The comparative results among the analytical (MATLAB), numerical (ADS), and the experimental models were obtained under several measurement conditions. A model E1003C5 toroidal ferrite core from Sontag Componentes Eletronicos ( was used in the experiments. Its geometric and electromagnetic data include an external diameter of 10 mm, an internal diameter of 5 mm, a width of 3 mm, relative permeability (µr) of 11, and an inductance factor (Al) of 4.2 nH/turns2 .11 It is specified for use in the frequency range from 500 kHz and 50 MHz. A five twists per centimeter line with 30AWG-conductor transmission line was used. At 130 MHz, this line has a 38-ohm characteristic impedance, phase factor (β) of 4.5501 rad/m, and propagation velocity (vp) of 1.7952 X 108 m/s. The optimum value of the characteristic impedance for a 50-ohm source impedance must be 100 ohms, given in Eq. 8, meaning a 0.38 times relationship. The k value for this difference and a insertion loss of 3 dB is 0.2207.

The first device was fabricated with four turns, resulting in a 9-cm line length. Figures 8, 9, and 10 show the frequency insertion loss behaviors for analytical, numerical, and experimental cases.

The table summarizes the main values, including insertion-loss results at maximum amplitude, at –3 dB frequencies (fmax, fi-3dB, and fs-3dB), the proper bandwidth (BW), and the percent frequency differences compared to model values. The differences among the analytical, numerical, and experimental results are very small, except at the maximum signal frequency. This can be attributed to limitations in the test system caused by noise and other parasitic effects in the measurement setup. Across the test frequency band with essentially constant amplitude, variations in signal level are imperceptible, perhaps accounting for differences in reporting the maximum signal amplitude frequency.

A second device was constructed with six turns and 11-cm line length. With the increase in the number of turns, the lower cutoff frequency decreases and the higher cutoff frequency is also lowered as a result of the increased line length. For the lower cutoff frequency, the values of the analytical and numerical results were as expected. However, the experimental responses do not agree exactly with the theoretical model. The values of the high-frequency responses are as expected, with good agreement among the three sets of values. Figures 11, 12, and 13 show insertion loss as a function of frequency for the analytical, numerical, and experimental cases (shown also in the table).

There is a small difference between the analytical and numerical results, due to the inherent limitations of the model. On the other hand, the experimental results prove the validity of the model, except at the low-frequency limit, where the largest error occurred. This is due to the fact that the theoretical model does not account for all of the parasitic elements of the components used in the constructed transformers.

For a further comparison, a transformer with eight turns and 14-cm line length was constructed. The results of the analytical, numerical, and experimental cases are summarized in the table and shown in Figs. 14, 15, and 16.

At the lower cutoff frequency, the analytical and numerical cases agree but the experimental results do not agree with the theoretical model. However, the higher cutoff frequency values are close to either other and to the expected results. As the number of turns was increased, the lower cutoff frequency was reduced; similarly, as the line length increased, the upper cutoff frequency was also reduced.