# Signal Amplification in a Time-Modulated Transmission Line and the Loss
Effect

Mohamed F. Hagag  Electronic Engineering Department, Military Technical
College, Cairo 11766, Egypt  Electronics and Communications Engineering
Department, American University in Cairo, Cairo 11835, Egypt
<EMAIL_ADDRESS>Thomas R. Jones  Elmore Family School
of Electrical and Computer Engineering, Purdue University, West Lafayette,
Indiana 47907, USA  Karim Seddik  Electronics and Communications Engineering
Department, American University in Cairo, Cairo 11835, Egypt  Dimitrios
Peroulis  Elmore Family School of Electrical and Computer Engineering, Purdue
University, West Lafayette, Indiana 47907, USA

###### Abstract

We investigate and simulate signal amplification in a transmission line (TL)
with time-modulated characteristic impedance $Z_{o}$. Periodically varying
$Z_{o}$ is achieved by loading TL with a sinusoidally time-modulated capacitor
(TMC). For a detailed study, three models are considered: a lossless L-C TL
lumped model loaded with shunt infinite quality factor (Q) TMC, a TL loaded
with a shunt infinite Q TMC, and finite Q TMC. By solving the eigenvalue
problem in all models, dispersion diagrams (DD) are plotted with a created
momentum band gap (MBG) at a modulation frequency double the signal frequency.
Within MBG, only imaginary frequencies are found and correlated to MBG width
and signal growth level. Using Harmonics Balance (HB) and Transient Simulation
(TS), signal amplification is confirmed, and the obtained results are
consistent with the DD outcomes. In the second model, the effect of TL length
on amplification is investigated and explained by studying the unit cell’s
Bloch impedance ($Z_{Bloch}$). The loss effect is considered by adding a
series resistance (Rc) to the third model’s TMC (finite Q). Decreasing
amplification levels, confirmed by circuit modeling, due to the increase of Rc
value is explained by studying real and imaginary DDs and the attenuation
constant.

##  1 Introduction

Temporal modulated media is an incoming technique that unleashes enormous
opportunities for artificial electromagnetic media [1, 2, 3]. It is realized
by varying the media characteristics periodically with time following a
specific modulation waveform. The interest in engineering artificial media and
materials with time-modulated properties has recently increased. One
attractive proposed property of time-modulated media is magnetless non-
reciprocity [4, 5]. Many magnetless non-reciprocal components are presented in
the literature, such as filters, circulators, and power dividers [6, 7, 8, 9,
10]. The non-reciprocity is created when a sub-component property (unit cell)
is time modulated with a speed lower than the signal speed with sufficient
successive phase shift, creating a spatiotemporal modulated component [11,
12]. Other interesting applications are proposed, such as creating an
effective magnetic field for photons [13] and synthetic dimensions [14, 15].

More interesting phenomena are discovered when modulating media properties in
time at high speed. This was first proposed as a photonic time crystal (PTC)
or generally time crystal (TC) [16, 17]. A photonic crystal (PC) is a periodic
structure at which the signal wave number (K) is engineered through space to
create a frequency band gap (FBG); within the FBG, the signal is forbidden to
propagate[18, 19]. As an analogy to PCs, TCs are created when periodic
structure properties are modulated in time and kept uniform in space. With a
sufficient time modulation speed, band gaps in momentum (MBG) are created
inside which a PTC possesses a non-Hermitian nature. As a result, the signal
is amplified exponentially in the MBG [14, 1] in contrast to the signal decay
in the FBG [18, 19]. To realize an MBG in a TC, its properties must be
modulated in time with a speed higher than the signal propagation speed inside
the TC. Realizing this in the microwave regime is often more attainable [20].
Within this regime, there are adaptive components such as varactors and photo-
diodes; the change in their properties may be able to catch up with the
modulation signal.

In this work, lossless transmission lines (TL) loaded with time-modulated
capacitors (TMCs), which act as 1-D TCs, are analyzed and discussed as a step
in realizing signal amplification in the microwave regime. First, the
eigenvalue problem is formulated and solved for a lossless TL-lumped model
with a TMC. The dispersion diagram is plotted, and the areas of the MBGs are
defined. To investigate the circuit performance within the MBG, S-parameter
(SP), Transient (TS), and Harmonic Balance (HB) simulations in Keysight
Advanced Design System (ADS) software are performed on 9 unit cells.
Amplification is observed, and there is an agreement between the DD and
simulation results. Second, a TL loaded with TMCs is considered. The above
analyses are applied, and agreement between the DD and simulation results is
achieved. The effect of varying the TL lengths on MBGs is discussed and
supported by plotting the unit cell Bloch impedance ($Z_{Bloch}$). Finally, to
study the loss effect, a series resistance (Rc) is introduced within the TMC
to realize a finite Q TMC. Real and complex DDs with different Rc (Q) values
are plotted, and weak interaction between harmonics is observed. The results
are confirmed using ADS TS and HB analysis.

##  2 Dispersion equation of a lossless TL lumped model with time-modulated
capacitor

This section constructs the dispersion equation of a lossless lumped model TL
loaded with a TMC. The unit cell of the loaded TL is shown in Fig. 1(a), which
is a T-shape symmetric unit cell constructed from two different elements,
series, and shunt. The shunt element (B) will always be the time-modulated
capacitor. The series element (A) will be a time-invariant inductor.

###  2.1 Transfer matrix of time-modulated capacitor

To construct the dispersion equation, transfer matrices of all elements were
computed. As in [21, 22, 23], a TMC with a period $T$ that follows the
function $C(\omega,t)=C(\omega,t+nT)\ $ with $n\in\mathbb{Z}$ can be expanded
into a complex Fourier series as follows

$C\left(\omega,t\right)=\sum_{s=-\infty}^{+\infty}{c_{s}(\omega)e^{js\omega_{M}t}}$
(1)

where $c_{s}(\omega)$ are complex coefficients and $\omega_{M}$ is the angular
modulation frequency. When the TMC is connected to a circuit operating at
angular signal frequency $\omega_{s}$, an infinite number of Floquet harmonics
are generated; the voltage or current across the TMC follows the series below
[22]

$F\left(t\right)=\sum_{n=-N}^{+N}{f_{n}e^{j\omega_{n}t}}$ (2)

where $f_{n}$ are the complex coefficients of voltage and current. $N$ is an
integer that represents half the number of considered harmonics in the problem
and depends on the electrical size of the unit cell. According to [22], it can
be approximated by

$N\ll\frac{\lambda_{s}}{d}$ (3)

where $\lambda_{s}$ is the signal’s wavelength, and $d$ is the length of the
unit cell. The harmonic frequencies can be given by

$\omega_{n}=\omega_{s}+n\omega_{M}$ (4)

where $\omega_{s}$ and $\omega_{M}$ are the main signal and modulation angular
frequencies, respectively. Eq. 1 depends mainly on the modulation waveform. In
this manuscript, the capacitance is modulated around its nominal value in a
sinusoidal manner following

$C\left(\omega,t\right)=C_{o}(\omega)\left(1+M_{D}\cos{(\omega_{M}t+\varphi_{M}}\right))$
(5)

where $M_{D}$ and $\varphi_{M}$ are the modulation depth and phase,
respectively. $C_{o}(\omega)$ is the capacitance nominal value. Using (5), (1)
is reduced to

$C\left(\omega,t\right)=C_{o}(\omega)+C_{-1}(\omega)e^{-j\omega_{M}t}+C_{1}(\omega)e^{j\omega_{M}t}$
(6)

where $C_{\pm 1}(\omega){=0.5\leavevmode\nobreak\ C_{o}(\omega)\ M_{D}e}^{\pm
j\varphi_{M}}$.

In general, as discussed in [22, 23], the TMC generated current and voltage
harmonics are related by

$i_{l}=\sum_{n=-\infty}^{\infty}{j\leavevmode\nobreak\
\omega_{r}\leavevmode\nobreak\ c_{l-n}(\omega_{l})\leavevmode\nobreak\ v_{n}}$
(7)

Consequently, the admittance matrix of the TMC is given by

$\bar{\bar{Y_{C}}}=j\bar{\bar{W}}P$ (8)

where
$\bar{\bar{W}}=diag(\omega_{-N},..,\omega_{-1},\omega_{s},\omega_{1},..,\omega_{N})$,
and $P$ in general is given by

$P=\left(\begin{matrix}\begin{matrix}c_{0}\left(\omega_{-N}\right)&c_{-1}\left(\omega_{1-N}\right)\\\
c_{1}\left(\omega_{-N}\right)&c_{0}\left(\omega_{1-N}\right)\\\
\end{matrix}&\cdots&\begin{matrix}c_{-2N}\left(\omega_{N}\right)\\\
c_{1-2N}\left(\omega_{N}\right)\\\ \end{matrix}\\\ \vdots&\ddots&\vdots\\\
\begin{matrix}c_{2N}\left(\omega_{-N}\right)\
&c_{2N-1}\left(\omega_{1-N}\right)\\\
\end{matrix}&\cdots&c_{0}\left(\omega_{N}\right)\\\ \end{matrix}\right)$ (9)

$P$ is $(2N+1)\times(2N+1)$ in size. Using a TMC that follows (6), the $P$
matrix is reduced to

$P=\left(\begin{matrix}\begin{matrix}c_{0}(\omega_{-N})&c_{-1}(\omega_{1-N})&0\\\
c_{1}(\omega_{-N})&c_{0}(\omega_{1-N})&c_{-1}(\omega_{2-N})\\\
0&c_{1}(\omega_{1-N})&c_{0}(\omega_{2-N})\\\
\end{matrix}&\begin{matrix}\cdots\\\ \cdots\\\ \cdots\\\
\end{matrix}&\begin{matrix}0\\\ 0\\\ 0\\\ \end{matrix}\\\
\begin{matrix}\vdots&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots&\ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ddots\\\ \end{matrix}&\ddots&\vdots\\\ \begin{matrix}0\ \ \ \ \
\ \ \ \ \ \ \ \ \ \ &0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ &0\\\
\end{matrix}&\cdots&c_{0}(\omega_{N})\\\ \end{matrix}\right)$ (10)

The transfer matrix of a TMC can be obtained as

$T_{C}=\left[\begin{matrix}\bar{\bar{Ones}}&\bar{\bar{Zeros}}\\\
\bar{\bar{Y_{C}}}&\bar{\bar{Ones}}\\\ \end{matrix}\right]$ (11)

where $\bar{\bar{Ones}}$ and $\bar{\bar{Zeros}}$ are the unit and zero
matrices, receptively. They have the same size of $\bar{\bar{Y_{C}}}$. This
makes the transfer matrix $T_{C}$ of size $(4N+2)\times(4N+2)$.

Figure 1: (a) General symmetric unit cell of an ideal transmission line. (b) A
unit cell of a lossless lumped model of a transmission line with a time-
modulated capacitor. A unit cell of a lossless TL with a length of
$m\lambda_{m}$ and $90\leavevmode\nobreak\ \Omega$ characteristic impedance
loaded with (c) infinite quality factor time-modulated capacitor, (d) finite
quality factor time-modulated capacitor. Figure 2: (a) Dispersion diagram of
the unit cell in Fig. 1(b) with $L=12.5$ nH, $C_{o}=5$ pF and $M_{D}=0.66$. In
the case of no modulation, the dispersion diagram only reduces to the red
curves. (b) Close-up look of the momentum band-gap area at different values of
$M_{D}$. (c) MBG-edge phase variation and harmonics ($\omega_{o}$ and
$\omega_{-1}$) mixing coefficient variation at different values of $M_{D}$.
(d) The imaginary part of complex frequency variation within the MBG at
$F_{s}/F_{m}=0.5$ for harmonics $\omega_{o}$ and $\omega_{-1}$. At
$M_{D}=0.66$, (e) Unit cell electrical length variation with the real and
imaginary parts (real is fixed at $F_{s}/F_{m}=0.5$ for imaginary part
variation) of the complex frequency, (f) The imaginary part of complex
frequency variation within the MBG at different real part values for harmonics
$\omega_{o}$ and $\omega_{-1}$, (g) For harmonics $\omega_{o}$ and
$\omega_{-1}$, The imaginary part of complex frequency variation with $M_{D}$
at $F_{s}/F_{m}=0.5$ and unit cell electrical length at nominal Capacitance
$C_{o}$ ($\beta d=0.81$).

###  2.2 Dispersion equation

In this section, the series element (A) in Fig. 1(a) is a time-invariant
inductor (TII). The impedance matrix of such an inductor in a time-invariant
system is given by

$\bar{\bar{Z_{L}}}=j\bar{\bar{W}}(L/2)$ (12)

where $L=\bar{\bar{Ones}}\times l_{o}$, and $l_{o}$ is the value of the
inductance. The transfer matrix of a TII can be obtained by

$T_{L}=\left[\begin{matrix}\bar{\bar{Ones}}&\bar{\bar{Z_{L}}}\\\
\bar{\bar{Zeros}}&\bar{\bar{Ones}}\\\ \end{matrix}\right]$ (13)

The total transfer matrix of the unit cell can be obtained using

$T_{t}=T_{L}\times T_{C}\times T_{L}$ (14)

The dispersion diagram can be plotted by first solving

$T_{t}X=qX$ (15)

where $X$ and $q$ are the eigenvectors and values, respectively. Once the
eigenvalues are obtained, the dispersion diagram is plotted using

$q=e^{i\gamma d}$ (16a) $\beta d=real[-iln(q)]$ (16b) $\alpha d=real[ln(q)]$
(16c)

where $\gamma$ is the propagation constant, $\beta$ is the phase constant,
$\alpha$ is the attenuation constant, and $d$ is the length of the unit cell.
Using (16c,15), plotting the dispersion variation against frequency results in
$(4N+2)$ curves; two for each harmonic plus two for the fundamental frequency.
This describes each frequency’s forward and backward propagation defined in
(4).

Figure 3: (a) Phase of $S_{21}$ for only one unit cell shown in Fig. 1(b) with
$L=12.5$ nH and $C_{o}=5$ pF (static), S-parameter simulation. For 9 unit
cells with 10 V input peak (1 Watt (rms)), TMC at $M_{D}=0.66$, $L=12.5$ nH,
$C_{o}=5$ pF, $F_{s}=0.5$ GHz, $F_{m}=1$ GHz loaded with a
$50\leavevmode\nobreak\ \Omega$ load impedance. (b) Output voltage (TS). (c)
Output voltage (HB simulation). (d) Close-up look for the harmonics at 0.5
GHz. (e) Close-up look for the harmonics at 1.5 GHz.

##  3  Signal Amplification in a lossless TL lumped model with time-modulated
capacitor

The DD is plotted in Fig. 2(a) for the unit cell shown in Fig. 1(b). To plot
Fig. 2(a), numerical values are chosen as follows: $F_{m}=1$ GHz, $L=12.5$ nH,
$C_{o}=5$ pF, and $M_{D}=0.66$. The unit cell has a 50 $\Omega$ characteristic
impedance at nominal capacitance $C_{o}$. Fig. 2(a) shows that only the
fundamental frequency $\omega_{s}$ exists in the absence of modulation; the
red curves describe the forward and backward propagation dispersion at
$\omega_{s}$ in case of no modulation. With modulation present, black curves
describe the dispersion of six considered harmonics (N=3) and the fundamental
frequency. All curves are labeled with the harmonic order and type of
propagation: forward (f) or backward (r). At $F_{s}=0.5$ GHz ($F_{m}=2F_{s}$),
strong interaction happens between the fundamental $\omega_{s}$ and the
harmonic $\omega_{-1}$, which operates at the same frequency. Due to the
strong coupling between harmonics mentioned above, a region of unstable K-gap
is created called MBG [1, 24, 25]. At this region, exponential amplification
occurs for the signal, and the circuit acts as a parametric amplifier [26,
22]. However, it is worth mentioning that the DD is not affected by varying
the modulation phase $\varphi_{M}$. As a result, contrary to parametric
amplification, no synchronization is needed between propagating and modulating
signals. In lossless media (or circuits), two main factors affect the
amplification level of time-modulated defined media (circuit): the modulation
depth and the loading impedance. Moreover, introducing loss to the system
limits the amplification, as illustrated in section 4.2. Fig. 2(b) shows a
close-up of the MBG region for different modulation depths. With a higher
value of $M_{D}$, a wider MBG can be obtained. The phases at the edges of each
band gap are written to be compared with the unit cell’s electrical length
(EL), which is discussed in the following paragraph. Fig. 2(c) shows the phase
variation at the edges of the MPG with $M_{D}$. In the case of no-modulation
($M_{D}=0$), the MBG edges match in phase, no MPG. However, as $M_{D}$
increases, the MBG edges’ phases deviate away, indicating an increase in the
gap width. Moreover, the phase deviation is not symmetric at both edges
because only part of the TL is a time-variant component, which makes the
relation between varying the TMC and unit cell Bloch impedance ($Z_{Bloch}$)
nonlinear. Based on edges’ phase variation and analogy to mode coupling
coefficient definition [27], the harmonics mixing coefficient ($\kappa$) is
plotted in Fig. 2(c) following

$\kappa=\frac{\left({\rm Phase}_{edge1}\right)^{2}-\left({\rm
Phase}_{edge2}\right)^{2}}{\left({\rm Phase}_{edge1}\right)^{2}+\left({\rm
Phase}_{edge2}\right)^{2}}$ (17)

Within the MPG, only complex frequencies associated with fundamental
$\omega_{s}$ and the harmonic $\omega_{-1}$ exist and are plotted in Fig.
2(d-g) by using the following equation instead of (4) in solving the
eigenvalue problem.

$\Omega=\omega_{n}-i\sigma_{n}$ (18)

As shown Fig. 2(d), imaginary DD is plotted considering harmonics
($\omega_{0,-1}$) at different values of $M_{D}$ with a fixed real frequency
$F_{s}=0.5$ GHz and varying the imaginary part. The positive imaginary
frequency causes the signal growth, while the negative imaginary frequency is
responsible for the signal decay. As shown, there is a direct relation between
imaginary part values, the MPG width, and the value of $M_{D}$. Within MPG, as
$M_{D}$ increases, the strength of the harmonic ($\omega_{-1}$) and the
imaginary frequency increase. Consequently, at a sufficient value of $M_{D}$,
signal growth starts. This sufficient value of $M_{D}$ depends on the unit
cells, their number, the matching at the terminals (load and source), and
$F_{m}$. Both real and imaginary ($F_{s}=0.5$ GHz real part) DDs are plotted
in Fig. 2(e) at $M_{D}=0.66$ to show that only within the MPG imaginary
frequencies exist. Deviating from the frequency real part value of $F_{s}=0.5$
GHz, the imaginary part gradually vanishes due to the existence of real
frequency solutions of the eigenvalue problem, as shown in Fig. 2(f). In other
words, time modulation causes the creation of MPGs at only frequency ratio
$F_{m}=2F_{s}$ [28]. Within the MPGs, only an imaginary frequency solution
exists. Now, focusing on the unit cell electrical length at $C_{o}$ ($0.81$
rad), imaginary part variation of the fundamental $\omega_{s}$ and the
harmonic $\omega_{-1}$, at fixed real part of $F_{s}=0.5$ GHz value, is
plotted in Fig. 2(g) with $M_{D}$. As shown, the harmonic $\omega_{-1}$
imaginary part is directly proportional to the value of $M_{D}$.

Figure 4: For 9 unit cells (Fig. 1(b)) with 10 V input peak (1 Watt(rms)),
$L=12.5$ nH, $C_{o}=5$ pF, $F_{s}=0.5$ GHz, $F_{m}=1$ GHz, loaded with a
$50\leavevmode\nobreak\ \Omega$ load impedance. (a) Output voltage (TS) at TMC
at $M_{D}=0.5$. (b) Output voltage (HB simulation) at TMC with $M_{D}=0.5$.
(c) Output voltage (TS) at TMC with $M_{D}=0.33$. (d) Output voltage (HB
simulation) at TMC with $M_{D}=0.33$. Figure 5: For 9 unit cells (Fig. 1(b))
with 10 V input peak (1 Watt(rms)), $L=12.5$ nH, $C_{o}=5$ pF, $F_{s}=0.5$
GHz, $F_{m}=1$ GHz, loaded with a $50\leavevmode\nobreak\ \Omega$ load
impedance. (a) Input instantaneous power (rms) in the absence of modulation
(TS). TS output power (rms) with TMC at (b) $M_{D}=0.66$, (c) $M_{D}=0.5$, and
(d) $M_{D}=0.33$.

SP, TS, and HB simulations in ADS are utilized for further investigation. Fig.
3(a) shows the phase of $S_{21}$ of the unit cell shown in Fig. 1(b) with
$L=12.5$ nH and $C=5$ pF (static), done by S-parameter simulation. The unit
cell has an absolute EL of $0.81$ radians at $F_{s}=0.5$ GHz. This phase lies
within the MBG illustrated in Fig. 2. In addition, to check the feasibility of
(3), for a unit cell that has an absolute EL of $0.81$ radians,
$\frac{\lambda_{s}}{d}$ gives $7.75$. So, choosing N=3 satisfies (3). On the
other hand, it can be seen in Fig. 2(a) that harmonics $\omega_{\pm 3}$ do not
play any role at $F_{s}=0.5$ GHz. A FBG occurs for these harmonics within the
0-1 GHz frequency range. By choosing N=2, the same DD of Fig. 2 can be
obtained. As a result, the correct value of N must satisfy (3), and it is also
the lowest value above which the DD no longer changes within the frequency
band of interest. Solutions to the eigenvalue problem do not come in order.
This makes relating each curve to its corresponding harmonic tricky.
Consequently, reducing the number of considered harmonics (N) to the lowest
possible value decreases the complexity of the DD and helps to identify the
harmonics’ curves. However, six harmonics (N=3) are considered in the whole
manuscript to account for any changes in the DD when the unit cell becomes
more complex.

For nine unit cells with TMCs at $M_{D}=0.66$ and $F_{m}=1$ GHz loaded with a
$50\leavevmode\nobreak\ \Omega$ load impedance, Fig. 3(b) shows the transient
simulation with an input of 10 V (peak) matched source (1 W (rms)) at
$F_{s}=0.5$ GHz. It is worth mentioning that 50 $\Omega$ matched voltage
sources are utilized here and connected in series with an ideal 50 $\Omega$
isolator. The output power after the isolator is 1 W (rms), and the output
voltage is 10 V peak. Within a 200 ns simulation period, the output voltage is
amplified, reaching a 65 V peak level. For the same circuit, HB results are
shown in Fig. 4(c-e). Figs. 4(d-e) are close-ups of the harmonics of interest.
It can be seen that there are two amplified harmonics landed at 0.5 GHz
($n=0,-1$) and another two harmonics ($n=-2,1$) landed at 1.5 GHz. There is an
agreement between the simulated amplification results obtained using HB and
TS. The output amplified voltage from TS is almost equal to the summation of
the two harmonics ($n=0,-1$) landed at 0.5 GHz obtained from HB. Usually, TS
is more accurate than HB in highly nonlinear circuits. However, HB shows the
output harmonics, which is very useful to verify the results depicted in Fig.
2. For lower values of $M_{D}$, Fig. 4(a-b) and (c-d) show the simulation
results at $M_{D}=0.5$ and $M_{D}=0.33$, respectively. At $M_{D}=0.5$, the
output voltage (TS, Fig. 4(a)) saturates after 12 ns at 18 V (peak), which is
close to the summation of the two harmonics (HB, Fig. 4(b)) landed at 0.5 GHz.
At $M_{D}=0.33$, the output voltage (TS, Fig. 4(c)) saturates after 8 ns at 13
V, which is exactly the summation of the two harmonics (HB, Fig. 4(d)) landed
at 0.5 GHz. Fig. 5 shows the TS instantaneous output powers in different
cases. Fig. 5(a) shows the input power with a value of 1 W. The same value of
input power will be used in the whole manuscript. When the TMC is utilized,
the instantaneous output power is amplified. Increasing the modulation depth
value causes higher output power values at the load. As shown in Fig. 5(b-d),
the output average power is saturated at 37 W, 3 W, and 1.4 W when modulating
with $M_{D}=0.66$, $M_{D}=0.5$, and $M_{D}=0.33$, respectively. The simulated
results confirm that power amplification is directly proportional to the
modulation width $M_{D}$.

##  4 Signal Amplification in a lossless TL loaded time modulated capacitor

In this section, the element (A) in Fig. 1(a) is replaced by a TL section with
an electrical length of $2m\pi$ radian at the modulation frequency and
$90\leavevmode\nobreak\ \Omega$ characteristic impedance ($Z_{o}$). Two
conditions are considered for element (B) in Fig. 1(a): Infinite quality
factor (Q) TMC and finite Q TMC.

###  4.1 Infinite quality factor time modulated capacitor

Figure 6: DD of the unit cell in Fig. 1(c) with $C_{o}=4$ pF, $M_{D}=0.5$,
$90\leavevmode\nobreak\ \Omega$ characteristic impedance ($Z_{o}$), and TL
sections with different lengths. (a) TL length $0.1\lambda_{m}$ ($m=0.1$), and
(b) TL length $0.15\lambda_{m}$ ($m=0.15$). (c) Close-up look of the MBGs.

In this subsection, element (B) will be an infinite Q TMC. The unit cell can
be seen in Fig. 1(c), and different cases are investigated at various values
of $m$. In order to plot the DD, (15) and (16c) are solved with the matrix
$T_{L}$ in (14) replaced by

$T_{tl}=\left[\begin{matrix}\bar{\bar{A}}&\bar{\bar{B}}\\\
\bar{\bar{C}}&\bar{\bar{D}}\\\ \end{matrix}\right]$ (19)

where

$\displaystyle\bar{\bar{A}}$ $\displaystyle=\cos(2\times\pi\times
m\times\frac{\bar{\bar{W}}}{\omega_{M}}))$ (20a) $\displaystyle\bar{\bar{B}}$
$\displaystyle=j\times Z_{o}\times\sin(2\times\pi\times
m\times\frac{\bar{\bar{W}}}{\omega_{M}}))$ (20b) $\displaystyle\bar{\bar{C}}$
$\displaystyle=j\times Y_{o}\times\sin(2\times\pi\times
m\times\frac{\bar{\bar{W}}}{\omega_{M}}))$ (20c) $\displaystyle\bar{\bar{D}}$
$\displaystyle=\cos(2\times\pi\times
m\times\frac{\bar{\bar{W}}}{\omega_{M}}))$ (20d)

Figure 7: For unit cell in Fig. 1(c) with $C_{o}=4$ pF, $M_{D}=0.6$,
$90\leavevmode\nobreak\ \Omega$ characteristic impedance ($Z_{o}$), TL section
lengths $0.1\lambda_{m}$ and $0.15\lambda_{m}$, at the fundamental frequency
(0.5 GHz). (a) Variation in Bloch impedances across time for one complete
cycle. (b) The absolute value of the reflection coefficient
($\lvert\Gamma\rvert$) within one complete cycle considering
$50\leavevmode\nobreak\ \Omega$ load (source).

Adding TL to the unit cell in Fig. 1(a) as element (A) allows us to
investigate the effect of varying the electrical length of the TL on the
amplification. Changing the electrical length of the time-modulated unit cell
will allow the control of the location of the FBG (space) in the DD of one of
the prominent harmonics ($\omega_{0,-1}$) relative to the MBG (time). The DD
(N=3) of the unit cell shown in Fig. 1(c) is plotted in Fig. 6 for different
lengths of the TL. $90\leavevmode\nobreak\ \Omega$ $Z_{o}$ TL sections are
utilized to match the unit cell at nominal capacitance $C_{o}$ to a
$50\leavevmode\nobreak\ \Omega$ load. Red curves in Fig. 6 represent the
dispersion curve of the unit cell without any modulation. In the no-modulation
case, it can be seen that varying the TL length changes the position of the
FBG that starts from and ends by phase $\pm\pi$ [29]. With time modulation,
the MBG appears at $F_{s}=0.5$ GHz, considering $F_{m}=1$ GHz, green circles
note the MBGs. Moreover, due to the interaction between harmonics, FBGs are
created at phase values different than $\pm\pi$. The FBGs are noted in blue
circles. We are interested in FBGs occurring by one of the prominent harmonics
($\omega_{0,-1}$). As shown in Fig. 6, as the FBG becomes closer to the MBG,
the MBG becomes wider, and the slope of the dispersion curves around the MBG
increases. Wider MBG and slopes indicate higher amplification values and
broader response around $F_{s}=0.5$ GHz.

Figure 8: For 9 unit cells (Fig. 1(c)) with 10 V input peak (1 Watt (rms)),
$C_{o}=4$ pF $F_{s}=0.5$ GHz, $F_{m}=1$ GHz, $M_{D}=0.6$, and the TL with
$90\leavevmode\nobreak\ \Omega$ characteristic impedance ($Z_{o}$) and
different lengths loaded with a $50\Omega$ load impedance. For
$0.1\lambda_{m}$ ($m=0.1$) TL: (a) output voltage (TS), (c) Fourier transform
(FT) of the output voltage (TS), (e) output voltage HB, and (g) power output
(TS). For $0.15\lambda_{m}$ ($m=0.15$) TL: (b) output voltage (TS), (d)
Fourier transform (FT) of the output voltage (TS), (f) output voltage HB, and
(h) power output (TS).

To further investigate the effect of TL length on the MBG width, the variation
across time of the unit cell Bloch impedances are plotted in Fig. 7 for the
fundamental frequency of 0.5 GHz. As in [29], Bloch impedance can be plotted
using

$Z_{Bloch}^{\pm}=\frac{-2B}{A-D\mp\sqrt{{(A+D)}^{2}-4}}$ (21)

where A, B, C, and D are the ABCD matrix elements of the unit cell at the
fundamental frequency ($\omega_{s}$). The $\pm$ solutions correspond to
$Z_{Bloch}$ for forward and backward traveling waves, respectively. Fig. 7(a)
shows the Bloch impedance variation for the unit cell of TL lengths
$0.1\lambda_{m}$ and $0.2\lambda_{m}$ sections within one complete cycle. It
can be seen that a unit cell of $0.2\lambda_{m}$ TL section has a slightly
wider impedance variation range of 41-70 $\Omega$ compared to the unit cell of
$0.1\lambda_{m}$ TL section, which shows 40-66 $\Omega$ impedance variation
range. This agrees with the results shown in Fig. 6; a unit cell of
$0.15\lambda_{m}$ TL section has a wider MBG than the unit cell of
$0.1\lambda_{m}$ section. As $M_{D}$ increases, a wider range of impedances
and MBGs are also expected. In addition, the difference between unit cells of
$0.1\lambda_{m}$ and $0.15\lambda_{m}$ TL sections in the MBG and impedance
variation range becomes more apparent. Fig. 7(b) shows the absolute value of
the reflection coefficient ($\lvert\Gamma\rvert$) within one complete cycle
considering a $50\leavevmode\nobreak\ \Omega$ load (source), which is computed
using

$\Gamma_{load\ (source)}=\frac{Z_{load\ (source)}-Z_{Bloch}}{Z_{load\
(source)}+Z_{Bloch}}$ (22)

As shown in Fig. 7(b), the values of $\lvert\Gamma\rvert$ are almost the same
except at the half of the cycle, where $\lvert\Gamma\rvert$ is higher for
$0.2\lambda_{m}$ TL unit cell compared to $0.1\lambda_{m}$ TL unit cell.
Increasing $\lvert\Gamma\rvert$ could enhance the amplification for
$0.2\lambda_{m}$ TL unit cell because the reflected signal can be added
constructively to the main signal and re-amplified [28]. However, increasing
the mismatch at the terminals can cause instability and oscillation.

TS are performed for 9 unit cells (Fig. 1(c)) with $50\leavevmode\nobreak\
\Omega$ matched source suppling 10 V peak (1W ), $C_{o}=4$ pF, $F_{s}=0.5$
GHz, $F_{m}=1$ GHz, $M_{D}=0.6$, and a TL with $90\leavevmode\nobreak\ \Omega$
characteristic impedance ($Z_{o}$) and different lengths loaded with a
$50\leavevmode\nobreak\ \Omega$ load impedance. Two TL lengths are chosen,
$0.1\lambda_{m}$ and $0.15\lambda_{m}$, to be almost matched at $C_{o}$. The
results are shown in Fig. 8, where signal amplification is observed. The unit
cell consists of ideal elements, and introducing loss will make the circuit
saturate at lower values and affect the system’s responsivity to the
modulation. Figs. 8(a,b) show the TS at TL lengths of $0.1\lambda_{m}$ and
$0.15\lambda_{m}$, respectively. Figs. 8(c,d) show Fourier transforms of the
results obtained in Figs. 8(a,b), receptively. As shown, as the TL length
increases, higher amplification values are obtained; the output voltage is
amplified to 18 V and 28 V using TL lengths of $0.1\lambda_{m}$ and
$0.15\lambda_{m}$, respectively. Figs. 8(e,f) show the HB simulation results,
which agree with the TS results shown in Figs. 8(a,b). The input power is the
same as shown in Fig. 5(a). The instantaneous output power at TL lengths
$0.1\lambda_{m}$ and $0.15\lambda_{m}$ with modulation are shown in Figs.
8(g,h), respectively. The instantaneous output power is amplified to 3.5 W and
9.2 W using TL lengths of $0.1\lambda_{m}$ and $0.15\lambda_{m}$,
respectively. The obtained simulation results match the analysis done using
the DD and Bloch impedance. Compared to the unit cell of $0.1\lambda_{m}$ TL
section, the unit cell of $0.15\lambda_{m}$ TL section has a wider MBG and
variation range of Bloch impedances. Consequently, as discussed, the unit cell
of $0.15\lambda_{m}$ TL section has higher voltage and power amplification
values.

###  4.2 Finite quality factor time modulated capacitor

In this subsection, element (A) in Fig. 1(a) is a TL with length
$0.15\lambda_{m}$ (m=0.15) and $90\leavevmode\nobreak\ \Omega$ characteristic
impedance ($Z_{o}$). Element (B) will be a TMC with series resistance (Rc).
The introduced resistance Rc limits the quality factor of the TMC. The unit
cell is illustrated in Fig. 1(d) and will be studied at different values of
Rc.

Figure 9:  For the unit cell in Fig. 1(d) with $C_{o}=4$ pF, $M_{D}=0.6$, and
a TL with $90\leavevmode\nobreak\ \Omega$ characteristic impedance ($Z_{o}$)
and length of $0.15\lambda_{m}$ (m=0.15). DD at (a) Rc = 5 $\Omega$ and (b) Rc
= 15 $\Omega$. Attenuation variation versus $F_{s}/F_{m}$ at (c) Rc = 5
$\Omega$ and (d) Rc = 15 $\Omega$. For various values of Rc (0, 5, 10, 15, 20)
$\Omega$, (e) Close-up look of the MBG, (f) The imaginary part of complex
frequency variation within the MBG at $F_{s}/F_{m}=0.5$ for harmonics
$\omega_{o}$ and $\omega_{-1}$ and (g) Close-up look of attenuation at the
MBG. (h) For Rc = 15 $\Omega$, the Unit cell electrical length variation with
the real and imaginary parts (real is fixed at $F_{s}/F_{m}=0.5$ for imaginary
part variation) of the complex frequency. (i) The real part of Bloch impedance
variation within one complete cycle. (j) The imaginary part of the Bloch
impedance variation within one complete cycle. (k) The absolute value of the
reflection coefficient ($\lvert\Gamma\rvert$) within one complete cycle
considering $50\leavevmode\nobreak\ \Omega$ load (source).

To plot the DD, the transfer matrix of the shunt element (B) should be
computed. The resistance matrix is given by

$\bar{\bar{Z_{Rc}}}=Rc\times\bar{\bar{Ones}}$ (23)

where Rc is the value of the resistance. The impedance matrix of the shunt
branch B can be calculated using

$\bar{\bar{Z_{B}}}=\bar{\bar{Z_{Rc}}}+{\bar{\bar{Y_{C}}}}^{-1}$ (24)

where $\bar{\bar{Y_{C}}}$ is given in (8). The transfer matrix of shunt branch
B is given by

$T_{B}=\left[\begin{matrix}\bar{\bar{Ones}}&\bar{\bar{Zeros}}\\\ \
{\bar{\bar{Z_{B}}}}^{-1}&\bar{\bar{Ones}}\\\ \end{matrix}\right]$ (25)

The total transfer matrix of the unit cell shown in Fig. 1(d) can be obtained
using

$T_{Rc}=T_{tl}\times T_{B}\times T_{tl}$ (26)

where $T_{tl}$ is given in (19) with m = 0.15.

The DDs are plotted in Fig. 9 utilizing (26) and (16c). In Figs. 9(a)(b), the
DD is plotted at Rc values of 5 and 15 $\Omega$, respectively. Compared to
Fig. 9(b), weak interaction between harmonics can be observed in the areas of
the FBGs, circled in blue, and the MBGs, circled in green. As the value of Rc
increases, the FBG and MBG areas shrink. Moreover, the attenuation constant
variations with frequency associated with the cases shown in Figs. 9(a)(b) are
shown in Figs. 9(d)(c), respectively. Focusing on the prominent harmonics
($\omega_{0,-1}$), the attenuation constant increases at all frequencies as
the Rc value increases except for $F_{s}=2F_{m}$; attenuation is always zero
as long as the MPG exists. To clearly show the effect of increasing the value
of Rc on the MBG and attenuation, a close-up look of the MBG and attenuation
at $F_{s}=2F_{m}$ at different values of Rc are plotted in Fig. 9(e),(f),
respectively. As shown in Fig. 9(e), the MPG shrinking is obvious as an effect
of increasing the value of Rc. On the other hand, as shown in Fig. 9(f),
despite hitting the zero value of attenuation at $F_{s}=2F_{m}$ regardless of
the value of Rc, attenuation at other frequencies increases with the increase
of Rc. It is worth mentioning that if Rc reaches a value that closes the MPG,
attenuation will not be zero anymore at $F_{s}=2F_{m}$. In Fig. 9(g),
imaginary DD is plotted at different values of Rc considering harmonics
($\omega_{0,-1}$) with a fixed real frequency $F_{s}=0.5$ GHz. As Rc
increases, the imaginary frequency moves more to the negative region,
explaining the expected drop in amplification levels. In Fig. 9(h), real and
imaginary (real at $F_{s}=0.5$) DDs are plotted at Rc= 15$\Omega$. The MBG
width is mainly defined by the phases at which the positive imaginary
frequency exists. In this case (Rc= 15$\Omega$) and within the MBG, the
imaginary frequency takes the values confined between the vertical blue dotted
lines because, outside this region, real solutions to the eigenvalue problem
exist.

Figure 10: For 9 unit cells (Fig. 1(d)), with 10 V input peak (1 Watt(rms)),
$C_{o}=4$ pF, $F_{s}=0.5$ GHz, $F_{m}=1$ GHz, $M_{D}=0.5$, a TL with
$90\leavevmode\nobreak\ \Omega$ characteristic impedance ($Z_{o}$) and length
of $0.15\lambda_{m}$ loaded with a $50\leavevmode\nobreak\ \Omega$ load
impedance. (a-b) Voltage output (TS) at Rc = 5 $\Omega$ and 15 $\Omega$,
respectively. (c-d) Fourier transforms for TS results at Rc = 5 $\Omega$ and
15 $\Omega$, respectively. (e-f) Voltage output (HB) at Rc = 5 $\Omega$ and 15
$\Omega$, respectively. (g-h) Power output (TS) at Rc = 5 $\Omega$ and 15
$\Omega$, respectively.

Figs. 9(i-k) show the effect of Rc on the Bloch impedance ($Z_{Bloch}$) and
$\lvert\Gamma\rvert$ at the terminals. The variation range of the real part of
$Z_{Bloch}$ shrinks as Rc increases (Fig. 9(i)). Moreover, the imaginary part
of $Z_{Bloch}$ increases which reflects the loss increase within the unit cell
(Fig. 9(j)). Moreover, the circuit gradually loses matching at the
$50\leavevmode\nobreak\ \Omega$ terminals at the quarter cycle (nominal
$C_{o}$) as Rc increases (Fig. 9(k)). Consequently, it can be concluded that
the value of Rc is inversely proportional to the signal amplification gain.

To confirm the observed weak interaction between harmonics and its relation to
Rc, TS is performed with 9 unit cells (Fig. 1(d)) with a
$50\leavevmode\nobreak\ \Omega$ matched source supplying 10 V peak (1 W ),
$C_{o}=4$ pF, $F_{s}=0.5$ GHz, $F_{m}=1$ GHz, $M_{D}=0.6$, a TL with
$50\leavevmode\nobreak\ \Omega$ characteristic impedance ($Z_{o}$) and length
of $0.15\lambda_{m}$ loaded with a $50\leavevmode\nobreak\ \Omega$ load
impedance. Two values of Rc are considered, 5 $\Omega$ and 15 $\Omega$, and
the results are plotted in Fig. 10. For Rc = 5 $\Omega$, the output voltage
saturates at the peak of 17 V (Fig. 10(a)(c)(e)), and the instantaneous output
power saturates at 3 W (Fig. 10(g)). Increasing the value of Rc to 15 $\Omega$
causes a drop in the output voltage to 7 V peak value (Fig. 10(b)(d)(f)), and
in the instantaneous output power to 0.6 W (Fig. 10(h)). As shown, when Rc
reaches 15 $\Omega$, there is no amplification.

##  5 Conclusion

Signal amplification in a TL with time-modulated $Z_{o}$ is investigated by
studying the eigenvalue problem and confirmed by circuit modeling. Three
models are considered: a lossless L-C TL lumped model with shunt TMC, a TL
loaded with shunt infinite Q TMC, and a TL loaded with shunt finite Q TMC to
study the loss effect. The eigenvalue problem is discussed in detail, and the
real and imaginary DDs are plotted for each model. While modulation, the Bloch
impedance of the unit cell and the reflection coefficient at the terminals are
plotted and discussed. SP, TS, and HB simulations are performed, and the
results are consistent with the DD, Bloch impedance variation of the unit cell
and reflection coefficient variation at the terminals. The third model
discusses the loss effect by plotting the attenuation variation with frequency
and real and imaginary DDs. In a lossy time-modulated media, if the loss is
low enough to allow the MBG to be created at $F_{s}=2F_{m}$, attenuation will
be zero at $F_{s}=2F_{m}$. However, the drop in amplification level happens
because of the movement of imaginary frequency within the MBG to the negative
region, causing the signal decay.

## Acknowledgements

This work has been completed under a research agreement between Purdue
University and The American University in Cairo.

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$\displaystyle|I_{2}^{r}|\leq\,C\bigg{(}\int_{0}^{t}\|\nabla({\bf{u}}_{r}-{\bf{u}})\|_{L^{2}(\mathbb{T}^{3})}^{2}d\tau\bigg{)}^{1/2},\quad|J_{2}^{r}|\leq\,C\|{\bf{U}}_{r}-{\bf{U}}\|_{L^{r}([0,T];L^{s}(\mathbb{T}^{3}))},$

where the constant $C$ depends on $\bf{U}$ and $\bf{u}$ only. Therefore, a
simple property of convolution reveals that

$\displaystyle\lim_{r\to\,0}I_{1}^{r}=\lim_{r\to\,0}I_{2}^{r}=\lim_{r\to\,0}J_{1}^{r}=\lim_{r\to\,0}J_{2}^{r}=0.$
(5.4)

Finally, letting $r\to 0$ in (5.3), and using (5.4), we obtain

$\begin{split}{\rm d}\left(\int_{\mathbb{T}^{3}}{\bf u}\cdot{\bf U}\,{\rm
d}x\right)&=\int_{\mathbb{T}^{3}}\left[-{\bf u}\cdot{\rm div}({\bf
U}\otimes{\bf U})+{{\bf u}\otimes{\bf u}}:\nabla_{x}{\bf u}\right]\,{\rm
d}x\,{\rm d}t\\\ &\qquad+\sum_{k\geq
1}\int_{\mathbb{T}^{3}}{\textbf{G}}_{k}({\bf U})\cdot{\textbf{G}}_{k}({{\bf
u}})\,{\rm d}x\,{\rm d}t-\int_{\mathbb{T}^{3}}\nabla_{x}{\bf U}:\nabla_{x}{\bf
u}\,{\rm d}x\,{\rm d}t+{\rm d}\mathcal{Q}_{1},\end{split}$ (5.5)

where $\mathcal{Q}_{1}(t)$ is a square integrable martingale given by

$\mathcal{Q}_{1}(t):=\int_{0}^{t}\int_{\mathbb{T}^{3}}{({\bf
u}\cdot\mathbb{G}(\bf U)+\bm{U}\cdot\mathbb{G}({\bf u})}\,{\rm d}x\,{\rm d}W.$

###  5.1 Proof of Theorem 2.19

We closely follow the strategy depicted in Subsection 4.2. To proceed, we
first introduce a stopping time

$\kappa_{L}:=\text{inf}\,\Big{\\{}t\in(0,T)\Big{|}\,\|{\bf
U}(t)\|_{L^{s}(\mathbb{T}^{3})}\,\geq\,L\Big{\\}}$

Since $\mathbb{E}\Big{[}\sup_{t\in[0,T]}\|{\bf
U}\|_{L^{s}(\mathbb{T}^{N})}\Big{]}\,\textless\,\infty$ by assumption, we have

$\mathbb{P}[\kappa_{L}\,\textless\,T]\,\leq\,\mathbb{P}\Big{[}\sup_{t\in[0,T]}\|{\bf
U}\|_{L^{s}(\mathbb{T}^{3})}\,\geq\,L\Big{]}\,\leq\,\frac{1}{L}\mathbb{E}\Big{[}\sup_{t\in[0,T]}\|{\bf
U}\|_{L^{s}(\mathbb{T}^{3})}\Big{]}\,\to\,0,$
$\mathbb{P}\bigg{[}\lim_{L\to\infty}\kappa_{L}=T\bigg{]}=1.$

Therefore, it is enough to show the result for a fixed $L$. We now make use of
the relative energy (5.1), the energy inequality (2.13), and (5.5), for both
solutions, to conclude that for all $t\in[0,T]$, $\mathbb{P}$-a.s.

$\displaystyle\mathrm{F}^{\mathrm{NS}}_{\mathrm{mv}}\left({\bf u}\Big{|}{\bf
U}\right)(t\wedge\kappa_{L})+\int_{0}^{t\wedge\kappa_{L}}\int_{\mathbb{T}^{3}}|\nabla_{x}{\bf(u-U)}|^{2}\,{\rm
d}x$ $\displaystyle\qquad=\,\frac{1}{2}\int_{\mathbb{T}^{3}}|{\bf
u}(t\wedge\kappa_{L})|^{2}\,{\rm d}x+\frac{1}{2}\int_{\mathbb{T}^{3}}|{\bf
U}(t\wedge\kappa_{L})|^{2}\,{\rm
d}x+\int_{0}^{t\wedge\kappa_{L}}\int_{\mathbb{T}^{3}}|\nabla_{x}{\bf
u}|^{2}\,{\rm d}xdt$
$\displaystyle\qquad\qquad+\int_{0}^{t\wedge\kappa_{L}}\int_{\mathbb{T}^{3}}|\nabla_{x}{\bf
U}|^{2}\,{\rm d}xdt-\int_{\mathbb{T}^{3}}{\bf u}(t\wedge\kappa_{L})\cdot{\bf
U}(t\wedge\kappa_{L})\,{\rm
d}x-2\int_{0}^{t\wedge\kappa_{L}}\int_{\mathbb{T}^{3}}\nabla_{x}{\bf
u}:\nabla_{x}{\bf U}\,{\rm d}xdt$
$\displaystyle\qquad\leq\mathrm{F}^{\mathrm{NS}}_{\mathrm{mv}}\left({\bf
u}(0)\Big{|}{\bf
U}(0)\right)+\int_{0}^{t\wedge\kappa_{L}}\int_{\mathbb{T}^{3}}\nabla_{x}{\bf
u}:({\bf(u-U)}\otimes{\bf U})\,{\rm d}xdt+\mathcal{Q}_{RE}(t\wedge\kappa_{L})$
$\displaystyle\qquad\qquad+\frac{1}{2}\int_{0}^{t\wedge\kappa_{L}}\int_{\mathbb{T}^{3}}|\mathcal{P}_{H}({\mathbb{G}}({\bf
u})-{\mathbb{G}}({\bf U}))|^{2}\,{\rm d}xdt$
$\displaystyle\qquad=\,\mathrm{F}^{\mathrm{NS}}_{\mathrm{mv}}\left({\bf
u(0)}\Big{|}{\bf
U}(0)\right)+\int_{0}^{t\wedge\kappa_{L}}\int_{\mathbb{T}^{3}}\nabla_{x}({\bf
u-U}):(({\bf u-U})\otimes{\bf U})\,{\rm
d}x+\mathcal{Q}_{RE}(t\wedge\kappa_{L})$
$\displaystyle\qquad\qquad+\frac{1}{2}\int_{0}^{t\wedge\kappa_{L}}\int_{\mathbb{T}^{3}}|\mathcal{P}_{H}({\mathbb{G}}({\bf
u})-{\mathbb{G}}({\bf U}))|^{2}\,{\rm d}xdt$
$\displaystyle\qquad\leq\,\mathrm{F}^{\mathrm{NS}}_{\mathrm{mv}}\left({\bf
u}(0)\Big{|}{\bf
U}(0)\right)+C\,\bigg{(}\int_{0}^{t\wedge\kappa_{L}}\int_{\mathbb{T}^{N}}|\nabla({\bf
u-U})|^{2}\bigg{)}^{1-1/r}\bigg{(}\int_{0}^{t\wedge\kappa_{L}}\|{\bf
U}\|_{L^{s}}^{r}\int_{\mathbb{T}^{3}}|{\bf u-U}|^{2}\,dxdt\bigg{)}^{1/r}$
$\displaystyle\qquad\qquad+\mathcal{Q}_{RE}(t\wedge\kappa_{L})+\frac{1}{2}\int_{0}^{t\wedge\kappa_{L}}\int_{\mathbb{T}^{3}}|{\mathbb{G}}({\bf
u})-{\mathbb{G}}({\bf U})|^{2}\,{\rm d}xdt$
$\displaystyle\qquad\leq\,\mathrm{F}^{\mathrm{NS}}_{\mathrm{mv}}\left({\bf
u}(0)\Big{|}{\bf
U}(0)\right)+\int_{0}^{t\wedge\kappa_{L}}\int_{\mathbb{T}^{3}}|\nabla_{x}{\bf(u-U)}|^{2}\,{\rm
d}x+C\,\bigg{(}\int_{0}^{t\wedge\kappa_{L}}\|{\bf
U}\|_{L^{s}}^{r}\int_{\mathbb{T}^{3}}|{\bf u-U}|^{2}\,dxdt\bigg{)}$
$\displaystyle\qquad\qquad+\mathcal{Q}_{RE}(t\wedge\kappa_{L})+\frac{D_{1}}{2}\int_{0}^{t\wedge\kappa_{L}}\int_{\mathbb{T}^{3}}{\bf|u-U|}^{2}\,{\rm
d}xdt$
$\displaystyle\qquad\leq\,\mathrm{F}^{\mathrm{NS}}_{\mathrm{mv}}\left({\bf
u}(0)\Big{|}{\bf
U}(0)\right)+\int_{0}^{t\wedge\kappa_{L}}\int_{\mathbb{T}^{3}}|\nabla_{x}({\bf
u-U})|^{2}\,{\rm
d}x+C(L)\bigg{(}\int_{0}^{t\wedge\kappa_{L}}\int_{\mathbb{T}^{3}}|{\bf
u-U}|^{2}\,dxdt\bigg{)}$
$\displaystyle\qquad\qquad+\mathcal{Q}_{RE}(t\wedge\kappa_{L})+\frac{D_{1}}{2}\int_{0}^{t\wedge\kappa_{L}}\int_{\mathbb{T}^{3}}{\bf|u-U|}^{2}\,{\rm
d}xdt,$

where

$\displaystyle\mathcal{Q}_{RE}(t)=\mathcal{Q}_{1}(t)+\int_{0}^{t}\int_{\mathbb{T}^{3}}\Big{(}{\bf
u}\,\mathcal{P}_{H}\mathbb{G}({\bf u})+{\bf U}\,\mathcal{P}_{H}\mathbb{G}({\bf
U})\Big{)}\,{\rm d}x\,{\rm d}W.$

Note that, in the above calculations, we have used the estimate (5.2), and
classical Young’s inequality

$\displaystyle xy\,\leq\,\frac{\gamma x^{a}}{a}+\frac{y^{b}}{b\gamma^{b/a}}$

for any $\gamma\,\textgreater 0,\,\frac{1}{a}+\frac{1}{b}=1$, and
$x,y\,\textgreater 0$. Therefore, we obtain for all $t\in[0,T]$,
$\mathbb{P}$-a.s.

$\mathrm{F}^{\mathrm{NS}}_{\mathrm{mv}}\left({\bf u}\ \Big{|}{\bf
U}\right)(t\wedge\kappa_{L})\leq\,\mathrm{F}^{\mathrm{NS}}_{\mathrm{mv}}\left({\bf
u}(0)\Big{|}{\bf
U}(0)\right)+C(L)\int_{0}^{t\wedge\kappa_{L}}{\mathrm{F}^{\mathrm{NS}}_{\mathrm{mv}}\left({\bf
u}\ \Big{|}{\bf U}\right)(s)}ds+\mathcal{Q}_{RE}(t\wedge\kappa_{L}).$

After taking expectation both side, we have

$\mathbb{E}\bigg{[}\mathrm{F}^{\mathrm{NS}}_{\mathrm{mv}}\left({\bf u}\
\Big{|}{\bf
U}\right)(t\wedge\kappa_{L})\bigg{]}\leq\,\mathbb{E}\bigg{[}\mathrm{F}^{\mathrm{NS}}_{\mathrm{mv}}\left({\bf
u}(0)\Big{|}{\bf
U}(0)\right)+C(L)\int_{0}^{t\wedge\kappa_{L}}{\mathrm{F}^{\mathrm{NS}}_{\mathrm{mv}}\left({\bf
u}\ \Big{|}{\bf U}\right)(s)}ds\bigg{]}.$

Since $\mathbb{E}\bigg{[}\mathrm{F}^{\mathrm{NS}}_{\mathrm{mv}}\left(\bf
u(0)\Big{|}\bf U(0)\right)\bigg{]}=0$, we may use Gronwall’s inequality to
conclude that

$\displaystyle\mathbb{E}\bigg{[}\mathrm{F}^{\mathrm{NS}}_{\mathrm{mv}}\left({\bf
u}\ \Big{|}{\bf U}\right)(t\wedge\kappa_{L})\bigg{]}=0.$

We can now pass to the limit as $L\to\infty$ to conclude that

$\displaystyle\mathbb{E}\bigg{[}\mathrm{F}^{\mathrm{NS}}_{\mathrm{mv}}\left({\bf
u}\ \Big{|}{\bf U}\right)(t)\bigg{]}=0,\,\,\text{for all}\,\,t\in[0,T].$

This implies that $\mathbb{P}$-almost surely, for all $t\in[0,T]$,

${\bf u}(x,t)={\bf U}(x,t),\,\,\,\text{for a.e.}\,\,x\in\mathbb{T}^{3}.$

This finishes the proof of the theorem.

## Acknowledgements

U.K. acknowledges the support of the Department of Atomic Energy, Government
of India, under project no.$12$-R$\&$D-TFR-$5.01$-$0520$, and India SERB
Matrics grant MTR/$2017/000002$.

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