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" " Here again we study a form of symmetrical T network in which Z1 and Z2 are pure reactances of opposite sign where, using the terminology of Fig. 202, " " Z1 jXC j=!C and Z2 jXL j!L " and where the network is terminated in a pure resistance of ZL RL ohms, as shown in Fig. 210.
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Note that at very LOW frequencies the reactances of the series capacitors are very " HIGH and the reactance of the shunt inductor is very LOW, so that V2 is very LOW at such frequencies. On the other hand, at very HIGH frequencies the reactances of the series
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" capacitors are very LOW and the reactance of the shunt inductor is very HIGH, so that V2 "1 at such frequencies. Thus (in just a general way) we see that Fig. is nearly EQUAL to V 210 constitutes a high-pass type of lter. But now let us get down to speci c details. To do this, we begin by noting that the two equations following Fig. 208 apply equally well to Fig. 210; thus, if you will now substitute into the second equation following Fig. 208 the values " Z1 j=!C and " Z2 j!L " and ZL RL and then, after doing this, multiply the numerator and denominator by j=!LRL , you should nd that " V2  V1 1 1 j !LRL 2!2 LC  1  1 L 4!2 C 2 C  360
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Now let s pause and try to decide upon a reasonable value for RL . To do this, we note that at very HIGH frequencies Fig. 210 would, for all practical purposes, become as shown in Fig. 211.
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Thus, for very HIGH values of ! the generator would see a pure resistance of RL ohms, " " in which V2 =V1 1, with zero phase shift between V2 and V1 . This would be the desired condition here, because Fig. 210 is to be a HIGH-PASS lter. " " With this in mind, and upon setting Z1 j=!C and Z2 j!L in eq. (340), we have that the characteristic impedance of the T-network of Fig. 210 is equal to r L 1 " 2 2 Z0 C 4! C in which, as you ll note, the value of the term 1=4!2 C 2 decreases rapidly in value as ! " increases; thus, at the p preferred HIGH frequencies the value of Z0 becomes, for practical purposes, equal to L=C ohms. It thus makes sense to let r L " 361 Z0 RL C because this will cause Fig. 210 to become equal to the desired condition of Fig. 211 at high frequencies. Next, let us try to express eq. (360) in terms of a dimensionless ratio !=!0 , as we did for the case of the low-pass lter (eq. (353)). To do this, let us begin by noting that the imaginary term in the denominator of eq. (360) can be written as     j 1 j 1 1 L 1 !LCRL 4!2 C !C RL 4!2 LC   j 1 p 1 362 ! LC 4!2 LC
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because, by eq. (361), j 1 j !C RL ! " V2  V1 1 r C j p LC 2 ! LC 1
Thus, upon substituting the j term (eq. (362)) into eq. (360), we have the desired form 1 1 j p 2!2 LC ! LC   1 1 4!2 LC  363
The above form is especially useful because it can readily be expressed in terms of the ratio of any frequency ! to a xed reference frequency !0 . This can be done by de ning that the reference frequency be equal to 1 !0 p 2 LC 364
p Thus, LC 1=2!0 and LC 1=4!2 , and upon making these substitutions into eq. 0 (363), then making the substitution h !=!0
365
(that is, !0 =! 1=h , and then multiplying the numerator and denominator by h , you should nd that eq. (363) becomes " h3 V2 V1 h h2 2 j2 1 h2 366
" If, now, we wish to investigate only the manner in which the magnitude of V2 =V1 varies with frequency, then eq. (366) becomes   " V2  h3 h3   q p V  4 h6 4h2 1 h2 h2 2 2 4 1 h2 2 p or, since 1= X 1=X 1=2 X 1=2 , we can write the above in the form   " V2    h3 4 h6 4h2 1=2 V  1 or, in decibels (see notes 19 and 22 in the Appendix and eqs. (319), (320), and (321)), the above equation becomes dB 60 log h 10 log 4 h6 4h2 367
Using your calculator, you can verify that the following table of values is correct for eq. (367), in which we ve rounded o the dB values to two decimal places.
h 0.2 0.4 0.6 0.8 1.0 1.1 dB 47.78 29.15 17.47 8.13 0.00 2.79 h 1.2 1.3 1.4 1.5 1.6 1.7 dB 3.87 3.68 3.10 2.51 2.02 1.63 h 1.8 1.9 2.0 2.5 3.0 4.0 dB 1.33 1.09 0.90 0.39 0.20 0.06
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