CHAPTER 9 Impedance Transformation

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the input impedance seen looking into terminals (1, 1) in Fig. 202 is " " "2 Z1 ZL Z " Z 2 " Zin 1 338 " 2 " "2 Z1 ZL Z 2 " " Now suppose that Z1 and Z2 are chosen to have the particular values that will make the INPUT IMPEDANCE EQUAL TO THE LOAD IMPEDANCE. To nd the required " " " " values of Z1 and Z2 needed to accomplish this, all we need do is replace Zin with ZL in eq. " (338) and then solve for ZL . In doing this, however, it has become customary to replace " " " " " " both Zin and ZL with the single symbol Z0 ; thus, setting Zin ZL Z0 in eq. (338) we have that " " " Z Z2 1 Z0 " Z 2 " Z0 1 339 " 2 Z " " Z2 1 Z0 2 " " " Thus, for the SPECIAL CONDITION represented by eq. (339), Zin ZL Z0 , Fig. 202 becomes Fig. 203.

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Having the above condition (input impedance equal to load impedance) has important advantages that we ll point out later on. To be able to produce this condition, however, we " " " must nd the relationship that must exist between Z0 and the two impedances Z1 and Z2 , "0 , as follows. which can be done by solving eq. (339) for Z First, multiplying both sides of eq. (339) by the denominator of the right-hand fraction puts the equation in the form Z1 Z1 Z1 Z1 Z0 Z2 Z0 Z2 Z0 Z0 Z2 2 2 2 2

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CHAPTER 9 Impedance Transformation

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Now multiplying as indicated, then collecting like terms, you should nd that " Z2 " " "2 Z0 Z1 Z2 1 4 thus we have " Z0 q "2 " " Z1 Z2 Z1 =4 340

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" The impedance Z0 given by eq. (340) is called the characteristic impedance of a " symmetrical T network. Or, if such a network already exists, the value of Z0 can be found experimentally by making open circuit and short circuit measurements, as indicated in Figs. 204 and 205.

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" In Fig. 204, Zp impedance looking into (1, 1) with (2, 2) OPEN-CIRCUITED, " whereas in Fig. 205, Zs impedance looking into (1, 1) with (2, 2) SHORT-CIRCUITED. By inspection, note that " " " Zp Z1 =2 Z2 and " " " " " Z Z Z Z =2 Z Z " Zs 1 " 1 2 " 1 " 1 2 " 2 2 Z1 =2 Z2 Z1 2Z2 " " Now, using these values of Zp and Zs , a careful multiplication will show that " Z2 " " " " Zp Zs 1 Z1 Z2 4 hence, upon making use of the relationship given just prior to eq. (340), we have that q " " " 341 Z0 Zp Zs Equation (341) is important because it provides a way to nd, by actual laboratory measurement, the characteristic impedance of a network known to be of the symmetrical T form. In using eqs. (340) and (341) we must remember that the addition and subtraction of complex numbers can be performed only in the rectangular form. On the other hand, to raise a complex number to a fractional power the number must be expressed in the trigonometric, polar, or exponential form (sections 6.6 and 6.7). Problem 174 Find the characteristic impedance of a low-pass symmetrical T network in which " " Z1 2 j5 and Z2 4 j3: Answer: 4:68 j2:03 ohms; approx: