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RF Electronics Chapter 7: RF Filters Page 242 2022, C. J. Kikkert, James Cook University, ISBN 978-0-6486803-9-0. For Butterworth filters, equation 7. 4 and 7.5 are: n Sin n Sin q q n 2 2 2 ) 1 2 ( 2 0 Eqn. 7.4 n i Sin n i Sin k ij 2 ) 1 2 ( 2 ) 1 2 ( 2 1 Eqn. 7.5 For a two-resonator filter n = 2, this gives q = 1.4142 and k = 0.7071 The reader is reminded that the small letters k and q denote the normalised values as would be obtained from table 7.1 and the capital letters K and Q are the de-normalised values, which relate to the normalised values by equations 7.6 and 7.7. Since S = 53 mm and f = 100, equation 7.18 gives the unloaded Q of the cavity as: Q 0 = 2363*0.053*10.0 = 1252. The normalised unloaded Q is q = 1259*1/100 = 12.52. From the filter table 7.1, the resulting filter will thus have close to 1 dB insertion loss. Since S = 53 mm, applying equation 7.15 gives the coil diameter as: d = 35 mm. From equation 7.17, the coil height is 53 mm and equation 7.19 indicates it requires 7.6 turns. The optimum wire diameter is thus 53/(2*7.6) = 3.49 mm. A one-eighth of an inch copper tube is 3.2 mm and is used for winding the coil. The minimum wire diameter of five times the skin depth (equation 7.20) is 0.033 mm. For practical windings, the minimum skin depth requirement is normally satisfied. From the filter tables the normalised values for Q and K are: q 1 = q n = 1.4142 and k 12 = 0.7071. De-normalising this gives Q 1 = Q n = 1.4142*100/1 = 141.42 and K 12 = 0.7071*1/100 = 0.007071. Q 1 and Q n are the loaded Q values, which are due to the input and output load impedance being coupled into the resonators and thus appearing as a load to the resonators. The doubly loaded input Q is defined as ½ Q 1 . This represents the total load seen by the resonator, if there is a very small insertion loss, as is the case here. The doubly loaded output Q is defined as ½ Q n . Q d is thus 70.71 for both the input and output. Putting this in the expression for R b /Z 0 gives: R b /Z 0 = 0.0105. Since we want the filter to be have 50 Ω input and output impedance, R tap = 50 Ω resulting in = 1.48 degrees. Since the coil wire length is 7.6*35*π = 842 mm and corresponds to a quarter wavelength at the centre frequency, or 90 degrees, the tapping point should be 1.48*842/90 = 13.9 mm from the bottom of the helix. In practice, the tapping point needs to be fine-tuned, to match the actual conditions after construction and include losses of the former used for winding the helix on. For this Butterworth filter, the tapping points for the input and output are the same. For Chebyshev or Bessel filters, or for lossy higher order Butterworth filters, the input and output tapping points will be different, and the above calculation needs to be performed for Q 1 and for Q n . The filter was constructed and performed exactly as expected. The measured bandwidth for the helical filter was 1 MHz and the insertion loss was 0.8 dB. A photograph of the filter is shown in figure 7.54 and the measured response is shown on figure 7.55. The tuning procedure for adjusting the tapping point for the input and output, and for verifying and if needed adjusting the coupling, is described on page 518 and figure 7.9.23 in Zverev [1]. This same procedure is used, to determine the coupling factors for interdigital filters as shown in figures 7.57, 7.58 and equations 7.29 and 7.30. RF Electronics: Design and Simulation 242 www.cadence.com/go/awr

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