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

Authors: Frédéric Broydé; Evelyne Clavelier
Journal: International Journal of Antennas and Propagation
 (2016)

---

## Abstract

A multiple-user-port antenna tuner having the structure of a multidimensionalπ-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.


---

## Body

## 1. 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 [1–3]. 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 [4–6]. Figure 1 shows how n antennas may be coupled to a MAPMUP antenna tuner having n antenna ports and m user ports (also referred to as “radio ports”) intended to be coupled to the radio device.Figure 1
An array ofn antennas coupled to a MAPMUP antenna tuner through n uncoupled 2-conductor transmission lines.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 matrixZ
S
a
n
t of size n
×
n and the user ports present an impedance matrix Z
U of size m
×
m. A MAPMUP antenna tuner comprises p 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
U, using a selection of the reactance values of its adjustable impedance devices, so as to obtain or approximate a wanted impedance matrix Z
U
W. This tuning capability differentiates a MAPMUP antenna tuner from a MAPMUP matching network [7–13]. 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 multidimensionalπ-network has recently been described [14–16]. This antenna tuner is scalable to any n
=
m
≥
1, unlike earlier MAPMUP antenna tuners. It is shown in Figure 2 for the case n
=
m
=
4. The case n
=
m
=
1 corresponds to a classical single-antenna-port and single-user-port (SAPSUP) antenna tuner having the structure of a π-network.Figure 2
The MAPMUP antenna tuner having the structure of a multidimensionalπ-network, in a configuration having n
=
4 antenna ports, labeled AP1 to AP4, and m
=
4 user ports, labeled UP1 to UP4.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 matchZ
U
=
Z
U
W, if such reactance values exist. For a lossless multidimensional π-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 π-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.


## 2. A Tuning Computation Problem

A circular antenna array is made up ofn
=
4 side-by-side parallel dipole antennas, each having a total length of 224.8 mm. The radius of the array is 56.2 mm. Each antenna is lossless and has a 60 mm long lossy feeder. The antenna array is intended to operate in the frequency band 700 MHz to 900 MHz. At the center frequency f
c
=
800 MHz, Z
S
a
n
t is approximately given by(1)
Z
S
a
n
t
≈
8.6
-
8.9
j
3.8
+
4.9
j
1.7
+
2.2
j
3.8
+
4.9
j
3.8
+
4.9
j
8.6
-
8.9
j
3.8
+
4.9
j
1.7
+
2.2
j
1.7
+
2.2
j
3.8
+
4.9
j
8.6
-
8.9
j
3.8
+
4.9
j
3.8
+
4.9
j
1.7
+
2.2
j
3.8
+
4.9
j
8.6
-
8.9
j
Ω
.At any frequency,Z
S
a
n
t is symmetric and circulant, as shown in (1) at f
c, so that Z
S
a
n
t 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.Figure 3
Entries ofZ
S
a
n
t versus frequency: R
e
(
Z
S
a
n
t
11
) is curve A; Im
⁡
(
Z
S
a
n
t
11
) is curve B; R
e
(
Z
S
a
n
t
12
) is curve C; Im
⁡
(
Z
S
a
n
t
12
) is curve D; R
e
(
Z
S
a
n
t
13
) is curve E; Im
⁡
(
Z
S
a
n
t
13
) is curve F.The MAPMUP antenna tuner of Figure2 is used to obtain (if possible) that, at any tuning frequency f
T in this frequency range, Z
U approximates the wanted impedance matrix Z
U
W, given by (2)
Z
U
W
=
r
0
1
4
,where r
0
=
50
Ω and where, for a positive integer q, we use 1
q to denote the identity matrix of size q
×
q.LetL be the inductance matrix of the coils shown in Figure 2. Let C
A and C
U 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 π-network MAPMUP antenna tuner. In this calculation, one of the matrices L, C
A, and C
U is arbitrary. Let us, for instance, posit(3)
L
≈
2.700
0.760
0.850
0.760
0.760
2.700
0.760
0.850
0.850
0.760
2.700
0.760
0.760
0.850
0.760
2.700
n
H
.The purpose of the tuning computation is to determine values ofC
A and C
U providing the ideal match Z
U
=
Z
U
W, 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
A and C
U. Each value of C
A 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: C
A
G for the 4 grounded adjustable impedance devices, C
A
N for 4 others, and C
A
F 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
U 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: C
U
G for the 4 grounded adjustable impedance devices, C
U
N for 4 others, and C
U
F for the 2 remaining ones. These 3 values are plotted in Figure 5, versus the tuning frequency.Figure 4
Capacitances of the adjustable impedance devices which realizeC
A in the lossless MAPMUP antenna tuner: C
A
G is curve A; C
A
N is curve B; and C
A
F is curve C.Figure 5
Capacitances of the adjustable impedance devices which realizeC
U in the lossless MAPMUP antenna tuner: C
U
G is curve A; C
U
N is curve B; and C
U
F is curve C.The tuning computation problem addressed in the next sections is the computation, for the MAPMUP antenna tuner of Figure2, in the presence of losses in the coils and in the adjustable impedance devices, of C
A and C
U providing the ideal match Z
U
=
Z
U
W at different frequencies.


## 3. Equations of the Tuning Computation Problem

In this section, we present the equations of the tuning computation problem. We useC
A and G
B to denote the capacitance and conductance matrices of the m
(
m
+
1
)
/
2 adjustable impedance devices coupled to one of the antenna ports, so that their admittance matrix is G
B
+
j
ω
C
A. We use C
U and G
V to denote the capacitance and conductance matrices of the m
(
m
+
1
)
/
2 adjustable impedance devices coupled to one of the user ports, so that their admittance matrix is G
V
+
j
ω
C
U. We use L and R to denote the inductance and resistance matrices of the coils, so that their impedance matrix is R
+
j
ω
L. Since C
A and C
U are adjustable, we consider that G
B depends on C
A and on the frequency and that G
V depends on C
U 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 useG
S
a
n
t and B
S
a
n
t to denote the conductance matrix and the susceptance matrix seen by the antenna ports so that Z
S
a
n
t
-
1
=
G
S
a
n
t
+
j
B
S
a
n
t. We see that Z
U is given by(4)
Z
U
=
Z
S
a
n
t
-
1
+
j
ω
C
A
+
G
B
-
1
+
j
ω
L
+
R
-
1
+
j
ω
C
U
+
G
V
-
1
.We want to solve the problem of findingC
A, L, and C
U such that(5)
Z
U
=
r
0
1
m
,where r
0 is a resistance. The equation to be solved is (6)
g
0
1
m
-
j
ω
C
U
-
G
V
=
Z
S
a
n
t
-
1
+
j
ω
C
A
+
G
B
-
1
+
j
ω
L
+
R
-
1
,where g
0
=
1
/
r
0. Let us introduce the real matrices G
T and B
T which satisfy(7)
G
T
+
j
B
T
=
Z
S
a
n
t
-
1
+
j
ω
C
A
+
G
B
=
G
S
a
n
t
+
G
B
+
j
B
S
a
n
t
+
ω
C
Aso that (6) becomes(8)
g
0
1
m
-
j
ω
C
U
-
G
V
=
j
ω
L
+
R
+
G
T
+
j
B
T
-
1
-
1
=
1
m
+
j
ω
L
+
R
G
T
+
j
B
T
G
T
+
j
B
T
-
1
-
1
=
G
T
+
j
B
T
1
m
+
R
G
T
-
ω
L
B
T
+
j
R
B
T
+
ω
L
G
T
-
1
.As shown in Appendix of [16], if M and N are two square real matrices of size m
×
m such that M
+
j
N and M are invertible, we have(9)
M
+
j
N
-
1
=
1
m
-
j
M
-
1
N
M
+
N
M
-
1
N
-
1
.It follows that (8) becomes(10)
g
0
1
m
-
j
ω
C
U
-
G
V
=
G
T
+
j
B
T
1
m
-
j
M
-
1
N
M
+
N
M
-
1
N
-
1
,where(11)
M
=
1
m
+
R
G
T
-
ω
L
B
T
N
=
R
B
T
+
ω
L
G
T
.Thus, (6) is equivalent to(12)
g
0
1
m
-
G
V
=
G
T
+
B
T
M
-
1
N
M
+
N
M
-
1
N
-
1and(13)
ω
C
U
=
G
T
M
-
1
N
-
B
T
M
+
N
M
-
1
N
-
1
.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
A and C
U. The left-hand sides of (12) and (13) depend only on C
U and their right-hand sides depend only on C
A. We see that if G
V was independent of C
U, (12) and (13) would be uncoupled because (12) could be solved to obtain C
A and (13) could be used to directly compute C
U. Thus, we can say that (12) and (13) are coupled because G
V depends on C
U. In the case m
=
1, (12) and (13) may be simplified as shown in Appendix A.


## 4. Iterative Tuning Computation Technique

An antenna tuner makes sense only if losses are small in the adjustable impedance devices, that is, only if⦀
ω
C
A
⦀
∞
≫
⦀
G
B
⦀
∞, ⦀
G
S
a
n
t
⦀
∞
≫
⦀
G
B
⦀
∞, ⦀
ω
C
U
⦀
∞
≫
⦀
G
V
⦀
∞, and g
0
≫
⦀
G
V
⦀
∞, 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
B
=
G
V
=
0 S (that is to say, for a lossless antenna tuner):(14)
B
T
=
ω
L
-
1
+
G
A
r
0
G
A
-
1
ω
L
-
2
-
1
m
1
/
2
.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 AppendixB, which gives a possible general solution of (12): (15)
B
T
=
ω
L
-
1
1
m
+
R
G
T
±
N
N
-
1
g
0
1
m
-
G
V
-
1
G
T
N
-
1
M
+
B
T
-
1
m
1
/
2
.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
T, because(i)
according to (11), M and N depend on B
T;
(ii)
G
V depends on C
U which by (13) depends on B
T;
(iii)
by (7), G
T depends on G
B, which depends on C
A, which again by (7) depends on B
T.The algorithm is shown in the box “Algorithm1.” At each step k
≥
0 in the iteration, there is no guaranty that C
A
(
k
) and C
U
(
k
) are real and that they are positive definite. At each step k
≥
0 in the iteration, we compute F
(
Z
U
(
k
)
), where we use Z
U
(
k
) to denote the value of Z
U given by (4) for C
A
=
C
A
(
k
), G
B
=
G
B
(
C
A
(
k
)
), C
U
=
C
U
(
k
), and G
V
=
G
V
(
C
U
(
k
)
) and where, for an arbitrary impedance matrix Z of size q
×
q, the return figure F
(
Z
) is defined by [14, § VI] (16)
F
Z
=
⦀
S
Z
⦀
2
,where(17)
S
Z
=
Z
+
r
0
I
q
-
1
Z
-
r
0
I
q
=
Z
-
r
0
I
q
Z
+
r
0
I
q
-
1and 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 F
(
Z
U
(
k
)
) ends the iteration. In Algorithm 1, the sufficiently small value is 10
-
4.Algorithm 1:Iterative algorithm for computing approximate nominal values of C
A and C
U.
Input: frequency of operation, Z
S
a
n
t.
Output: approximate nominal values of C
A and C
U, maximal value of the index k, and resulting F
(
Z
U
(
k
)).
(1) compute C
A
(
0
) as the value of C
A given by (14) and then (7) in which R
=
0 Ω and G
B
=
G
V
=
0
S
(2) compute C
U
(
0
) as the value of C
U given by (7), (11) and (13) in which R
=
0 Ω, G
B
=
0 S and C
A = C
A
(
0
)
(3) k
←
0
(4) compute F(Z
U
(
0
))
(5) while (k
<
25) ∧ (F(Z
U
(
k
)) >
10
-
4)
(6)  k
←
k
+
1
(7)  compute C
A
(
k
) as the value of C
A given by (7), (11) and (15) in which G
B = G
B
(
C
A
(
k
-
1
))
andG
V
=
G
V
(
C
U
(
k
-
1
)
), and in which, except in the left hand side of (15), C
A
=
C
A
(
k
-
1
)
(8)  compute C
U
(
k
) as the value of C
U given by (7), (11) and (13) in which G
B = G
B
(
C
A
(
k
-
1
)
) and C
A
=
C
A
(
k
)
(9)  compute F
(
Z
U
(
k
))
(10) compute the nominal value of C
A corresponding to C
A
(
k
) at the frequency of operation
(11) compute the nominal value of C
U corresponding to C
U
(
k
) at the frequency of operation
(12) return said nominal value of C
A, said nominal value of C
U, k and F
(
Z
U
(
k
))A setting of each adjustable impedance device corresponds to frequency dependent valuesC
A, G
B, C
U, and G
V. At the end of the algorithm, we compute nominal values of C
A and C
U, which are frequency independent matrices which correspond to a setting of each adjustable impedance device such that the computed values of C
A and C
U 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 m
=
1.


## 5. First Tuning Computation

We use a lossy coil model based on the equivalent circuit shown in Figure6(a). According to this model, a winding of nominal inductance L
N has an impedance which is given by(18)
Z
L
=
1
1
/
j
ω
L
N
+
R
S
+
j
ω
C
+
1
/
R
P
.Figure 6
Equivalent circuits (a) for lossy winding of nominal inductanceL
N and (b) for a lossy voltage-controlled capacitor of nominal capacitance C
N.
(a)
(b)In this section, we useL
N
=
2.7 nH, R
S
≈
119
m
Ω, R
P
≈
20.7 kΩ, and C
≈
48.8 fF, these parameters being realistic for a high-Q coil. Our model of coupled lossy windings has an impedance matrix Z
L in which the diagonal entries are impedances given by (18) and the nondiagonal entries are produced by frequency independent mutual inductances. We have used(19)
Z
L
=
j
ω
L
+
R
=
Z
L
1
m
+
j
ω
0
0.76
0.85
0.76
0.76
0
0.76
0.85
0.85
0.76
0
0.76
0.76
0.85
0.76
0
n
H
.Figure7 shows the quality factor Q
L of the diagonal entries of the impedance matrix Z
L. 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.Figure 7
Quality factorQ
L of the diagonal entries of the impedance matrix Z
L and quality factor Q
C of all entries of the admittance matrix Y
C, for the antenna tuner of Section 5.We use an adjustable impedance device model which is based on the equivalent circuit shown in Figure6(b). According to this equivalent circuit, a capacitor of nominal capacitance C
N has an admittance which is given by (20)
Y
C
=
C
N
1
/
j
ω
+
ω
P
+
1
/
ω
Sin which 1
/
R
P
=
ω
P
C
N and 1
/
R
S
=
ω
S
C
N. Thus, the resulting quality factor is independent of the capacitance value. In this section, we use ω
P
=
9
×
10
6 rd/s and ω
S
=
3
×
10
12 rd/s, these parameters being reasonable for high-Q varactors. Figure 7 shows the quality factor Q
C of all entries of the admittance matrix Y
C. In the frequency range 700 MHz to 900 MHz, this quality factor satisfies 284
<
Q
C
<
289.The losses in the coils and adjustable impedance devices are moderate in this frequency range.To solve the problem set out in Section2, we have used the iterative algorithm of Section 4 (Algorithm 1) to obtain the nominal values of C
A and C
U, our program ending the iteration when F
(
Z
U
) is less than −80 dB. Each computed nominal value of C
A and C
U is symmetric and circulant. The nominal capacitances C
A
G, C
A
N, and C
A
F which realize the nominal value of C
A, as explained in Section 2, are plotted in Figure 8, versus the tuning frequency. The nominal capacitances C
U
G, C
U
N, and C
U
F which realize the nominal value of C
U, 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 k) at each tuning frequency is shown in Figure 10, and the corresponding return figure F
(
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
U below 800 MHz. This indicates that moderate losses may strongly modify the results of the tuning computation.Figure 8
Nominal capacitances of the adjustable impedance devices which realizeC
A in the lossy MAPMUP antenna tuner of Section 5, versus the tuning frequency: C
A
G is curve A; C
A
N is curve B; and C
A
F is curve C.Figure 9
Nominal capacitances of the adjustable impedance devices which realizeC
U in the lossy MAPMUP antenna tuner of Section 5, versus the tuning frequency: C
U
G is curve A; C
U
N is curve B; and C
U
F is curve C.Figure 10
Number of iterations used in the tuning computation.Figure 11
The return figure at the end 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 useC
A
=
C
A
(
0
) and C
U
=
C
U
(
0
) as the initial value of an optimization of the capacitances C
A
1
,
…
,
C
A
h of the capacitors which produce C
A and of the capacitances C
U
1
,
…
,
C
U
h of the capacitors which produce C
U, to minimize F
(
Z
U
) and obtain(21)
C
A
1
,
…
,
C
A
h
,
C
U
1
,
…
,
C
U
h
=
a
r
g
m
i
n
C
A
i
>
0
,
C
U
j
>
0
F
Z
U
.In the present case, since the symmetry of our problems requires thatC
A and C
U are symmetric and circulant, we only need to consider an optimization of the 6 parameters C
A
G, C
A
N, C
A
F, C
U
G, C
U
N, and C
U
F 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 F
(
Z
U
)
=
0, but it finds an approximate solution for which F
(
Z
U
) 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.


## 6. 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 matrixZ
S being connected to the user ports. The insertion gain of the antenna tuner evaluated for Z
S
=
r
0
1
m, denoted by G
I, is given by(22)
G
I
=
P
o
u
t
P
W
A
T
Z
S
=
r
0
1
m
,where P
o
u
t is the power delivered by the antenna ports of the antenna tuner to the antennas and P
W
A
T 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 forZ
S
=
r
0
1
m, denoted by G
T, is defined as (23)
G
T
=
P
o
u
t
P
a
v
a
Z
S
=
r
0
1
m
,where P
a
v
a denotes the power available from the multiport source. The antenna tuner being passive, we have G
T
≤
1. At the tuning frequency, if Z
U 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 P
a
v
a. In this case, G
T is the efficiency of the antenna tuner.We may define a mismatch factor without the antenna tuner evaluated forZ
S
=
r
0
1
m, denoted by M
W
A
T and given by(24)
M
W
A
T
=
P
W
A
T
P
a
v
a
Z
S
=
r
0
1
m
.Here,G
I, G
T, and M
W
A
T are functions of the column vector of the open-circuit voltages of the source, denoted by V
0. More precisely G
I, G
T, and M
W
A
T are ratios of Hermitian forms of V
0. In the case where V
0 is known, G
I, G
T, and M
W
A
T 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 G
I, G
T, and M
W
A
T. It directly follows from (22)–(24) that(25)
G
I
=
G
T
M
W
A
T
.Using an antenna tuner is advantageous from the insertion gain standpoint if and only ifG
I
>
1. Since G
T
≤
1, by (25) the use of an antenna tuner only makes sense if M
W
A
T is sufficiently low. In Figure 12, we have plotted G
I, G
T, and M
W
A
T 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 (26)
V
01
=
2
0
0
0
V
;
V
02
=
0
2
0
0
V
;
V
03
=
0
0
2
0
V
;
V
04
=
0
0
0
2
Vor 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.Figure 12
Insertion gain (curve A), transducer power gain (curve B), and mismatch factor without the antenna tuner (curve C).Forα
∈
{
1
,
…
,
4
}, let E
0
α be the electric field radiated by the antenna array in a configuration where Z
S
=
r
0
1
m and V
0 is equal to V
0
α given by (26), in which rms values are used. A plot of the average radiation intensity of E
0
α in the far field, as a function of an angle, may be referred to as a radiation pattern of user port α. Let us use a spherical coordinate system having an origin at the center of the dipole centers and a z-axis parallel to the direction of the parallel dipoles, θ being the zenith angle (i.e., the angle with respect to the z-axis) and φ being the azimuth angle, with respect to the first antenna. Thus, the dipole centers are in the plane θ
=
π
/
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, π
/
2, π, and 3
π
/
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 θ
=
π
/
2, versus φ.Figure 13
Radiation patterns of the user ports, at 800 MHz, in the planeθ
=
π
/
2, versus the azimuth angle φ: 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.LetP
R be the matrix of the self- and cross and complex powers radiated by the antenna array over all values of θ and φ, defined as follows: for α
∈
{
1
,
…
,
4
} and β
∈
{
1
,
…
,
4
}, the entry P
R
α
β of P
R is given by (27)
P
R
α
β
=
1
η
0
∫
0
π
∫
0
2
π
E
0
α
∗
E
0
β
r
2
sin
⁡
θ
d
φ
d
θ
,where η
0
≈
376.7
Ω is the intrinsic impedance of free space, where E
0
α and E
0
β are regarded as column vectors, where the star denotes the Hermitian adjoint, and where the integration is carried out at a large distance r from the antennas lying in free space. Using a numerical integration, at 800 MHz, we obtain(28)
P
R
≈
10.872
3.350
-
0.320
3.350
3.350
10.872
3.350
-
0.320
-
0.320
3.350
10.872
3.350
3.350
-
0.320
3.350
10.872
m
W
.Here, the available power of 20 mW, corresponding to any one of theV
0
α 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 G
T 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), P
R
α
β is an inner product [17, § 0.6.4] of the beams E
0
β and E
0
α, we can define the beam cosines ρ
α
β
=
P
R
α
β
/
(
P
R
α
α
P
R
β
β
)
1
/
2 of P
R. At 800 MHz, the matrix of these beam cosines is given by (29)
ρ
α
β
≈
1.000
0.308
-
0.029
0.308
0.308
1.000
0.308
-
0.029
-
0.029
0.308
1.000
0.308
0.308
-
0.029
0.308
1.000
.Let us now consider that the antenna tuner and the antennas are used for reception, a multiport load of impedance matrixZ
L 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
L
=
r
0
1
m and a plane wave of electric field amplitude 1 V/m rms impinges on the antenna array, from the direction (
θ
,
φ
) with an electric field lying in a plane containing the z-axis. A plot of the average power delivered by the user port α, as a function of an angle, may be referred to as a reception pattern of user port α. Figure 14 shows the reception patterns of the user ports, for a tuning frequency of 800 MHz and at this frequency, in the plane θ
=
π
/
2, versus φ. 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.Figure 14
Reception patterns of the user ports, at 800 MHz, in the planeθ
=
π
/
2, versus the azimuth angle φ: 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.LetP
D be the matrix of the self- and cross and complex powers delivered by the user ports, averaged over all values of θ and φ, defined as follows: for α
∈
{
1
,
…
,
4
} and β
∈
{
1
,
…
,
4
}, the entry P
D
α
β of P
D is given by (30)
P
D
α
β
=
1
4
π
r
0
∫
0
π
∫
0
2
π
U
0
α
U
¯
0
β
sin
⁡
θ
d
φ
d
θ
,where the bar indicates the complex conjugate and where U
0
α and U
0
β are entries of U
0. Thus, it is possible to write(31)
P
D
=
1
4
π
r
0
∫
0
π
∫
0
2
π
U
0
U
0
∗
sin
⁡
θ
d
φ
d
θ
.Using a numerical integration of (31), we get (32)
P
D
≈
16.125
4.969
-
0.475
4.969
4.969
16.125
4.969
-
0.475
-
0.475
4.969
16.125
4.969
4.969
-
0.475
4.969
16.125
μ
W
.The trace ofP
D 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), P
D
α
β is an inner product of the beams U
0
α and U
0
β, we can define the beam cosines P
D
α
β
/
(
P
D
α
α
P
D
β
β
)
1
/
2 of P
D. In our configuration, the beam cosines of P
D are equal to the beam cosines of P
R, 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 to1
4, as requested by the theory presented in [21, 22].


## 7. Second Tuning Computation

We now use different parameters in the loss models of Section5: L
N
=
2.7 nH, R
S
=
0.36
Ω, R
P
≈
10.1
k
Ω, C
≈
93.8
f
F, ω
P
=
37
×
10
6
r
d
/
s, and ω
S
=
650
×
10
9
r
d
/
s.Figure15 shows the quality factor Q
L of the diagonal entries of the impedance matrix Z
L. 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 Q
C of all entries of the admittance matrix Y
C. 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.Figure 15
Quality factorQ
L of the diagonal entries of the impedance matrix Z
L and quality factor Q
C of all entries of the admittance matrix Y
C, for the antenna tuner of Section 7.We have again used the iterative algorithm of Section4 (Algorithm 1) to obtain C
A and C
U, our program ending the iteration when F
(
Z
U
) 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 F
(
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.Figure 16
Number of iterations used in the tuning computation.Figure 17
The return figure at the end of the tuning computation.In Figure18, we have plotted G
I, G
T, and M
W
A
T 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 G
I and G
T 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.Figure 18
Insertion gain (curve A), transducer power gain (curve B), and mismatch factor without the antenna tuner (curve C).


## 8. 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 Section5. 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 F
(
Z
U
) at 800 MHz, for a tuning frequency of 800 MHz.Since, at the tuning frequency,F
(
Z
U
) 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 F
(
Z
U
) 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 F
(
Z
U
) at the tuning frequency, as a function of a relative capacitance variation of a single adjustable impedance device contributing to C
U. All curves of Figure 19 have a corner point at zero relative capacitance variation. The same phenomenon occurs in plots of F
(
Z
U
) at the tuning frequency, as a function of a relative capacitance variation of a single adjustable impedance device contributing to C
A. 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.Figure 19
F
(
Z
U
) at the tuning frequency, versus a relative capacitance variation of a single adjustable impedance device contributing to C
U: curve A if its initial value is C
U
G; curve B if its initial value is C
U
N; curve C if its initial value is C
U
F.Figures20 and 21 show F
(
Z
U
) in dB, at the tuning frequency, as a function of a relative capacitance variation of a single adjustable impedance device contributing to C
A or to C
U.Figure 20
F
(
Z
U
) in dB at the tuning frequency, versus a relative capacitance variation of a single adjustable impedance device contributing to C
A: curve A if its initial value is C
A
G; curve B if its initial value is C
A
N; curve C if its initial value is C
A
F.Figure 21
F
(
Z
U
) in dB at the tuning frequency, versus a relative capacitance variation of a single adjustable impedance device contributing to C
U: curve A if its initial value is C
U
G; curve B if its initial value is C
U
N; curve C if its initial value is C
U
F.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 deviationσ
C. At the tuning frequency, we have determined the statistic of F
(
Z
U
) in dB, the mean of F
(
Z
U
) in dB, denoted by m
F, and the corrected sample standard deviation of F
(
Z
U
) in dB, denoted by σ
F. The histogram of Figure 22 shows the relative frequency of F
(
Z
U
) in dB obtained for σ
C
=
1
%, with 10000 samples (of the MAPMUP antenna tuner). The assumption σ
C
=
1
% could, for instance, correspond to a ±3% tolerance defined by a 3-sigma deviation. For the statistics shown in Figure 22, we have m
F
≈
-
27.22dB and σ
F
≈
2.79dB. The minimum value of F
(
Z
U
) is −39.96 dB. It is remarkable that this minimum value is not closer to the lowest possible value of about −85.68 dB shown in Figure 11 for 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 F
(
Z
U
) is −19.00 dB, whereas for a normal distribution the probability of having F
(
Z
U
) greater than m
F
+
3
σ
F
≈
-
18.86dB would be about 1.35
×
10
-
3, corresponding to an expectation of about 13.5 for 10000 samples. Figure 23 shows m
F and σ
F, as a function of σ
C, obtained with 1000 samples for each value of σ
C.Figure 22
Histogram of the relative frequency ofF
(
Z
U
) in dB, for σ
C
=
1
%, obtained with 10000 samples, and normal distribution having the same mean and the same standard deviation (dashed curve).Figure 23
m
F and σ
F versus σ
C and linear regression lines based on the values obtained for 0.01
%
≤
σ
C
≤
10
%.Based on the foregoing, we may conclude that a specified maximum return figure ofm
F
+
3
σ
F
≈
-
18.86dB at the tuning frequency can reliably be obtained with σ
C
≤
1
%. This value of m
F
+
3
σ
F is less impressive than the return figures below −80 dB shown in Figure 11. However, this specified maximum return figure can be compared to a reasonable design target −10 dB for a wireless transmitter and to the return figure achievable with an alternative antenna tuner made up of 4 independent and uncoupled SAPSUP antenna tuners each having the structure of a π-network. Such an antenna tuner corresponds to the schematic diagram of Figure 2, with the constraint that C
A,L, and C
U are diagonal matrices. Using the same L
N and Y
C 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 C
A
G and C
U
G, 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 [4–6], 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 open-loop control, like the ones considered in [24–26]. With open-loop control, the effects of component value variations are crucial, because they are not inherently compensated.


## 9. Conclusion

We have studied the design of a MAPMUP antenna tuner having the structure of a multidimensionalπ-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 π-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 multidimensionalπ-network antenna tuner may be able to provide an ideal match Z
U
=
r
0
1
m 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 π-network and multidimensional π-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 [24–26], 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 ofπ-network and multidimensional π-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 π-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 π-network antenna tuners, with arbitrary losses.



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*Source: 4758486-2016-11-20.xml*