# Dual-Frequency Decoupling Networks for Compact Antenna Arrays

Authors: Jacob Carl Coetzee
Journal: International Journal of Microwave Science and Technology
 (2011)

---

## Abstract

Decoupling networks can alleviate the effects of mutual coupling in antenna arrays. Conventional decoupling networks can provide decoupled and matched ports at a single frequency. This paper describes dual-frequency decoupling which is achieved by using a network of series or parallel resonant circuits instead of single reactive elements.


---

## Body

## 1. Introduction

The adverse effects of mutual coupling on the performance of multiport antennas are well known [1]. The effects can be countered by using a decoupling network which provides an additional signal path to effectively cancel the external coupling between array elements to yield decoupled ports [2–4].In its simplest form, the decoupling network consists of reactive elements connected between neighbouring array elements, but this approach only applies when the mutual admittances between elements are all purely imaginary [2]. The design of decoupling networks for arrays with arbitrary complex mutual admittances has been described [3–5]. In [6], closed-form design equations for the decoupling network elements of symmetrical 2-element and 3-element arrays were presented. This concept was extended to the decoupling of larger, circulant symmetric arrays through repeated decoupling of the eigenmodes [7]. However, these methods are only applicable to the decoupling of arrays over a small bandwidth at a single frequency. This paper describes dual-frequency decoupling of arrays. The procedure is based on the methods described in [6, 7], but where each reactive element in the single-frequency decoupling network is replaced with either a series or parallel combination of an inductor and a capacitor in order to achieve simultaneous decoupling and matching at two frequencies.


## 2. Theory

The procedures described in [6] or [7] may be employed to design a network at two distinct frequencies, f1 and f2. Subsequently, the relations provided in [8] can be used to design L-section impedance matching networks which match the decoupled port impedances to the system impedance Z0 at f1 and f2. Refer, for example, to the decoupling and matching networks for a 2-element array, shown in Figure 1.  X11 and B21 are the elements of the decoupling network at frequency f1, while X12 and B22 are the corresponding values at frequency f2. The element values are obtained in closed form from [6]. Ports 1′ and 2′ will thus be decoupled at both frequencies, but the port impedances Z1′=R1′+jX1′ and Z2′=R2′+jX2′ are not matched to Z0. The matching networks shown in Figure 1 are for the case where R1′>Z0 and R2′>Z0. B31 and X41 and are the matching network elements at frequency f1, whereas B32 and X42 are the elements at frequency f2.Figure 1
Decoupling network (DN) and matching network (MN) for a 2-element array at frequencyfj, j=1,2.In order to realise the required values of the elements of the decoupling and matching networks, each reactive element in Figure1 can be replaced with a circuit consisting of either a series or a parallel combination of a capacitor and an inductor. The component values of the capacitors and inductors can be computed using the relations provided in Table 1. The chosen topology in Figure 2 sees the series and parallel elements in Figure 1 replaced by series and parallel L-C circuits, respectively. Note that the design equations in [6–8] provide several possible solutions for the element values of the decoupling and matching networks in Figure 1. A number of these options would result in nonphysical, negative values for the capacitors and inductors in Figure 2. Care should therefore be taken to only select those options which yield positive values for each capacitor and inductor in the implementation.Table 1
Design equations for calculation of the parallel or seriesL-C circuit implementations of reactive elements in the decoupling and matching networks. Here, ωj=2πfj represents the angular frequency.
Network elementsParallel/seriesComponent valuesXi1, Xi2ParallelCi=(ω1/Xi1-ω2/Xi2)/(ω22-ω12)Li=(ω1/Xi1+ω12Ci)-1SeriesLi=(ω2Xi2-ω1Xi1)/(ω22-ω12)Ci=(ω12Li-ω1Xi1)-1Bi1, Bi2ParallelCi=(ω2Bi2-ω1Bi1)/(ω22-ω12)Li=(ω12Ci-ω1Bi1)-1SeriesLi=(ω1/Bi1-ω2/Bi2)/(ω22-ω12)Ci=(ω12Li+ω1/Bi1)-1Figure 2
Implementation of the decoupling and matching networks for a 2-element array using parallel or seriesL-C circuits.


## 3. Example

To verify the theory, dual-frequency decoupling and matching networks were designed for the same 2-element monopole array used in [6]. The array elements were wires measuring λ/4 in length and λ/40 in diameter at 2.5 GHz, and an element spacing of λ/10 was used. A system impedance of Z0=50Ω was assumed. This array was decoupled and matched at frequencies f1=2.4 GHz and f2=2.6 GHz. The scattering parameters, decoupling and matching network elements, and the components of the implementation are shown in Table 2. The calculated scattering parameters of the decoupled array are shown together with those of the original array in Figure 3. The results clearly illustrate the validity of the theory, since the array is both decoupled and matched at f1 and f2.Table 2
Decoupling and matching network elements and component values of the circuit in Figure2.
Array scattering parametersDecoupling network elementsMatching network elementsFrequency =f1S11a=-9.27dB∠137.6°X11=-29.7052ΩB31=-0.0132Ω-1S12a=-4.95dB∠-22.3°B21=-0.0279Ω-1X41=-47.3233ΩFrequency =f2S11a=-7.50dB∠89.5°X12=-9.9391ΩB32=0.0101Ω-1S12a=-5.53dB∠-45.2°B22=0.0219Ω-1X42=54.5235ΩL1=7.2337 nHL3=0.4352 nHImplementationC1=0.4778 pFC3=9.2297 pFL2=0.2038 nHL4=40.638 nHC2=19.731 pFC4=0.1005 pFFigure 3
Scattering parameters of the 2-element array, decoupled and matched atf1=2.4 GHz and f2=2.6 GHz.It should be noted that the decoupling/matching circuit is sensitive to capacitor and inductor tolerances. The decoupled 2-element monopole array was analysed repeatedly with element values varying randomly within a tolerance of ±1% and ±5%. The results are shown in Figures4 and 5, respectively. Although some frequency shifting is apparent in Figure 4, the performance of the system is still adequate. However, in the case of the ±5% element tolerance in Figure 5, the overall performance has deteriorated beyond acceptable standards. Element tolerances will therefore be an important consideration in the practical implementation of such circuits.Figure 4
Scattering parameters of the decoupled and matched 2-element array with capacitor and inductor tolerance of ±1%.Figure 5
Scattering parameters of the decoupled and matched 2-element array with capacitor and inductor tolerance of ±5%.


## 4. Conclusion

Dual-frequency decoupling of tightly coupled arrays was described. The approach involves the design of decoupling and matching networks at two distinct frequencies and implementing them simultaneously using a ladder network of parallel or seriesL-C circuits. The approach was illustrated for a 2-element array, but is equally applicable to larger arrays. Single-frequency decoupling networks are usually characterised by narrow bandwidths. By selecting the two frequencies of a dual-frequency decoupling network close to the desired operating frequency, the bandwidth for a single-frequency application can possibly also be improved.



---

*Source: 249647-2011-03-17.xml*