Measuring Array Mutual Impedances Using Embedded Element Patterns

The radiation pattern of an antenna element embedded in a multiport antenna such as a phased array depends on the loads connected to the array element ports. If the embedded element patterns (EEPs) are measured using at least two known loading conditions, the patterns can be used to determine the array mutual impedance matrix. In previous work, this result has been derived with the simplifying assumption that the impedance of the source connected to the driven element changes along with the load impedances connected to the nondriven elements. In a practical test configuration, the source impedance cannot be readily changed. We analyze the case of EEPs measured with a fixed source impedance and changing impedances on the nondriven elements. The transformation from one set of EEPs to another with fixed source impedance is more complex than in the case of a source impedance that changes with the load impedances. The transformation depends on the coupling between elements and is only weakly sensitive to the element self-impedances. With measured EEPs for an array of identical elements, the impedance matrix can be found up to a scale factor. We demonstrate the method experimentally by measuring the patterns of an antenna array terminated with one loading condition and repeating the pattern measurements with a different loading condition. The mutual impedance matrix extracted from the pattern measurements compared to network analyzer mutual impedance measurements is accurate to within 1– $2~\Omega $ for most of the mutual impedances.

large arrays is a tedious and time-consuming process that is prone to error. The goal here is to explore a method for extracting mutual impedances from element patterns with the hope of some advantages. This would provide an alternative method to measure impedances for large antenna array applications and effectively allow an antenna range to function as a network analyzer.
Coupling between elements causes perturbations in the element radiation pattern. When one element in the array is excited, the resulting radiated field is referred to as the embedded element pattern (EEP). The relationship between mutual coupling and EEPs has been explored from many perspectives [1]. In particular, EEPs depend on the loads on the nondriven elements. Kelley [2] showed that mutual impedances can be determined from measurements of the EEPs with different sets of loads on the nondriven elements, where open-circuit (OC) and short-circuit (SC) loaded EEPs were used to obtain the array impedance matrix from EEPs based on numerical simulations. This relationship was generalized to include EEPs with any loading impedance condition in [3].
In previous work, the assumption was that the impedance of the generator that excites the driven elements when EEPs are measured changes along with the load impedances that terminate the nondriven element ports. This assumption considerably simplifies the procedure for extracting impedances from EEPs. Consider a single, isolated antenna. If the antenna is driven by a source with a given internal impedance and the radiated field measured, and then driven by another source with a different internal impedance and the second radiated field measured, the fields radiated in the two cases at any point around the antenna depend in a simple, algebraic way on the two source impedances. If the generator impedance changes along with the loads for the nondriven elements, as is assumed in [3], the input current that excites the driven element changes as well, leading to a significant change in the EEPs from which element impedances can be relatively easily determined.
In practice, however, this is not how EEPs are obtained. In a typical antenna test configuration, the driven element is excited with a source matched to the system impedance (50 ). The loads on nondriven elements can be changed from one impedance to another (e.g., from an open to a short), but the source impedance of the driven element remains unchanged.
This simple practical issue has surprising ramifications for the relationship between EEPs and the array mutual impedance matrix. In this article, we obtain the mutual impedance matrix from EEPs by changing loading conditions of the nondriven 0018-926X © 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
elements while exciting the driven element with a Thévenin source and a fixed generator impedance. Transformations between EEPs with different loading conditions were given in [4]. In that work, constant current and constant voltage sources were used to excite the driven elements. These transformations are used as a starting point for the more complicated EEP loading condition transformations where Thévenin sources are assumed for the driving elements. With the generator impedance at the driven element unchanged, the EEPs are only affected through the change in loads on the nondriven elements. The nondriven elements act as parasitic antennas with switched load impedances that indirectly affect the radiation pattern of the driven element. After deriving the required EEP transformations, we demonstrate the proposed method by experimentally measuring EEPs with a fixed-impedance Thévenin source. Calibration is not required as long as the source remains the same for each measurement. Antenna range EEP measurements were performed with OC, SC, and matched loads on the nondriven elements and converted to OC EEPs. A candidate impedance is used to transform the measured SC-loaded EEPs to OC-loaded EEPs. The error between the transformed OC-loaded EEPs and the measured OC-loaded EEPs is used as a cost function and the mutual impedance matrix is solved iteratively using numerical optimization. The EEP loading condition transformations are only weakly sensitive to the element self impedances Z A,nn . In the case of identical array elements, this means that the impedance matrix obtained from the numerical optimization differs from the actual impedance matrix by a scale factor. A single port impedance measurement can be used to scale the numerically-determined impedance matrix. The resulting impedance matrix extracted from EEPs together with a singleport impedance measurement agrees to within a few ohms per matrix element with network analyzer measurements at the ports of the array. These results show that measured EEPs with different element loads can be used to find the port-toport mutual impedance matrix of an antenna array.

II. EEPS AND LOADING CONDITION TRANSFORMATIONS
Before deriving the cases with Thévenin driven sources, we will briefly review the basic EEP loading condition transformations using the notation from [4] and the key result of [3] for extracting the array mutual impedance matrix from EEPs.
Electric field sample measurements of the EEPs can be stored in the N × M array Each column is a far-field measurement in a given polarization for the N array elements in the array at position r m , m = 1, . . . , M. Multiple polarizations can be included by appending in the array as where p 1 and p 2 represent orthogonal far-field polarization components of the radiated electric field.
EEPs measured with a given set of element loads represented by the diagonal matrix Z L = Z L I are denoted by E Z L . The Thévenin equivalent generator voltage at the driven element is V 0 . The transformation from EEPs measured with an arbitrary element loading condition Z L to OC loaded EEPs is [4] where Z A is the antenna array mutual impedance matrix and I 0 is the reference antenna port input current used to obtain the OC loaded EEPs in the array E oc . Using this relationship, it is relatively straightforward to extract the impedance matrix from two sets of EEPs with different loading conditions [3]. Applying the transformation (3) twice, the transformation between EEPs with two arbitrary loading conditions is where V 0 1 , Z L 1 are the Thévenin equivalent source voltage and load impedances used for the first set of EEP measurements and V 0 2 , Z L 2 for the second. Because E Z L 1 would usually be nonsquare (greater than N field measurements) and thus overdetermined, its inverse does not exist. The Moore-Penrose pseudoinverse E Z + L 1 [5] can be used to find the least-squares solution If we define the solution can be written in the simpler form This formula for extracting the mutual impedance matrix expresses the key result of [3] in the notation used in this article.

A. Thévenin Driven Sources
The result of [3] reviewed in Section II is relatively straightforward from a theoretical perspective but difficult to apply in practice. In a typical antenna range, the excitation is produced by a network analyzer port and with impedance fixed at the system impedance, usually 50 . The system impedance and the addition of cables and connectors can be collectively considered the source impedance Z 0 . As illustrated in Fig. 1, the source impedance does not change along with the nondriven element loads. To develop a practical method for obtaining the mutual impedance matrix from measured EEPs, we must consider the case of element excitations with Thévenin source impedance fixed at Z 0 .
We will use OC-loaded EEPs with a constant current source as a basis for the desired Thévenin source excited EEPs. The advantage of using OC-loaded EEPs with a constant current source is that if we can determine the input currents at each EEPs measured with Thévenin sources for which the generator impedance does not change along with the nondriven element loading conditions. element port with an arbitrary loading condition and source impedances, the EEPs for the arbitrary case can be found from a linear combination of the OC loaded EEPs with a constant current source. We will denote the OC-loaded EEPs with a constant current source at the driven element as E oc .
With a Thévenin source with impedance Z 0 at element n and loads Z L,n at the nondriven elements, the input currents at the array element ports are given by the column vector where v g,n = [0, . . . , 0, V 0 , 0, . . . , 0] T is a zero vector with the generator voltage V 0 at the nth position.
} is a diagonal matrix with Z L at positions corresponding to nondriven elements and Z 0 in the nth position. Using the currents found in (8), the desired EEP with loading condition Z L,n where the nth element is excited with a Thévenin source with impedance Z 0 is If we let C n = (Z A +Z L,n ) −1 and observe that v g,n in (8) has only one non-zero element, C n is reduced to its nth column and the EEP becomes We can create an array of all the EEPs for each element in the array by stacking the row vectors e n to obtain This result can be simplified by separating C n into an impedance matrix and a perturbation matrix, so that where n is a diagonal matrix of all zeros but with Z 0 − Z L at the nth position. This has the form of a rank one perturbation to an invertible matrix. Using the Sherman-Morrison formula (see Appendix) on this form for each C n , a perturbation matrix can be extracted from (11) and we are left with where F is a diagonal perturbation matrix and equals Note that the nth diagonal value of F contains the nth value of the diagonal of (Z A + Z L ) −1 .
Using the matrix F, the transformation (13) from Thévenindriven EEPs to the OC-loaded EEPs with constant current source can be written in a form similar to (3) In principle, these formulas hold for nonreciprocal arrays, which account for the transpose operation in the matrix formulas. In Section II-B, we will specialize this transformation to the particular nondriven element loads that will be used in the experimental demonstration of finding the impedance matrix from EEPs.

B. Special Cases: OC, SC, and Matched Loads
When measuring EEPs experimentally, three cases for the loads on nondriven elements are most common: OC, SC, and matched loads. These can be realized with fabricated or commercially available connectorized terminators. We reduce the general result in Section II-A to the EEP transformations for each of these cases. In the remainder of this article, we assume that the array is reciprocal, which eliminates the need for matrix transposes in the formulas.
We first consider the transformation from EEPs with the three loading conditions with Thévenin sources to EEPs with OC loads and constant current excitations. For the OC loading condition with Thévenin driven sources, Z L = ∞ and (15) is difficult to reduce, but careful analysis of the multiport network yields the transformation where Z A,d is a matrix of the main diagonal elements of Z A and zeros elsewhere, I is the identity matrix, and Y 0 = 1/Z 0 . The transformation with SC loading conditions is reduced from (15) where Z L = 0, and becomes where Y A,d is a matrix of the diagonal elements of the mutual admittance matrix Y A = Z −1 A . For the matched loading condition, (15) reduces to (3). To emphasize the symmetry between the three cases, we write the third transformation in the form Each of these transformations takes EEPs with Thévenin sources on the driven element and an open, short, or matched load on the nondriven elements to the reference EEP with OC loads and a constant current I 0 at the driven element.
To determine the array mutual impedance matrix from EEPs, we require transformations between each pair of the open, short, and matched load cases. Equating any two of the transformations above and solving for one set of EEPS, we can obtain These formulas are the key theoretical result of the article. From these transformations for Thévenin source-driven EEPs, it can be seen that solving for the impedance matrix Z A analytically is difficult. Extraction of the impedance matrix from EEPs must be done numerically. Moreover, two of the transformations are a perturbation of the identity matrix. This observation is critical to understanding the behavior of the numerical results for extracting the impedance matrix from measured EEPs in Section III.
If the impedances of the sources used to excite driven element changes along with the EEP loading condition, the resulting EEP transformations are similar to (3), which is not a perturbation of the identity. In that case, the impedance matrix can be easily determined from the EEPs. With the more practical case of Thévenin sources with fixed impedances, the transformations in (19) are insensitive to the element self impedances. This is because the loading conditions are only changed for nondriven elements, which act like parasitic elements in their effect on the EEPs. Consequently, the structure of the coupling between array elements can be found numerically from measured EEPs, but the element selfimpedances are more difficult to determine.

III. EXPERIMENTAL RESULTS
EEPs for a 2 × 2 array of half-wavelength spaced microstrip patch antennas with center frequency 10.25 GHz were measured in the Chalmers University compact antenna range with the setup shown in Fig. 2. The antenna array is shown in Fig. 3. Twelve field measurements were performed based on permutations with each of the four ports driven and the nondriven ports terminated with OC, SC, or 50 loads. An identical source was used for each measurement so that the voltage and current scalars in the transformations above cancel out and no calibration is needed. Far-field samples were obtained in the hemisphere above the plane of the array at intervals of 10 • in azimuth and 1 • in elevation. The frequency  We estimate the array impedance matrix from the EEPs using numerical optimization. An iterative guess for the mutual impedance matrix is used together with (19) to transform the EEPs with one loading condition to the EEPs with a second loading condition. The error between the transformed EEPs and the measured EEPs is used as a cost function. By minimizing the cost function using an optimization algorithm, an approximation for the array impedance matrix can be found.
For the case of OC and SC loads, the cost function iŝ where M(E sc,th ; Z A ) is the transformation (19a). We performed a global search to find the impedance matrix that minimizes the cost function using a genetic algorithm. This was followed by a local search to refine the solution using the Nelder-Mead simplex method [6]. To speed computation, symmetry in the impedance matrix was assumed to reduce the dimensions of the search space. Due to the insensitivity of the transformations in (19) to the array element self impedances, the output of the optimizer can yield an estimateẐ A that is different from the true mutual S-parameters measured directly with a VNA compared to S-parameters computed from the impedance matrix extracted from measured EEPs.
impedance matrix Z A . In the case of an array with similar selfimpedances, the numerical optimization gives the impedance matrix within a complex scale factor, so that where α is undetermined. With highly accurate numerically simulated EEPs, this scalar can be found with a further optimization step. With measured EEPs, the scale factor is more difficult to determine. To remove the remaining scale factor ambiguity, we used a direct measurement of the selfimpedance of one of the array elements to determine the scale factor, so that The resulting impedance matrix extracted from the measured EEPs was where reciprocity was assumed in the optimization process, so the resulting matrix is symmetric.
The mutual impedance matrix measured using a vector network analyzer (VNA) was The mutual impedances extracted from the measured EEPs are within 1-2 of the directly measured mutual impedances for the four-element array.
The array of mutual impedances were computed using the same procedure at each frequency for which EEPs were measured. The mutual impedance matrices were transformed to S-parameters 1 and compared to the scattering matrix measured with a VNA as shown in Fig. 4. The results show good agreement for strongly coupled elements. Smaller coupling, such as between elements 2 and 3 which are diagonal to each other in the array, leads to a larger error between the S-parameters found from EEPs and the directly measured array S-parameters.
The experimental results given here are for the EEPs measured with open and SC loads. We were unable to extract the impedance matrix successfully from the EEPs with matched loads. A possible reason is that the transformations (19b) and (19c) are perturbations of the identity and the diagonal elements are unity, which means that the transformations involving the matched load case are less sensitive to the array mutual impedances than the SC to OC transformation.

IV. CONCLUSION
We have developed a method for determining the array mutual impedance matrix from EEPs and validated the method experimentally. The theoretical development needed for the experimental demonstration is the extension of EEP transformation formulas to the case for which the excitation is a Thévenin source with matched generator impedance. When compared to the simpler EEP transformations developed in past work for constant current and constant voltage excitations, the assumption of a Thévenin excitation for the driven elements leads to an interesting consequence. When using practical Thévenin sources to excite the driven element, loading condition changes for nondriven elements influence the EEPs through parasitic effects. This is reflected in the mathematical form of the EEP loading condition transformations. Experimental tests suggest that there is insufficient information to fully extract the mutual impedance matrix using numerical optimization. In the case where the element self-impedances are identical, the scaling problem can be overcome using single-port impedance measurements to scale the optimizerderived mutual impedance matrix. After scaling, the results give good agreement with the matrix measured directly from a network analyzer.
In essence, the method allows an antenna range to be used as a network analyzer. Rather than measuring O(N 2 ) pairs of mutual impedances, N EEP measurements are required (although a minimum of O(N) angle points are required per EEP). As long as the Thévenin sources remain the same between measurements, no source calibration is necessary except for a single-port measurement for scaling. The practical tradeoff is between connecting a VNA to each pair of element ports in the traditional method, and switching between open and SC loads while measuring EEPs in the proposed method. The proposed method may lend itself to measurement setups using electronically switched loads at nondriven ports, allowing for rapid mutual coupling measurements in an antenna range.
In future work, it would be of interest to validate the method for larger arrays with more antenna elements for massive MIMO, 5G arrays, and other applications. To reduce the number of degrees of freedom and the computation time for the numerical optimization, subarrays of the larger array could be treated separately.

APPENDIX DERIVATION OF PERTURBATION MATRIX
We wish to decompose each C n in (11) to pull out the common part (Z A + Z L ) and be left with the perturbation matrix F. This can be achieved by using the Sherman-Morrison formula [7]. This formula is a method to allow rank 1 updates to invertible matrices and can be stated as where A is the invertible matrix and uv H is the rank 1 update expressed as the outer product of two vectors.
This final equation is a scalar multiplied against a column of A −1 . The transposed stacking in (11) allows us to pull out that scalar and form the diagonal perturbation matrix F in (13) and (14).