, both the time function and frequency functions are periodic. In general, if the time-sequence is real-valued, then the DFT will have real components which are even and imaginary components that are odd. Simi- larly, for an imaginary valued time sequence, the DFT values will have an odd real component and an even imaginary component. The FFT can be used to a obtain the power spectrum of a signal, b do digi- tal filtering, and c obtain the correlation between two signals.

The vector x is truncated or zeros are added to N, if necessary. The sampling interval is ts. Its default value is 1. The spectra are plotted versus the digital frequency F. Solution a From Equation 8. With the sampling interval being 0. The duration of g t is 0. The am- plitude of the noise and the sinusoidal signal can be changed to observe their effects on the spectrum. Math Works Inc. Using the FFT algorithm, generate and plot the frequency content of g t.

Assume a sampling rate of Hz. Find the power spectrum. Diode circuit analysis techniques will be discussed. The electronic symbol of a diode is shown in Figure 9. Ideally, the diode conducts current in one direction. The cur- rent versus voltage characteristics of an ideal diode are shown in Figure 9. The characteristic is divided into three regions: forward-biased, reversed- biased, and the breakdown. If we assume that the voltage across the diode is greater than 0.

The following example illustrates how to find n and I S from an experimental data. Example 9. Figure 9. The thermal voltage is directly propor- tional to temperature. This is expressed in Equation 9. The reverse satura- tion current I S increases approximately 7. T1 and T2 are two different temperatures. Assuming that the emission constant of the diode is 1. We want to determine the diode current I D and the diode volt- age VD. There are several approaches for solving I D and VD.

In one approach, Equations 9. This is illustrated by the following example. Assume a temperature of 25 oC. Then, from Equation 9. Using Equation 9. The iteration technique is particularly facilitated by using computers. It consists of an alternat- ing current ac source, a diode and a resistor. The battery charging circuit, explored in the following example, consists of a source connected to a battery through a resistor and a diode.

Use MATLAB a to sketch the input voltage, b to plot the current flowing through the diode, c to calculate the conduction angle of the diode, and d calculate the peak current. Assume that the diode is ideal. The output of the half-wave rectifier circuit of Figure 9. The smoothing circuit is shown in Figure 9. When the amplitude of the source voltage VS is greater than the output volt- age, the diode conducts and the capacitor is charged.

When the source voltage becomes less than the output voltage, the diode is cut-off and the capacitor discharges with the time constant CR. The output voltage and the diode cur- rent waveforms are shown in Figure 9. Therefore, the output waveform of Figure 9. When v S t is negative, diode D1 is cut-off but diode D2 conducts.

The current flowing through the load R enters it through node A. The current entering the load resistance R enters it through node A. The output voltage of a full-wave rectifier circuit can be smoothed by connect- ing a capacitor across the load.

The resulting circuit is shown in Figure 9. The output voltage and the current waveforms for the full-wave rectifier with RC filter are shown in Figure 9. The capacitor in Figure 9. Solution Peak-to-peak ripple voltage and dc output voltage can be calculated using Equations 9. I ZM is the maximum current that can flow through the zener without being destroyed. A zener diode shunt voltage regulator circuit is shown in Fig- ure 9. Con- versely, if R is constant and VS decreases, the current flowing through the zener will decrease since the breakdown voltage is nearly constant; the output voltage will remain almost constant with changes in the source voltage VS.

Now assuming the source voltage is held constant and the load resistance is decreased, then the current I L will increase and I Z will decrease. Con- versely, ifVS is held constant and the load resistance increases, the current through the load resistance I L will decrease and the zener current I Z will increase.

In the design of zener voltage regulator circuits, it is important that the zener diode remains in the breakdown region irrespective of the changes in the load or the source voltage. From condition 1 and Equation 9. I Z ,max 9. Solution Using Thevenin Theorem, Figure 9. In addition, when the source voltage is 35 V, the output voltage is The zener breakdown characteristics and the loadlines are shown in Figure 9.

Lexton, R. Shah, M. Angelo, Jr. Sedra, A. Beards, P. Savant, Jr. Ferris, C. Ghausi, M. Warner Jr. Assume a temperature of 25 oC, emission coef- ficient, n , of 1. Both intrinsic and extrinsic semicon- ductors are discussed. The characteristics of depletion and diffusion capaci- tance are explored through the use of example problems solved with MATLAB. The effect of doping concentration on the breakdown voltage of pn junctions is examined.

Electrons surround the nucleus in specific orbits. The electrons are negatively charged and the nucleus is positively charged. If an electron absorbs energy in the form of a photon , it moves to orbits further from the nucleus. An electron transition from a higher energy orbit to a lower energy orbit emits a photon for a direct band gap semiconductor. The energy levels of the outer electrons form energy bands.

In insulators, the lower energy band valence band is completely filled and the next energy band conduction band is completely empty.

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