\documentclass[fleqn,10pt]{wlscirep}
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\title{ Design Theory of Compact Power Divider with Reconfigurable Power Division and Negative Group Delay Characteristics}

\author{Rekha G Nair$^*$, and Natarajamani S}
\affil{Department of Electronics and Communication Engineering }

\affil{Amrita School of Engineering-Coimbatore, Amrita Vishwa Vidyapeetham, India}

\affil[*]{g-rekha@cb.students.amrita.edu}



%\keywords{Keyword1, Keyword2, Keyword3}

\begin{abstract}
This article presents the combined analysis of reconfigurable power division  and Negative Group Delay (NGD) in a power divider. A novel composite transmission line (CT) based reconfigurable power divider with high power division ratio, variable negative group delay and lower characteristic impedance is presented in this work. The impedance transformation in composite transmission lines controls both NGD and power division. This power divider possesses a wide range of power division ratios from 1 to 39, adequate isolation, impedance matching, and NGD of -3.4 ns to -4.7 ns in the reconfigurable transmission path. The NGD is achieved without using any additional group delay circuits. Theoretical equations corresponding to the low characteristic impedance of the transmission line sections and that of isolation elements are derived. The measurement results justifies the attainment of high tuning of the power division ratio, and NGD . Isolation and return loss are higher than -15dB at the centre frequency of 1.5GHz. The major contributions of this design can be listed as the wide reconfigurable power division along with negative group delay and reduced size.
\end{abstract}
\begin{document}

\flushbottom
\maketitle
% * <john.hammersley@gmail.com> 2015-02-09T12:07:31.197Z:
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%  Click the title above to edit the author information and abstract
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\thispagestyle{empty}

\section*{Introduction}
\section{Introduction}

 Wireless communication systems require equal, unequal, and reconfigurable power dividers as feeding networks for the antenna arrays. In reconfigurable power dividers operating band or power-dividing ratio is controlled by  DC voltages and lumped elements like varactor diodes. The capacitance of varactor diodes can be controlled by DC voltage, resulting in bandwidth or power-dividing ratio change.  \cite{@1118977529}$^,$\cite{quddious2018use}$^,$ \cite{7601284}. In addition to reconfigurability, NGD is required to overcome beam squint problems in antenna array elements. 
Some reconfigurable power dividers  are presented in \cite{9694458}$^,$\cite{7470611}$^,$\cite{8842611}$^,$ \cite{9756942}.
Switchable inverters are used for getting fully reconfigurable power divisions in\cite{9694458}. However, the presence of twenty PIN diodes makes the system complex. In \cite{7470611}, only two varactor diodes are used, but tunability is from 1:1 to 1:2.4 only. Reconfigurable PDs with continuous division ratio are designed in \cite{8842611}, but the power division range of -12.4 to 14.8 dB and -8.6 to -22.5 dB only. PD with a more comprehensive division from −1.25 dB to 20 dB is realized in \cite{9756942}, but the circuit uses a large number of different characteristic impedances. Planar magic-T-based voltage-controlled variable power divider with a maximum power division range of 14 dB is presented in \cite{9395618}. However, the circuit has a phase difference of around 15 degrees. 
Power splitting architecture of SWIPT system is studied in \cite{Nair}$^,$\cite{9588824} and 
reconfigurability based on cascaded varactor diode and a reflective phase-shifter method is presented in  \cite{Skyworks}$^,$ \cite{9726449} $^,$\cite{leinovel}. The isolation section with additional transmission lines is shown in  \cite{wang2014trantanella}$^,$\cite{8913621} to eliminate the undesirable coupling occurring while tuning. 

Phased array antennas suffer from beam-squint problems, which leads to unwanted perturbation in the shape and direction of the radiation pattern. A power divider with tunable PDR and NGD will be beneficial to overcome this design challenge \cite{chaudhary2019gd}. Power dividers based on controlling insertion loss and having a wide range dividing ratio and narrow bandwidth are presented in \cite{zarghami2022narrow}. However, GD analysis is not done in this work. The conventional power dividers designed for antenna arrays using NGD circuits suffer from small fractional bandwidths (FBWs) and poor reconfigurability in PDR. Combining power division reconfigurability and NGD will compensate for circuit group delay. In RF circuits, the NGD phenomenon is observed within a narrow frequency band, it will increase linearity resulting in enhanced performance of a wireless communication system \cite{9038500} as in the case of squint-free series-fed antenna arrays \cite{7100962}. A distributed passive band pass NGD circuit cascaded with a microwave amplifier is presented in \cite{9427558}. All works mentioned earlier, are successful in realizing functions like tunable bandwidth, frequency, power division, or NGD. A work simultaneously addressing power division reconfiguration and negative group delay is but not yet reported In literature, 

 This article presents a new design technique to achieve negative group delay and tunable power division ratio without additional group delay circuits. The impedance transformation in composite transmission lines controls the negative group delay and power division. In the proposed circuit, the transmission path between 3 and 1 provides NGD characteristics, and positive group delay is provided in the transmission path between 2 and 1. The proposed power divider can be used in envelope-tracking applications or dynamic power amplifiers. The positive group delay path can be directly linked to the RF path, and the negative group delay path can be connected to the detector path to make up for the time mismatch between the signal envelope and the dynamic power supply. The circuit also maintains proper impedance matching and adequate isolation. The paper is arranged with the theoretical analysis and design techniques in section II and section III analyzes and investigates results in different scenarios and compares this work with the recent state-of-the-art designs in the literature. The work is then concluded in section IV.


\section{Design Techniques and mathematical analysis}
The concept of CT lines is used in this design to attain: reconfigurability in power division, smaller characteristic impedance, size reduction, and NGD. The isolation network has additional transmission lines and lumped elements to improve port matching and reconfigurability. The block diagram of the proposed power divider is shown in Fig.1. The transmission matrix of a conventional quarter wave length (QWL) section is equated to that of the power divider's modified branches (CT lines). The power flowing through the arms will be proportional to the impedance of the dividing arms. 

The S-matrix relation between composite transmission lines and quarter wavelength line can be given as:

\begin{pmatrix}
M_1  \\

\end{pmatrix}
\
*
\begin{pmatrix}

M_{C_1}\\

\end{pmatrix}
=
\begin{pmatrix}
M_Q_W_L\\

\end{pmatrix} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{pmatrix}
Cos\theta_1 & jZ_1 Sin\theta_1   \\
jY_1 Sin\theta_1 & Cos\theta_1
\end{pmatrix}*\begin{pmatrix}
1 & - \frac{j}{\omega C_1}\\
0& 1
\end{pmatrix}= \begin{pmatrix}
0 & - {j}{\sqrt2Z_0}\\
\frac{j}{\sqrt{2}Z_0} & 0
\end{pmatrix}    


%%%%%%%%%%%%%%%%%

$M_1$ is the transmission matrix of a microstrip transmission line and $ M_C_1 $ and $ M_Q_W_L$ are that of capacitor and quarter wavelength line respectively. By solving the matrices the value of tuning element ( $C_1$) can be found.
 
The equivalent circuit model of the PD for even mode analysis is shown in Fig.2a and based on that we get (1),(2),(3) Here $\theta_1$, $\theta_2$ and $\theta_3$ represent the electrical length and $Z_1$, $Z_2$, and $ Z_3 $ corresponds to characteristic impedance. 
\begin{equation}
Z_{Even1}= Z_1 \frac{2Z_{0}+j2Z_{1}tan\theta_{1}}{Z_{1}+j2Z_{0}tan{\theta_{1}}}         
\end{equation}

\begin{equation}
Z_{Even2}=
\frac{{Z__{Even1}}{Z_{Even3}}}{Z_{Even1}+Z_{Even3}} 
\end{equation}



\begin{figure}[htp]
	\centering
	\includegraphics[width=9cm]{Fig1.png}
	\caption{ The block diagram of power divider with reconfigurable PDR }
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htp]
	\centering
	\includegraphics[height=4.5cm,width=9cm]{Fig.2 (1).jpg}
	\caption{ Equivalent Circuits (a) Odd mode (b)Even mode }
\end{figure}
%%%%%%%%%%%%%%%%%%%%%


   
    
  
 % \begin{equation}
% Z_{Even3}= \frac {Z_3}{jtan\theta_3}
%\end{equation}


where,$Z_{Even3}$= $Z_3/jtan \theta _3$
\begin{equation}
Z_{EvenTOT}= Z_2 \frac{Z_{Even2}+jZ_{2}tan\theta_{2}}{Z_2+jZ_{Even2}tan{\theta_{2}}}
\end{equation}


For impedance matching  LHS of (3), can be equated to port impedance $Z_0$ , giving:


$Z_0$= $Z_2$[($Z_{Even2}$+j$Z_{2}tan\theta_{2}$)/[$Z_2+$j$Z_{Even2}tan\theta_{2}$]]--3(a)  


 
 

 
 
 
 Modifying (2) using (1) and later applying  in (3 --- a) leads to the values of $Z_2$ and $Z_3$. 
 
 %%%%%%%%%%%
 \begin{equation}
Z_2=\sqrt{\frac{{(2Z_0^2+Z_1^2+4(Z_0^2/P))-(P2Z_0^2) -(PZ_1^2)}}{2+2/P}}
\end{equation}

\begin{equation}
Z_3=\frac{Z_1 Z_2 tan\theta_3[2Z_2tan\theta_2 - Z_1tan\theta_1]}{Z_1 Z_2 + (Z_1^2 - 2Z_2^2)tan\theta_1 tan\theta_2}
\end{equation}
 %%%%%%%%%%
 Here P= $tan^2\theta_1$. 
 Odd-mode analysis is used to find the  values of lumped  elements in the isolation circuit. In Fig.2 $Z_{IN}$ corresponds to the impedance of the isolation network.
  
From the odd mode circuit in Fig.2 we get:

$Z_O_d_d_1$= j $Z_1 tan\theta_1$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%
and
\begin{equation}
Z_O_d_d_3= Z_3\left[\frac{(RLC_s_e_r_i_e_s/2) +jZ_3tan\theta_3}{Z_3+j(RLC_s_e_r_i_e_s/2)tan\theta_3} \right]
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
Z_O_d_d_2= \frac {Z_O_d_d_1.Z_O_d_d_3} { Z_O_d_d_1+Z_O_d_d_3}
\end{equation}
 Substitute for $ Z_O_d_d_3 $ and $Z_O_d_d_1$ in (7). The modified value of $Z_O_d_d_2 $ is then applied in (8), which leads to (8-a). On separating the real and imaginary parts of (9) after this modification gives (10).

\begin{equation}
Z_O_D_D_T_O_T = Z_2\frac{ Z_O_d_d_2+jZ_2tan\theta_2}   {Z_2+jZ_O_d_d_2tan\theta_2    }
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


($Z_O_D_D_T_O_T$ = [$Z_O_d_d_3 /$ 2 $sin^2\theta_1$] +[j$Z_2/tan\theta_2 $]) --- 8 (a)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$ Z_{Odd3}$ in isolation network consists of added transmission line section $Z_3$ and  series RLC. So $Z_O_D_D_3$ term in (8 -- a) is modified using  $Z_3$ and series RLC resulting in (9).

\begin{equation}
Z_O_D_D_T_O_T= \frac{2Z_3+R}{2sin^2\theta_1} +j \left[ \frac{\omega L - \frac{1}{\omega  C_3}}{2sin^2\theta_1} +\frac{Z_2}{tan\theta_2} \right]
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%
To get perfect matching the real part can be equated to $Z_0$ and imaginary part to zero, this results in (10), (11)
\begin{equation}
R=Z_0*2Sin^2\theta_1-2Z_3
\end{equation}
and 
\begin{equation}
[\omega L] - [1/\omega  C_3] = 2Z_2 \frac{sin^2\theta_1}{tan\theta_2} 
\end{equation}

The criteria for matching is that the sum of the impedance of the CT lines should be equal to the impedance of the bridge separating the arms,    
%\begin{equation}
$(Z_{Odd3}=Z_{IN2}+Z_{IN3})$.
%\end{equation}
Substituting for $ Z_I_N_2$ and $Z_I_N_3$ based on CT lines shows that 
    \(C_2=2C_3\).
 The impedance of the lower CT line can be transformed w.r.t. reconfigurable isolation network based on this condition, and this results in different ratios in the power division. 
 
 
 
 
 
 The GDs associated with the magnitudes of transmission coefficients at a power division ratio $k^2$ and operating frequency f
can be evaluated based on \cite{7931604}. Hence from equations (12 ) and (13), we can calculate the group delay between ports 2 and 1 and between ports 3 and 1.  

\begin{equation}
\tau_2_1 = \frac{1+2k^2 +\sqrt{1+k^2}+k^2\sqrt{1+k^2}}{8fk^2\sqrt{1+k^2}}
\end{equation}



\begin{equation}
\tau_3_1=\frac{1+2k^2}{4f2k\sqrt{1+k^2}}
\end{equation}




 %\b%%%%%%%%%%%%%%%%%%%

   
  


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
	\centering
		\includegraphics[height=6cm,width=7cm]{fig3.png}
	\caption{ Schematic of the design }
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{ Design procedure} %%%%
The design layout is shown in Fig.3. It has three different transmission line sections: $Z_1$, $ Z_2$, $ Z_3$, and an isolation network.
Impedance $Z_1$ corresponds to the characteristic impedance of composite transmission lines placed in the dividing arms. Tuning the varactor diodes associated with composite transmission lines results in transforming the corresponding impedance and hence the power flowing through the dividing arms. Based on even mode analysis the characteristic impedance $Z_2$ and $Z_3$ are found. Here $Z_2$ is the impedance of transmission line sections leading to the output arms, and $Z_3$ is that of the extended transmission line sections placed in the isolation arm. These values are given by equations (4) and (5). Odd mode analysis is done to find the values of the lumped elements in the isolation arm. The lumped element values are found using equations (10) and (11). The design parameters are listed in table-1. The group delay associated with output ports can be calculated from (12) and (13). 

\begin{table*}[hbt]
\centering
\caption{Design parameters and dimension
 }
\label{tab:my-table}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
\textbf{Impedance (ohms)}  &  \textbf{ Electrical  length (degrees)} &      \textbf{ lUMPED Elements} & \textbf{ Length(mm)} & \textbf{Width(mm)} \\ \hline
$Z_1$=65   & 55 & R= 150ohms   &  L_0 =2.5 & W_0= 2.2    \\
\hline



$Z_2$ =60&  30    &  L= 3.3nH  &  L_1 =15.5   & W_1=1.51   \\ \hline

$Z_3$=45& 7  & C_1= 4.7pF& L_2 =8.56  &    W_2= 1.51    \\ \hline

   &     &   C2,C3(VD=SMV2019)   &  L_3 =1.87   &   W_3= 1.51  \\ \hline





\end{tabular}


\end{table*}

The design procedure is summarized as follows:
\begin{enumerate}
   \item Specify the centre frequency f and PDR.
    \item Replace the  conventional quarter wave length lines by composite transmission lines, based on the value of $Z_1$ and lumped  elements obtained  by solving the transmission matrix.   
   \item Find the values of impedance $Z_2,Z_3$ as per (4) and (5).
    \item Determine the value of components present in the reconfigurable isolation network as per (10) and (11).
     \item Realize the capacitors $C_2$ and $C_3$ by varactor diode SMV 2019LF from Skyworks solutions. 
  \item Calculate GD at output ports as per (12) and (13).
    \item Vary the biasing of varactor diodes (VD) to vary the capacitance in the ratio  \(C_2=2C_3\) and hence vary the arm impedances which in turn  results in variation in PDR and NGD. 
    \item  Compare the attained and calculated results. 
  
    \end{enumerate}
The design parameters are shown in Table-1




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ Analysis of results} 
 
 The prototype is fabricated with a Roger RT/5880 substrate having $\epsilon _r $ = 2.2 and a height of 0.8mm. The proposed work is based on the impedance transformation of the composite transmission line sections. The composite transmission line sections are having low impedance than quarter wave length transmission lines and hence overcomes the fabrication limitations. By trtansforming the impedance of composite transmission line we can vary the power flow through the line. This controls the insertion loss (IL) in the pass band. The ratio of the power available at the output ports can be mathematically calculated based on the magnitude of insertion loss. While tuning, the arm of the power divider corresponding to port three significantly affects its impedance. So by controlling the impedance using varactor diodes results in variation in the power-dividing ratio. When equal power is needed at output ports, the varactor diodess are placed in destructive mode, ie, unbiased. The R and L values of the isolation network are calculated as R = 120 ohms and L=3.3 nH to attain perfect isolation at the center frequency. A  power divider with a power division ratio of 1:39 , 30 percentage size reduction, and a transmission path with reconfigurable negative group delay is designed at a center frequency of 1.5 GHz. 
\begin{figure}
	\centering
		\includegraphics[width=12cm]{Fig 5 .png}
	\caption{(a) S11 at different biasing voltages (b)S32 at different biasing voltages }
\end{figure}


\begin{figure}\centering
\includegraphics[width=12cm]{Fig6.png}
\caption{Variation in PDR w.r.t. biasing voltages : (a) PDR=39 (Vdc=0V) (b)
PDR=15 (Vdc=7) (c)PDR=7 (Vdc=20V)}
\label{fig}
\end{figure}


\begin{figure}[htp]
	\centering
	\includegraphics[width=9cm]{Fig 7 amp phase imb.png}
	\caption{ Amplitude and phase imbalance between out put ports}
	\end{figure}


The simulation is carried out using ANSYS-HFSS software. Hyper abrupt junction varactor diode SMV 2019LF is used for tuning the capacitance $C_2$ and $C_3$ . Tuning the varactor diodes for different biasing voltages provides impedance transformation  which in turn results in reconfigurability in power division and NGD. Transmission line sections with short electrical length in the isolation network compensate for the limited capacitance range of commercial varactor diodes. This improves the overall tuning range. Biasing voltage across the varactor diode is applied through RF choke inductors. The schematic of the designed PD is shown in Figure 3,  The performance of the circuit is validated by measuring the scattering  parameters and group delay using a Keysight E5C vector network analyser. When the control voltage changes from (0-20)V, the capacitance of the varactor diode varies from (2.2-0.3) pF. This transforms arm impedance and results in a variation in power flow from -1.47 dB to -16.4 dB, resulting in a PDR up to 39. The simulated and measured input return loss is around -19.dB  at the centre frequency and is almost constant regardless
of the tuning of the capacitance values. Tuning of the varactor diodes without altering the impedance matching accounts for this. Isolation also is greater than -15dB at the centre frequency (1.5 GHz). Fig.4.(a) shows the magnitude of S11 and Fig 4.(b) shows the magnitude of isolation (S32) for different biasing voltages corresponding to different power division ratios. Power division ratio at the output is calculated mathematically from the results of insertion loss. The measurement and simulation results for power division corresponding to  three different power division ratios are illustrated in Fig.5. The amplitude and phase difference corresponding to a power division ratio of 39 is shown in Figure.6. The phase difference is around 1.25 degrees only. The  unequal power division nature of the circuit accounts for the high amplitude imbalance .
 Power division is controlled by using impedance transformation of composite transmission line; there is a wider range to divide power unequally over a narrow range of DC voltage. So a (0-20)V DC voltage source is sufficient. 

\begin{table*}[hbt]
\centering
\caption{ Fractional Power Divisiion Ratio in different cases
 }
\label{tab:my-table}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
\textbf{Cases}  &  \textbf{ MPDR} &      \textbf{IL Max } & \textbf{ ILMin} & \textbf{ FPDR} \\ \hline
1.   & 7& 11.1  &  1.8  &  78  \      \\
\hline



2. &  15  & 12.2  & 1.8 & \ 80    \\ \hline

3. & 39  & 19.6 &   \  1.8       &   \ 87     \\ \hline

\end{tabular}
\end{table*}





The fractional power division ratio (FPDR) is a significant performance parameter for power dividers with a reconfigurable power division ratio. The FPDR can be evaluated from (14) based on the values of the Maximum power division ratio (MPDR) and maximum and minimum values of insertion loss. From FPDR, the reconfigurability of the proposed power divider can be found as 87 percent. This parameter is not analysed in the case of most of the reported reconfigurable power dividers. The FPDR value for three different power division ratios is shown in Table 2. The FPDR value can be seen as increasing with the power division ratio.
\begin{equation}
FPDR= \frac{(MPDR-1) × (ILmax-ILmin)*100}{MPDR * ILmax}
\end{equation}





 From the analytical study it can be seen that the group delay associated with the transmission paths between 3 and 1 is negative, and that associated between transmission paths 2 and 1 is positive. The transmission path with a series-connected composite transmission line gives PGD, and the transmission path with parallel connected composite transmission line provides NGD. The path of NGD can be used to compensate for the non-linearity in time between the signal envelope and dynamic power supply, 
whereas the path with PGD can be linked directly to the RF path. So the proposed power divider is a promising design that can be used in the dynamic power or 
envelope tracking power amplifiers. The almost constant phase difference observed in a bandwidth of 200 MHz around the centre frequency is an added justification for this application. The NGD corresponding to different power division ratios of 7, 15, and 39 are shown in Fig.7. a,b,c. The  negative group delay and  phase at the centre frequency is shown in Fig 7,d. The graph shows that NGD w.r.t different PDRs varies from -3.6 ns to -4.6 ns in paths 1 to 3. Hence path 3 can be used to make up the non linearity in time between the dynamic power supply and signal envelope.  As the value of the impedance associated with arm three is varied to lower values the value of S31 and NGD also decreases. But group delay at port 2 and S21 remains almost constant. So port 2 can be directly linked to the RF path. The experimental group delay values are verified with the corresponding theoretical values as per equations (12) and (13). The limitation in working bandwidth is due to the trade off between the operating band width and maximum achievable NGD .The prototype and measurement setup are shown in Fig.8.



%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htp]
	\centering
	\includegraphics[width=15cm]{7 15 merg.jpg}
	
	

\centering
	\includegraphics[width=15cm]{c,d Fig 7.jpg}
	\caption{ Group delay (GD) for different PDRs: (a) 7 (b) 15 (c) 39 (d) (d) phase corresponding to NGD at port three. }
	\end{figure}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{table*}[hbt]
\centering
\caption{Table of performance comparison of the proposed PD with related
state-of-the-art power dividers.
 }
\label{tab:my-table}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
\textbf{Ref.}  &  \textbf{ $f_c$(GHz)} &      \textbf{ Size(\lambda $_g^2 $)  } & \textbf{ No. of LE, AE} & \textbf{RL(dB), Isln(dB)} & \textbf{IL(dB)}  &  \textbf  {PDR} \\ \hline
\cite{@1118977529}   & 2.4 & 0.67*0.8   &  5 and 1     &    \ge-20 , \ge12    &    -2/-14   &1:28   \\
\hline



\cite{quddious2018use}&  2.4      &0.7*0.81     & 3 and 2    & \ge -22,-15      &  -2.7/-4.2   & 1:10 \\ \hline

\cite{7470611}  & 1.2      & 0.25*0..02   &   \  4 and  2       &   \ge-10 , -10      & -2.2/- 6  &1:2.4 \\ \hline

\cite{8842611}   & 2.45    & 0.66*0.81      &  5 and 2    & \ge-20 ,-20      & -1.23/-1.11     & 1:14\\ \hline

\cite{leinovel} &  2.45      & 0.6*0.75 &\ 5 and 2  & \ge-19 ,-20        &  -1.3/-0.97    &
1:29 \\ \hline


\cite{8913621}  &1.2      & 0.2*0.19  &\ 2 and 5     &   \ge-20,-15        & -2.2/-4  & 1:1.5 \\ \hline



Proposed work   &    1.5    & 0.38*0.74  
& 3 and 2     & \ge-15,-14       & -2.1/-19.4    &1:39\\ \hline
\end{tabular}

*$f_c$=Centre frequency, LE= Lumped Elements, AE= Active elements, RL=Return Loss, Isln=Isolation, IL=Insertion Loss, PDR=Power Division Ratio
\end{table*}


\begin{figure}
	\centering
		\includegraphics[width=9cm]{Fig 4 fabrn and measurement.jpg}
	\caption{Prototype  and measurement set up of the power divider}
\end{figure}












































 The novelty of the design technique and the novelty of the application oriented  results are the key features of this design. 
 Table 3 demonstrates the advantages of this design based on  comparison with other recent publications. The table reveals the wide and high PDRs achieved by this design, along with the compact size and NGD path without using additional NGD circuits. A power divider simultaneously performing reconfiguration of power division ratio and NGD is not yet reported in literature. Such a power divider is a promising candidate for dynamic power or 
envelope tracking power amplifiers. This design can also be used in squint free antenna arrays. The band width of the proposed design is limited due to the trade off between maximum attainable value of NGD and band width.  Cascading  more sections with slight difference in  center frequencies may enhance the band width.







\section{Conclusion}
This paper presents a novel method of designing a reconfigurable power divider with high  power division ratios and negative group delay. The high range of power division, the existence of a negative group delay path, lower number of tuning elements, low characteristic impedance (< quarter wave length lines ), compact size, and high fractional power division ratio are the significant advantages of the proposed design. This design is a novel solution to simultaneously achieve high reconfigurable PDR and negative group delay which is suitable for dynamic power or 
envelope tracking power amplifiers and also in squint free antenna arrays. The design equations, along with experimental results, have been presented. The composite transmission lines reduce the impedance values and help to attain negative group delay without additional GD circuits. Small electrical length transmission lines in the isolation arm improve the tuning range of varactor diodes. 

.

\bibliography{sample}

\noindent 


\section*{Author contributions statement}
Design and conceptualization: R.G.N and N.S. Writing original draf: R.G.N and N.S. Analyzing results: R.G.N and N.S. All authors reviewed the final manuscript.

 



\end{document}
\begin{figure}[htp]
	\centering
	\includegraphics[width=15cm]{fig 8.jpg}
	\caption{ Group delay for different PDRs and phase corresponding to NGD at port three. }
	\end{figure}
	