A Tuning Computation Technique for a Multiple-Antenna-Port and Multiple-User-Port Antenna Tuner

. A multiple-user-port antenna tuner having the structure of a multidimensional 𝜋 -network has recently been disclosed, together with design equations which assume lossless circuit elements. This paper is about the design of this type of antenna tuner, when losses are taken into account in each circuit element of the antenna tuner. The problem to be solved is the tuning computation, the intended results of which are the reactance values of the adjustable impedance devices of the antenna tuner, which provide an ideal match, if such reactance values exist. An efficient iterative tuning computation technique is presented and demonstrated.


Introduction
A multiple-antenna-port and multiple-user-port (MAPMUP) antenna tuner is intended to be inserted between several antennas and a radio device which uses these antennas simultaneously in the same frequency band [1][2][3].The radio device may be a receiver, a transmitter, or a transceiver for single-user MIMO radio communication, and the antenna tuner is typically adjusted automatically [4][5][6].Figure 1 shows how  antennas may be coupled to a MAPMUP antenna tuner having  antenna ports and  user ports (also referred to as "radio ports") intended to be coupled to the radio device.
In a frequency band of intended operation, with respect to the antenna ports and the user ports, the antenna tuner must behave as a passive linear device and its losses should be as low as possible.The antenna ports see an impedance matrix Z Sant of size  ×  and the user ports present an impedance matrix Z  of size  × .A MAPMUP antenna tuner comprises  adjustable impedance devices, each of which may be any component having two terminals which behave as the terminals of a passive linear two-terminal circuit element and which present a reactance which is adjustable by mechanical or electrical means.The function of the antenna tuner is to allow an adjustment of Z  , using a selection of the reactance values of its adjustable impedance devices, so as to obtain or approximate a wanted impedance matrix Z  .This tuning capability differentiates a MAPMUP antenna tuner from a MAPMUP matching network [7][8][9][10][11][12][13]. Some aspects of the tuning capability of MAPMUP antenna tuners are covered in [14], which shows that the need for a suitable tuning capability (such as a full tuning capability) makes the design of a MAPMUP antenna tuner completely different from the design of a MAPMUP matching network.
A MAPMUP antenna tuner having the structure of a multidimensional -network has recently been described [14][15][16].This antenna tuner is scalable to any  =  ≥ 1, unlike earlier MAPMUP antenna tuners.It is shown in Figure 2 for the case  =  = 4.The case  =  = 1 corresponds to a classical single-antenna-port and singleuser-port (SAPSUP) antenna tuner having the structure of a -network.
This paper is about the design of this type of MAPMUP antenna tuner.The problem to be solved is the tuning computation, the intended results of which are the adjustable impedance device reactance values which provide an ideal match Z  = Z  , if such reactance values exist.For a lossless multidimensional -network antenna tuner, a set of closed-form tuning computation formulas was proven in [16].In this paper, we shall present an efficient iterative tuning computation technique, which can be used when losses are present in the circuit element of the antenna tuner.Section 2 presents a tuning computation problem and the solution obtained for a lossless multidimensional -network antenna tuner.Sections 3 and 4 explain the theory of the iterative algorithm.Sections 5 and 7 present solutions to the problem of Section 2, for different losses in the circuit elements.Sections 6 and 8 explore the main characteristics of the MAPMUP antenna tuner obtained in Section 5.

A Tuning Computation Problem
A circular antenna array is made up of  = 4 side-byside parallel dipole antennas, each having a total length of  ) ) ) Ω. ( At any frequency, Z Sant is symmetric and circulant, as shown in (1) at   , so that Z Sant is fully determined by the first three entries of its first row.These entries are plotted in the frequency range 700 MHz to 900 MHz, in Figure 3.
The MAPMUP antenna tuner of Figure 2 is used to obtain (if possible) that, at any tuning frequency   in this frequency range, Z  approximates the wanted impedance matrix Z  , given by where  0 = 50 Ω and where, for a positive integer , we use 1  to denote the identity matrix of size  × .
Let L be the inductance matrix of the coils shown in Figure 2. Let C  and C  be the capacitance matrices of the adjustable impedance devices shown on the left and on the right, respectively, in Figure 2. In [16], formulas for the tuning computation were derived, for a lossless multidimensional network MAPMUP antenna tuner.In this calculation, one of ( The purpose of the tuning computation is to determine values of C  and C  providing the ideal match Z  = Z  , as a function of the tuning frequency.If we ignore losses in the circuit element of the antenna tuner, we can use (3) and formulas (7) and ( 9) of [16], to obtain C  and C  .Each value of C  is symmetric and circulant.Thus, it corresponds to 3 values of the capacitances of the 10 adjustable impedance devices coupled to one of the antenna ports:   for the 4 grounded adjustable impedance devices,   for 4 others, and   for the 2 remaining ones.These 3 values are plotted in Figure 4, versus the tuning frequency.In the same way, each value of C  is symmetric and circulant, so that it corresponds to 3 values of the capacitances of the 10 adjustable impedance devices coupled to one of the user ports:   for the 4 grounded adjustable impedance devices,   for 4 others, and   for the 2 remaining ones.These 3 values are plotted in Figure 5, versus the tuning frequency.
The tuning computation problem addressed in the next sections is the computation, for the MAPMUP antenna tuner of Figure 2, in the presence of losses in the coils and in the adjustable impedance devices, of C  and C  providing the ideal match Z  = Z  at different frequencies.

Equations of the Tuning Computation Problem
In this section, we present the equations of the tuning computation problem.We use C  and G  to denote the capacitance and conductance matrices of the ( + 1)/2 adjustable impedance devices coupled to one of the antenna ports, so that their admittance matrix is G  + C  .We use C  and G  to denote the capacitance and conductance matrices of the ( + 1)/2 adjustable impedance devices coupled to one of the user ports, so that their admittance matrix is G  +C  .We use L and R to denote the inductance and resistance matrices of the coils, so that their impedance matrix is R+L.Since C  and C  are adjustable, we consider that G  depends on C  and on the frequency and that G  depends on C  and on the frequency.Since different choices of coils at the design stage, to obtain different values of L, entail different losses, we may consider that R depends on L and on the frequency.We use G Sant and B Sant to denote the conductance matrix and the susceptance matrix seen by the antenna ports so that Z −1 Sant = G Sant + B Sant .We see that Z  is given by We want to solve the problem of finding C  , L, and C  such that where  0 is a resistance.The equation to be solved is where  0 = 1/ 0 .Let us introduce the real matrices G  and B  which satisfy International Journal of Antennas and Propagation so that (6) becomes As shown in Appendix of [16], if M and N are two square real matrices of size  ×  such that M + N and M are invertible, we have It follows that (8) becomes where Thus, ( 6) is equivalent to and At this stage, we have separated the nonlinear complex matrix equation ( 6) into the coupled nonlinear real matrix equations (12) and (13).For a known L, R is also known, so that the unknowns are C  and C  .The left-hand sides of ( 12) and ( 13) depend only on C  and their right-hand sides depend only on C  .We see that if G  was independent of C  , (12) and ( 13) would be uncoupled because (12) could be solved to obtain C  and (13) could be used to directly compute C  .Thus, we can say that ( 12) and ( 13) are coupled because G  depends on C  .In the case  = 1, (12) and ( 13) may be simplified as shown in Appendix A.

Iterative Tuning Computation Technique
An antenna tuner makes sense only if losses are small in the adjustable impedance devices, that is, only if where ‹A‹ ∞ is the maximum row sum matrix norm of a matrix A [17, § 5.6.5].Appendix B explains in detail the tuning computation algorithm which will now be concisely presented.
We shall need an equation which gives a possible solution of (12) in the special case R = 0 Ω and G  = G  = 0 S (that is to say, for a lossless antenna tuner): This equation is a consequence of (B.8) of Appendix B, but it is also equation ( 26) of [16].
We shall also need an equation derived in Appendix B, which gives a possible general solution of ( 12): ) .
In ( 14) and ( 15) and in Appendix B, the power 1/2 of a matrix denotes any square root [18, § 6.4.12], such as the primary matrix function associated with a suitable choice of square root in C, this principal matrix function being defined for any nonsingular matrix [18, § 6.2.14].We note that, unlike ( 14), ( 15) cannot be used to directly compute B  , because (i) according to (11), M and N depend on B  ; (ii) G  depends on C  which by ( 13) depends on B  ; (iii) by ( 7), G  depends on G  , which depends on C  , which again by (7) depends on B  .
The algorithm is shown in the box "Algorithm 1."At each step  ≥ 0 in the iteration, there is no guaranty that C  () and C  () are real and that they are positive definite.At each step  ≥ 0 in the iteration, we compute (Z  () ), where we use Z  () to denote the value of Z  given by (4) for ) , and G  = G  (C  () ) and where, for an arbitrary impedance matrix Z of size  × , the return figure (Z) is defined by [14, § VI] where and the spectral norm ‹A‹ 2 of a square matrix A is the largest singular value of A [17, § 5.6.6].A sufficiently small value of (Z  () ) ends the iteration.In Algorithm 1, the sufficiently small value is 10 −4 .A setting of each adjustable impedance device corresponds to frequency dependent values C  , G  , C  , and G  .At the end of the algorithm, we compute nominal values of C  and C  , which are frequency independent matrices which correspond to a setting of each adjustable impedance device such that the computed values of C  and C  are obtained at the frequency of operation.
The selection of different square roots in ( 14) and ( 15) can be used to obtain different solutions.Appendix C shows how the iterative algorithm can be simplified in the special case  = 1.
Input: frequency of operation, Z Sant .Output: approximate nominal values of C  and C  , maximal value of the index , and resulting (Z  () ).
(1) compute C  (0) as the value of C  given by ( 14) and then (7) in which R = 0 Ω and G  = G  = 0 S (2) compute C  (0) as the value of C  given by ( 7), ( 11) and ( 13) in which R = 0 Ω, G  = 0 S and computeC  () as the value of C  given by ( 7), ( 11) and (15) in which and ), and in which, except in the left hand side of (15), computeC  () as the value of C  given by ( 7), ( 11) and ( 13) in which

First Tuning Computation
We use a lossy coil model based on the equivalent circuit shown in Figure 6(a).According to this model, a winding of nominal inductance   has an impedance which is given by In this section, we use   = 2.7 nH,   ≈ 119 mΩ,   ≈ 20.7 kΩ, and  ≈ 48.8 fF, these parameters being realistic for a high-Q coil.Our model of coupled lossy windings has an impedance matrix Z  in which the diagonal entries are impedances given by ( 18) and the nondiagonal entries are produced by frequency independent mutual inductances.We have used Figure 7 shows the quality factor   of the diagonal entries of the impedance matrix Z  .This quality factor is not very high, since it varies from about 94 to about 117 in the frequency range 700 MHz to 900 MHz.
We use an adjustable impedance device model which is based on the equivalent circuit shown in Figure 6(b).According to this equivalent circuit, a capacitor of nominal capacitance   has an admittance which is given by

International Journal of Antennas and Propagation
in which 1/  =     and 1/  =     .Thus, the resulting quality factor is independent of the capacitance value.In this section, we use   = 9 × 10 6 rd/s and   = 3 × 10 12 rd/s, these parameters being reasonable for high-Q varactors.Figure 7 shows the quality factor   of all entries of the admittance matrix Y  .In the frequency range 700 MHz to 900 MHz, this quality factor satisfies 284 <   < 289.
The losses in the coils and adjustable impedance devices are moderate in this frequency range.
To solve the problem set out in Section 2, we have used the iterative algorithm of Section 4 (Algorithm 1) to obtain the nominal values of C  and C  , our program ending the iteration when (Z  ) is less than −80 dB.Each computed nominal value of C  and C  is symmetric and circulant.The nominal capacitances   ,   , and   which realize the nominal value of C  , as explained in Section 2, are plotted in Figure 8, versus the tuning frequency.The nominal capacitances   ,   , and   which realize the nominal value of C  , as explained in Section 2, are plotted in Figure 9 versus the tuning frequency.The number of iterations used in the algorithm (i.e., the maximum value of ) at each tuning frequency is shown in Figure 10, and the corresponding return figure (Z) is shown in Figure 11.A comparison of Figure 8 with Figure 4 and of Figure 9 with Figure 5 shows large differences for C  below 800 MHz.This indicates that moderate losses may strongly modify the results of the tuning computation.
Instead of an iterative algorithm, we can also consider a brute force numerical optimization to perform the tuning computation.For instance, it is possible to use and C  = C  (0) as the initial value of an optimization of the capacitances  1 , . . .,  ℎ of the capacitors which produce C  and of the capacitances  1 , . . .,  ℎ of the capacitors which produce C  , to minimize (Z  ) and obtain ( 1 , . . .,  ℎ ,  1 , . . .,  ℎ ) = arg min In the present case, since the symmetry of our problems requires that C  and C  are symmetric and circulant, we only need to consider an optimization of the 6 parameters   ,   ,   ,   ,   , and   defined above.We have also applied this approach to our example, using a commercially available nonlinear numerical solver, at a single frequency, because the convergence is slow.At this frequency (800 MHz), the solver does not reach the exact solution (Z  ) = 0, but it finds an approximate solution for which (Z  ) is about −64 dB.
Moreover, if the problem did not have a rotational symmetry, the optimization approach would become unpractical, because a 20-parameter optimization would be needed (subject to the so-called "curse of dimensionality").In contrast, the proposed iterative algorithm is unaffected by a lack of symmetry.

Performance of the Antenna Tuner
Let us first consider that the antenna tuner and the antennas are used for emission, a multiport source of internal impedance matrix Z  being connected to the user ports.The insertion gain of the antenna tuner evaluated for Z  =  0 1  , denoted by   , is given by where  out is the power delivered by the antenna ports of the antenna tuner to the antennas and  WAT is the power received by the antennas if the antenna tuner is not present, that is, if the antennas are directly connected to the multiport source.
The insertion gain is a significant measure of the benefits of the antenna tuner.
The transducer power gain of the antenna tuner evaluated for Z  =  0 1  , denoted by   , is defined as where  ava denotes the power available from the multiport source.The antenna tuner being passive, we have   ≤ 1.
At the tuning frequency, if Z  closely approximates a wanted impedance matrix providing maximum power transfer (as is the case in Section 5, according to Figure 11), then the power received by the user ports of the antenna tuner is near  ava .In this case,   is the efficiency of the antenna tuner.
We may define a mismatch factor without the antenna tuner evaluated for Z  =  0 1  , denoted by  WAT and given by Here,   ,   , and  WAT are functions of the column vector of the open-circuit voltages of the source, denoted by V 0 .More precisely   ,   , and  WAT are ratios of Hermitian forms of V 0 .In the case where V 0 is known,   ,   , and  WAT can be computed.In the case where V 0 is not known, it may be considered as random complex vectors.In this case, if we had suitable information on the statistics of V 0 , we could derive the expectations of   ,   , and  WAT .It directly follows from ( 22)- (24) that Using an antenna tuner is advantageous from the insertion gain standpoint if and only if   > 1.Since   ≤ 1, by (25) the use of an antenna tuner only makes sense if  WAT is sufficiently low.In Figure 12, we have plotted   ,   , and  WAT as a function of the tuning frequency, for the tuning solution shown in Figures 8 and 9, and for V 0 equal to V 01 or V 02 or V 03 or V 04 given by or to any multiple of any one of these vectors.The relationship ( 25) is visible in Figure 12.Since the insertion gain is greater than 0 dB in most of the frequency band 700 MHz to 900 MHz, this antenna tuner does not seem ridiculous from the insertion gain standpoint.
For  ∈ {1, . . ., 4}, let E 0 be the electric field radiated by the antenna array in a configuration where Z  =  0 1  and V 0 is equal to V 0 given by (26), in which rms values are used.A plot of the average radiation intensity of E 0 in the far field, as a function of an angle, may be referred to as a radiation pattern of user port .Let us use a spherical coordinate system having an origin at the center of the dipole centers and a -axis parallel to the direction of the parallel dipoles,  being the zenith angle (i.e., the angle with respect to the -axis) and  being the azimuth angle, with respect to the first antenna.Thus, the dipole centers are in the plane International Journal of Antennas and Propagation  = /2 orthogonal to the dipole antennas, which is their plane of maximum far field radiation, and the azimuths of the centers of antennas 1 to 4 are 0, /2, , and 3/2, respectively.
Figure 13 shows the radiation patterns of the user ports, for a tuning frequency of 800 MHz and at this frequency, in the plane  = /2, versus .Let P  be the matrix of the self-and cross and complex powers radiated by the antenna array over all values of  and , defined as follows: for  ∈ {1, . . ., 4} and  ∈ {1, . . ., 4}, the entry   of P  is given by where  0 ≈ 376.7 Ω is the intrinsic impedance of free space, where E 0 and E 0 are regarded as column vectors, where the star denotes the Hermitian adjoint, and where the integration is carried out at a large distance  from the antennas lying in free space.Using a numerical integration, at 800 MHz, we obtain Here, the available power of 20 mW, corresponding to any one of the V 0 given by ( 26), leads to a power of about 15.756 mW at the output of the antenna tuner (in line with the value of   shown in Figure 12), but only 10.872 mW reaches the antennas and is radiated, because of feeder loss.The feeder loss of about 1.61 dB is much larger than the product of the feeder attenuation constant and the feeder length, equal to 0.20 dB at 800 MHz.This effect is similar to, but more complex than, the additional feeder loss caused by the high standing-wave ratio associated with the use of SAPSUP antenna tuners [19, ch. 3].
Since, by (27),   is an inner product [17, § 0.6.4] of the beams E 0 and E 0 , we can define the beam cosines   =   /(    ) 1/2 of P  .At 800 MHz, the matrix of these beam cosines is given by ) . ( Let us now consider that the antenna tuner and the antennas are used for reception, a multiport load of impedance matrix Z  being connected to the user ports.Let U 0 be the column vector of the rms voltages delivered by the user ports in a configuration where Z  =  0 1  and a plane wave of electric field amplitude 1 V/m rms impinges on the antenna array, from the direction (, ) with an electric field lying in a plane containing the -axis.A plot of the average power delivered by the user port , as a function of an angle, may be referred to as a reception pattern of user port . Figure 14 shows the reception patterns of the user ports, for a tuning frequency of 800 MHz and at this frequency, in the plane  = /2, versus .The shapes of the reception patterns of the user ports correspond to those of the radiation patterns of the user ports.This is because, for each user port, the reception pattern and the radiation pattern correspond to the directivity pattern of a single-port antenna made up of all items shown in Figure 1, plus 50 Ω resistors connected to the other user ports.
Let P  be the matrix of the self-and cross and complex powers delivered by the user ports, averaged over all values of  and , defined as follows: for  ∈ {1, . . ., 4} and  ∈ {1, . . ., 4}, the entry   of P  is given by where the bar indicates the complex conjugate and where  0 and  0 are entries of U 0 .Thus, it is possible to write Using a numerical integration of (31), we get The trace of P  is the power delivered by the user ports, averaged over all directions of arrival of a plane wave of electric field amplitude 1 V/m rms impinging on the antenna array.
Since, by (30),   is an inner product of the beams  0 and  0 , we can define the beam cosines   /(    ) 1/2 of P  .In our configuration, the beam cosines of P  are equal to the beam cosines of P  , so that they are, at 800 MHz, given by (29).We note that these beam cosines are similar to, but different from, the correlation coefficients given by ( 29) or (30) of [20].The beam cosines are sometimes referred to as "orthogonality coefficients."The values of the beam cosines indicate that the beams are not far from being orthogonal, but they are not orthogonal either.
We have checked that if we remove the losses of the feeders and perform a new tuning computation for a lossless antenna tuner, we obtain a matrix of the beam cosines equal to 1 4 , as requested by the theory presented in [21,22].
Figure 15 shows the quality factor   of the diagonal entries of the impedance matrix Z  .Unfortunately, this quality factor is not high, since it varies from about 31.6 to about 39.5 in the frequency range 700 MHz to 900 MHz. Figure 15 also shows the quality factor   of all entries of the admittance matrix Y  .In the frequency range 700 MHz to 900 MHz, this quality factor is close to 66. Thus, the losses in the coils and adjustable impedance devices are higher than in Section 5 and very substantial.This is good for testing our iterative algorithm, because higher losses are less favorable for convergence.
We have again used the iterative algorithm of Section 4 (Algorithm 1) to obtain C  and C  , our program ending the iteration when (Z  ) is less than −80 dB.The number of iterations needed to obtain this required accuracy at each tuning frequency is shown in Figure 16, and the corresponding return figure (Z) is shown in Figure 17.A comparison of Figure 10 with Figure 16 shows that the number of iterations has increased.However, the algorithm still converges rapidly.
In Figure 18, we have plotted   ,   , and  WAT as a function of the tuning frequency, for the tuning solution obtained here, and for V 0 equal to V 01 or V 02 or V 03 or V 04 given by (26) or to any multiple of any one of these vectors.A comparison of Figure 12 with Figure 18 shows that   and   have decreased.Since the insertion gain is below 0 dB below about 790 MHz, this antenna tuner is not satisfactory from the insertion gain standpoint.

Effects of Capacitance Deviations
To ensure that the tuning computation technique is applicable to real components, we now study the criticality of the calculated values, for the MAPMUP antenna tuner designed in Section 5. We consider deviations of one or more of the capacitances of its 20 adjustable impedance devices, from the computed ones, and we investigate the effects of the deviations on the return figure (Z  ) at 800 MHz, for a tuning frequency of 800 MHz.Since, at the tuning frequency, (Z  ) practically reaches its minimum value of 0, it would, if it was differentiable, have all its partial derivatives with respect to the capacitances practically equal to zero, at the tuning frequency.Thus, each of the 20 absolute sensitivities of (Z  ) with respect to the capacitance of one of the adjustable impedance devices would practically be zero, at the tuning frequency [23, ch. 68]. Figure 19 shows (Z  ) at the tuning frequency, as a function of a relative capacitance variation of a single adjustable impedance device contributing to C  .All curves of Figure 19 have a corner point at zero relative capacitance variation.The same phenomenon occurs in plots of (Z  ) at the tuning frequency, as a function of a relative capacitance variation of a single adjustable impedance device contributing to C  .For much smaller capacitance variations, the corner points vanish, but no useful absolute or relative sensitivity with respect to the capacitances of the adjustable impedance devices can be defined.
Figures 20 and 21 show (Z  ) in dB, at the tuning frequency, as a function of a relative capacitance variation of a single adjustable impedance device contributing to C  or to C  .
To obtain a better idea of the effect of simultaneous deviations of the capacitance of the 20 adjustable impedance devices, we have assumed independent normally distributed capacitance deviations, with a zero mean deviation from the computed values, and a specified relative standard deviation   .At the tuning frequency, we have determined the statistic of (Z  ) in dB, the mean of (Z  ) in dB, denoted by   , and the corrected sample standard deviation of (Z  ) in dB, denoted by   .The histogram of Figure 22 shows the relative frequency of (Z  ) in dB obtained for   = 1%, with 10000 samples (of the MAPMUP antenna tuner).The assumption   = 1% could, for instance, correspond to a ±3% tolerance   independent and uncoupled SAPSUP antenna tuners each having the structure of a -network.Such an antenna tuner corresponds to the schematic diagram of Figure 2, with the constraint that C  , L, and C  are diagonal matrices.
Using the same   and   as in Section 5, we obtain, for this alternative antenna tuner, a minimum return figure of −5.90 dB at 800 MHz, obtained for nominal values of   and   , respectively, equal to about 25.36 pF and 24.02 pF.Thus, the specified maximum return figure is much better than the return figure achievable with the alternative antenna tuner using optimum capacitances.
The antenna tuners considered in this paper are typically intended to be adjusted automatically.In an automatic tuning system using closed-loop control, like the ones considered in [4][5][6], deviations of the adjustable impedance device capacitances caused by manufacturing tolerances, temperature, or other external causes may be automatically compensated.This is not the case in an automatic tuning system using openloop control, like the ones considered in [24][25][26].With openloop control, the effects of component value variations are crucial, because they are not inherently compensated.

Conclusion
We have studied the design of a MAPMUP antenna tuner having the structure of a multidimensional -network, when losses are taken into account in all circuit elements of the antenna tuner.Our work also covers a SAPSUP having the structure of a -network.The main aspect that we have addressed is the tuning computation, the intended results of which are the adjustable impedance device reactance values which provide an ideal match of the antenna tuner, if such reactance values exist.
An efficient iterative tuning computation technique was explained and demonstrated in two examples.It can be used with circuit element losses depending in any way on the frequency and on the reactance of the circuit element.
In the examples used to demonstrate the iterative tuning computation algorithm, we have chosen to use moderate losses in the first example and higher losses in the second example.The algorithm converges rapidly in both cases, but the speed decreases with increased losses.We have found that moderate losses may substantially modify the results of the tuning computation.In other words, the adjustable impedance device reactance values which provide an ideal match may, in the presence of moderate losses, be significantly different from what they would be if losses were not present.
The fact that the iterative tuning computation technique converges shows that the multidimensional -network antenna tuner may be able to provide an ideal match Z  =  0 1  which realizes decoupling and matching.In the investigated examples, this is obtained at any frequency in a wide frequency band of intended operation.The iterative tuning computation technique can be used as a tool to design network and multidimensional -network antenna tuners.In addition, it can be used in automatic tuning systems based on an open-loop control scheme, such as the ones described in [24][25][26], to obtain the tuning control signals which determine the reactance of each of the adjustable impedance devices.
The design of a MAPMUP antenna tuner is a new field, with many unanswered questions.For instance, the selection of the inductance values of -network and multidimensional -network antenna tuners has not been addressed in this paper, since we have solved ( 12) and ( 13) for a known L. The literature in this area is limited to SAPSUP -network antenna tuners, and available design techniques (except optimization using a general-purpose circuit optimizer, which is extremely slow) ignore losses in the circuit elements of the antenna tuner or do not fully take them into account [27,28].We believe that this work could also be used to solve this problem, for multidimensional -network antenna tuners, with arbitrary losses.port sees an impedance  Sant and the user port presents an impedance   .We use   and   to denote the capacitance and the conductance of the adjustable impedance device coupled to the antenna port, so that its admittance is   +   .We use   and   to denote the capacitance and the conductance of the adjustable impedance device coupled to the user port, so that its admittance is   +   .We use  and  to denote the inductance and the resistance of the coil, so that its impedance is +.We consider that   depends on   and on the frequency,   depends on   and on the frequency, and  depends on  and on the frequency.
We use  Sant and  Sant to denote the conductance and the susceptance seen by the antenna port, so that 1/ Sant =  Sant +  Sant .Here, (4)

B. Derivation of a Tuning Computation Technique for a MAPMUP Antenna Tuner
In the case of a multidimensional -network MAPMUP antenna tuner, the results of Section 4 may be established as follows.
Assuming that losses are small in the adjustable impedance devices, an iterative algorithm could in principle be used to solve (12) and ( 13): at each iteration of this algorithm, C  and C  are determined using (12) and (13), respectively, in which G  and G  are regarded as constants which are updated at each iteration, based on the values of C  and C  determined at the previous iteration, G  and G  being initially set to the null matrix of size  × .
A difficulty is that we are not able to easily compute a value of C  which is a solution of (12), for arbitrary values of G  and G  .To address this question, (12) can be manipulated to obtain Thus, a possible solution of (12) may satisfy in which, according to (11), N unfortunately depends on B  .However, it may easily be shown that, in the special case where R is the null matrix, using (11) we get

Figure 1 :Figure 2 :
Figure 1: An array of  antennas coupled to a MAPMUP antenna tuner through  uncoupled 2-conductor transmission lines.

Figure 3 :
Figure 3: Entries of Z Sant versus frequency: Re(Z Sant 11 ) is curve A; Im(Z Sant 11 ) is curve B; Re(Z Sant 12 ) is curve C; Im(Z Sant 12 ) is curve D; Re(Z Sant 13 ) is curve E; Im(Z Sant 13 ) is curve F.

Figure 4 :
Figure 4: Capacitances of the adjustable impedance devices which realize C  in the lossless MAPMUP antenna tuner:   is curve A;   is curve B; and   is curve C.

Figure 5 :
Figure 5: Capacitances of the adjustable impedance devices which realize C  in the lossless MAPMUP antenna tuner:   is curve A;   is curve B; and   is curve C.

Figure 6 :
Figure 6: Equivalent circuits (a) for lossy winding of nominal inductance   and (b) for a lossy voltage-controlled capacitor of nominal capacitance   .

Figure 7 :
Figure 7: Quality factor   of the diagonal entries of the impedance matrix Z  and quality factor   of all entries of the admittance matrix Y  , for the antenna tuner of Section 5.

Figure 8 :
Figure 8: Nominal capacitances of the adjustable impedance devices which realize C  in the lossy MAPMUP antenna tuner of Section 5, versus the tuning frequency:   is curve A;   is curve B; and   is curve C.

Figure 9 :Figure 10 :Figure 11 :
Figure 9: Nominal capacitances of the adjustable impedance devices which realize C  in the lossy MAPMUP antenna tuner of Section 5, versus the tuning frequency:   is curve A;   is curve B; and   is curve C.

Figure 12 :
Figure 12: Insertion gain (curve A), transducer power gain (curve B), and mismatch factor without the antenna tuner (curve C).

Figure 13 :
Figure 13: Radiation patterns of the user ports, at 800 MHz, in the plane  = /2, versus the azimuth angle : the pattern of user port 1 is curve A; the pattern of user port 2 is curve B; the pattern of user port 3 is curve C; the pattern of user port 4 is curve D; and the average of curves A to D is curve E.

Figure 14 :
Figure 14: Reception patterns of the user ports, at 800 MHz, in the plane  = /2, versus the azimuth angle : the pattern of user port 1 is curve A; the pattern of user port 2 is curve B; the pattern of user port 3 is curve C; the pattern of user port 4 is curve D; and the average of curves A to D is curve E.

Figure 15 :Figure 16 :Figure 17 :Figure 18 :
Figure 15: Quality factor   of the diagonal entries of the impedance matrix Z  and quality factor   of all entries of the admittance matrix Y  , for the antenna tuner of Section 7.

Figure 20 :
Figure20: (Z  ) in dB at the tuning frequency, versus a relative capacitance variation of a single adjustable impedance device contributing to C  : curve A if its initial value is   ; curve B if its initial value is   ; curve C if its initial value is   .

Figure 11 .Figure 22 :Figure 23 :
Figure 22: Histogram of the relative frequency of (Z  ) in dB, for   = 1%, obtained with 10000 samples, and normal distribution having the same mean and the same standard deviation (dashed curve).
Figure21: (Z  ) in dB at the tuning frequency, versus a relative capacitance variation of a single adjustable impedance device contributing to C  : curve A if its initial value is   ; curve B if its initial value is   ; curve C if its initial value is   .definedbya3-sigmadeviation.For the statistics shown in Figure22, we have   ≈ −27.22 dB and   ≈ 2.79 dB.The minimum value of (Z  ) is −39.96dB.It is remarkable that this minimum value is not closer to the lowest possible value of about −85.68 dB shown in Figure11for 800 MHz.This is because each sample comprises 20 normally distributed capacitances, so that the probability of having all of these capacitances close to their respective computed values is very small.The maximum value of (Z  ) is −19.00 dB, whereas for a normal distribution the probability of having (Z  ) greater than   + 3  ≈ −18.86 dB would be about 1.35 × 10 −3 , corresponding to an expectation of about 13.5 for 10000 samples.Figure23shows   and   , as a function of   , obtained with 1000 samples for each value of   .Based on the foregoing, we may conclude that a specified maximum return figure of   +3  ≈ −18.86 dB at the tuning frequency can reliably be obtained with   ≤ 1%.This value of   + 3  is less impressive than the return figures below −80 dB shown in becomes  = ((( −1 Sant +   +   ) −1 +  + )We want to solve the problem of finding   , , and   such that   is equal to a wanted resistance  0 .The equation to be solved is 0 −   −   = (( −1 Sant +   +   ) −1 +  + )Let us introduce the reals   and   which satisfy  +   = 1  Sant +   +   =  Sant +   +  [ Sant +   ] .−   ) 2 + (  +   ) 2 , (A.4)   =  ( 2  +  2  ) −   (1 +   −   )2 + (  +   ) 2(A.5) which replace (12) and (13).