# Derivation of Antenna $Q$-factor based on Antenna Scattering-Matrix Theory

Julien de Rosny, and François Sarrazin Manuscript created March 14,
2024.Julien de Rosny is with ESPCI Paris, PSL Research University, CNRS,
Institut Langevin, Paris, France.François Sarrazin is with Univ Rennes, INSA
Rennes, CNRS, IETR-UMR 6164, F-35000 Rennes.This work was support in part by
the French “Agence Nationale de la Recherche" (ANR) under Grant
ANR-22-CPJ1-0070-01.

###### Abstract

A radio antenna is primarily designed to convert electromagnetic waves into
electrical current and vice versa. However, a part of the incident wavefield
is scattered due to structural effects and reflected power at the antenna’s
electrical port. Because the latter depends on the load impedance, an antenna
can also be referred to as a loaded scatterer. Its interaction with
electromagnetic waves is defined by means of absorption and scattering cross-
sections (ACS and SCS). When immersed in a diffuse field such as the one
generated within a reverberation chamber (RC), the impact of the load antenna
results from averaged properties over incident angles. One fundamental
quantity is the antenna quality factor (Q-factor) directly releated to the
averaged ACS. Current formulations of Q-factor are based on different power
budget analysis which do not enable the consideration of wave interferences
between the ingoing and outgoing fields. Additionally, the structural
component is always neglected in the current formulations. In this paper, we
introduce a rigorous formulation of the antenna Q-factor which takes into
account the aforementioned effects. The antenna is modeled using the
scattering-matrix theory which linearly links the ingoing and outgoing waves
in terms of spherical harmonics expansion. The derived theory is validated
using several numerical simulations based on a Method-of-Moment code. Then,
the capability to take advantage of this model to retrieve antenna properties
from multiple Q-factor estimations in an RC is demonstrated. All results are
compared to the existing Q-factor formulations.

##  I Introduction

Reverberation chambers (RCs) are now widely used in a large range of
applications such as electromagnetic compatibility testings [1] and antenna
characteristic measurement. The latter includes not only average quantities
such as radiation efficiency [2, 3] but also line-of-sight contributions such
as radar cross-section [4, 5] and gain pattern [6, 7, 8]. An important
parameter to characterize an RC is its quality factor ($Q$-factor) which
describes the capability of a chamber to store energy. This $Q$-factor is
often referred as a composite $Q$-factor as it can be decomposed over a wide
range of loss mechanisms including wall losses, loaded objects losses,
aperture losses and antenna losses, each of those mechanisms being related to
a corresponding $Q$-factor [9]. In particular, antennas can not be avoided for
any electromagnetic measurement and their contribution to the overall losses
can be significant, especially in the lower frequency range [10]. Therefore,
it is important to have models to describe their absorbing properties in such
an environment. Also, such models are of interest to perform contactless
antenna measurements within RCs [11].

An antenna illuminated by incoming electromagnetic waves can be seen as a
loaded scatterer, i.e., an antenna just terminated by a passive load on its
electrical port. From a general point of view, the interaction of a scatterer
with an incident wave can be characterized through the scattering and
absorption cross sections (SCS and ACS, respectively). In a large RC, the
interaction of the scatterer with the diffuse field, i.e., a random
superposition of incident waves coming from many different directions, is
related to the averaged SCS (ASCS) [12] and the averaged ACS (AACS) [13] over
all the possible incident angles. Rather than relying on this last, the
absorption of an antenna in an RC is predominantly characterized by its
Q-factor which is inversely proportional to the AACS but a directly
measureable quantity. A first model of antenna $Q$-factor has been derived by
D. Hill in 1998 [9] considering that losses were only due to the power
dissipated by the antenna termination load. This model has been refined in
2018 [14] in order to better take into account the dissipating effect due to
the antenna radiation efficiency. However, both models do not take into
account the _structural_ component of the antenna absorption properties.
Indeed, following pioneer works on antenna scattering in the mid-20th century
[15, 16], it has been well established that the field scattered by an antenna
is made of two contributions labelled as the _structural mode_ and the
_antenna mode_ (also referred as the radiation mode) [17, 18]. On the one
hand, the structural mode describes the behavior of an object independently of
the fact that the considered object is designed to be an antenna. Fundamental
considerations imply that a structural component depends on a reference
because it is defined as the response of the antenna under an arbitrary load
condition. While a short-circuit, an open-circuit and a matched load have all
been suggested as a reference (and are equally valid), the most-widely used
formulation is based on the conjugate matched load introduced by R. Green
[19]. On the other hand, the antenna mode is directly related to the antenna
radiation and impedance properties and the reference load. Such decomposition
has been extensively studied in the antenna scattering community and found
applications not only for antenna radar cross-section (RCS) reduction [20] but
also in antenna measurement where antenna parameters can be retrieved from
backscattering measurement, therefore in a contactless manner [4, 21, 22].

The objective of this paper is to introduce a complete and rigorous model of
antenna $Q$-factor within an RC that takes into account both structural and
antenna modes. To that end, the antenna is represented using a scattering
matrix where the incoming and outgoing waves are defined according to the
spherical wave expansion (SWE). The antenna $Q$-factor is then derived from
the computation of the AACS of an antenna immersed within a diffuse field.
This new derivation highlights the existence of not only a structural and an
antenna mode but also the interference between the two, which can not be
modeled through a power budget analysis only. Such model will help to better
understand the loss mechanisms induced by an antenna within an RC and can also
be of interest for antenna characterization based on $Q$-factor estimations.

The paper is organized as follows. Section II briefly recalls the current
formulation of antenna $Q$-factor based on a power budget analysis and their
inherent limitations. Then, Section III introduces the new formulation
starting from the $Q$-factor definition in terms of AACS and deriving this
last from the antenna scattering-matrix representation. This section ends by a
discussion of the inherent assumptions made in the previous models. Section IV
presents a numerical validation of the introduced antenna $Q$-factor model by
considering a half-wavelength dipole antenna of various radiation
efficiencies. Finally, Section V exhibits the capability to retrieve antenna
radiation efficiency and input impedance from antenna $Q$-factor estimations
for multiple load conditions.

##  II Current formulations of antenna $Q$-factor within an RC

Figure 1: Description of the power dissipation process of a loaded scatterer
as stated in [9] and [14]. Red color indicates power losses whereas blue color
indicates transmitting power.

An antenna illumated by incoming waves can be seen as a loaded scatterer,
i.e., an antenna (input impedance $Z_{\mathrm{A}}$) terminated by an load
impedance $Z_{\mathrm{L}}$ as it can be seen in Fig. 1. In terms of antenna
properties, such a system is characterized by two parameters, namely the
reflection coefficient $\Gamma_{\mathrm{L}}$ and the antenna radiation
efficiency $e_{\mathrm{r}}$. On the one hand, the reflection coefficient
quantifies the reflection that occurs between the antenna and the load and is
given by
$\Gamma_{\mathrm{L}}=(Z_{\mathrm{L}}-Z_{\mathrm{A}}^{*})/(Z_{\mathrm{L}}+Z_{\mathrm{A}})$
where the superscript ∗ stands for the complex conjugate. On the other hand,
the antenna radiation efficiency, denoted here $e_{\mathrm{r}},$ is defined
(in the transmitting case) as the ratio of the total power radiated by an
antenna to the net power accepted by the antenna from a transmitter of
impedance $Z_{\mathrm{L}}$ that would be connected to the antenna port [23],
and is therefore independent of $\Gamma_{\mathrm{L}}$. Based on these two
antenna characteristics, a first model of antenna $Q$-factor within the
diffuse field of an RC has been derived by D. Hill in [9] such as

$\textrm{{D.\,Hill\,[9]\,\,\,}
}\frac{Q_{0}}{Q_{\mathrm{a,Hill}}}=e_{\mathrm{r}}(1-\left|\Gamma_{\mathrm{L}}\right|^{2})$
(1)

where $Q_{0}=16\pi^{2}V/\lambda^{3}$ is the $Q$-factor of a perfectly-matched
and ideally-efficient (lossless) antenna, $V$ is the RC volume and $\lambda$
is the wavelength. This model of antenna $Q$-factor has been widely used in
the literature to evaluate the losses introduced by antennas and other
receivers during RC measurements. It considers that the losses brought by an
antenna are solely due to the power that is absorbed by its load. Indeed,
according to the losses mechanism labelled \raisebox{-.9pt} {2}⃝ in Fig. 1,
the power dissipated by the load is equal to
$\left<P_{\mathrm{r}}\right>e_{\mathrm{r}}(1-\left|\Gamma{}_{\mathrm{L}}\right|^{2})$
where $\left<P_{\mathrm{r}}\right>$ is the average power received by the
antenna within the RC. As we can see in Fig. 1 and pointed out in [14], this
model does not properly take into account all the losses mechanisms. For
example, a poorly efficiency antenna ($e_{\mathrm{r}}\longrightarrow 0$)
should in practice induce high losses within the chamber but would be
considered as transparent according to (1) [9]. Indeed, when
$e_{\mathrm{r}}\rightarrow 0$, $Q_{\mathrm{a,Hill}}\rightarrow\infty$. This
definition has therefore been refined in 2018 [14] taking into account three
dissipation processes illustrated in Fig. 1. Part of the average received
power is first dissipated because of the antenna radiation efficiency (event
\raisebox{-.9pt} {1}⃝). Then, some of the power received at the antenna load
is dissipated depending on $\Gamma_{\mathrm{L}}$ (event \raisebox{-.9pt}
{2}⃝). Finally, the power reflected by the load is transmitted back to the RC,
through some losses due, again, to the radiation efficiency (event
\raisebox{-.9pt} {3}⃝). This led to a new antenna $Q$-factor formulation as

$\textrm{{A.\,Cozza\,[14]\,\,}}\frac{Q_{0}}{Q_{\mathrm{a,Cozza}}}=1-e_{\mathrm{r}}^{2}\left|\Gamma_{\mathrm{L}}\right|^{2}.$
(2)

This refined model has been indirectly validated by experimentally showing
that the estimated radiation efficiency does not depend on the load impedance
[14].

However, these models rely on a power balance approach that takes into account
losses during receptions, reflections on the load, and re-emission as
independent processes. First, it totally neglects the losses due to the
antenna structure, i.e., losses which do not depend on $\Gamma_{\mathrm{L}}$
and $e_{\mathrm{r}}$, while, according to antenna scattering theory, and
related works on antenna RCS [20], the antenna structure has an impact on the
overall losses brought by the antenna. To illustrate this idea, let’s consider
a horn antenna on which a piece of absorbing material has been sticked on the
outside of the waveguide. Such an absorber would modify neither the antenna
radiation efficiency nor its reflection coefficient; still, it would
drastically increase the losses brought by such an antenna within an RC.
Second, this approach fails to consider potential interference phenomena
between the various contributions, which could significantly alter the overall
power budget. Indeed, because of these interference effects, a dipole antenna
still strongly interact with the incident field when short-circuited
($\Gamma_{\mathrm{L}}=-1$) and is almost transparent when open-circuited
($\Gamma_{\mathrm{L}}=1$) whereas both models expect identical losses for
these two values of $\Gamma_{\mathrm{L}}$.

##  III Derivation of antenna $Q$-factor

This section aims at deriving a new antenna $Q$-factor model. It starts from
the antenna $Q$-factor definition and its relation to the antenna AACS. Then,
after a brief recall of the SWE formalism, the antenna is described in terms
of a scattering matrix where the ingoing and going waves are spherical
harmonics. From this representation, the antenna ACS is derived, following by
its spatially-averaged value. A new antenna $Q$-factor model is then computed
from the AACS highlighting the existence of a structural component. Finally,
the last part of this section is dedicated to a comparison with the two
previous models and the inherent assumptions that were made.

###  III-A Antenna $Q$-factor definition

If an antenna is placed within an electrically-large cavity such as an RC, its
$Q$-factor $Q_{\mathrm{a}}$ is defined as

$Q_{\mathrm{a}}=\omega U/P_{\mathrm{d}}$ (3)

where $\omega$ is the angular frequency, $U$ is the steady-state energy in the
cavity and $P_{\mathrm{d}}$ is the power dissipated by the antenna. It has
been shown in [24] that the antenna $Q$-factor can be related to the antenna
average absorption cross-section (AACS) $\bar{\sigma}_{\mathrm{abs}}$ as

$\frac{Q_{0}}{Q_{\mathrm{a}}}=\frac{8\pi\bar{\sigma}_{\mathrm{abs}}}{\lambda^{2}}$
(4)

where $\lambda$ is the wavelength. Therefore, the objective of the following
subsections is to derive $\bar{\sigma}_{\mathrm{abs}}$.

###  III-B Vectorial field spherical harmonic decomposition

This subsection briefly recalls the SWE formalism used throughout this paper
and described in [25]. The vectorial electric field $\mathbf{E}$ at a position
$\mathbf{r}$ can be decomposed over converging $\mathbf{E}_{-}$ and diverging
$\mathbf{E}_{+}$ fields expressed in terms of spherical harmonics such as

$\mathbf{E}(\mathbf{r})=\underbrace{\sum_{smn}c_{smn}\mathbf{F}_{smn}^{(4)}(\mathbf{r})}_{\mathbf{E}_{-}}+\underbrace{\sum_{smn}d_{smn}\mathbf{F}_{s,m,n}^{(3)}(\mathbf{r})}_{\mathbf{E}_{\mathrm{+}}}$
(5)

where $\mathbf{F}_{smn}^{(3)}(\mathbf{r})$ and
$\mathbf{F}_{smn}^{(4)}(\mathbf{r})$ are the spherical Hankel wave functions
of first (diverging) and second (converging) kind, of degree
$n=\\{1,2,3,...,N\\}$ and of order $m=\\{-n,-n+1,...,0,...,n-1,n\\}$,
$c_{smn}$ and $d_{smn}$ are the complex coefficients of the wave functions,
$s$ is the index indicating whether it is a TE-wave ($s=1)$ or a TM-wave
($s=2)$ coefficient. Note that
$\mathbf{F}_{smn}^{(3)*}(\mathbf{r})=(-1)^{m}\mathbf{F}_{s,-m,n}^{(4)}(\mathbf{r})$
(see Eq. (A1.54) in [25]). For sake of simplicity, a new linear index $j$ is
introduced which combines the three indexes $s$, $m$ and $n$ in a univoke
manner as $j=2\\{n(n+1)+m-1\\}+s$, so that (5) can be written as

$\mathbf{E}(\mathbf{r})=\underbrace{\sum_{j=1}c_{j}\mathbf{F}_{j}^{(4)}(\mathbf{r})}_{\mathbf{E}_{-}}+\underbrace{\sum_{j}d_{j}\mathbf{F}_{j}^{(3)}(\mathbf{r})}_{\mathbf{E}_{\mathrm{+}}}$
(6)

###  III-C Antenna scattering matrix

We consider an antenna scattering system as first introduced by R. H. Dicke in
1947 [15] whose representation is given in Fig. 2. The antenna is fed from a
single transmission line or waveguide supporting only one propagating mode.
The incident wave (towards the terminal plane $P_{1})$ and reflected wave
(towards the load impedance $Z_{\mathrm{L}}$) in the feed line have amplitudes
$a$ and $b$, respectively. The field outside a sphere of radius $r=r_{0}$
enclosing the antenna can be expanded into converging and diverging
propagating spherical modes with complex coefficients $c_{j}$ and $d_{j}$,
respectively, according to the formalism presented in subsection III-B.
Because of the linearity, the input channels are linearly related to the
output ones trough the scattering matrix $\mathbf{S}$ defined as

$\left[\begin{array}[]{c}b\\\ d_{1}\\\ \vdots\\\
d_{J}\end{array}\right]=\underbrace{\left[\begin{array}[]{cccc}S_{00}&S_{01}&\cdots&S_{0J}\\\
S_{10}&S_{11}&\cdots&S_{1J}\\\ \vdots&\vdots&\ddots&\vdots\\\
S_{J0}&S_{J1}&\cdots&S_{JJ}\end{array}\right]}_{\mathbf{\mathbf{S}}}\left[\begin{array}[]{c}a\\\
c_{1}\\\ \vdots\\\ c_{J}\end{array}\right].$ (7)

Please note that because reciprocity hold, $\mathbf{S}$ is symmetrical, that
is $S_{ij}=S_{ji}$.

Figure 2: Antenna scattering matrix representation.

From (7), $b$ can be expressed as the sum of the reflected wave at the
terminal impedance and the contribution of the external converging waves such
as

$b=S_{00}a+\sum_{j=1}^{J}S_{0j}c_{j},$ (8)

where $J$ is the maximum index sum that is roughly equal to $2\left\lfloor
kr_{0}\right\rfloor^{2}$, $k=2\pi/\lambda$ being the wavenumber. Here, the
antenna is driven by external waves and a linear load $Z_{\mathrm{L}}$ is
attached to the port, then $a=\Gamma b$ where $\Gamma$ is the reflection
coefficient which is defined according to an arbitrary reference impedance
$Z_{0}$ as $\Gamma=(Z_{\mathrm{L}}-Z_{0}^{*})/(Z_{\mathrm{L}}+Z_{0})$.
Therefore, (8) can be rewritten as

$b=\frac{1}{(1-S_{00}\Gamma)}\sum_{j=1}^{J}S_{0j}c_{j}.$ (9)

Following the same process, $d_{i}$ can be expressed from (7) as

$d_{i}=S_{i0}\Gamma b+\sum_{j=1}^{J}S_{ij}c_{j}.$ (10)

Replacing $b$ by its expression from (9), it comes

$d_{i}=\frac{S_{i0}\Gamma}{(1-S_{00}\Gamma)}\sum_{j=1}^{J}S_{0j}c_{j}+\sum_{j=1}^{J}S_{ij}c_{j}.$
(11)

Please note that $S_{00}$ is the antenna reflection coefficient defined
according to the reference impedance $Z_{0}$ as
$S_{00}=(Z_{\mathrm{A}}-Z_{0}^{*})/(Z_{\mathrm{A}}+Z_{0})$. For sake of
simplicity but without loss of generality, we consider in the following that
$Z_{0}=Z_{\mathrm{A}}$, therefore $S_{00}=0$. Also, we can define
$\Gamma_{\mathrm{L}}$ as equivalent to $\Gamma$ when $Z_{0}=Z_{\mathrm{A}}$,
i.e.,
$\Gamma_{\mathrm{L}}=(Z_{\mathrm{L}}-Z_{\mathrm{A}}^{*})/(Z_{\mathrm{L}}+Z_{\mathrm{A}})$,
it comes

$d_{i}=S_{i0}\Gamma_{\mathrm{L}}\sum_{j=1}^{J}S_{0j}c_{j}+\sum_{j=1}^{J}S_{ij}c_{j}=\sum_{j=1}^{J}S_{ij}^{\prime}c_{j}$
(12)

where

$S_{ij}^{\prime}=S_{i0}\Gamma_{\mathrm{L}}S_{0j}+S_{ij}.$ (13)

###  III-D Antenna absorption cross-sections

The ACS $\sigma_{\mathrm{abs}}^{p}(\Omega_{0})$ of any object illuminated by a
plane wave of polarization $p$ coming from the incident solid angle
$\Omega_{0}$ is defined as the ratio between the power absorbed by this object
$P_{\mathrm{abs}}$ and the intensity $I_{0}$ of the incident plane wave. The
absorbed power is deduced here from the difference between the ingoing power
$P_{\mathrm{in}}$ and the outgoing power $P_{\mathrm{out}}$ so that

$\sigma_{\mathrm{abs}}^{p}(\Omega_{0})=\frac{P_{\mathrm{abs}}}{I_{0}}=\frac{P_{\mathrm{in}}-P_{\mathrm{out}}}{I_{0}}.$
(14)

The ingoing power can be computed as the sum of all the converging field
components, $P_{\mathrm{in}}$ is therefore given by

$P_{\mathrm{in}}=\frac{1}{2\eta}\oint_{S}\left\|E_{-}(\mathbf{r})\right\|^{2}\mathrm{d}^{2}r,$
(15)

where $\eta$ is the wave impedance and the integral is performed over a close
surface $S$ surrounding the antenna. Assuming that the far-field condition is
fulfilled, the ingoing power can be expressed in a very simple way as (see
Section 2.2.4 in [25])

$P_{\mathrm{in}}=\frac{1}{2\eta k^{2}}\sum_{j=1}^{J}\left|c_{j}\right|^{2}.$
(16)

In the case of a plane wave, the expansion coefficients $c_{j}$ are known to
be equal to (see Section A1.6 in [25])

$c_{j}=c_{smn}=\frac{1}{2}\sum_{smn}(-1)^{m}\sqrt{4\pi}\,\mathrm{i}\,\mathbf{E}_{0}\cdot\mathbf{K}_{s,-m,n}(\Omega_{0})$
(17)

where $\mathbf{E}_{0}$ is the incident electric field and
$\mathbf{K}_{s,-m,n}$ are the far-field pattern functions. Please note that
the expansion coefficients depend on $\Omega_{0}$ . Following the same steps
than for $P_{\mathrm{in}}$, the power $P_{\mathrm{out}}$ is given by

$P_{\mathrm{out}}=\frac{1}{2\eta}\oint_{S}\left\|E_{\mathrm{div}}(\mathbf{r})\right\|^{2}\mathrm{d}^{2}r=\frac{1}{2\eta
k^{2}}\sum_{j=1}^{J}\left|d_{j}\right|^{2}.$ (18)

Noticing that the incident intensity can be expressed as
$I_{0}=\left\|\mathbf{E}_{0}\right\|/2\eta$, injecting (16) and (18) into
(14), and replacing $d_{j}$ by its expression from (12), it comes

$\sigma_{\mathrm{abs}}^{p}(\Omega_{0})=\frac{1}{k^{2}\left\|\mathbf{E}_{0}\right\|}\sum_{i=1}^{J}\left(\left|c_{i}\right|^{2}-\left|\sum_{j=1}^{J}S_{ij}^{\prime}c_{j}\right|^{2}\right).$
(19)

###  III-E Average absorption cross-section

The average absorption cross-section (AACS) $\bar{\sigma}_{\mathrm{abs}}$ is
computed from the averaging of $\sigma_{\mathrm{abs}}^{p}(\Omega_{0})$ over
the 2 polarizations and the incident solid angle $\Omega_{0}$, i.e.,

$\bar{\sigma}_{\mathrm{abs}}=\frac{1}{4\pi}\frac{1}{2}\sum_{p=1}^{2}\int\sigma_{\mathrm{abs}}^{p}(\Omega_{0})\mathrm{d}\Omega_{0}.$
(20)

Because of the orthogonality of the far-field pattern functions (as
demonstrated in Appendix VII-A),

$\displaystyle\sum_{p=1}^{2}\int
c_{j}(\Omega)c_{j^{\prime}}^{*}(\Omega)\mathrm{d}\Omega$ $\displaystyle=$
$\displaystyle 4\pi^{2}\left\|\mathbf{E}_{0}\right\|^{2}\delta_{jj^{\prime}},$

where $\delta_{jj^{\prime}}$ is the Kronecker delta. As a consequence, the
antenna AACS is given by

$\bar{\sigma}_{\mathrm{abs}}=\frac{\pi}{2k^{2}}\sum_{i=1}^{J}\left(1-\sum_{j=1}^{J}\left|S_{ij}^{\prime}\right|^{2}\right).$
(21)

###  III-F Antenna $Q$-factor including a structural mode

The $Q$-factor $Q_{\mathrm{{a}}}$ of an antenna within an RC can be determined
from (4) and (21) so that

$\frac{Q_{0}}{Q_{\mathrm{a}}}=\frac{8\pi\bar{\sigma}_{\mathrm{abs}}}{\lambda^{2}}=\sum_{i=1}^{J}\left(1-\sum_{j=1}^{J}\left|S_{ij}^{\prime}\right|^{2}\right).$
(22)

Replacing $S_{ij}^{\prime}$ by its expression from (13), and noticing that the
antenna radiation efficiency
$e_{\mathrm{r}}=\sum_{i}\left|S_{i0}\right|^{2}=\sum_{j}\left|S_{0j}\right|^{2}$,
the previous equation can be rewritten:

$\displaystyle\frac{Q_{0}}{Q_{\mathrm{a}}}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{J}\left(1-\sum_{j=1}^{J}\left|S_{ij}\right|^{2}\right)$
(23) $\displaystyle-
e_{\mathrm{r}}^{2}\left|\Gamma_{\mathrm{L}}\right|^{2}-2\sum_{i=1}^{J}\sum_{j=1}^{J}\Re\left(S_{i0}\Gamma_{\mathrm{L}}S_{0j}S_{ij}^{*}\right).$

By definition, and according to the convention introduced in [19], the
structural mode of the antenna $Q$-factor corresponds to the antenna
$Q$-factor when $\Gamma_{\mathrm{L}}=0$ . Therefore, we introduce the
structural antenna $Q$-factor $Q_{\mathrm{s}}$ that is given by

$\frac{Q_{0}}{Q_{\mathrm{s}}}\equiv\sum_{i=1}^{J}\left(1-\sum_{j=1}^{J}\left|S_{ij}\right|^{2}\right).$
(24)

Equation (23) can therefore be rewritten as

$\frac{Q_{0}}{Q_{\mathrm{a}}}=\frac{Q_{0}}{Q_{\mathrm{s}}}-e_{\mathrm{r}}^{2}\left|\Gamma_{\mathrm{L}}\right|^{2}-2\sum_{i=1}^{J}\sum_{j=1}^{J}\Re\left(S_{ij}^{\mathrm{L}}S_{ij}^{*}\right),$
(25)

where $S_{ij}^{\mathrm{L}}=S_{i0}\Gamma_{\mathrm{L}}S_{0j}$ is the scattering
matrix including only the interaction of the modes with the load impedance.
Therefore, it is demonstrated here that the antenna $Q$-factor is composed of
three distinct parts:

  * •

the first one is the contribution from the antenna structure which dissipate
energy independently of the antenna radiation and impedance properties, i.e.,
the structural mode,

  * •

the second one is solely due to the antenna radiation and impedance
properties, i.e., the antenna mode,

  * •

the third one is the interference between both antenna and structural modes,
i.e., the interference mode. The latter is expressed here as a correlation
between the scattering component related to the antenna load
$S_{ij}^{\mathrm{L}}$ and the scattering component related to the structure
$S_{ij}^{*}$.

Equation (25) constitutes the main result of this paper.

###  III-G Inherent assumptions of previous models

If we consider that the antenna is coupled to only one electromagnetic mode,
i.e., $J=1$, therefore (23) can be simplified as

$\frac{Q_{0}}{Q_{\mathrm{a}}}=\left(1-\left|S_{11}\right|^{2}\right)-e_{\mathrm{r}}^{2}\left|\Gamma_{\mathrm{L}}\right|^{2}-2\Re\left(S_{10}\Gamma^{\prime}_{\mathrm{L}}S_{01}S_{11}^{*}\right).$
(26)

Note that this mode has not necessary to be a harmonic spherical mode but it
can be any mode that can be obtained by a unitary transformation. Then, if we
also consider that the antenna structure is not responsible for any losses
within the RC, i.e., if we assume that $S_{11}=0$, it comes

$\frac{Q_{0}}{Q_{\mathrm{a}}}=1-e_{\mathrm{r}}^{2}\left|\Gamma_{\mathrm{L}}\right|^{2},$
(27)

which turns out to be exactly the formula from (2) [14]. Therefore, the model
from [14] is only valid if we consider that the antenna is coupled to one
electromagnetic mode ($J=1$) and if the losses due to the structural mode are
neglected ($Q_{\mathrm{s}}=Q_{0}$).

As stated in section II and in [14], Hill’s formulation is not consistent as
it does not consider the radiation efficiency as a loss mechanism but solely
as a kind of aperture efficiency, i.e., what is captured by the antenna but
not transmitted to the load is not dissipated by the antenna. In other words,
it is based on the assumption that the radiation efficiency is equal to 100 %.
Therefore, if we consider the specific case of $e_{\mathrm{r}}=1$, the two
models from [9] and [14] lead to the same result:

$\frac{Q_{0}}{Q_{\mathrm{a}}}=1-\left|\Gamma_{\mathrm{L}}\right|^{2}.$ (28)

##  IV Numerical validation

This section is dedicated to the validation of the proposed model using
numerical simulations. First, the home-made simulation code is briefly
introduced and validated through the analysis of the absorption and scattering
properties of a half-wavelength dipole antenna. Then, the accuracy of the
proposed model is presented and compared to the two previous models.

###  IV-A AACS and ASCS of a dipole antenna

Figure 3: Dipole antenna in the receiving mode (left), and in the transmitting
mode (right).

A Python code based on the Method of Moment (MoM) described in Orfanidis’s
online book [26] and the associated Matlab toolbox has been implemented in
order to validate the proposed model using numerical results. We consider a
dipole antenna whose length is 0.48 $\lambda$ and diameter is $5\times
10^{-4}\,\lambda$ at the frequency of 300 MHz. The antenna is discretized over
149 segments and a resistance per unit length $R_{\Omega}$ is considered in
order to introduce losses. For simulations in the transmitting mode (Fig. 3
(right)), the antenna is fed by a delta-gap voltage source. The antenna
radiation efficiency as well as the reflection coefficient are numerically
obtained by solving the Hallén’s equations. For simulations in the receiving
mode, the dipole antenna is terminated by a load impedance (Fig. 3 (left)) and
illuminated by an incoming plane wave of polarization
$\mathbf{E}_{\mathrm{inc}}$. The scattered far-field
$\hat{\mathbf{E}}_{\mathrm{s}}(\Omega)$ and the current distribution are
provided by the numerical inversion of the Pocklington’s equation.

The ACS, $\sigma_{\mathrm{abs}}$ is computed using (14) where the power
dissipated by ohmic losses is deduced from the current distribution, the real
part of the load impedance and $R_{\Omega}$. To validate the accuracy of the
simulation, we also compute two other cross-sections, the scattering
($\sigma_{\mathrm{sca}}$) and extinction ($\sigma_{\mathrm{ext}}$) cross-
sections. They are respectively deduced from [27]

$\displaystyle\sigma_{\mathrm{sca}}$ $\displaystyle=$
$\displaystyle\frac{\int_{4\pi}\left\|\hat{\mathbf{E}}_{\mathrm{sca}}(\Omega)\right\|^{2}d\Omega}{\left\|\mathbf{E}_{\mathrm{inc}}\right\|^{2}},$
(29) $\displaystyle\sigma_{\mathrm{ext}}$ $\displaystyle=$
$\displaystyle\frac{4\pi\Im\left(\mathbf{E}_{\mathrm{inc}}^{*}\cdot\hat{\mathbf{E}}_{s}(\Omega=\Omega_{\mathrm{in}c})\right)}{\left\|\mathbf{E}_{\mathrm{inc}}\right\|^{2}k}.$
(30)

Because of the circular symmetry and the fact that the dipole interacts only
with an incident field that is, at least partially, vertically polarized, it
is sufficient to integrate over the elevation angle only with a 1-degree step
to estimate the averaged absorption ($\bar{\sigma}_{\mathrm{abs}}$),
scattering ($\bar{\sigma}_{\mathrm{sca}}$) and extinction
($\bar{\sigma}_{\mathrm{ext}}$) cross-sections, respectively.

Fig. 4 (top) presents the 3 different averaged cross-sections as a function of
a purely-real load impedance $R_{\mathrm{L}}$ for an ideally-efficient antenna
($R_{\Omega}=0$ $\Omega\mathrm{m}^{-1}$). In that case, $e_{\mathrm{r}}=1$ and
$Z_{\mathrm{A}}=72.1\,\Omega+\mathrm{i}0.43\,\Omega$. It is observed that the
sum of $\bar{\sigma}_{\mathrm{abs}}$ and $\bar{\sigma}_{\mathrm{sca}}$
overlaps with $\bar{\sigma}_{\mathrm{ext}}$, as expected theoretically (see
e.g., [27]) which confirms that our simulation is fully consistent. Also,
$\bar{\sigma}_{\mathrm{abs}}$ and $\bar{\sigma}_{\mathrm{sca}}$ are equal when
the antenna is matched ($R_{\mathrm{L}}\sim 72\,\Omega$), confirming that
under matched condition, the dipole antenna scatters as much energy as it
absorbs. Finally, the short-circuit and the open-circuit cases lead to the
same level of absorption. Fig. 4 (center) presents the same results but for a
lossy antenna ($R_{\Omega}=100$ $\Omega\mathrm{m}^{-1}$). In that case,
$e_{\mathrm{r}}=0.75$ and
$Z_{\mathrm{A}}=96.9\,\Omega-\mathrm{i}2.75\,\Omega$. Unlike the previous
case, it is observed that $\bar{\sigma}_{\mathrm{abs}}$ of a short-circuited
dipole antenna is not zero and therefore different from the one of the open-
circuited dipole antenna. Such behavior could not be modeled by considering
the previous $Q$-factor derivations as they only depend on the modulus of the
reflection coefficient, i.e., $\left|\Gamma_{\mathrm{L}}\right|$. Finally,
Fig. 4 (bottom) presents the same results but for a highly lossy antenna
($R_{\Omega}=1000$ $\Omega\mathrm{m}^{-1}$). In that case,
$e_{\mathrm{r}}=0.22$ and $Z_{\mathrm{A}}=304\,\Omega-\mathrm{i}65.8\,\Omega$.
In that scenario, it is seen that $\bar{\sigma}_{\mathrm{{ext}}}$ is much
lower than for the two previous cases and the absorption dominates whatever
the load impedance.

Figure 4: Average scattering, absorption and extinction cross-sections as a
function of $R_{\mathrm{L}}$ for three cases: (top) $R_{\Omega}=0$
$\Omega\mathrm{m}^{-1}$ ($e_{\mathrm{r}}=1$), (center) $R_{\Omega}=100$
$\Omega\mathrm{m}^{-1}$ ($e_{\mathrm{r}}=0.75$) and (bottom) $R_{\Omega}=1000$
$\Omega\mathrm{m}^{-1}$ ($e_{\mathrm{r}}=0.22$).

###  IV-B Comparison of the three models

This subsection is dedicated to evaluating the accuracy of the introduced
antenna $Q$-factor model using MoM numerical results and how it compares with
the two previous models. Regarding the two models from [9] and [14], only
three parameters are required, i.e., the antenna radiation efficiency
$e_{\mathrm{r}}$ and the real and imaginary parts of the reflection
coefficient $\Gamma{}_{\mathrm{L}}$, which can be computed from simulations in
the transmitting mode. Regarding the introduced model from (25), it can be
rewritten here as

$\frac{Q_{0}}{Q_{\mathrm{a}}}=\frac{Q_{0}}{Q_{\mathrm{s}}}-e_{\mathrm{r}}^{2}\left|\Gamma_{\mathrm{L}}\right|^{2}-2\left(\Re\left(\Gamma_{\mathrm{L}}\right)\Re\left(C\right)-\Im\left(\Gamma_{\mathrm{L}}\right)\Im\left(C\right)\right)$
(31)

where

$C=\sum_{i=1}^{J}\sum_{j=1}^{J}S_{i0}S_{0j}S_{ij}^{*}.$ (32)

Therefore, one needs to estimate three more parameters:
$Q_{0}/Q_{\mathrm{s}}$, $\Re\left(C\right)$ and $\Im\left(C\right)$. To that
end, we compute the antenna $Q$-factor for three different load conditions. If
$Z_{\mathrm{L}}=Z_{\mathrm{A}}^{*}$, it comes $\Gamma_{\mathrm{L}}=0$ and
therefore

$\frac{Q_{0}}{Q_{\mathrm{s}}}=\frac{Q_{0}}{Q_{\mathrm{a,\Gamma_{\mathrm{L}}=0}}}.$
(33)

If $Z_{\mathrm{L}}\rightarrow\infty$, it comes $\Gamma_{\mathrm{L}}=1$, and
therefore $\Re\left(C\right)$ can be evaluated through

$\Re\left(C\right)=-\frac{1}{2}\left[\frac{Q_{0}}{Q_{\mathrm{a,\Gamma_{\mathrm{L}}=1}}}-\frac{Q_{0}}{Q_{\mathrm{s}}}+e_{\mathrm{r}}^{2}\right].$
(34)

Finally, if
$Z_{\mathrm{L}}=(Z_{\mathrm{A}}^{*}+\mathrm{iZ_{A}})/(1-\mathrm{i})$, it comes
$\Gamma_{\mathrm{L}}=\mathrm{i}$ , and $\Im\left(C\right)$ can be estimated as

$\Im\left(C\right)=\frac{1}{2}\left[\frac{Q_{0}}{Q_{\mathrm{a,\Gamma_{\mathrm{L}}^{{}^{\prime}}=\mathrm{i}}}}-\frac{Q_{0}}{Q_{\mathrm{s}}}+e_{\mathrm{r}}^{2}\right].$
(35)

Fig. 5 presents the antenna $Q$-factor as a function of
$\Re\left(\Gamma_{\mathrm{L}}\right)$ for the same three cases than
previously. Results obtained from the MoM simulation are compared to the three
different models. For the lossless case in Fig. 5 (top), the three models give
the same result and agree with the MoM simulation. Indeed, for that particular
lossless case, both the structural mode and the radiation efficiency do not
induce any losses (the mathematical proof is given in Appendix VII-B).
Regarding the two other lossy cases, it is observed, on one hand, a very good
agreement between the MoM results and the introduced model over the entire
range of $\Re\left(\Gamma_{\mathrm{L}}\right)$. On the other hand, both older
models give very inaccurate results.

Figure 5: Antenna $Q$-factor as a function of the real part of the reflection
coefficient for three cases: (top) $R_{\Omega}=0$ $\Omega\mathrm{m}^{-1}$
($e_{\mathrm{r}}=1$), (center) $R_{\Omega}=100$ $\Omega\mathrm{m}^{-1}$
($e_{\mathrm{r}}=0.75$) and (bottom) $R_{\Omega}=1000$ $\Omega\mathrm{m}^{-1}$
($e_{\mathrm{r}}=0.22$).

##  V Antenna characteristics retrieval

In this section, we take benefit of the introduced model in order to retrieve
antenna characteristics (i.e., radiation efficiency and input impedance),
thanks to multiple $Q$-factor evaluations for various $Z_{\mathrm{L}}$. We
consider the case of the lossy antenna with $R_{\Omega}=100$
$\Omega\mathrm{m}^{-1}$. A set of $K=10$ numerical simulations is performed
where the load impedance is set purely real ranging from 0.1 $\Omega$ to 1
k$\Omega$. Then, a non-linear iterative minimization search algorithm has been
implemented in order to minimize the function $F_{\mathrm{Model}}$ defined as

$F_{\mathrm{Model}}=\sum_{k=1}^{K}(\tilde{Q}_{\mathrm{MoM}}^{-1}(Z_{\mathrm{L}}^{k})-\tilde{Q}_{\mathrm{Model}}^{-1}(Z_{\mathrm{L}}^{k}))$
(36)

where $Z_{\mathrm{L}}^{k}$ is the $k^{\mathrm{{th}}}$ load impedance,
$\tilde{Q}_{\mathrm{MoM}}$ and $\tilde{Q}_{\mathrm{Model}}$ are the estimated
antenna $Q$-factors using the MoM algorithm and the considered model,
respectively. In the cases of D. Hill and A. Cozza theories, there are 3
parameters to work out: the antenna efficiency $e_{\mathrm{r}}$ and the real
and imaginary part of the antenna input impedance $Z_{\mathrm{A}}$. Three more
parameters are considered for the proposed model: the antenna structural
$Q$-factor $Q_{0}/Q_{\mathrm{s}}$, and the real and imaginary part of $C$
defined in (32). Fig. 6 presents the antenna $Q$-factor as a function of
$\left|Z_{\mathrm{L}}\right|$. Results using the MoM simulation are compared
with the best fit that has been obtained using the three models. It is shown
that the new model exhibits a much higher accuracy than the two other models
as both the MoM results and the new model overlap.

Figure 6: Estimated antenna $Q$-factor from a set of 10 purely-real
$Z_{\mathrm{L}}$ ranging from 0.1 $\Omega$ to 1 k$\Omega$.

The relevant extracted parameters are presented in Table I. It has to be noted
that the structural antenna $Q$-factor cannot be estimated directly through
MoM simulation, therefore, it is computed from (33) using the case
$Z_{\mathrm{L}}=Z_{\mathrm{A}}^{*}$. Also, the models from D. Hill and A.
Cozza do not take into account this structural mode leading to
$Q_{0}/Q_{\mathrm{s}}=1$. It is observed that the extracted antenna radiation
efficiency and input impedance using the introduced model are consistent with
the ones obtained from the MoM simulation, although the relative error
regarding $\Im\left(Z_{\mathrm{A}}\right)$ is quite large (about 25 %). As
expected, both previous models do not allow retrieving relevant parameters,
even exhibiting some non-physical results such as $e_{\mathrm{r}}>1$.

| $Q_{\mathrm{\mathrm{0}}}/Q_{\mathrm{\mathrm{s}}}$ | $e_{\mathrm{r}}^{2}$ | $\Re\left(Z_{\mathrm{A}}\right)$ | $\Im\left(Z_{\mathrm{A}}\right)$
---|---|---|---|---
MoM | 0.93 | 0.55 | 96.9 | -2.72
New model | 0.93 | 0.55 | 96.9 | -2.02
Cozza | 1 | 0.40 | 17.6 | -0.48
Hill | 1 | 1.64 | 16.3 | 0.00
Table I: Estimated antenna parameters from a set of 10 purely-real
$Z_{\mathrm{L}}$ ranging from 0.1 $\Omega$ to 1 k$\Omega$.

A second study is performed where the load impedance $Z_{\mathrm{L}}$ is set
complex by adding a transmission line of length linearly varying from 0 m to
0.4 m between the antenna and the real part of the load (still ranging from
0.1 $\Omega$ to 1 k$\Omega$). Fig. 7 presents the antenna $Q$-factor obtained
from the MoM simulation and compared to the three models after processing the
minimization search algorithm. Once again, the new model overlaps with the MoM
results whereas the two older models lead to inaccurate results. The relevant
extracted parameters are presented in Table II. All parameters are well
retrieved using the new model. In particular, it is shown that
$\Im\left(Z_{\mathrm{A}}\right)$ is better estimated than previously thanks to
the set of complex-valued $Z_{\mathrm{L}}$.

Figure 7: Estimated antenna $Q$-factor from a set of 10 complex-valued $Z_{\mathrm{L}}$. | $Q_{\mathrm{\mathrm{0}}}/Q_{\mathrm{\mathrm{s}}}$ | $e_{\mathrm{r}}^{2}$ | $\Re\left(Z_{\mathrm{A}}\right)$ | $\Im\left(Z_{\mathrm{A}}\right)$
---|---|---|---|---
MoM | 0.93 | 0.55 | 96.9 | -2.72
New model | 0.93 | 0.55 | 96.9 | -2.71
Cozza | 1 | 0.19 | 117 | -89.4
Hill | 1 | 1.18 | 527.5 | 13.2
Table II: Estimated antenna parameters from a set of 10 complex-valued
$Z_{\mathrm{L}}$.

##  VI Conclusion

This paper introduced a new model of antenna $Q$-factor based on antenna
scattering matrix theory where the ingoing and outgoing waves are dealt with
using the spherical wave expansion. Considering the plane wave integral
representation of fields within the reverberation chamber, the antenna
$Q$-factor is derived by averaging over incident angles its absorption cross-
section. The introduced model exhibits three parts: the structural mode, the
antenna mode and the interference mode. The structural mode includes all
losses that are due to the structure of the antenna, i.e., not related to its
radiation properties. On the contrary, the antenna mode describes all losses
that are purely due to the antenna radiation properties, i.e., radiation
efficiency and reflection coefficient. Finally, the interference mode is
related to the inevitable interaction between the two previous modes. Such a
model is compliant with the usual decomposition of antenna backscattering with
the presence of a structural mode which leads to either constructive or
destructive interferences with the antenna mode.

The introduced model has been validated through MoM-based numerical
simulations. A half-wavelength dipole antenna has been considered with three
different lossy conditions ($e_{\mathrm{r}}=\left\\{1,0.75,0.22\right\\}$) and
it has been shown that the new model and the MoM results are in very good
agreement, regardless of the reflection coefficient (varying from $-1$ to
$1$). Results have been compared with the two models that can be found in the
literature (from D. Hill [9] and A. Cozza [14]) which have been found to be
ineffective to properly model the antenna $Q$-factor for real cases, i.e.,
$e_{\mathrm{r}}\neq 1$. Finally, the capability to retrieve relevant antenna
characteristics from multiple antenna $Q$-factor estimations for various
loading conditions, has also been shown. Therefore, such a model not only
enable to better estimate the antenna losses within an RC but also opens new
possibilities in the field of contactless antenna characterization.

Future works include an experimental validation of the proposed model within
an RC. Such measurements might be delicate as it relies the discrimination of
the antenna $Q$-factor from the overall RC composite $Q$-factor. Moreover, it
requires several RC $Q$-factor estimations using various known complex loads.

##  VII Appendices

###  VII-A Orthogonality of the far-field pattern functions

In this appendix, we demonstrate the orthogonality of the far-field pattern
function defined from (A1.5) in [25] as

$\displaystyle\mathbf{K}{}_{smn}(\theta,\phi)$ $\displaystyle=$
$\displaystyle\lim_{kr\rightarrow\infty}\sqrt{4\pi}kre^{-ikr}\text{$\mathbf{F}$}_{smn}^{(3)}(r,\theta,\phi).$
(37)

To that end, we take the benefit of (A1.69) in [25]

$\oint\left\\{\left(\mathbf{F}_{smn}^{(3)}(r,\theta,\phi).\hat{\boldsymbol{\text{$\theta$}}}\right)\left(\mathbf{F}_{s^{\prime}m^{\prime}n^{\prime}}^{(4)}(r,\theta,\phi).\hat{\boldsymbol{\text{$\theta$}}}\right)\right.\\\
\left.+\left(\mathbf{F}_{smn}^{(3)}(r,\theta,\phi).\hat{\text{$\boldsymbol{\phi}$}}\right)\left(\mathbf{F}_{s^{\prime}m^{\prime}n^{\prime}}^{(4)}(r,\theta,\phi).\hat{\text{$\boldsymbol{\phi}$}}\right)\right\\}d\Omega\\\
=\delta_{ss^{\prime}}\delta_{m,-m^{\prime}}\delta_{nn^{\prime}}(-1)^{m}R_{sn}^{(3)}(kr)R_{sn}^{(4)}(kr)$
(38)

where $R_{sn}^{(c)}(kr)$ are the radial functions. In the far-field limit, the
expression of the spherical Hankel function of the first kind can be directly
deduced from (37)

$\displaystyle\mathbf{F}_{smn}^{(3)}(r,\theta,\phi)$
$\displaystyle\underset{kr\gg 1}{=}$
$\displaystyle\frac{\mathbf{K}_{smn}}{\sqrt{4\pi}}\frac{e^{ikr}}{kr}$

Moreover, from (A1.14) and (A1.16) [25], it appears that the far-field
expression is given by

$\displaystyle R_{sn}^{(3)}(kr)$ $\displaystyle\underset{kr\gg 1}{=}$
$\displaystyle(-i)^{n}(-i)^{2-s}\frac{e^{ikr}}{kr}$

As a consequence,

$\displaystyle\mathbf{F}_{smn}^{(3)}(r,\theta,\phi)$
$\displaystyle\underset{kr\gg 1}{=}$
$\displaystyle\frac{R_{sn}^{(3)}(kr)(i)^{n}i^{2-s}}{\sqrt{4\pi}}\mathbf{K}_{smn}(\theta,\phi)$
(39)

and

$\mathbf{F}_{smn}^{(4)}(r,\theta,\phi)\underset{kr\gg 1}{=}\\\
(-1)^{m}\frac{R_{sn}^{*(3)}(kr)(i)^{-n}i^{s-2}}{\sqrt{4\pi}}\mathbf{K}_{smn}^{*}(\theta,\phi)$
(40)

Because
$\mathbf{F}_{smn}^{*(3)}(r,\theta,\phi)=(-1)^{m}\mathbf{F}_{s^{\prime}-m^{\prime}n^{\prime}}^{(4)}(r,\theta,\phi)$
(see (A1.54) in [25]),

$\oiint\mathbf{F}_{smn}^{(3)}(r,\theta,\phi).\mathbf{F}_{s^{\prime}m^{\prime}n^{\prime}}^{*(3)}(r,\theta,\phi)=\\\
\oiint(-1)^{m}\mathbf{F}_{smn}^{(3)}(r,\theta,\phi).\mathbf{F}_{s^{\prime}-m^{\prime}n^{\prime}}^{(4)}(r,\theta,\phi).$
(41)

Because of Eqs. (38), (39), (40) and (41), it finally demonstrates the
orthogonality of the far-field pattern functions

$\displaystyle\oiint\oint\mathbf{K}_{smn}(\theta,\phi).\mathbf{K}_{s^{\prime}m^{\prime}n^{\prime}}^{*}(\theta,\phi)d\Omega$
$\displaystyle=$ $\displaystyle
4\pi\delta_{ss^{\prime}}\delta_{mm^{\prime}}\delta_{nn^{\prime}}.$

###  VII-B Antenna $Q$-factor model in the lossless case

We consider here the case of a lossless system, i.e., $R_{\Omega}=0$ which
leads to a radiation efficiency $e_{\mathrm{r}}=1$. Due to energy
conservation, it is possible to write
$\sum_{j=0}^{J}\left|S_{ij}\right|^{2}=1\,\forall i$. By pulling out the term
corresponding to $j=0$, it comes
$\sum_{j=1}^{J}\left|S_{ij}\right|^{2}=1-\left|S_{i0}\right|^{2}\,\forall i$.
Then, according to the definition of the structual $Q$-factor in (24), one can
show that

$\frac{Q_{0}}{Q_{\mathrm{s,lossless}}}=\sum_{i=1}^{J}\left|S_{i0}\right|^{2}=e_{\mathrm{r}}=1.$

By taking benefit of the unitary of the $S$-matrix for a lossless system, the
intereference mode can be expressed as the following

$2\Re\left\\{\Gamma{}_{\mathrm{L}}\sum_{i=1}^{J}\left[S_{i0}\sum_{j=1}^{J}\left(S_{0j}S_{ij}^{*}\right)\right]\right\\}=\\\
2\Re\left\\{\Gamma{}_{\mathrm{L}}\sum_{i=1}^{J}S_{i0}\left[\delta_{\mathrm{0i}}-S_{00}S_{i0}^{*}\right]\right\\}$
(42)

Because $S_{00}=0$, it comes

$\displaystyle
2\Re\left\\{\Gamma{}_{\mathrm{L}}\sum_{i=1}^{J}\left[S_{i0}\sum_{j=1}^{J}\left(S_{0j}S_{ij}^{*}\right)\right]\right\\}$
$\displaystyle=$ $\displaystyle 0.$

Therefore, it is proved here that the antenna $Q$-factor model in the lossless
case is equivalent to the two models from [9] and [14] in the theoretical case
of $e_{\mathrm{r}}=1$.

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