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RF Electronics Chapter 7: RF Filters Page 226 2022, C. J. Kikkert, James Cook University, ISBN 978-0-6486803-9-0. conversely the output impedance is determined by the coupling out of the last resonator. For LC coupled resonator filters, the resonators consist of parallel LC circuits. The coupling between LC resonators can be done by connecting capacitors between the resonators, this is called capacitive coupling. The coupling can also be done by connecting inductors between the resonators, this is called inductive coupling. Capacitive coupling is the cheapest as standard capacitors can be used, while the inductors often need to be wound individually. Equation 7.3 together with normalised impedance values from Filter tables, or equations 7.5 for Butterworth filters, give the required coupling factors (K) for the desired filter. The loading required to achieve the desired first and last resonator Q values, can be obtained from the Q values of the filter tables, from equation 7.3 or for Butterworth filters, equation 7.4. In many cases, filters with 50 input and output impedances are required, while the desired loaded Q values of the resonators require very different impedances. A capacitive impedance transformer can be used to obtain the required impedance transformations. Equations 7.3 and 7.4 only apply for lossless resonators. The K and Q filter tables in Zverev allow the filter to be designed to accommodate the finite Q values of the LC resonators, which result in the filter having an insertion loss. Table 1 shows the K and Q value table from Zverev for Butterworth filters. De-normalising the filter coefficients: c ij ij F BW k K BW F q Q c n n Eqn. 7.6 Zverev shows that the coupling capacitor is given by: �� �� �� � � � �� �� �� � � � � � Eqn. 7.7 where C i and C j are the resonator capacitors for the i th and j th resonator and K ij is the de- normalised coupling coefficient, k ij is the normalised coupling coefficient from the tables, F c is the centre frequency and BW is the filter bandwidth. Note the coupling capacitors are to be included in the total capacitance of the resonator so that for the resonator j the resonating capacitance is given by: � � ��� � �� � �� Eqn. 7.8 where C res is the total resonating capacitance as determined from the resonant frequency and inductance for the resonator. Using coupling inductors results in: �� � � � �� � � �� � � � � �� ��� � � � Eqn. 7.9 Where L i and L j are the resonator inductors. In a similar manner, the coupling inductors must be considered part of the inductance of the resonator, so that the total resonator inductance is made up the inductance of the resonator in parallel with the coupling inductors on either side. jk ij res j L L L L 1 1 1 1 Eqn. 7.10 where L res is the total resonating inductance as determined from the resonant frequency and capacitance for the resonator. To facilitate the design of these filters, Figure 7.29 shows the equations, which can be incorporated into a Cadence AWR DE project, to allow the component values for the filter shown in figure 7.30 to be calculated. RF Electronics: Design and Simulation 226 www.cadence.com/go/awr

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