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RF Electronics Chapter4: Transmission Line Transformers and Hybrids Page 82 2022, C. J. Kikkert, James Cook University, ISBN 978-0-6486803-9-0. For the analysis, consider figures 4.26 and 4.28. When an input is applied at port 1, port 4 is isolated and has no voltage at that port. Since lines TL3 and TL4 are quarter wavelength long, no current flows in line TL4 at port 3, so that TL4 can be disconnected from the circuit without any effect. Similarly, no current flows in TL3 at port 1, so that TL3 can also be disconnected, resulting in figure 4.28. If a normalised input power of 1 is applied at port 1 and we want an output power of P to occur at port 3, then a power of (1-P) will be available at port 2. At port 2, the load impedance is Z 0 , which typically is 50 . If the voltage at port 2 is V2, then the power split at port 2 is given by: 0 2 2 ) 1 ( Z P PZ V P so that: P Z P Z P 0 ) 1 ( Eqn. 4.15 Where Z P is the impedance of port 3 transformed through TL3 and seen at port 2 looking into TL1. Since this is obtained by impedance transformation through the quarter wave long transmission line TL1, the impedance required for TL1 is given by: � � � � � � � � ����� � Eqn. 4.16 The impedance Z e seen at the port 2 end of TL1 is Z 0 in parallel with Z P and is thus: � � � � � � � � � � � � � ������ � � � � � � � ������ � Eqn. 4.17 The line impedance Z 1 to transform this to Z 0 is thus: � � � √ ��� Eqn. 4.18 In many cases an equal power split is required, so that P=0.5. Typically Z 0 = 50 . Substituting this in equations 4.16 and 4.18 results in Z 2 = Z 0 = 50 and Z 1 = Z 0 /2 = 35.36 . For a 10 dB coupler P = 0.1, so that Z 2 = 3Z 0 = 150 and Z 1 = 0.949Z 0 = 47.43 . These can just be made using Microstrip circuits, however 150 results in a very thin track and any lower amount of coupling is extremely difficult to make. Figure 4.29. Ideal Branchline coupler amplitude response. RF Electronics: Design and Simulation 82 www.cadence.com/go/awr

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