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RF Electronics Chapter 3: Transformers and Hybrids Page 52 2022, C. J. Kikkert, James Cook University, ISBN 978-0-6486803-9-0. At low frequency, the Magnetising inductance Lm shunts the output load. The lower cut off frequency is thus when jL M = R L . At high frequencies, the Leakage inductance is in series with the output. The upper cut off frequency is thus when jL S = R L . For the model of figure 3.3, the leakage inductance can most easily be measured by short- circuiting the secondary winding of the transformer. Under those conditions the input impedance of the transformer is R s + jL S . When the transformer is open circuited, the input impedance is R s + jL S + jL M . Since L m is typically 1000 times larger than L s , the leakage inductance can be ignored as part of the lower frequency calculation. This simple model does not include inter-winding and self-capacitances, but for many applications, it is accurate enough as R s often governs the upper cut-off frequency (F high ). If one normally wants to operate the transformer at a frequency F c then it is desirable to make this high low c F F F where F low and F high are the upper and lower cut off frequencies of the transformer. As described above, the upper and lower cut off frequencies are determined by the leakage and magnetising inductance respectively. The input impedance of the transformer at the centre frequency is thus: � � � �� �� � � � � Eqn. 3.2 For a 50 Ω system, one wants this Z c to be 50 Ω. Under those conditions R s + jL S is much smaller than R L and jL M is much larger than R L , so that virtually all the input power is transferred to the output. Since the inductance is proportional to the number of turns squared, the characteristic impedance of the transformer is also proportional to the number of turns squared. The number of turns chosen for the transformer winding is such that the characteristic impedance of the transformer matches the system impedance. Since in many cases the detailed properties of the ferrite may not be known, the characteristic impedance is most easily determined by winding a trial-winding on the transformer and using the measured open and short circuited impedances to then calculate the correct number of turns required. In practice, the magnetising inductance has some losses associated with it and the resistive losses of the windings are in series with the leakage inductance. The resistive losses are normally very small. Since the inductance is proportional to the frequency, at the upper and lower frequency limits the inductance dominates, so that in the calculation for the upper and lower frequency limits, the resistive part of the measured input impedance can be ignored. Example 3.1: RF Transformer Design An RF isolation transformer operating at 1 MHz and a 50 Ω impedance is required. To determine the number of turns required on a ferrite toroid, wind a bifilar trial winding on it. At the desired centre frequency, determine the leakage inductance of the transformer by measuring the input impedance with the secondary winding short-circuited. Then determine the magnetising inductance by measuring the input impedance with the secondary winding open-circuited. For an 11 turn bifilar trial winding, at 1 MHz, the short circuit impedance is Z Ls = j0.4 , corresponding to a leakage inductance of 64 nH. The open circuit input impedance is Z Lm = j400 , corresponding to a magnetising inductance of 64 H. From equation 3.2, the characteristic impedance is thus Z c = (400 x 0.4) = 12.65 . RF Electronics: Design and Simulation 52 www.cadence.com/go/awr

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