Network Theory and the Array Overlap Integral Formulation

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1 Chapter 7 Network Theory and the Array Overlap Integral Formulation Classical array antenna theory focuses on the problem of pattern synthesis. There is a vast body of work in the literature on methods for designing array excitation coefficients to achieve desired patterns for linear arrays, planar arrays, arrays with large numbers of elements, sparse arrays, random arrays, and many other types of multiantenna systems. In recent years, the focus in this field has shifted to digital beamforming, real time processing, adaptive methods for dynamic pattern control, and multiple input multiple output (MIMO) systems. The pattern synthesis methods of the previous chapter make the assumption that the array element radiation patterns are identical, and mutual coupling and interaction effects are neglected. A more rigorous analytical framework is needed to account for these effects and accommodate the more sophisticated signal processing methods that are used in modern array applications. To facilitate this transition from classical array pattern synthesis techniques to modern array signal processing theory, we will introduce the concepts of the embedded element radiation pattern and the array pattern overlap integral matrix, and we will develop a network model for the array and its associated driving electronics. In the short term, the network and overlap integral formulations will provide a convenient way to work analytically with array radiation patterns and directivity. As the treatment becomes more sophisticated, in later chapters the overlap integral matrix will be used to explore connections with array signal processing theory theory and noise analysis. We will begin with transmitting arrays in this chapter, and consider receiving arrays in Chapter Embedded Element Radiated Field Patterns The key approximation used in the previous chapter to simplify the analysis of array antennas was that interactions between elements are negligible. Each element in the array was assumed to have an identical radiation pattern, with the exception of phase and amplitude shifts due to the element driving current and the element s position offset relative to the coordinate system origin. To develop a more rigorous and accurate formulation for array antennas, we need to relax this assumption, and develop a notation that allows each element in the array to have a different radiation pattern. This leads to the concept of the embedded element radiated field pattern. For an arbitrary array, let the field radiated by the nth element excited by an input current I 0 with the terminals of the other elements open circuited be denoted as E n (r). If the element patterns are known for a loading condition different from open circuit, network theory can be used to transform to the open circuit loaded patterns. Other loading conditions can be used to define the embedded element radiation patterns, such as short circuits at the non-driven element terminals or a matched load at the system impedance, but 79

2 ECEn 665: Antennas and Propagation for Wireless Communications 80 we will choose the open circuit loading condition as our convention for defining the patterns. We will refer to E n as the open circuit loaded embedded element radiated field pattern, or more succinctly, as the element pattern. E n can be measured, numerically simulated, or analytically approximated. If it is convenient, the formulation in this section can be modified to use embedded element radiation field patterns for a different loading condition than an open circuit. 7.2 Element Interactions and Mutual Coupling Embedded element radiation patterns are typically different from the patterns radiated by an element in the absence of the other elements in the array, or the isolated element patterns. When one element is excited, the radiated fields induce currents on neighboring elements, which leads to a perturbation in the radiation pattern. Because the embedded element pattern of a given element is affected by the presence of other neighboring elements that scatter fields radiated by the driven element, the embedded patterns of elements near the edge of an array can be different from the patterns of elements in the interior of the array. For a regular array, the geometrical pattern of neighboring elements is the same for any interior element, and the embedded elements are nearly the same for all interior elements. Edge elements have fewer neighboring elements, and have different embedded patterns. For this reason, interactions between array elements are said to cause edge effects. In some cases, the perturbation on the element patterns caused by the other elements can be neglected, so that E n can be approximated by the isolated element radiated field. For a half wave dipole, an open circuit load interrupts the current induced on the dipole by radiation from the excited element, so that scattering by the other elements is smaller than would be the case for other loads. The two halves of the dipole become quarter wave dipoles with short circuit loads, which still perturb the fields radiated by the excited element, but the effect is not as strong as if the load were different. An antenna that does not perturb the fields radiated by other nearby antennas is known as a minimum scattering antenna. A short dipole with an open circuit load is approximately minimum scattering. For dipoles, this means that the open circuit loading condition is a natural choice, because the embedded element patterns with open circuit loads on other elements is reasonably well approximated by the isolated dipole pattern. In addition to perturbing the element patterns, induced currents on neighboring elements generally couple to the neighboring element ports, which means that the network scattering parameter matrix or mutual impedance matrix is not diagonal. A current impressed at the input of one element can induce a voltage at the terminals of other nearby elements. As a consequence, some of the power accepted by the input port of one element is fed as a reverse wave back into the feeding transmission lines connected to other element ports instead of radiating to the far field. This effect is referred to as mutual coupling between array elements. Embedded element pattern perturbations and nonzero mutual impedances are both often loosely referred to as mutual coupling, but in a rigorous sense, mutual coupling refers to port coupling or nonzero mutual impedance. It is possible for interactions between elements to cause induced currents that flow on neighboring elements but do not induce any voltage at the neighboring element terminals, so the currents can perturb the embedded element patterns without causing nonzero mutual impedances. Interactions between elements that influence the embedded patterns and mutual coupling associated with the element ports are not exactly identical, but the effects are closely related. These two effects, perturbation of the element patterns and element feed port coupling, are closely related, since the induced currents on the non-driven elements produce both the perturbation to the radiation pattern and the induced voltage at the element terminals. The first effect, scattering from nearby elements is already incorporated into the definition of the embedded element patterns E n. The second effect, coupling between element ports, can be modeled using microwave network theory.

3 ECEn 665: Antennas and Propagation for Wireless Communications Transmit Array Network Model If we view the input terminals of the elements in an array as ports of a microwave network, we can model mutual coupling using a scattering matrix (S-matrix or S-parameters) or an impedance matrix (Z-matrix). S-parameters are commonly used when dealing with measurements of system properties using a microwave network analyzer, and we will find that Z-parameters are particularly convenient when working with antenna arrays. The two formulations are equivalent, and we will show how to convert S-parameters into Z-parameters or impedance matrices. For a two-port network, the S-matrix is defined by [ ] [ ] [ ] b1 S11 S = 12 a1 (7.1) b 2 S 21 S 22 a 2 where a n and b n represent the forward and reverse voltage wave amplitudes at the nth port, respectively. The mutual impedance formulation is [ ] [ ] [ ] v1 Z11 Z = 12 i1 (7.2) v 2 Z 21 Z 22 i 2 where the total voltage is related to the forward and reverse waves by v n = Z 0 (a n + b n ) and the total current is i n = (a n b n )/ Z 0. Z 0 is the characteristic impedance of the transmission line at the nth port. This is a matrix generalization of the phasor form of Ohm s law. The S-matrix and impedance matrix formulations are related by S = (Z + Z 0 I) 1 (Z Z 0 I) (7.3) which is a matrix generalization of the single-port reflection coefficient formula. For an N element array, (7.2) becomes v A = Z A i A (7.4) where v A is a vector of voltages across the element ports and i A is a vector of the element port input currents. In either the S-matrix or Z-matrix formulation, the off-diagonal matrix elements represent mutual coupling between array elements. Z A,mn is equal to the voltage induced at the open circuited terminals of the mth element when the nth element is excited by an input current of 1 A. Mutual coupling also perturbs the the diagonal matrix elements, so that the embedded element self impedances Z A,nn are different from the isolated element input impedance Z in. For the block diagram shown in Figure 6.1 in Chapter 6, we can arrange the source impedances into a diagonal matrix Z g I and model the generators as an N-port network with impedance matrix Z g = Z g I and N open circuit phasor voltages given by the vector v g. In most cases, the coupling between sources can be ignored, but if the coupling is significant, then Z g is nondiagonal. For the source network, we must add a generator voltage term to the voltage/current relationship, so that (7.2) becomes v A = Z g i A + v g (7.5) The negative sign arises because the impedance matrix is defined with respect to currents flowing into the ports, and i A flows out of the sources and into the array ports. Combining (7.4) and (7.5) leads to i A = (Z g + Z A ) 1 v g (7.6) If the generator phasor voltages are given, the array input port currents can be used to find the radiated fields using a linear combination of the open circuit loaded element patterns as in (7.9). Conversely, if we want to achieve a given radiation pattern, the required element excitation currents must be found first and then (7.6) used to find the required generator voltages.

4 ECEn 665: Antennas and Propagation for Wireless Communications Reference Planes Voltages and currents in microwave networks are defined at groups of nodes, ports, or terminals at a specified physical location on each connecting transmission line. A collection of nodes, ports, or terminals is referred to as a reference plane. In a network that represents a system with inputs, signal handling components such as amplifiers, filters, or mixers, and outputs, voltages and currents in the network can be specified at any reference plane along the transmission lines that connect each component. If the S-parameters or Z-parameters of each component are known, then the voltages at any given reference plane completely determine the voltages at other reference planes. While voltages and currents can be specified at any reference plane in the system, there are a few reference planes that we will use most commonly when working with array antennas. Voltages at the output of a transmit beamforming network before signal conditioning and amplification are clearly important, since these are the signals that are created by the beamforming network and which are designed using the techniques of Chapter 6. These signals could instead be represented by the Thévenin equivalent open circuit voltages or Norton equivalent short circuit currents that represent the power amplifiers that drive the antenna elements. This is the vector v g in (7.5). The array antenna excitations can also be represented in terms of the element input currents i A, which is useful when dealing with antennas for which the radiated fields are parameterized in terms of input currents. 7.5 Beamformer Weight Vector In array signal processing theory, the excitations that drive each array element are commonly arranged into a beamformer weight vector of the form w 1 w 2 w = (7.7). The array pattern synthesis methods considered in Chapter 6 are techniques for designing the values of the beamformer weight vector w to achieve a given radiation pattern shape or figure or merit, such as directivity. The beamformer weights are complex, as they represent phasor voltages or currents and include a magnitude and phase. The beamformer weight vector can be defined at various reference planes in the transmitting array system. When the focus of an analysis is on signal processing, it is often left unspecified where in the system the beamformer weights are defined. Mutual coupling and other network effects must then be added to the treatment as an extra correction step. When the array design itself is the focus of the analysis, a common choice to think of the beamformer weights w as the input currents into each element in the array, since antennas are often modeled as equivalent currents relative to a given driving current phasor at the element input port. Another possibility is to define w as the vector of voltages at the inputs to the amplifiers and electronics that drive the array. In a digitally beamformed array, this might represent the output voltages of digital to analog converters that transform digital values in a DSP system to analog voltages. Another possibility is to consider the open circuit voltages in a Thévenin equivalent for each driving amplifier as the beamformer weights. These are the generator voltages in Figure 6.1. If mutual coupling is ignored, the transformation in (7.6) is a diagonal matrix, and if the elements in the array are identical, then the transformation is a scaled identity matrix. In this approximation, the vector of open circuit generator voltages is proportional to the vector of input currents, and the issue of which reference plane is used to define the beamformer weight vector is unimportant. This approximation is often made in the array signal processing literature. In a more rigorous treatment of array antennas that w N

5 ECEn 665: Antennas and Propagation for Wireless Communications 83 accommodates mutual coupling effects, the transformation in (7.6) is nondiagonal, and the reference plane for the beamformer weight vector must be specified. Among the various possibilities for the reference plane to choose when defining w, the most convenient for our purposes is the vector of input currents into each array element. Most of the results we have developed for antenna radiation and other properties are given in terms of the input current phasor. This means that formulas for array antennas are simplest when given in terms of a beamformer weight vector that consists of element input currents. Using network theory, we can transform excitations from other reference planes (typically digital values in DSP or amplifier open circuit voltages) to input currents. In this chapter, we will define the array beamformer weights for a transmitting array as the column vector I 1 I w = 2. where I n is the phasor input current into the nth array element. We refer to the vector w as the array beamformer weight vector or simply weight vector. The elements of the vector are equal to the complex conjugates of the driving currents for each element in the array, so that w n = I n in terms of elements of the vectors. If the currents are arranged into a vector i = [I 1, I 2,..., I N ] T, then w = i. In the array signal processing literature, for receive arrays, the beamformer weight vector w is defined to be the complex conjugates of the coefficients that are applied to array output voltages before summation. This allows receive array output quantities to be expressed as vector inner products (e.g., the receive array output voltage can be expressed in the form w H v, where v is a vector of voltages at the array outputs). For transmit arrays, there less motivation to use the complex conjugate in the definition of w, but in Chapter 8 we will develop many relationships based on reciprocity between an array excited as a transmitter and then used as a receiver, and these formulas are simpler if we use the conjugate in the definition (7.8) for the transmit beamformer weight vector as well. I N (7.8) 7.6 Array Pattern Overlap Integral Formulation In terms of the embedded element patterns E n and the beamformer weight vector w, the total radiated electric field for a transmitting array is E(r) = 1 I 0 N n=1 w ne n (r) (7.9) In this expression, we must divide each embedded element pattern by the current I 0 used to simulate or measure E n and then multiply by the driving current wn to scale the field radiated to the proper level for each element port. The total radiated power is P rad = 1 2η = 1 I η = 1 I 0 2 E(r) 2 r 2 dω N N m=1 N m=1 n=1 w m w me m (r) N w n E n(r)r 2 dω n=1 1 E m (r) E 2η n(r)r 2 dω w n (7.10) }{{} Overlap integral

6 ECEn 665: Antennas and Propagation for Wireless Communications 84 Motivated by this expression, we define the element pattern overlap matrix A with elements given by A mn = 1 E m (r) E 2η n(r)r 2 dω (7.11) If all the elements are identical and we neglect array edge effects, then the diagonal elements A nn are equal and represent the power radiated by the array with input current I 0 into the nth element port and the other ports open circuited. In matrix notation, the total radiated power is P rad = 1 I 0 2 wh Aw (7.12) In some treatments, the overlap matrix is normalized so that the diagonal elements are equal to one, but here it is useful to retain the physical units of power as determined by (7.11). A matrix-vector product of the form of (7.12) is referred to variously as a matrix inner product or a quadratic form. To compute the directivity of an array, we also need the radiated power density in a given direction. The radiated power density can be placed into matrix form using S r (r) = 1 2η E(r) 2 1 = 2η I 0 2 w1e 1 + w2e wne N 2 1 = 2η I 0 2 wme m w n E n = 1 I 0 2 m,n m w m n 1 2η E m E n }{{} B mn(r) w n = 1 I 0 2 wh B(r)w (7.13) where the directivity is in the direction of the far-field point r. By combining the total radiated power and the power density, we can express the directivity as D(Ω) = 4πr2 w H B(r)w w H Aw (7.14) This is referred to as a ratio of quadratic forms, since the numerator and denominator are quadratic in the elements of the weight vector w. This expression is convenient for designing array weights because we only need to compute A and B once, and we can find the gain for any set of array input currents or weights by simply evaluating two quadratic forms. The matrix elements of A are integrals of the elements of B over a sphere, so we can express the directivity in the alternate form D(Ω) = m,n w me m E nw n Em E (7.15) ndω w n m,n w m 1 4π directly in terms of the array embedded element patterns.

7 ECEn 665: Antennas and Propagation for Wireless Communications 85 Partial directivity. We can modify (7.14) to be the partial directivity with respect to the polarization ˆp by changing the numerator to B p can be expressed in the form where E p is the column vector S r,p (r) = 1 ˆp E(r) 2 2η = 1 wm ˆp E m w n ˆp E n 2η m = w 1 m 2η E m,pen,pw n m,n n = w H B p w (7.16) B p = 1 2η E pe H p (7.17) E p = [ˆp E 1 ˆp E 2 ˆp E N ] T (7.18) Since the matrix can be expressed as an outer product of vectors, it follows that B p is a rank one matrix. The partial directivity is D p (Ω) = 4πr2 w H B p (r)w w H Aw (7.19) If the array elements are identically polarized and if ˆp is aligned with the polarization of the array elements, then the partial directivity is equal to the directivity Computing Overlap Integrals For some types of antenna arrays, scattering of the radiated fields from one element by other elements can be neglected, and we can approximate the embedded element patterns with the isolated element pattern. For simple antennas, the isolated element pattern is available as a closed form approximation, which allows the overlap integrals to be evaluated analytically. For more complex patterns, the overlap integrals can be computed numerically using a quadrature rule. In the case of identical elements and neglecting array edge effects, the equivalent currents for the elements of the array with unit strength excitations are J n (r) = J(r r n ) (7.20) where r n is the spatial offset of the element location with respect to a reference element with equivalent current J(r). The far field of the nth element is E n (r) = e jk rn E(r) (7.21) where E(r) is the far field of the reference element with input current I 0 and k = kˆr. The overlap integrals are A mn = 1 e jk (r m r n ) E(r) E (r)r 2 dω 2η = e jk (r m r n ) S r (r)r 2 dω (7.22)

8 ECEn 665: Antennas and Propagation for Wireless Communications 86 where S r (r) is the far-field radiated power density due to one element in the array. If we approximate the elements as isotropic radiators, the overlap integral is A mn = e jk (r m r n ) P rad 4πr 2 r2 dω = P rad e jk r mn dω 4π where r mn = r m r n is the shift vector between the two elements. Since we are integrating r over a complete sphere, we can rotate r arbitrarily without changing the value of the integral. Let us rotate r so that the z axis is aligned with k, in which case θ becomes the angle between k and r mn. The overlap integral is If we let u = cos θ, this becomes A mn = P rad 4π = P rad 2 2π π 0 π 0 0 A mn = P rad 2 e jkr mn cos θ sin θ dθ dϕ e jkrmn cos θ sin θ dθ (7.23) 1 1 e jkr mnu du = P rad sin(kr mn ) kr mn (7.24) From this, it can be seen that the elements of the overlap matrix for isotropic radiators consists of samples of a sinc function. The diagonal elements of the overlap matrix are the total power radiated by each element with an input current of I 0. If the element spacing is λ/2, then the off-diagonal elements of the overlap matrix are zero. For isotropic radiators, we can also find the elements of the matrix B in closed form. If the total radiated power from one element is P rad, then the electric field strength is 2ηPrad E 0 = 4πr 2 (7.25) The radiated field amplitude from the nth element in the array is then E n = E 0 e jk r n (7.26) where r n is the location of the element and k = kˆr. The elements of the matrix B are then B mn = E me n /(2η). For more complicated array elements, the overlap integrals are of the form I = 2π π 0 0 f(θ, ϕ) sin θ dθ dϕ (7.27) where f(θ, ϕ) is the inner product of a pair of element patterns. This integral can be computed numerically using the midpoint quadrature rule. If the number of sample points or quadrature points for the integral along ϕ is N, then the integration step sizes are ϕ = 2π, θ = ϕ (7.28) N

9 ECEn 665: Antennas and Propagation for Wireless Communications 87 The midpoints for each integration step are ϕ m = (m 1/2) ϕ, m = 1, 2,..., N θ n = (n 1/2) θ, n = 1, 2,..., N/2 where we have assumed that N is even. The integral can be approximated using I N/2 N f(θ n, ϕ m ) sin θ n θ ϕ (7.29) m=1 n=1 The larger the separation between the elements, the more rapidly f(θ, ϕ) varies with angle. Because of this, the number of integration points must increase with the maximum dimension of the array. An empirical rule is N = kd + 20 (7.30) where is the next largest integer or ceiling operation and d is the longest linear dimension of the array. A more efficient integration rule can be used by making the transformation u = cos θ and evaluating the u integral using Gauss-Legendre quadrature Directivity Optimization The directivity is proportional to the quantity Q(w) = wh Bw w H Aw (7.31) which is a ratio of quadratic forms. The goal is to find the array weights that lead to the highest gain by maximizing Q(w) over w. We can maximize this quantity by taking derivatives with respect to the real parts and imaginary parts of the elements of w separately and setting all derivatives to zero. A handy trick for complex optimization is that the same thing can be done with less algebra by taking the derivative with respect to the complex conjugate of the variables and setting that to zero. If we take the derivative of the quadratic form w H Bw with respect to w n, we obtain w n w H Bw = [Bw] n (7.32) which is the nth element of the column vector Bw. Using this result, the derivative of Q(w) with respect to w n is Q(w) w n = [Bw] n w H Aw wh Bw[Aw] n (w H Aw) 2 (7.33) Setting this expression to zero to find the maximum leads to ( w H ) Bw w H Aw = Bw (7.34) Aw }{{} λ which is a generalized eigenvalue problem of the form λaw = Bw (7.35)

10 ECEn 665: Antennas and Propagation for Wireless Communications 88 By inspection of (7.34), the largest possible value of Q(w) is the largest generalized eigenvalue. This result can be used to optimize the directivity of an array over the excitation currents. The excitations which maximize directivity therefore can be found from the vector w which satisfies λ max Aw = Bw (7.36) According to (7.8), the excitation currents that achieve the maximum directivity are given by the elements of the weight vector w. Rank one case. For arrays with identically polarized elements, from (7.13) B is a rank one matrix. B is also rank one if we are considering partial directivity, since B p = 1 2η E pe H p. If B is rank one, all of the generalized eigenvalues of (7.36) are zero except for one. The corresponding eigenvector can be found from which shows that the optimal excitations are λ max Aw = E p 1 2η EH p w }{{} Scalar (7.37) w = A 1 E p (r) (7.38) where we have dropped a scale factor since directivity is independent of the common scaling of the excitations. The r dependence is included as a reminder that E p is a column vector of radiated fields for each element in the array evaluated at some far field point and the resulting excitations maximize the directivity pattern in the direction ˆr of that far field point. Conjugate field match (CFM). If the array elements are spaced far enough apart that the off-diagonal overlap integrals are small, then A is approximately a scaled identity matrix. In this case, the excitations for maximum directivity are given by w = E p (r) (7.39) Since the excitation currents i = w at each array element input port are equal to the vector of complex conjugates E p of the radiated fields with each element excited in turn with input current I 0, this is called the conjugate field match solution. If A is not a scaled identity, then the directivity must be less than that obtained with the optimal weights (7.38). For a ULA, if the element spacing is greater than about 0.4 λ, the directivity with CFM is close to the optimal directivity, since A is strongly diagonal. Physically, for an electronically scanned beamforming array, the conjugate field match solution produces beamformer weights with phase that match the propagation delay along each path from the array elements to the desired steered beam direction. The vector E p represents the embedded element patterns evaluated a given direction, and the difference in patch length to each element in the array is reflected as a phase shift in E p. Those phases become phase shifts in the beamformer weight vector w such that the formed beam is steered in the desired direction. We can find the resulting directivity by inserting the CFM array weights into (7.14). The directivity pattern is E H p (r 0 )E p (r) 2 D(Ω) = 1 4π E H p (r 0 )E p (r ) 2 dω (7.40) where Ω is the spherical angle of r. If A is a scaled identity matrix, so that the elements in the array radiate identical fields (except for a phase shift due to the locations of the elements in the array) and the off-diagonal overlap integrals are zero, then the maximum directivity is D = 1 2η EH p E p P el /(4πr 2 ) (7.41)

11 ECEn 665: Antennas and Propagation for Wireless Communications 89 where P el is the power radiated by one of the elements. Each term in the inner product is equal, so that D = N 1 2η E p,el(r) 2 P el /(4πr 2 ) = ND el(ˆr 0 ) (7.42) where D el (ˆr 0 ) is the directivity of one array element in the ˆr 0 direction. The factor of N is the peak value of the directivity estimate for a ULA of isotropic radiators obtained in (6.29). This result only holds if the overlap matrix is a scaled identity. Otherwise, (7.40) must be used to find the directivity. The physical interpretation of the CFM weights is that we evaluate the field radiated by each element with a unit strength excitation at some far field point, and then to maximize the directivity in that direction, we excite the array elements with currents equal to the conjugates of those fields. The phase of the excitation compensates for the phase change due to the differential propagation delays across the array, and the fields add in phase in the desired main beam direction. Superdirectivity. If the elements of an array are close together, from (7.24) it can be seen that the elements of the overlap matrix are all close in value. In this case, A is nearly singular and has a very small eigenvalue. If the excitations are chosen to align with the corresponding eigenvector, then the total radiated power in the denominator of (7.14) is small and the directivity becomes large. This is a superdirective solution. The beam pattern consists of a narrow main lobe in the ˆr direction and low sidelobes, even though the total antenna size is electrically small. There are several practical problems with superdirective arrays. First, since w corresponds to a small eigenvalue of A, from (7.12) the currents must be large in order to radiate an appreciable amount of power. This leads to large ohmic losses in the antenna elements, and hence superdirectivity does not necessarily produce supergain. Second, superdirectivity is sensitive to the precise magnitudes and phases of the excitation currents. If there are errors in the driving current magnitudes or phases, w no longer aligns with the proper eigenvector, and the directivity decreases. A similar effect occurs when the operating frequency is shifted away from the design center frequency. From a circuit point of view, a superdirective array has a high quality factor and a narrow bandwidth. For these reasons, superdirectivity has limited practical application Mutual Resistance and the Overlap Matrix We can use conservation of energy to develop a relationship between the array mutual impedance matrix and the overlap matrix. The total power flowing into the element ports is P in = 1 2 Re[iH A v] = 1 2 Re[iH A Z A i A ] = 1 4 (ih A Z A i A + i H A Z H A i A ) = 1 4 ih A (Z A + Z H A )i A If the array is reciprocal, so that Z A is a symmetric matrix, then we can simplify this to P in = 1 2 ih A Re[Z A ]i A (7.43) By conservation of energy, the input power is either dissipated in the array or radiated to the far field, which means that P in = P rad + P loss (7.44) Using (7.12) and (7.43) in this relationship leads to 1 2 ih A Re[Z A ]i A = 1 I 0 2 ih A Ai A ih A R A,loss i A (7.45)

12 ECEn 665: Antennas and Propagation for Wireless Communications 90 where we have used the fact that A is symmetric and R A,loss is the part of the impedance matrix Z A that represents ohmic resistances or dielectric losses in the array. Since equality must hold for all input currents i A, it follows that A = I R rad (7.46) where R rad = Re[Z A ] R A,loss (7.47) For a lossless array, A = I 0 2 Re[Z A ]/2, which means that the radiation field pattern overlap matrix is proportional to the real part of the mutual impedance matrix or mutual resistance matrix. We can also derive an equivalent result in terms of the S-parameters. The total input power is P in = 1 2 (ah a b H b) = 1 2 (ah a a H S H A S A a) = 1 2 ah (I S H A S A )a where a and b are column vectors of the forward and reverse voltage wave amplitudes at each element port, respectively. The forward voltages are related to the input currents by i A = a b Z0 = 1 Z0 (I S A )a (7.48) In the case of lossless elements, P in must be equal to (7.12) for all possible input voltage waves, from which it follows that 2 (I S H A )A(I S A ) = I S H A S A Z 0 (7.49) Solving for the overlap matrix, A = I Z 0(I S H A ) 1 (I S H A S A )(I S A ) 1 (7.50) for a lossless array. Equation (7.46) has several important implications. The pattern overlap matrix has real elements, which is not obvious from the definition. If the array is not mutually coupled (so that S A is diagonal), then the overlap matrix is also diagonal. Conversely, if the overlap matrix is not diagonal, then the array must be mutually coupled. Therefore, even if we approximate the open circuit loaded element patterns with the isolated element patterns, in general the off-diagonal pattern overlap integrals are nonzero, which means that we can still take into account mutual coupling effects in the system by taking Re[Z A ] to be given by (7.46) and (7.47) in terms of the overlap matrix Radiation Efficiency The radiation efficiency of a transmitting array antenna can be expressed using the overlap integral formulation and mutual impedance matrix. Using the radiated power given by (7.12) and the input power (7.43), the radiation efficiency is η rad = P rad P in = 2 I 0 2 w H Aw w H Re[Z A ]w = wh R rad w w H Re[Z A ]w (7.51) where R rad is the radiation part of the mutual resistance matrix as expressed in (7.47). We have also used the fact that the input power and mutual resistance matrix Re[Z A ] are both real.

13 ECEn 665: Antennas and Propagation for Wireless Communications Active Impedances and Active Reflection Coefficients The ratio of voltage to current at a given port is only equal to one of the elements of Z A if the other ports are open-circuited, so that the currents are zero. If the driving currents are nonzero at all the ports, then the voltage to current ratio at each port is a function of multiple elements of the impedance matrix as well as the currents at other element ports. This ratio is called the driving point impedance or active impedance. For a two-element array, the active impedances are Z act,1 = v A,1 i A,1 = Z A,11 + Z A,12 i A,2 i A,1 Z act,2 = v A,2 i A,2 = Z A,22 + Z A,21 i A,1 i A,2 For an N element array, the active impedance of the mth element is Z act,m = 1 i A,m In terms of the beamformer weight vector w = i A the active impedances are Z act,m = 1 w m N Z A,mn i A,n (7.52) n=1 N Z A,mn wn (7.53) By converting the active impedance to a reflection coefficient, we can obtain a similar formula for the active reflection coefficient at each element port. The ramification of this analysis is that the active impedances and active reflection coefficients actually depend on the excitation currents at each input port. This means that the impedances depend on the beamformer coefficients and hence the direction of beam steering. Impedance matching can be challenging, as there is not one port impedance to which the amplifiers or driving transmission lines must be matched, but the impedance changes as the beam is scanned. Fortunately, for many arrays, mutual coupling is small, and the active impedances are approximately equal to the self impedances, and the perturbation caused by mutual coupling can often be neglected in the array design process. Scan blindness. One effect observed in very large arrays with steered beams is scan blindness. As the progressive phase shift across the array is increased to steer the main beam away from broadside, a steering angle is eventually reached at which the active reflection coefficients become close to one in magnitude and the power accepted by the array decreases. This corresponds to nearly pure imaginary active impedances, which from (7.53) means that the input current is a combination of eigenvectors of Re[Z A ] with small eigenvalues and the input power (7.43) must be small Characterizing Mutual Coupling Mutual coupling in terms of S-parameters or the impedance matrix can be measured using a network analyzer. Analytically, mutual coupling can be approximated using the induced EMF method, which for a z-directed dipole leads to the integral n=1 v A,2 1 l/2 E z,21 (z)i 2 (z) dz (7.54) I 0 l/2

14 ECEn 665: Antennas and Propagation for Wireless Communications 92 where v A,2 is the voltage induced at the terminals of antenna 2 due to antenna 1, with antenna 1 excited by an input current of i A,1. E z,21 is the field radiated by antenna 1 at the location of antenna 2 if antenna 2 were absent and I 2 (z) is an equivalent current model for antenna 2 with input current I 0. The mutual impedance is Z A,21 = v A,2 /i A,1. Using the sinusoidal current model, E z,21 can be approximated and the integral for the coupling evaluated. For more accurate results, the method of moments with a thin-wire integral equation can be used. Characterizing mutual coupling for complex array antennas require numerical models based on surface integral equations, volume integral equations, the finite difference time domain method, the finite element method, or other numerical algorithms. The same simulation can be used to obtain the embedded element patterns. Modeling an array with many elements can be computationally challenging, due to the large number of required field or current sample points, mesh elements, or degrees of freedom in the simulation. For very large arrays, a periodic boundary condition can be implemented, so that only one unit cell in the array must be modeled numerically. This approach ignores array edge effects. Simulating large, finite arrays including edge effects is an important problem in the field of computational electromagnetics.

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