In a conventional impedance transformer, the energy transference between the primary and secondary windings occurs mainly through magnetic coupling which also accounts for the transformer's capability of providing good low-frequency response. Assuming a lossless ferrite core and purely resistive load and source impedances, and considering only the influence of its magnetization inductance, the simplified low-frequency model for the transformer can be represented by the structure in Fig. 2.6,7 The low-frequency model response under maximum power transference condition is determined by the device's insertion loss:
Pg = the maximum available power of the source,
Pc = the power delivered to the load,
Rg = the source impedance, and
Xm = the magnetization reactance. This last parameter is determined by the operating frequency, f, and the core magnetization inductance, Lm, through:
The value of Lm depends on the number of turns for the primary winding and a core inductance factor, Al. Usually, this factor is specified by ferrite core manufacturers in nanoHenries/turns squared (nH/turns2). Therefore, the magnetization inductance in nH is:
By applying this parameter within the corresponding reactance formula, and substituting the result of that calculation into the insertion loss equation, the lower cutoff frequency of the transformer can be determined by putting the relationship in Eq. 2. Therefore:
This value decreases as the number of turns for the primary winding increases. This formula can also be applied for a given desired cutoff frequency in order to calculate the proper number of turns for the primary winding. The factor of 109 is used so that the inductance specification ca be presented in nH.
The electrical coupling between the primary and secondary windings of a transmission-line transformer improves the transfer of high-frequency energy. Figure 3 shows the high-frequency model for a transmission-line 1:4 unun transformer, considered without losses because of its short length.5 In this idealized model, the source and load impedances are assumed to be pure resistances. The high-frequency model response is quantified also by its insertion loss. Again, the ratio between the available source power and the secondary load power is:
Rg = the source impedance,
Rc = the load impedance,
Zo = the transmission-line characteristic impedance,
β / = the phase factor, and
l = kλ = the length of the transmission line with k a fraction of wavelengthλ.
Equation 5 shows the importance of an optimum Zo value to achieve good wideband high-frequency response. The power transference is nullified for a half-wavelength (λ/2) line length, and will be 1 dB less than the maximum value for a quarter-wavelength (λ/4) line length. From this is can be seen that shorter lengths of line result in broader-bandwidth high-frequency responses. The optimum line characteristic impedance and the load impedance for maximum power transmission are:
A 1:4 transformation is necessary between the source and load impedances for impedance matching. With this result, a relationship between line characteristic impedance and the source and load impedances can be established: