Power divider network for dual-fed adaptive antenna

Abstract Exposing the near field of an antenna to varying dielectric environment causes changes of the antenna input impedance and, thus, unwanted feed mismatch. Feeding such an antenna at different points, and selecting an appropriate feed for best match at a given scenario, may solve the problem. For the case of two scenarios of different dielectric environments and an antenna with two feed points, this work presents a passive power divider network, which keeps the antenna matched to the source in either scenario. Specific impedance transformations in the two branches of the divider network realize power transfer in a first scenario from the source to complex feed impedance at the first antenna feed, while in a second scenario, with now different antenna feed impedances, matched power transfer is from the source to the second antenna feed. Analytical formulae are derived for the design of the divider network. An experiment uses an example antenna with two feeds and a microstrip divider network, connected to a common 50 ohm port. Measurements are conducted with the antenna radiating, first, in air and, secondly, into butter. The measurements show antenna match at 1 GHz in either case and agree well with the analytical results.

proposed power divider concept realizes a particular case of unequal power split, with different splits for two respective sets of complex load impedances at its output ports (i.e. four different, freely-chosen output load impedances). It is optimally adapted to the use case of a dual-fed antenna operating in varying electromagnetic environment.
The proposed technique is fully passive and theoretically lossless, and can be realized in standard transmission line (TL) technology, such as microstrip. Thereby, it is reciprocal, and all statements related to a transmit scenario apply equally to a receiver scenario.
In section "Power divider design", the power divider design is given. It comprises four subsections. In the first one, a most simplified case is presented which shows the principle. In the second subsection, a lumped element-based TL model is introduced, which is the basis for the impedance transformation network (ITN) given in the third subsection. Two of these ITNs form the two branches of the power divider network (PDN), which is discussed in the fourth subsection. Section "Example" shows an example with experimental verification. Two monopoles with unequal lengths, mounted very close to each other, act as a single, dual-fed antenna. They are fed with the PDN. The last section contains the conclusion, consisting of a summary, a classification of this work with regard to RFID technology, and an outlook on future extensions.
Field simulation results are obtained from the finite-element method-based solver in CST Microwave Studio.

Most simplified case
This section explains the principle of this work in a most simplified case. The circuit of Fig. 2, i.e. just two resistors R 1 and R 2 , in parallel with source voltage V i and source resistor R i , is considered. There are two scenarios assumed, where R 1 and R 2 have different values, while R i remains constant. For a particular valueset of R 1 and R 2 , which is scenario 1, matching between the source, and the two resistors are obtained. In scenario 2 the values of R 1 and R 2 are interchanged in such a way that the source is still matched to the load. In Fig. 3 the normalized (to the maximum available power of the source) power dissipated in R 1 (red dashed line "p 1 ") and in R 2 (blue dotted line "p 2 ") are shown. Thereby, the value R 1 varies from 10 to 450 ohm while R 2 varies from 450 to 10 ohm as indicated by the two x-axes. The two vertical solid lines indicate the two scenarios. In addition, the reflected power (black dash-dotted line) is plotted. In these scenarios, the reflected power is zero. In scenario 1 the dissipated power in R 1 = R 1,max (≈ 57 ohm) is 87.6 % of the maximum power that the source can deliver. Then, the dissipated power in R 2 = R 2,min (≈ 403 ohm) is 12.4 %. In scenario 2 it is the other way around, as can be seen in Fig. 3. Consequently, this behavior can be seen as a natural power switch that flips in dependency on the load. It is also clear that, for a particular scenario, one of the resistors should have a value that is as close as possible to the value of the source resistance. If this is the case, the value of the other resistor must be very high in order to achieve matching.
In this work, a PDN based on TLs is developed which transforms (for both scenarios) two impedances in such a way that for scenario 1 most of the power is transferred to the first complex impedance and for scenario 2 to the second complex impedance. Therefore, it is important to note that in the general case, the four complex loads can be chosen arbitrarily (contrary to the aforementioned most simplified case).

Transmission line model
TLs shall constitute the power divider. For simplicity, lossless TLs are assumed. The approach is based on the well-known equivalent   Portions of the maximum power, which the source can deliver: reflected power |Γ| 2 (black dash-dotted line), the power dissipated in R 1 (p 1 red dashed line) and the power dissipated in R 2 (p 2 blue dotted line). 2 Serafin B. Fischer and Jan Hesselbarth circuit for a TL as given in [10]. This lumped-element equivalent circuit of a small length of a TL is continued periodically leading again to a lumped-element equivalent circuit and is depicted in Fig. 4(a). It is used to derive the design equations. This circuit comprises a series reactance, a transformer and a shunt reactance. It is equivalent to a TL of length L at a given frequency under conditions of (1).
Here, β denotes the phase constant, u is the transformer's turns ratio and sgn the sign function.

Impedance transformation network
The impedance transformation network is a series circuit of two TLs ( Fig. 4(b)). The load impedance Z L (i.e. antenna feed impedance) can take two different values. It has the following properties: (i) Maximum power transfer through the ITN, with respect to the source impedance Z m , for the load impedance Z L = Z L,max . (ii) Minimum power transfer through the ITN, with respect to the source impedance Z m , for the load impedance Z L = Z L,min . In the first case, if Z L = Z L,max , the ITN provides conjugate-complex match for maximum power transfer. Considering (1) and the circuit of Fig. 4(b), then (2) leads to Since (2) leads to two equations (if split in real and imaginary part), but the network itself comprises four parameters, two of them are freely selectable indicated by x and y (with the unit ohm).
It is stated that the input impedance Z in for Z L = Z L,min (i.e. Z in | Z L =Z L, min ) must be as high as possible to minimize the power in Z L,min . From here, the real part of Z in is considered only, because the imaginary part will be compensated when two ITNs are combined forming the power divider (see subsection "Power divider network"). Figure 5 colormap shows the real part of Z in if the ITN is terminated in Z L,min for different values of x and y. Dark red gives the maximum value, whereas dark blue indicates a low value with the unit ohm. Here, example values for Z L,min and Z L,max are chosen deliberately. It is calculated with (3) and consequently fulfils (2). To find the maximum values, the following equation must be met ∇ x,y indicates the gradient with respect to x and y. Both resulting equations are fulfilled for a certain dependency between x and y   (2) is fulfilled for all values of x and y.
where terms T i , i = 1, 2, …, 6 depend on Z m , Z L,max , and Z L,min and are given in Appendix. Inserting (5) into (3) gives the solution that fulfils the aforementioned properties of the ITN. For the same example of Fig. 5 the relation between x and y of (5) is plotted in Fig. 6. The curve can be recognized in Fig. 5 as the maxima. Furthermore, by using (1), the characteristic impedances as well as the electrical lengths of the two TLs, which form the ITN, can be obtained.

Power divider network
Two of the ITN described above form a three-port PDN. As shown in Fig. 7(a), the two scenarios (with source and PDN unchanged) have different loads, Z L1 and Z L2 . These loads are transformed through the ITNs, and the simplified circuits of Fig. 7(b) are obtained. Here, the subscript "max" indicates the load where the dissipated power is maximized, and the subscript "min" indicates the load where the dissipated power is minimized. For example, Z in,max denotes the input impedance of the ITN connected to the load Z L = Z L,max for the scenario where the maximum power should be delivered to this load. From the circuits of Fig. 7(b), conjugate-complex match to the source impedance Z i gives which can be written as The two equations of (7) have unknowns Z in1,max , Z in2,max , Z in1,min , Z in2,min . In addition, (2) and (4) relate Z in,max and Z in,min for each of the two ITNs. Therefore, this system of equations can be solved numerically and parameters of the TLs can be calculated.
The PDN presented here is a way of solving the problem of two different environments around an antenna. Intuitively, an alternative approach would be to use only one feed point of the antenna which is then matched with only one ITN for two load impedances (two different environments). However, the design of such a network is not straightforward due to the impedance matching domain problem as discussed in [11] for LC ladder networks. Thus, this single-feed approach is not suitable for a generalized case. On the other hand, the proposed dual-fed approach is straightforward, as shown in the following example.

Example
The procedure developed in section "Power divider design" is applied to a practical example of an adaptive antenna. The idea is to use the PDN to feed two monopoles (diameter of 2 mm) of different length, very close to each other (≈ λ/15) at 1 GHz ( Fig. 8(a)). Here, "M1" indicates monopole 1 (length 72 mm ≈ λ/4) and "M2" monopole 2 (length 35 mm ≈ 0.116λ). In the first scenario, the monopoles are placed in free space. In the second scenario, the monopoles are fully immersed in a lossy organic material (namely, butter). This example setup of the two coupled monopole antenna was chosen because it allows for an easy and clear separation of the electromagnetic environment of the antenna (namely, "air" versus "butter") from the PDN (the microstrip circuit in the opposite side of the antenna ground plane). The PDN matches the monopoles, in both scenarios, to a single 50 ohm source. From inspection of field simulation field plots, it can be noted that in the first scenario (free space), most of the power is radiated via the first monopole, while in the second scenario (butter), mostly the second monopole radiates.
The dielectric properties of butter are measured. Transmission through a cylindrical cavity resonator, at first air-filled and then filled with butter ( Fig. 8(b)), is measured and modeled in a field  simulator, allowing to obtain permittivity and loss tangent by fitting. Permittivity ε r,butter = 4.13 and tanδ butter = 0.04 is found at 1 GHz.
Field simulation provides the respective input impedances for the two monopoles M1, M2 for the two scenarios, as listed in Table 1 under notation "1 st simulation". The indices are chosen such that M1 is primarily fed to radiate in the free-space scenario, whereas M2 is primarily fed to radiate in the "butter scenario". For illustration, Fig. 9 shows plots of the variation of the monopole input impedances while linearly varying permittivity and loss tangent from the values of the first scenario to their values of the second scenario. It should be noted that strong coupling between the two monopoles (also expected for most dual-fed antenna structures) leads to some change of feed impedance of one monopole while the other monopole's feed is connected to a varying load (active impedance). The PDN is derived from the impedances, which in turn are obtained from a field simulation assuming simple, e.g. 50 ohm, port impedance. This PDN, however, then provides different impedances to the antenna, changing the active impedances of the antenna network at its ports. In an iterative procedure, these updated antenna impedances allow to derive an updated PDN. This process converges quickly. Note that additional field simulations are not required, as the scattering parameter matrix (or impedance matrix) of the antenna (for each scenario) needs to be obtained only at the beginning. Table 1 lists under notation "converged" the respective input impedances of the two monopoles M1, M2 for the two scenarios. Using the input impedances of Table 1, equations (1)-(7) permit the calculation of impedances and electrical lengths of the TLs forming the ITNs (Fig. 10, Table 2). These calculations are executed with Matlab. For each ITN, a remaining degree of freedom allows to select one parameter. In this example, the parameter y = −5 ohm is chosen for each ITN (this choice is governed by the resulting, "realizable" TL impedances). Table 2 lists the obtained parameters of the TLs of the PDN. The values for the PDN of the first iteration (notation "1 st simulation") do not differ much from the values obtained from the converged iterative design process (notation "converged"). In particular, in a practical realization of the PDN in microstrip technology, the differences will be negligible. However, for an evenstronger coupling between the ports of the dual-fed antenna, the changes during the iterative design procedure may become larger.
In Fig. 11 the simulated (here, TLs are lossless) portions of the maximum power that can be delivered from the source of the reflected power |Γ| 2 (black dash-dotted line), the power dissipated in Re{Z L1 } (p 1 red dashed line), and the dissipated power in Re {Z L2 } (p 2 blue dotted line) are plotted. Thereby, the input impedance changes over the relative permittivity and the loss tagent is considered (see Fig. 9). In the air-scenario, 99.8% of the maximum power is consumed by monopole M1, while only 0.2% is dissipated in monopole M2. If the monopoles are exposed in butter, the reflected power is about 0.2%, the power in M1 is 9.5 %, and in M2 is 90.3%.    International Journal of Microwave and Wireless Technologies 5 So far, optimum operation was required at two different scenarios, as illustrated in Fig. 11, showing the reflected power at source becoming zero for ε r = 1 and ε r = 4.13. A modification of the usecase may require good performance (i.e. small reflected power) over a range of dielectric property of the electromagnetic environment. Such use-case may represent somewhat varying material property and/or varying geometry of the environment of the antenna, as it can be expected for many RFID-like applications.
Exposing the pair of coupled monopole radiators to three other pairs of environmental dielectric scenarios, Fig. 12 illustrates the simulated resulting reflection coefficient magnitude. The four permittivity-pairs related to Fig. 12 are: (i) ε r = {1, 4.13}, (ii) ε r = {1.5, 3.5}, (iii) ε r = {2, 3}, (iv) ε r = {2.25, 2.75}. Obviously, the dielectric property of the electromagnetic environment can always vary by a small amount, without resulting in large mismatch. Once the two "optimally matched" permittivity values are close-enough to each other (as in cases (iii) and (iv)), a range of electromagnetic environment, leading well-matched input impedance, is obtained. Similar behavior is common to any ITN.
The microstrip line parameters are readily found for the circuit realized on Rogers RT6002 substrate (thickness 0.762 mm, ε rel = 2.94, tanδ = 0.0012). Figure 13 shows the realization of the two monopole radiators and the PDN on a microstrip circuit board. The ground layer has openings of 4 mm diameter for the monopoles. The monopole metal rods pass through the board and are soldered to the microstrip traces. The feed reflection magnitude at the coax connector, measured for the two scenarios, is shown in  In the scenario where the monopoles are placed in air (ε r = 1, tanδ = 0) the reflected power is zero and the dissipated power p 1 in Re{Z L1 } is about 99.8% of the maximum power that can be delivered from the source. Then the power p 2 in Re{Z L2 } is 0.2%. In the scenario of butter (ε r = 4.13, tanδ = 0.04) | Γ| 2 is 0.2%, p 1 is 90.3%, and p 2 is 9.5%.  Good match is achieved for both scenarios (free-space and butter). Furthermore, the measured curves are in good agreement with the field simulations.

Conclusion
Analytical formulae for a power divider feed network for a dual-fed antenna are presented. If the antenna operates in two electromagnetically different environments, called scenarios, its feed impedances will change. The proposed power divider feed network directs the power from a matched source to the first antenna feed in the first scenario, and to the second antenna feed in the second scenario. The proposed power divider feed network is passive and assumed lossless. In a practical demonstration, two strongly coupled monopole radiators of largely different length are fed by an accordingly designed power divider feed network. This structure radiates with perfect source match (50 ohm) into free space, but also when completely immersed into butter (ε r, butter = 4.13, tanδ butter = 0.04). The measurements confirm the derived theoretical formulae. The presented PDN can be used with RFID tags to adapt the antenna to two different environments. It ensures an energy-efficient operation of the RFID tag without much power being reflected at the antenna input.
From a theoretical point of view, the number of different environmental scenarios, as well as the number of antenna feed points, can both be increased to a number larger than two. If N scenarios shall be covered with an antenna having N feed points, then the extension of the approach discussed in this paper (here N = 2) shows that each of the N arms of the power divider requires N sections of TL to realize the appropriate impedance transformation. However, already for N = 3, these nine lines contribute to loss and increase the physical size of the circuit likely beyond a practical limit.