4.2 Mutual impedanceBasic theoryRadiating elements
(dipoles,
loops,
etc.) are usually grouped into an antenna array and fed from a common source in order to primarily reach
the desired
directivity pattern
of the radiation.
Designing the feeding system of an antenna array, even the mutual influence among the antenna elements has to be considered and a true value of
the impedance at the input of each antenna element has to be determined. Knowledge of these entries enables to determine input current of each
antenna element and even currents on the feeding ports of the array. Moreover, the structure of the feeding system can be modified to reach
a suitable value of the
input impedance
of the antenna array in its whole.
In this chapter, we concentrate on the method of calculating input impedance of a linear antenna
(dipole)
by the
method of induced electromotoric forces.
The approach is then going to be generalized in order to compute impedances of single elements in the antenna array.
Radiation impedance
Z_{Σ} of the antenna can be considered as a proportionality constant between the complex power P_{Σ}, which is radiated
by the antenna, and the squared value of the related current I

${P}_{\Sigma}={Z}_{\Sigma}{\leftI\right}^{2}={Z}_{\Sigma vst}{\left{I}_{vst}\right}^{2}={Z}_{\Sigma m}{\left{I}_{m}\right}^{2}$.

( 4.2A.1 )

As a related current, we consider the input current of the element I_{vst} or the value of the current I_{m}
in the maximum of the
standing wave.
Since currents I_{vst} and I_{m} of the same antenna (under identical conditions) usually differ, but the radiated
power stays the same, even the values of the
radiation impedance
differ  the value of Z_{Σvst} related to the input current I_{vst} and the value of Z_{Σm}
related to the current in its maximum I_{m} are not the same.
Neglecting losses of the antenna, the radiation impedance Z_{Σvst} equals to the
input impedance
of the antenna Z_{vst} computed as a ratio of input voltage U_{vst} and input current I_{vst}

${U}_{vst}={Z}_{vst}{I}_{vst}$.

( 4.2A.2 )

The abovedefined values of input impedance Z_{vst} are usually provided for basic types of antennas. These values are
valid for antennas placed in free space, i.e. in satisfactorily long distance from other antennas or objects. A detailed algorithm of
computing Z_{vst}, which is based on the radiated power, is given in the layer B.
In an antenna array, single antenna elements mutually influence themselves and impedance of each element depends on the type of elements
in the surrounding, on the way of their positioning and feeding. In order to evaluate and compute the mutual influence,
we change our view to the equation (4.2A.2).
The current I is assumed to be excited in the antenna any way. The current I is characterized by the current I_{vst}
and by a function of
current distribution.
Antenna radiates and creates a given intensity of electric field E in its surrounding, which is proportional to the current on the antenna.
A certain field intensity E_{t} appear even on the surface of the radiating antenna. At the same moment, antenna acts as the
receiving antenna, and the receive results in a given voltage at the input port of the antenna. Voltage U_{vst} in eqn.
(4.2A.2) can be therefore considered as a voltage, produced on the antenna by the receive of the own radiation. This
own radiation is proportional to the current magnitude I_{vst} at its input and the quantity Z_{vst}
(selfimpedance)
plays the role of the proportionality constant.
The abovedescribed consideration can be simply applied to the whole antenna array. Even in the array, there is a certain field intensity
E_{t} on the surface of each antenna element. This field is not created not only by the selfradiation of the antenna element but
too by the radiation of the other elements.

Fig. 4.2A.1  Array of n dipoles 

In fig. 4.2A.1, the antenna array consisting of n elements
(dipoles),
which are fed by currents I_{vst i} on their input terminals, is depicted. In analogy to (4.2A.2),
input voltages of antenna elements are given by the following set of equations

$\begin{array}{c}{U}_{vst1}={Z}_{11}{I}_{vst1}+{Z}_{12}{I}_{vst2}+\mathrm{........}+{Z}_{1n}{I}_{vstn}\\ {U}_{vst2}={Z}_{21}{I}_{vst1}+{Z}_{22}{I}_{vst2}+\mathrm{........}+{Z}_{2n}{I}_{vstn}\\ \vdots \\ {U}_{vstn}={Z}_{n1}{I}_{vst1}+{Z}_{n2}{I}_{vst2}+\mathrm{........}+{Z}_{nn}{I}_{vstn}\end{array}$.

( 4.2A.3 )

Impedance coefficients Z_{jk} (4.2A.3) express the mutual coupling between jth
and kth element and are called
mutual impedance.
Mutual impedance is a complex quantity fulfilling Z_{jk}= Z_{kj}. Its magnitude depends on the shape, on
the dimensions and on the mutual position of antenna elements, and even on their current distribution.
Coefficient Z_{jj} is called
selfimpedance
and determines relation between the current and the voltage at the input of antenna elements out of the array and equals to the radiation
impedance of a separated jth antenna element in free space.
Typical dependency of the components of the mutual impedance Z_{jk} = R_{jk}+ jX_{jk} on the distance d
(multiplied by wavenumber k = 2π/λ) between two parallel dipoles of the same length is depicted in fig.
4.2A.2.

Fig. 4.2A.2  Mutual impedance between two identical parallel dipoles in the distance d. Length of one half of the dipole is l = λ/4. 

Magnitude of
mutual impedance
depends on dimensions and on the distance of antenna elements. Real and imaginary component of
mutual impedance
can be both positive and negative and its maximum values decrease when the distance between elements d rises (the influence of very
distant elements negligible).

Fig. 4.2A.3  Couple of parallel dipoles 

Values of
mutual impedance,
which are provided for technical needs in charts, are usually related to the current in maximum. Recomputation to input terminals can be done
considering the sine
current distribution
on the antenna

${Z}_{jkvst}=\frac{{Z}_{jkm}}{{\mathrm{sin}}^{2}\left(kl\right)}$,

( 4.2A.4 )

where kl = (2π/λ) l, if antenna length l is not close to the integral multiple of λ/2.
More accurate values of
mutual impedance
can be computed using the program from the layer C. The program computes values of
mutual impedance
Z_{jk}, related to the maximum current or to the input one for a couple of two parallel dipoles of the same length and of the same
standing wave on the wire of the dipoles. The length of the dipole l, the distance between dipoles d and the magnitude
of the axial shift of the feeding ports of the dipoles h (fig. 4.2A.3) have to be entered when multiplied by
wavenumber
k = 2π/λ. Graphic representation of the magnitude of both components of mutual impedance R_{jk}
and X_{jk} on the variation of a elected quantity (l, d or h) is suitable when the influence of the spatial
arrangements of the antenna array to the impedance relations is investigated.
The same way, components of the
radiation impedance
Z_{ii} at the input of an isolated antenna element can be computed if the distance between elements d
is put to be equal to the radius of the antenna wire a.
Computing voltages and currents in the system of n antenna elements, 2n independent quantities (n voltages and n
currents) have to be determined. Considering the way of feeding, other (n1) equations can be built. For elected magnitude of a single
voltage (current), the (2n1) unknown quantities can be computed. In antenna arrays, where single antenna elements are fed by the system
of transmission lines, computations are complicated by the fact that impedance of each element depends on currents in other elements.
That way, relations on transmission lines and radiators are mutually coupled. In case of single fed element, computations are rather simple.
Deriving equations (4.2A.3) for a given situation, we have to respect the fact that orientation of voltage
U_{vst i} and of a current I_{vst i} correspond to the feeding of an element by a generator. In case of a passive
element (voltage on the input port is caused by the current I_{vst} flowing through the load Z of the input port)
is the orientation of the voltage opposite, and therefore, the sign has to be changed in (4.2A.3). For two typical
situations, corresponding sets of equations are given in fig. 4.2A.4.

$\begin{array}{l}{U}_{vst1}={Z}_{11}{I}_{vst1}+{Z}_{12}{I}_{vst2}\\ {U}_{vst2}={Z}_{21}{I}_{vst1}+{Z}_{22}{I}_{vst2}\\ {U}_{vst1}={U}_{vst2}\end{array}$
a) 
$\begin{array}{l}{U}_{vst1}={Z}_{11}{I}_{vst1}+{Z}_{12}{I}_{vst2}\\ {U}_{vst2}={Z}_{21}{I}_{vst1}+{Z}_{22}{I}_{vst2}\\ {U}_{vst1}=Z{I}_{vst1}\end{array}$
b) 
Obr. 4.2A.4  Voltages and currents in antenna systems. a) two dipoles fed by a transmission line (d=λ/2), b) system consisting passive element 

Set (4.2A.3) enables to compute the ratio between any couple of unknown quantities. Input impedance of ith element
Z_{vst i} is obtained when dividing ith eqn. (4.2A.3) by input current of ith element
I_{vst i}

${Z}_{vsti}=\frac{{U}_{vsti}}{{I}_{vsti}}={Z}_{i1}\frac{{I}_{vst1}}{{I}_{vsti}}+{Z}_{i2}\frac{{I}_{vst2}}{{I}_{vsti}}+\dots \dots +{Z}_{ii}+\dots \dots +{Z}_{ni}\frac{{I}_{vstn}}{{I}_{vsti}}$.

( 4.2A.5 )

Impedance of every element of an antenna array is therefore given by the sum of its own impedance Z_{ii}
and of submissions of the other elements, depending on the product of
mutual impedances
Z_{jk} and respective currents (amplitudes and phases) flowing through the input ports of those elements. The change of feeding
any antenna element therefore causes the change of impedances of all the antenna elements in the array. As an example, let us consider results
obtained for the arrays depicted in fig. 4.2A.4. In the case of the couple of dipoles fed by the crossed transmission
line of the length d = λ/2 (fig. 4.2A.4a), currents of the same magnitude and the same phase flow in both
the dipoles, and both of the dipoles are of the input impedance

${Z}_{vst1}={Z}_{vst2}={Z}_{11}+{Z}_{12}\frac{{I}_{vst2}}{{I}_{vst1}}={Z}_{11}+{Z}_{12}$.

( 4.2A.6 )

Impedance on the port of the dipole "1" equals to the half of the value Z_{vst1}. For the couple of the
halfwavelengthlong dipoles (kl = 90°), we ge Z_{11} = (73,1 + j42,5) Ω
and Z_{12} = (12,5 + j30) Ω. Assuming inphase feeding
(fig. 4.2A.4a), the resultant value of the impedance of both the dipoles equals to
Z_{1} = (60,6 + j12,5) Ω. Considering antiphase feeding, the resultant input
impedance equals to the difference of Z_{11} and Z_{12}, which yields the result
Z_{1} = (85,6 + j72,5) Ω. The same result is obtained for the dipole in the distance
λ/4 from the planar reflector. Impedance on the port "1" in fig. 4.2A.4a equals to the
half of the computed value Z_{1}, i.e. Z_{soust} = (30,3 + j6,2) Ω.
Input impedance of the dipole "2" in fig. 4.2A.4b is given by the relation

${Z}_{vst2}={Z}_{22}\frac{{Z}_{21}^{2}}{Z+{Z}_{11}}$.

( 4.2A.7 )

Substituted values of both the impedances Z_{11} and Z_{12} have to be related to the input current of the dipole
I_{vst} in both the cases.
