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and so on, for eqs. (521) through (523). Next, d indicates the determinant value of a basic matrix; thus    z11 z12    dz det z   z11 z22 z12 z21  z21 z22     y11 y12    dy det y   y11 y22 y12 y21  y21 y22  and so on, in the same way, dh det h h11 h22 h12 h21 dg det g g11 g22 g12 g21 da det a a11 a22 a12 a21
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CHAPTER 11 Matrix Algebra. Networks
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The chart is useful because it allows us to nd the value of any parameter in terms of any other parameter. For example, suppose, in a certain case, that we wish to work in terms of, say, the z parameters, but are given only the values of, say, the h parameters. From the de nition of equal matrices in section 11.1, inspection of the chart then shows that values of the z parameters are calculated from corresponding values of the h parameters by means of the formulas z11 dh=h22 ; z12 h12 =h22 ; z21 h21 =h22 ; z22 1=h22
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You may be interested in learning how the various entries listed in the chart were arrived at. This was done by making use of the basic equations summarized in eqs. (519) through (523). For example, let s begin with, say, the basic eq. (521); thus ! ! ! V1 I1 h11 h12 h21 h22 V2 I2 which becomes, after multiplying both sides by the inverse matrix h 1 , I1 V2 ! h11 h21 h12 h22 ! 1 V1 I2 !
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and comparison of this last equation with eq. (522) shows that " # " # 1 g11 g12 h11 h12 g21 g22 h21 h22 g h 1
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Now nd the inverse of the 2 2 h-parameter matrix as indicated. (In connection with nding the inverse of the 2 2 matrix, you may wish to review problem 232 in section 11.3.) Doing this, you should nd that 3 2 h22 h12 ! 6 dh g11 g12 dh 7 7 525 g 6 4 h g21 g22 h11 5 21 dh dh Equation (525) veri es that the g-matrix g can be written in terms of the h-parameters in the form shown in the chart. From the condition required for equal matrices given in section 11.1, inspection of eq. (525) shows that h-parameters can be converted into equivalent g-parameters by means of the formulas g11 h22 =dh g12 h12 =dh g21 h21 =dh g22 h11 =dh
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Problem 247 Suppose the h-parameter values for an unknown network inside the box in Fig. 277 are found to be h11 850 ohms h21 26 h12 8 10 3 h22 4 10 4 mhos
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Using the matrix conversion chart, nd the equivalent values of the g parameters.
Matrix Algebra. Networks
Problem 248 Making use of the de ning eqs. (494) and (495), and (506) and (507), prove that the values of the z-parameters in terms of the h-parameters, as given in the rst row of the conversion chart, are correct. Problem 249 Convert the following g-parameter values into equivalent z-parameter values: g11 0:068 mhos g21 228 g12 0:073 g22 8755 ohms
Problem 250 Prove that h g 1 by making use of eqs. (508) and (511) and section 11.3. Problem 251 Convert the h-parameter values in problem 247 into equivalent z-parameter values.
Matrix Operations for Interconnected Two-Ports
In circuit design work it s often possible to consider a complex system as being composed of an interconnection of separate, individual two-ports. This approach can greatly simplify the work, because it is usually much easier to deal with each such building block individually, and then connect them together to form the whole, than it is to deal with the whole complex system as a single unit. There are ve basic ways of interconnecting individual two-ports to form a single equivalent two-port. These ve con gurations are known as the SERIES, the PARALLEL, the SERIES-PARALLEL, the PARALLEL-SERIES, and the CASCADE connections. We take up each of the ve modes in this section, beginning with the series connection. The parameter (z, y, h, g, or a) that will be used in any given case will depend upon the type of connection (series, parallel, and so on).
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