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Impedance matching devices are commonly used in high frequency circuits, typically to match the impedance of a device or component to the characteristic impedance of a circuit or system. In some circuits, it is desirable that impedance matching achieve frequency coverage over several octaves accompanied by low insertion loss. To help designers work with impedance transformations, this article examines the design of wideband unbalanced-to-unbalanced (unun) impedance transformers with an impedance ratio of 1:4. Such transformers are useful in radio communication systems, typically in hybrid circuits, signal combiners and splitters, and for interstage coupling of amplifier chains.

Such broadband impedance transformers are also useful for test circuits, optical receiver systems,^{1}Microwave circuits with broadband impedance matching,^{2}and antenna coupling.^{3}Modern computer programs that can be used for the design and simulation of high-frequency circuits contain this device in their toolboxes.^{4}A wide band unequal impedance transformer consists of a toroidal ferrite core wrapped with a twisted bifilar transmission line, with the wires insulated by enamel foil. By combining design elements from conventional and transmission line impedance transformers it is possible to create a true broadband component. The design offers high efficiency with an impedance transformation ratio of 1:4 (Abb. 1).^{5}

## Conventional

In a conventional impedance transformer, the energy transfer between the primary and secondary windings occurs primarily through magnetic coupling, which is also responsible for the transformer's ability to provide good low-frequency response. Assuming a lossless ferrite core and purely resistive load and source impedances, and considering only the influence of its magnetizing inductance, the simplified low-frequency model for the transformer can be represented by the structure inAbb. 2.^{6,7}

The low-frequency model response at maximum power transfer is determined by the insertion loss of the device:

Wo:

P_{G}= the maximum available power of the source,

P_{C}= the power delivered to the load,

R_{G}= the source impedance and

X_{M}= the magnetizing reactance. This last parameter is determined by the operating frequency f and the core magnetizing inductance L_{M}, through:

The value of L_{M}depends on the number of turns of the primary winding and a core inductance factor A_{l}. This factor is usually specified by ferrite core manufacturers in nanoHenries/turns squared (nH/turns).^{2}). Therefore the magnetizing inductance in nH is:

By applying this parameter in the appropriate reactance formula and substituting the result of this calculation into the insertion loss equation, the lower limit frequency of the transformer can be determined by using the relationship in Eq. 2. Therefore:

This value decreases as the number of turns in the primary winding increases. This formula can also be used, given a desired cutoff frequency, to calculate the correct number of turns for the primary winding. The factor 10^{9}used so that the inductance specification can be displayed in nH.

The electrical coupling between the primary and secondary windings of a transmission line transformer improves the transmission of radio frequency energy.Figure 3shows the high frequency model for a transmission line 1:4 Unun transformer, which is considered lossless 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 also quantified by its insertion loss. Again, the ratio between available source power and secondary load power is:

Wo:

R_{G}= the source impedance,

R_{C}= the load impedance,

Z_{Ö}= the characteristic impedance of the transmission line,

β / = the phase factor and*l*= kλ = the length of the transmission line, where k is a fraction of the wavelength λ.

Equation 5 shows the importance of an optimal Z_{Ö}value to achieve good broadband high frequency response. The power transfer is negated for a half-wavelength (λ/2) line length and is 1 dB less than the maximum value for a quarter-wavelength (λ/4) line length. From this it can be seen that shorter line lengths lead to broader bandwidth high-frequency responses. The optimal line characteristic impedance and load impedance for maximum power transfer are:

A 1:4 transformation between the source and load impedances is required for impedance matching. With this result, a connection can be established between the line characteristic impedance and the source and load impedance:

## Twisted transmission line

The use of a twisted pair transmission line in a transformer makes it possible to adjust the characteristic impedance to near an optimal value for a desired passband by varying the number of twists per unit length of the transmission line. Increasing the number of twists per unit length leads to a decrease in the characteristic impedance.

Figure 4* *Figure 12 illustrates the behavior of insertion loss as a function of k for optimized and non-optimized values of the characteristic impedance. In a non-optimized characteristic impedance case, the insertion loss increases and bandwidth decreases relative to an optimized characteristic impedance case.

Thus, the use of a twisted pair transmission line is easily justified in order to obtain optimal values of the characteristic impedance.^{8,9}

For comparison, simulated performance was predicted using Agilent Technologies' Advanced Design System (ADS®) computer-aided engineering (CAE) software suite (www.agilent.com), while design prototypes were measured with a commercial microwave vector network analyzer (VNA). The analysis showed the relationship between the load and the source power.

In order to determine the low frequency response of a transformer, the properties of the ferrite core must be known, since the inductance factor A_{l}is determined relative to a specific frequency. Knowing this as well as the internal impedance of the source (R_{G}), a designer can use the low-frequency cut-off frequency (f_{I}) and using Eq. 4, can calculate the required number of turns (N_{P}) for the primary winding. To determine the high frequency response, information about the transmission line is needed, such as: B. their characteristic impedance (Z_{Ö}), the speed of propagation (v_{P}) and the phase factor (β), all at the desired operating frequency. Together with the values of the source impedance (R_{G}) and the load impedance (R_{C}), the optimal theoretical value of the wave resistance (Z_{opt}) can be obtained by applying Eq. 6. Know the characteristics of the transmission line, the high-frequency cut-off frequency (f_{S}) and the true characteristic impedance of the transmission line, Z_{Ö}, the propagation speed (v_{P}) and the phase factor (β). With the value of the true wave impedance Z_{Ö}, the difference between it and Z_{opt}can be verified and the final insertion loss for f_{S}be specified. Figure 4 shows the determination of the values of k as a function of the true wave impedance (Z_{Ö}) and insertion loss. With values for k, v_{P}, and f_{S}, the line length (*l*) required to achieve the previous specifications can be calculated by:

MATLAB mathematical analysis software by 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 responses (Eq. 1) and high frequency responses (Eq. 5) were combined. The desired target values were plugged into the MATLAB equations to obtain the final broadband transformer response. To achieve an electrical simulation of the numerical response of the MATLAB model, the ADS modeling software was used. The software has a useful internal model for the source called XFERRUTH with variable parameters including the number of turns (N), the inductance factor (AL), the characteristic impedance of the line (Z) and the electrical length of the transmission line (E) and the reference frequency (F) for calculating the transmission line length.

In order for ADS to perform a device simulation on the transformer with the responses as scattering parameters (S-parameters), it uses its S_Param modeler, which adjusts the beginning (start) and the ending (stop) sweep frequencies according to a given 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 used in the ADS simulation.

The measurements were performed on a commercial Advantest VNA (www.advantest.com), a 300 kHz to 3.8 GHz model R3765CG. The analyzer has unbalanced measurement connections with a connection impedance of 50 ohms. Since the broadband Unun impedance transformer has unbalanced terminals and a 1:4 turns ratio, another device with a 4:1 turns ratio was needed to perform the impedance conversion needed to match the device to the measurement equipment.numbers 6 and 7* *Show all terminal connections.

Both test terminals and all cables used with the VNA have been calibrated to minimize their error contributions. The insertion loss and the transmission behavior were analyzed using the transmission coefficient S_{21}Measurements presented in log-size form.

## measurement conditions

The comparison results between analytical (MATLAB), numerical (ADS) and experimental models were obtained under several measurement conditions. A model E1003C5 toroidal ferrite core from Sontag Componentes Eletronicos (www.sontag.com.br) was used in the experiments. Its geometric and electromagnetic specifications include an outer diameter of 10 mm, an inner diameter of 5 mm, a width of 3 mm, relative permeability (µ_{R}) of 11 and an inductance factor (A_{l}) of 4.2 nH/rev^{2}.^{11}It is specified for use in the frequency range from 500 kHz to 50 MHz. Five twists per centimeter wire was used with 30 AWG conductor transmission line. At 130 MHz, this line has a characteristic impedance of 38 ohms, a phase factor (β) of 4.5501 rad/m and a propagation speed (v_{P}) from 1.7952x10^{8}MS. The optimal value of the characteristic impedance for a 50 ohm source impedance must be 100 ohms, given in Eq. 8, which means a 0.38-fold relationship. The k-value for this difference and an insertion loss of 3 dB is 0.2207.

The first device was made with four turns, resulting in a lead length of 9 cm.numbers 8,9, and 10* *show the frequency insertion loss behavior for analytical, numerical and experimental cases.

TheTisch* *summarizes key values, including insertion loss results at maximum amplitude, at -3 dB frequencies (f

_{max}, fi-3dB and f

_{s-3dB}), the correct bandwidth (BW) and the percentage frequency differences compared to the model values. The differences between the analytical, numerical and experimental results are very small, except at the maximum signal frequency. This is due to limitations in the test system caused by noise and other parasitic effects in the measurement setup. Over the substantially constant amplitude test frequency band, variations in signal level are imperceptible, possibly explaining differences in reporting the maximum signal amplitude frequency.

A second device was constructed with six turns and a cable length of 11 cm. As the number of turns increases, the lower cut-off frequency decreases and the upper cut-off frequency is also reduced due to the increased line length. For the lower cut-off frequency, the values of the analytical and numerical results were as expected. However, the experimental answers do not agree exactly with the theoretical model. The high frequency response values are as expected, with good agreement between the three sets of values.figures 11, 12, and 13* *show the insertion loss as a function of frequency for the analytical, numerical, and experimental cases (also shown 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 because the theoretical model does not account for all the parasitic elements of the components used in the designed transformers.

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

At the lower cut-off 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 each other and the expected results. As the number of turns increased, the lower limit frequency was reduced; Similarly, as the line length increased, the upper cut-off frequency was also reduced.

## Summary

Although the three sets of results were obtained using different methods, they appear to be satisfactory and in good agreement with each other. The analytical (MATLAB) and numerical (ADS) model responses and the experimental responses (VNA measurements) were closely compared. The analytical and numerical cases gave approximately the same values, with some differences compared to the experimental results. This is best explained by the fact that the theoretical models do not fully take into account the complex properties of the components used in the manufacture of the transformers, which have been modeled as more or less "ideal" components.

These model equations represent a simplified equivalent circuit model for a coil transformer. Recent studies have shown the need to use a more sophisticated model that includes resistive and reactive effects as a function of increasing frequency and number of turns.^{12}

These advanced models also include the effects of interwinding capacitance, which lowers the inductor's natural resonant frequency. Nonetheless, these simplified design equations can provide reasonable results and replace the more complicated empirical processes normally associated with the design of a 1:4 impedance transformer. As these simplified equations show, they can be used to design transformers over a wide frequency range (three octaves) with low insertion loss and at low cost.

REFERENCES

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*Microwave & RF*, March 2005, Vol. 45, No. 1. - IJ Bahl, "Broadband and compact impedance transformers for microwave circuits", I
*EEE Microwave Magazine*, August 2006, Bd. 7, Nr. 4, S. 56-62. *The ARRL Handbook for Radio Communications*, 82. Aufl., The American Radio Relay League Inc., Newington, CT, 2005.- Advanced Design System (ADS) 2004C, Agilent Technologies (www.agilent.com), 2004.
- C.L. Ruthroff, „Einige Breitbandtransformatoren“,
*Procedure of the IRE*, August 1959, Bd. 47, Nr. 8, S. 13371342. - J. Sevic,
*transmission line transformers*, 4. Aufl., Noble Publishing, Atlanta, GA, 2001. - J. Sevick, "A Simplified Analysis of the Transformer for Broadband Transmission Lines",
*high frequency electronics*, February 2004, pp. 48-53. - AA Ferreira Jr. and W.N.A. Pereira, "Experimental tests with broadband RF transformers", in
*Proceedings of the World Congress On Computer Science, Engineering and Technology Education-WCCSETE'2006*, Santos, Sao Paulo, Brasilien, 2006, S. 473-477. - AA Ferreira, Jr., "Design of a high-frequency impedance transformer with passband control", Master's thesis, Instituto Nacional de Telecomunicacoes, Santa Rita do Sapucai, Minas Gerais, Brazil, 2006.
- MatLab 7, The MathWorks, Inc. (www.mathworks.com), 2005.
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*EDN*, September 2001, S. 67-74.

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