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CHAPTER 11 Matrix Algebra. Networks
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NETWORK for the unknown contents of the box, GIVEN only the four measurable external quantities shown in Fig. 277. To do this, we must rst write suitable equations for the contents of the box, in terms of the four measurable external quantities shown in Fig. 277. This can be done as follows. In Fig. 277 we can, for example, choose any two of the four external quantities to be independent variables, leaving the other two as dependent variables (two independent variables will require two simultaneous equations). For example, if we choose I1 and I2 to be the independent variables, the two equations will be of the form V1 z11 I1 z12 I2 V2 z21 I1 z22 I2 494 495
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where the zs are constant coe cients whose values are to be found by actual measurement on any given two-port. Inspection shows that the z coe cients are impedances, and so are measured in ohms. For this reason, eqs. (494) and (495) are said to represent the IMPEDANCE form of an equivalent circuit for the contents of the box in Fig. 277. In matrix form the equations become ! ! ! z11 z12 I1 V1 496 V2 z21 z22 I2 Note that the double subscripts used with the constant z coe cients, are purposely written, in the two equations (494) and (495), so that they give the proper location, row and column, in the matrix of eq. (496). The above matrix equation is often written in the abbreviated form V z I 497
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or even more simply as V zI, if it is understood in the discussions that V and I represent the column matrices of eq. (496), and where ! z11 z12 z z z21 z22 this being the impedance matrix for the equivalent network. Now let s pause here, brie y, to discuss how the values of the foregoing z coe cients can be found experimentally in the laboratory. Remember that we do not necessarily know what is actually inside the box in Fig. 277; all we can do, in the laboratory, is measure the EXTERNAL values of V1 , V2 , I1 , and I2 . Take, for example, the coe cient z11 . To nd the value of z11 for a given two-port, we apply a known voltage V1 to terminals 1; 1 in Fig. 277, and measure the value of current I1 with the output terminals 2; 2 open-circuited. For this condition I2 0, and therefore, setting I2 0 in eq. (494), we have z11 V1 I1 I2 0 498
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which de nes the experimental procedure for nding the value of z11 . Next suppose we wish to nd, by experimental means, the value of, say, z12 . To do this we can apply any convenient voltage to terminals 2; 2 in Fig. 277 and measure the value of the current I2 with terminals 1; 1 open-circuited. This condition makes I1 0 and,
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putting I1 0 in eq. (494), we have that z12
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Matrix Algebra. Networks
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V1 I2 I1 0
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499
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which thus provides an experimental procedure for nding the value of z12 . In the same manner, making use of eq. (495), expressions can be found that will allow us to nd the values of z21 and z22 by experiment in the laboratory. In regard to the above, we must remember that, except for purely resistive circuits, impedance is a complex number of the form z r jx, and likewise the resulting currents will be complex numbers of the form I Ia jIb . For this reason, experimental determination of the true values of the coe cients requires precision measuring equipment and adequate skill and understanding on the part of the experimenter. Next recall that admittance, which is denoted by y, is the reciprocal of impedance, y 1=z. Thus I V=z yV, and the equations describing the behavior of the unknown network inside the box can also be written in the ADMITTANCE form, thus I1 y11 V1 y12 V2 I2 y21 V1 y22 V2 or, in matrix form, I1 I2 or, more simply, I y V where y y11 y21 y12 y22 ! 503 500 501
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