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qr code vb.net library Matrix Algebra. Networks in Visual Studio .NET
Matrix Algebra. Networks Code 128 Code Set A Recognizer In .NET Framework Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Code128 Creator In Visual Studio .NET Using Barcode creation for .NET framework Control to generate, create Code 128 Code Set B image in Visual Studio .NET applications. Now try the following problems. As you may note, some of the problems would be as easy, or easier, to work without using matrices. But this is, of course, beside the point, as our object here is to provide practice in thinking in terms of matrices. Matrix algebra is a shorthand method of manipulating systems of simultaneous equations; its real value becomes evident in the analysis of complex networks represented by such equations. It allows us to study systems of interconnected blocks of elements without having to write out the mass of individual equations associated with the system. Digital computer programs for solving matrix equations are available, and are used to provide actual numerical answers if this is required. Problem 257 For eq. (545) show that I1 Z22 Z V1 Z12 Z V2 dz Z11 Z22 Z12 Z21 Z Code 128 Code Set C Recognizer In Visual Studio .NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in Visual Studio .NET applications. Bar Code Generator In VS .NET Using Barcode creator for .NET framework Control to generate, create barcode image in VS .NET applications. where dz Z11 Z22 Z12 Z21 Problem 258 Write eq. (545) in terms of the hparameters of the transistor. Problem 259 Can eq. (504), in section 11.6, be applied directly to eq. (543) Problem 260 Solve eq. (548) for the matrix Vo Io by taking the inverse of the coe cient matrix. (Note: the above will be easy if you take advantage of the special formula for nding the inverse of a 2 2 matrix given in the solution to problem 256.) Next, in the basic Fig. 277, suppose a load impedance of ZL ohms is connected to the output terminals, as in Fig. 307, and that the PROBLEM is to nd the value of the output load current IL . ! Read Bar Code In VS .NET Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in .NET framework applications. Create Code 128 Code Set C In Visual C#.NET Using Barcode generation for Visual Studio .NET Control to generate, create Code128 image in VS .NET applications. Fig. 307
Code 128C Creator In .NET Using Barcode maker for ASP.NET Control to generate, create ANSI/AIM Code 128 image in ASP.NET applications. ANSI/AIM Code 128 Generator In Visual Basic .NET Using Barcode generation for VS .NET Control to generate, create Code128 image in .NET framework applications. * At junction point a in Fig. 307, by Kirchho s current law, I2 IL 0; that is, I2 IL .
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GS1128 Generation In .NET Framework Using Barcode drawer for VS .NET Control to generate, create USS128 image in VS .NET applications. Create 2/5 Standard In VS .NET Using Barcode creation for .NET Control to generate, create 2/5 Industrial image in .NET applications. Problem 261 For the network of Fig. 307 inside the box, suppose it is found that h11 400 ohms h21 20 h12 0:100 h22 0:002 mhos DataMatrix Creator In Java Using Barcode creator for BIRT Control to generate, create Data Matrix 2d barcode image in Eclipse BIRT applications. Data Matrix Creator In .NET Using Barcode creator for ASP.NET Control to generate, create ECC200 image in ASP.NET applications. Given that V1 12 volts, nd the value of load current if ZL RL 150 ohms. (Answer: IL 0:2927 amps) Problem 262 Rework problem 261, this time beginning with the matrix equation (514) in section 11.6. Problem 263 Two identical transistors, operating in commonemitter mode, are connected in cascade as shown in Fig. 308. Paint ANSI/AIM Code 39 In None Using Barcode generation for Excel Control to generate, create Code39 image in Microsoft Excel applications. Draw Bar Code In None Using Barcode maker for Font Control to generate, create bar code image in Font applications. Fig. 308
Making GS1  12 In None Using Barcode drawer for Office Excel Control to generate, create UPC Symbol image in Microsoft Excel applications. DataMatrix Creator In None Using Barcode generation for Online Control to generate, create Data Matrix image in Online applications. In the gure, let it be given that the transistor hparameter values are h11 1000 ohms h21 40 h12 0:004 h22 0:0005 mhos Draw Bar Code In C# Using Barcode encoder for .NET Control to generate, create barcode image in VS .NET applications. Matrix 2D Barcode Printer In Visual Studio .NET Using Barcode drawer for ASP.NET Control to generate, create 2D Barcode image in ASP.NET applications. If it is given that R 500 ohms and RL 900 ohms, nd the output voltage VL if the input voltage V1 is 0.001 volt. (Again, as in problem 262, let us begin with eq. (514) in section 11.6.) (Answer: 0.3196 volts) Problem 264 Write the set of simultaneous eqs. (455), (456), and (457), in Chap. 10, in the form of a single matrix equation. Problem 265 Note that the answer to problem 264 says that 3 1 2 3 2 3 2 A A1 1 1 1 6 7 6 7 6 7 2 4 A2 5 4 a a 1 5 4 B 5 C a2 a 1 A0 As an exercise in matrix manipulation, verify that the above expression does produce eqs. (460), (463), and (466) in Chap. 10. Our nal example, which follows, will provide further practice in matrix manipulation and will also bring to light an interesting fact concerning power in unbalanced threephase systems. In doing this we ll freely make use of our previous work in threephase theory Matrix Algebra. Networks
Fig. 309
in sections 10.7 through 10.11. Let us begin with the threephase generator depicted in Fig. 309. In the gure, Va , Vb , and Vc represent the rms values of three unbalanced phase voltages, with Ia , Ib , and Ic representing the rms values of the three corresponding unbalanced phase currents (also the line currents here), as shown. In the work here we wish to concentrate our attention on the POWER produced in the unbalanced condition, ESPECIALLY in regard to expressing the power in terms of the SYMMETRICAL COMPONENTS of the unbalanced system. To begin, let PT denote the total true power produced by the above unbalanced generator. From inspection of the gure it s clear that PT is equal to the SUM OF THE POWERS produced by the three individual phases of the generator; thus (see note 29 in Appendix) in terms of the actual phase voltages and currents the value of PT is equal to the sum of the real parts (srp) in the expression " " " PT srp : Va Ia Vb Ib Vc Ic 549 " " " in which the overscore in Ia ; Ib ; Ic denotes the CONJUGATE of the quantity represented by the letter. (It s understood that the Vs and Is are, in general, complex numbers.) Note that eq. (549) is expressed in terms of the actual phase voltages and currents; but we, however, wish to express the power in terms of the SYMMETRICAL COMPONENTS of the phase voltages and currents. To do this, let us start by writing eq. (549) in matrix notation; thus 2 3 " Ia 6" 7 PT srp: Va Vb Vc 4 Ib 5 550 " Ic Let us now rst work on the above current matrix, as follows. From inspection of Fig. 309 we have Ia I1 I2 Io Ib aI1 a2 I2 Io Ic a2 I1 aI2 Io Now take the CONJUGATES of the above equations. Remembering that the conjugate of the sum of a number of complex numbers is the sum of the conjugates and that the

