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" which allows us to nd the value of Z3 . Equation (301) is thus the rst of the three " " equations we require; equations for nding the values of Z1 and Z2 can now be found as follows. " Substitute, into the left-hand side of eq. (297), the value of Z3 just found in eq. (301); "1 ; thus doing this gives us the value of Z " " ZA ZB " Z1 " 302 " " ZA ZB ZC

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" which is the second of the three equations we require. Next, substitute the value of Z3 , from eq. (301), into the left-hand side of eq. (299) to get " " ZB ZC " Z2 " 303 " " ZA ZB ZC which is the third and nal equation that we require. Thus eqs. (301), (302), and (303) are the three equations needed to convert a given pi network into an equivalent T network. Again, it must be noted that, if the three equations are used to calculate de nite values of R, L, and C at a speci c frequency, then the T network, thus found, is equivalent to the given pi network only at that speci c frequency (see discussion following eq. (284) in section 9.2). Now consider the opposite problem; that is, suppose we must convert a given T network into its equivalent pi network. In such a case it can be shown that the required three equations are as follows: " " " " " " Z Z Z1 Z3 Z2 Z3 " ZA 1 2 304 "2 Z " " " " " " Z Z Z1 Z3 Z2 Z3 " ZB 1 2 "3 Z " " " " "1 Z2 Z1 Z3 Z2 Z3 " Z " ZC " Z1 Problem 159 Have a try at proving that eqs. (304), (305), and (306) are correct. Problem 160 Given the pi network of Fig. 180, nd the R, L, and C values for the equivalent T network for operation at 500 kilohertz ( denotes ohms ). 305 306

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Problem 161 The T network of Fig. 181 is composed of pure reactances having the values as shown for the frequency of operation. Find the values of the reactances required for the equivalent pi network.

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CHAPTER 9 Impedance Transformation

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Impedance Transformation by T and Pi Networks

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In section 9.1 we studied the use of the L network as an impedance-transforming device. At the conclusion of that section we mentioned that it will often be necessary to use a T or a pi network in place of the simpler L network. This is especially true in certain applications where the suppression of harmonic frequencies* is important, such as in the output stage of a radio transmitter. With this in mind, let us now look into the possibility of using a T or a pi network as an impedance-changing device; this can be done with the aid of Fig. 182 for the T case.

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In this section we ll assume the actual load to be a pure resistance of R ohms, as shown in the gure. We will also assume that we wish to see a pure resistance of Rin ohms looking into the input terminals (1, 1) as shown in the gure. This requirement can be stated mathematically for Fig. 182 by noting that, looking to the right into terminals (1, 1), we " " " see that Z1 is in series with the combination of Z3 in parallel with Z2 R; thus, looking into terminals (1, 1) we have that " " Z R Z3 " Rin Z1 " 2 " Z2 Z3 R 307

" " Or, upon multiplying both sides of the above equation by (Z2 Z3 R), you should verify that " " " " " " " " " " RRin Rin Z2 Z3 Z1 Z2 Z1 Z3 Z2 Z3 R Z1 Z3