Twisted transmission line
The use of twisted transmission line in a transformer makes it possible to set the characteristic impedance almost at an optimum value for a desired passband by varying the number of twists per unit length of transmission line. An increase in the number of twists per unit length results in a decrease in the characteristic impedance.
Figure 4 plots insertion loss behavior as a function of k for optimized and non-optimized values of characteristic impedance. For a case with non-optimized characteristic impedance, the insertion loss increases and bandwidth decreases relative to a case using optimized characteristic impedance. Thus, the use of twisted transmission line is readily justified for obtaining optimum values of characteristic impedance.8,9
As a comparison, simulated performance was predicted using the Advanced Design System (ADS®) computer-aided-engineering (CAE) software suite from Agilent Technologies (www.agilent.com) while design prototypes were measured with a commercial microwave vector network analyzer (VNA). The analysis indicated the relationship between the load and source power.
In order to determine a transformer's low-frequency response, the characteristics of the ferrite core must be known since the inductance factor Al is determined relative to a specific frequency. Knowing this as well as the source's internal impedance (Rg), a designer can establish the low-frequency cutoff frequency (fi) and, using Eq. 4, can calculate the required number of turns (Np) for the primary winding. To determine the high-frequency response, information about the transmission line is required, such as its characteristic impedance (Zo), the propagation velocity (vp), and the phase factor (β), all at the desired operating frequency. Along with the values of the source impedance (Rg) and the load impedance (Rc), the optimum theoretical value of the characteristic impedance (Zopt) can be determined by applying Eq. 6. Knowing the characteristics of the transmission line, the high-frequency cutoff frequency (fs), and the true characteristic impedance of the transmission line, Zo, it is possible to calculate the propagation velocity (vp) and the phase factor (β). With the value of the true characteristic impedance, Zo, the difference between it and Zopt can be verified and the definitive insertion loss for fs be specified. Figure 4 shows the determination of values of k as a function of true characteristic impedance (Zo) and insertion loss. With values for k, vp, and fs , the line length (l) required for achieving the previous specifications can be calculated through:
MATLAB mathematical analysis software from The MathWorks (www.mathworks.com) was used to analyze the response of the transformer device model.10 In this analysis, the insertion-loss responses for the individual low-frequency (Eq. 1) responses and high-frequency (Eq. 5) responses were combined. The desired target values were substituted into the MATLAB equations to obtain the final response of the wideband transformer. To achieve an electrical simulation of the MATLAB model's numerical response, the ADS modeling software was used. The software features a useful internal model for the source, called XFERRUTH, with variable parameters that include the number of turns (N), the inductance factor (AL), the characteristic impedance of the line (Z), the electrical length of the transmission line (E), and the reference frequency (F) for the calculation of the transmission-line length.
In order for ADS to performance a device simulation on the transformer with the responses as scattering parameters (S-parameters), it employs its S_Param modeler, adjusting the initial (start) and the final (stop) sweep frequencies according to a specified step size. and the graduation step size. The source and load impedances are represented by a specific termination called Term, with an impedance value of Z. Figure 5 shows the circuit that was used in the ADS simulation.
Measurements were performed on a commercial VNA from Advantest (www.advantest.com), a 300-kHz-to-3.8GHz model R3765CG. The analyzer features unbalanced test ports with terminal impedance of 50 ohms. Since the wideband unun impedance transformer has unbalanced terminals and a transformation ratio of 1:4, another device with 4:1 transformation ratio was needed to perform the impedance conversion needed to adapt the device to the measurement equipment. Figures 6 and 7 show all the terminals connections. Both test terminals and all cables used with the VNA were calibrated in order to minimize their error contributions. The insertion loss and the passband response were analyzed by means of transmission coefficient S21 measurements presented in log-magnitude form.