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CHAPTER 9 Impedance Transformation
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Fig. 174. T network.
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Fig. 175. Pi (p) network.
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The values of the three unknown impedances in Fig. 174 have to be found by making either impedance calculations or actual measurements at the terminals of the actual network it is desired to replace. Since we must nd the values of THREE unknown impedances, we must make at least three di erent, independent measurements on the actual network. The most convenient measurements to make are called the OPEN-CIRCUIT measurements and the SHORT-CIRCUIT measurements these measurements are made on the actual network, as follows. " First, disconnect the generator and the load impedance ZL from the actual network. Then make the following measurements or calculations at the terminals of the actual network. " Z1O impedance looking into terminals 1,1 with terminals 2,2 OPEN-CIRCUITED. " Z1S impedance looking into terminals 1,1 with terminals 2,2 SHORT-CIRCUITED. " Z2O impedance looking into terminals 2,2 with terminals 1,1 OPEN-CIRCUITED. " Z2S impedance looking into terminals 2,2 with terminals 1,1 SHORT-CIRCUITED.
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The relationship of the above measurements (which, remember, are to be made on the actual network) to the values of the three elements of the hypothetical equivalent T network can be found with the aid of Fig. 176.
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Fig. 176
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Referring to Fig. 176, note that operation of the switches will provide us with the following information: Switch #1 open open open closed Switch #2 open closed open open " " " Z1O Z1 Z3 " " Z Z " " Z1S Z1 " 2 3 " Z2 Z3 " " " Z2O Z2 Z3 " " Z Z " " Z2S Z2 " 1 3 " Z1 Z3 278 279 280 281
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CHAPTER 9 Impedance Transformation
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" " Z Z " " " Now subtract eq. (279) from (278): Z1O Z1S Z3 " 2 3 . Now multiply both " Z2 Z3 " " sides of this equation by Z2 Z3 ), then make use of eq. (280); this should give you " Z3 q " " " Z2O Z1O Z1S
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in which, as is shown, it s customary to use just the positive value of the square root. Thus, making use of this equation and also eqs. (280) and (278), we get the following relationships " Z3 q " " " Z2O Z1O Z1S 282 283 284
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" " " Z2 Z2O Z3 " " " Z1 Z1O Z3
These are the three equations required to convert any actual four-terminal network into " " " an EQUIVALENT T NETWORK. Note that the values of Z1O , Z1S , and Z2O are found by making three calculations or measurements at the terminals of the ACTUAL NETWORK. It should be understood that if the three equations are used to calculate de nite values of R, L, and C at a given frequency, then the T network, so found, is equivalent to the actual network only at that one frequency. Practically speaking, however, such an equivalent network can satisfactorily replace the actual network over a band of frequencies usually extending a few percent above and below the center frequency. Now let s turn our attention to the equivalent pi network of Fig. 175. The equations for nding the values of an equivalent pi network are found in the same general way as those for the equivalent T network, and are as follows: " Z0 " ZA " " Z2O Z 00 " " " ZB Z 0 =Z 00 " Z0 " ZC " " Z1O Z 00 where Z prime and Z double prime have the following values: " " " Z 0 Z2O Z1S q " " " " Z 00 Z2O Z1O Z1S 288 289
285 286 287
" " where Z1O through Z2S are found in the same way as for the equivalent T, as de ned just prior to Fig. 176. In this regard, let us now replace the T network in Fig. 176 with the pi network of Fig. 175. Upon doing this, operation of the two switches will now produce the
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