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\title{Using Neural Networks to Design Transistor Amplifier Circuits}
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\begin{document}

             \author[1]{Thomas L.  Hemminger}

             \affil[1]{  Penn State Erie, The Behrend College}

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\date{\small \em Received: 11 December 2017 Accepted: 2 January 2018 Published: 15 January 2018}

\maketitle


\begin{abstract}
        


This paper is an extension of previous work that addressed the application of bipolar transistor amplifier design using neural networks. That work addressed the design of common emitter amplifiers by first mathematically determining specific output parameters from a large selection of biasing resistors. Once the outputs had been determined, a neural network was trained, using the aforementioned results as inputs and the biasing resistors as outputs. This was initially performed with ideal emitter bypass capacitors, but was then followed-up by employing several non-ideal capacitors, making it much more interesting and useful. This paper focuses on the common collector and the common base configurations. Bipolar junction transistor amplifier parameters often include voltage gain, input impedance, output impedance, and the voltage difference between the collector and emitter. These will be addressed in this paper as before. There are several methods that can provide a suitable solution for each design, however the objective of this work is to indicate which external resistors are necessary to yield useful results by employing neural networks.

\end{abstract}


\keywords{feedforward neural networks, bipolar junction transistor circuits, common collector amplifier, common base amplifier.}

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\textcolor{GJBlue}{\footnotesize \\ Artificial Intelligence formulated this projection for compatibility purposes from the original article published at Global Journals. However, this technology is currently in beta. \emph{Therefore, kindly ignore odd layouts, missed formulae, text, tables, or figures.}}
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\let\tabcellsep& 	 	 		 
\section[{I. INTRODUCTION}]{I. INTRODUCTION}\par
his paper addresses two amplifier topologies that use bipolar junction transistors (BJTs). In a previous paper a common emitter amplifier was evaluated and it was determined that the value of the bypass capacitor played a significant role in the voltage gain and other parameters due to the low emitter resistance of the transistor \hyperref[b0]{[1]}. It was also established through simulations and analysis that the coupling capacitors did not affect the results of required amplifier parameters to any great extent. Here we address the common collector amplifier, otherwise known as the emitter follower, and the common base amplifier. Both configurations employ coupling capacitors, so for this work they will be considered as ideal. The output impedance of the source and the input impedance of the load can be factored in after developing the initial model.\par
For either amplifier design, and for a given transistor, the designer works through a set of calculations to determine the biasing resistor values, then often has to modify those values to achieve the proper voltage gain (A v ), input impedance (R in ), output impedance (R o ), and voltage difference between the collector and emitter (V ce ). This latter parameter can affect whether the amplifier enters saturation or cutoff. For the common collector amplifier the voltage gain is just under unity, as shown later.\par
The goal of this work is primarily intended for engineers, but others may want to evaluate specific amplifier designs based on the results illustrated in this paper. For any specific amplifier design, the neural network should be able to provide reasonably close biasing resistor values, provided that the parameters input by the user are within appropriate limits. It is hoped that with the addition of these two amplifier designs to the common emitter circuit shown in a previous paper, the user will be able to approximate the appropriate resistor values in a streamlined manner.\par
This paper is organized as follows. First, the design procedure for the dc equivalent common collector circuit is introduced along with some of its defining equations. Next, the expressions needed to solve for the ac equivalent circuit are developed. This is followed by a brief discussion of the neural network architecture and results. The next section addresses the common base ac equivalent circuit and the corresponding results. Finally, some conclusions will be discussed and some thoughts for further work. When analyzing both of these amplifiers there are the dc bias values and the ac signals to contend with, but one must look at each of them separately in order to compute the proper operating points. 
\section[{II. THE COMMON COLLECTOR AMPLIFIER}]{II. THE COMMON COLLECTOR AMPLIFIER}\par
The common collector circuit is one of the basic configurations introduced when studying the BJT \hyperref[b1]{[2]}- \hyperref[b3]{[4]}. It is considered as a voltage buffer providing a high input impedance and a low output impedance, which is useful for impedance matching with other circuits. As stated above, the voltage gain of this circuit is approximately equal to one.An example of the common collector amplifier is shown in Fig. \hyperref[fig_0]{1} where the 2N3904 NPN transistor is used with ?=160. The ac input is Vi while the output is taken from the emitter terminal.For this amplifier the neural network is designed to accept three input parameters: input resistance (R in ), output resistance (R out ), and the voltage across the collectoremitter junction (V ce ). The objective of the network is to provide the biasing resistors R b1 , R b2 , and R e , needed to achieve the desired results.\par
Initially, the dc circuit is analyzed with all capacitors considered as open circuits in order to find the currents and voltages from the power supply and biasing resistors. The coupling capacitors isolate the dc component where its circuit equivalent is shown in Fig. \hyperref[fig_1]{2}.As in the previous work, the dc biasing voltage was 15 volts for both circuits described in this paper. All input signals were set to 3 kHz sinusoids.  To determine the dc biasing values, the base resistors and source are replaced with their Thevenin equivalents as shown here.?? ??? = ?? ???? ? ?? ?? 2 ?? ?? 1 +?? ?? 2 ? and ?? ?? = ?? ??1 ||?? ??2\textbf{(1)}\par
This allows a single loop circuit to be analyzed as in \hyperref[b1]{(2)} where the base current, B I , can be determined.\par
??? ??? + ?? ?? ?? ?? + 0.7 + (?? + 1)?? ?? ?? ?? = 0\par
After the dc bias values have been found those sources are set to zero and only the ac components are considered. Recall that the dc voltage sources become short circuits to ground when set to zero. The ac equivalent of the transistor circuit is shown in Fig. \hyperref[fig_2]{3} using the hybrid -? model \hyperref[b3]{[4]}. The ac values of input and output resistance, along with V ce are computed from the dc bias currents. Most of the ac equivalent expressions can be found in texts on microelectronics so they are only summarized here. As shown below, the total collector current C i is ascertained where be v is the ac base to emitter voltage and T V is the thermal voltage, usually approximated at 25 mV yielding:?? ?? ? ?? ?? ?1 + ?? ???? ?? ?? ? = ?? ?? + ? ?? ?? ?? ?? ? ?? ????\textbf{(3)}\par
From now on the ac component is of interest and the DC sources are shut down. By looking at the far right term from equation 3 it can be seen that the ac equivalent is:be T C c v V I i ? ? ? ? ? ? ? ? = (4)\par
The component in parenthesis is referred to as the transconductance, with symbol m g so:T C m V I g / =\textbf{(5)}\par
therefore:be m c v g i =\textbf{(6)}\par
The ac input resistance of the transistor itself is given as( ) m be m be b be g v g v i v r ? ? ? = = =\textbf{(7)}\par
Or alternatively( ) ( ) B T T B T C I V V I V I r = = = ? ? ? ? (8)\par
Note from Fig. \hyperref[fig_2]{3} that:?? ?? = (?? + 1)?? ??\textbf{(9)}\par
So the output voltage can be computed as:?? ?? = ?? ?? (?? + 1)(?? ?? ||?? ?? )\textbf{(10)}\par
where ?? ?? is the resistance seen looking into the collector and is often calculated as 100/?? ?? . By writing a KVL equation around the baseemitter loop we arrive at:?? ???? = ?? ?? [?? ?? + (?? + 1)(?? ?? ||?? ?? )]\textbf{(11)}\par
The small signal input impedance results from dividing the expression above by the base current yielding:?? ???? = ?? ???? ?? ?? = ?? ?? + (?? + 1)(?? ?? ||?? ?? )\textbf{(12)}\par
Taking into account the two biasing resistors ?? ??1 and?? ??2 which are in parallel with?? ???? the overall input impedance of the amplifier is:?? ???? = ?? ??1 ||?? ??2 ||?? ????\textbf{(13)}\par
As one might expect, the input impedance of this amplifier is significant, resulting in a very low loading effect on stages that might precede it. By combining equations ( \hyperref[formula_9]{10}) and ( \hyperref[formula_11]{12}) the voltage gain is determined to be:?? ?? = (??+1)(?? ?? ||?? ?? ) ?? ?? +(?? +1)(?? ?? ||?? ?? ) ? 1 (14)\par
Computing the output impedance,?? ?? , is somewhat involved, so only the result will be presented here where it can be seen as being very low due to the ?? + 1 factor. It can be on the order of tens of ohms.?? ?? = ?? ?? ?? +1 ||?? ?? ||?? ??\textbf{(15)} 
\section[{III. NEURAL NETWORKS}]{III. NEURAL NETWORKS}\par
Neural networks are most commonly considered as pattern recognition systems. This author has used them to develop a method of impedance matching using feed-forward neural networks \hyperref[b4]{[5]} and also in the design of common emitter amplifiers \hyperref[b0]{[1]}. They are non-linear systems and are often employed to separate input patterns by setting up a set of hyperplanes in ndimensional space \hyperref[b5]{[6]} - \hyperref[b7]{[8]}.\par
In order to train the neural networks in this project a set of "for" loops was created in MATLAB ? for the three biasing resistors. For all of the tests, the resistor values ranged as shown in table I. Since the voltage gain is near unity for the common collector amplifier it was not addressed in this work. The values of R in , R o , and V ce were calculated for all of the resistor combinations. Once this was completed a neural network was trained using the new input values of the three parameters above to compute the three biasing resistor values. In developing the network, the inputs and outputs were normalized to ensure convergence. For this circuit there were 7,500 training patterns, limited to realistic output values. For example, V ce was held to the range of 2 volts to 12 volts. The test sets consisted of a larger number of patterns, none of which had been used in training.\par
The neural network package in MATLAB ? was employed to train the networks, by utilizing the Levenberg-Marquardt algorithm, using one hidden layer of 14 sigmoidal (Tanh) neurons each \hyperref[b8]{[9]}. As noted in the previous work, a lesser number of nodes yielded unacceptable results, and more nodes or more than one hidden layer did not provide any improvement in performance. The network was trained for 2000 epochs resulting in a mean-squared error (mse) of 2.45x10 -8\par
. Further training did not seem improve performance. A comparison between the neural network results and those by direct calculation is shown in table II. Fig.  {\ref 4} shows the architecture of the neural network. This network employs hyperbolic tangent activation functions to map the transistor parameters to the values of the resistors. The number of patterns changes with all of the training and testing scenarios because as the values of the biasing resistors change, the outputs can fall out of the ranges specified earlier. Only those that fall within those ranges are employed in the tests. When using the res istance values illustrated in table I the output parameters have the ranges shown in table III. It is not required that these ranges be followed precisely but it is likely a good practice to stay within them when considering an input set. Year 2018 ( ) D Fig.  {\ref 4}: Architecture of the neural network with 14 hidden nodes. It is important to realize that not all input parameter combinations are feasible. For example, if the base bias resistors are kept to a low value the collector and emitter currents can be greater, resulting in a smaller value of V ce . In this case it would not be appropriate to set a small value of dc input resistance and a large value of V ce , since they can be mutually exclusive. However, by judiciously choosing realistic inputs the results can be close to the desired values. Some comparisons of the neural network out against calculation are shown in table IV. During the testing phase the network was asked to determine the resistor values for three different ac parameters. The input impedances, output impedances, V ce , and the values of the resistors predicted by the network were recorded. The resistor values from the network were then used to calculate those same ac values directly. The results were remarkably good as shown in table  {\ref IV}. The resistor values from the last trial from table IV were used in a P-Spice simulation. The values were? = k R b 96 . 7 1 , ? = k R b 98 . 9 2 and ? = k R e 07 . 1\par
. The results are summarized in table V along with the percent errors.  
\section[{IV. THE COMMON BASE AMPLIFIER}]{IV. THE COMMON BASE AMPLIFIER}\par
The second circuit addressed in this paper is the common base amplifier. The hybrid -? model of the transistor is also used. The DC analysis of this circuit is essentially the same as for the previous one except that it has collector resistor, so it will not be addressed here. The ac equivalent of a common base amplifier is shown in Fig. \hyperref[fig_4]{5} in which the base is at signal ground. The input is at the emitter and the output is at the collector. A summary of the voltage gain, current gain, input impedance, and output impedance is presented here. As in the other configurations, the output impedance of the signal source is not included, since that parameter would be unknown for the individual amplifiers.?? ?? = ð??"ð??" ?? ?? ??\textbf{(16)}\par
The current gain can be approximated by:?? ?? ? ?? ?? +1 = ?? < 1\textbf{(17)}\par
The input resistance of the amplifier ?? ???? is quite low where the input signal sees the emitter resistance ?? ?? :?? ???? = ?? ?? ?? +1 = ?? ??\textbf{(18)}\par
The output resistance ?? ?? is approximated as:?? ?? = ?? ?? ||?? ??\textbf{(19)}\par
One may wonder about the significance of this circuit but since the output impedance looking back into the collector is very large. However due to this characteristic it behaves almost like an ideal current source. It is sometimes referred to as a current buffer.  As in the common collector circuit, the values of Rin, Ro, and Vce were calculated for all of the resistor combinations. Again, V ce was held to a range of 2 volts to 12 volts.\par
Here the network consisted of one hidden layer of 14 sigmoidal neurons. The network was trained for 1904 epochs resulting in a msetargetof1.0x10 -8\par
. A comparison of the results obtained from the network against the true values of resistance are revealed in table VII for several different test sets. The number of patterns changes with all of the training and testing scenarios because as the values of the biasing resistors change, the can fall out of the ranges specified earlier. Only those that fall within those ranges were employed in the tests. When using the resistance values illustrated in table VI the output parameters have the ranges shown in table VIII. It is not required that these ranges be followed precisely but it is likely a good practice to stay within them when considering an input For the common base amplifier the voltage gain was the primary focus, however the output impedance and V ce were also determined. The input impedance was more problematic. For this amplifier, several combinations of resistor values yielded the same input impedance values. Therefore, when training from the input impedance to find the bias resistors, the network experienced multiple targets meaning that the network could not converge. Several experiments that included input impedance were conducted, but the results were unsuccessful. For this amplifier the input impedance is very low, and from the bias resistors contained in table VI, Rin only ranged from about 4.7? to 10.6?. In this case it might be prudent to approximate the value as 7.5? for most applications. The results between the neural network and those by direct calculation are shown in table IX. The resistor values from the last trial from table IX were used in a P-Spice simulation with ?? ??1 = 9 ???,?? ??2 = 4.5???, ?? ?? = 1.1???, and ?? ?? = 600?. The results are summarized in table X along with the percent errors.  
\section[{V. CONCLUSIONS AND FURTHER WORK}]{V. CONCLUSIONS AND FURTHER WORK}\par
It was remarkable how well the network results compared against those from direct calculation and from P-Spice. Several experiments were conducted more than once, and the output values checked frequently, to be certain there were no errors.\par
This has been an interesting research project as a follow-up to previous work. Here, two classical BJT amplifiers have been analyzed, then synthesized by a set of neural networks. In both cases the capacitors did not play a significant role and were considered to be ideal. This neural network paradigm should be useful to engineers and faculty members when looking for solutions to various designs. The research described in this paper can help to resolve and verify transistor solutions for both configurations and can illustrate the efficacy of neural networks as a tool for this branch of amplifier design. It is the intension of this author to look into other, more complex amplifier designs, such as the cascode circuit which is a multistage system, consisting of a common emitter amplifier driving a common base amplifier.\begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Fig. 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2}\includegraphics[]{image-3.png}
\caption{\label{fig_1}Fig. 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3}\includegraphics[]{image-4.png}
\caption{\label{fig_2}Fig. 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{5}\includegraphics[]{image-5.png}
\caption{\label{fig_4}Fig. 5 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{P{0.14623655913978495\textwidth}P{0.4204301075268817\textwidth}P{0.13709677419354838\textwidth}P{0.14623655913978495\textwidth}}
Resistor\tabcellsep \multicolumn{3}{l}{Start value Step Value Stop Value}\\
Rb1\tabcellsep 4 k?\tabcellsep 250 ?\tabcellsep 10 k?\\
Rb2\tabcellsep 4 k?\tabcellsep 250 ?\tabcellsep 10 k?\\
Re\tabcellsep 400 ?\tabcellsep 100 ?\tabcellsep 1.5 k?\end{longtable} \par
 
\caption{\label{tab_0}Table 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2} \par 
\begin{longtable}{P{0.33674121405750795\textwidth}P{0.11134185303514377\textwidth}P{0.1330670926517572\textwidth}P{0.1303514376996805\textwidth}P{0.13849840255591053\textwidth}}
Data type\tabcellsep Number of patterns\tabcellsep Upper base resistor Rb1(mse)\tabcellsep Lower base resistor Rb2(mse)\tabcellsep Emitter resistor Re(mse)\\
Training set\tabcellsep 7500\tabcellsep 0.784\tabcellsep 6.00\tabcellsep 0.0131\\
Test Set \#1\tabcellsep 8316\tabcellsep 1.48\tabcellsep 7.80\tabcellsep 0.0225\\
Test Set \#2\tabcellsep 10440\tabcellsep 1.39\tabcellsep 7.20\tabcellsep 0.020\\
Test Set \#3\tabcellsep 11160\tabcellsep 1.23\tabcellsep 6.71\tabcellsep 0.018\\
Test Set \#4\tabcellsep 13728\tabcellsep 1.41\tabcellsep 7.24\tabcellsep 0.020\\
\multicolumn{5}{l}{The training set was included for comparative purposes.}\end{longtable} \par
 
\caption{\label{tab_1}Table 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3} \par 
\begin{longtable}{P{0.1829113924050633\textwidth}P{0.33354430379746836\textwidth}P{0.33354430379746836\textwidth}}
Parameter\tabcellsep Minimum value\tabcellsep Maximum Value\\
Rin\tabcellsep 1.94 k ?\tabcellsep 4.90 k?\\
Ro\tabcellsep 1.4 ?\tabcellsep 10.5?\\
Vce\tabcellsep 5.1 V\tabcellsep 11.6 v\end{longtable} \par
 
\caption{\label{tab_2}Table 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4} \par 
\begin{longtable}{P{0.1251937984496124\textwidth}P{0.27674418604651163\textwidth}P{0.09883720930232558\textwidth}P{0.17790697674418604\textwidth}P{0.02635658914728682\textwidth}P{0.1449612403100775\textwidth}}
Rin (?)\tabcellsep \multicolumn{4}{l}{\%error Ro (?) \% error Vce}\tabcellsep \%error\\
2402\tabcellsep 0.083\tabcellsep 2.634\tabcellsep 0.076\tabcellsep 6.67\tabcellsep 0.075\\
3101\tabcellsep 0.032\tabcellsep 3.386\tabcellsep \multicolumn{2}{l}{0.003 5.695}\tabcellsep 0.004\\
4310\tabcellsep 0.001\tabcellsep 3.583\tabcellsep \multicolumn{2}{l}{0.002 7.547}\tabcellsep 0.013\end{longtable} \par
 
\caption{\label{tab_3}Table 4 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{5} \par 
\begin{longtable}{P{0.15235849056603773\textwidth}P{0.425\textwidth}P{0.13632075471698113\textwidth}P{0.13632075471698113\textwidth}}
Paramet ers\tabcellsep \multicolumn{3}{l}{Neural Network P-Spice \% difference}\\
Rin\tabcellsep 4310?\tabcellsep 4274?\tabcellsep -0.835\\
Ro\tabcellsep 3.583?\tabcellsep 3.546?\tabcellsep -1.033\\
Vce\tabcellsep 7.547V\tabcellsep 7.549V\tabcellsep 0.027\end{longtable} \par
 
\caption{\label{tab_4}Table 5 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.5292452830188679\textwidth}P{0.11226415094339622\textwidth}P{0.17641509433962266\textwidth}P{0.03207547169811321\textwidth}}
R b1\tabcellsep R b2\tabcellsep R e\\
Hidden\tabcellsep \tabcellsep \\
Layer\tabcellsep \tabcellsep \\
R in\tabcellsep R o\tabcellsep V ce\\
Year 2018\tabcellsep \tabcellsep \\
28\tabcellsep \tabcellsep \\
)\tabcellsep \tabcellsep \\
D\tabcellsep \tabcellsep \\
(\tabcellsep \tabcellsep \\
\tabcellsep \tabcellsep Vi\\
\tabcellsep \tabcellsep Re\tabcellsep Rc\end{longtable} \par
 
\caption{\label{tab_5}Table}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{6} \par 
\begin{longtable}{P{0.12686567164179102\textwidth}P{0.21567164179104478\textwidth}P{0.25373134328358204\textwidth}P{0.25373134328358204\textwidth}}
Rb1\tabcellsep 4 k?\tabcellsep 250 ?\tabcellsep 10 k?\\
Rb2\tabcellsep 4 k?\tabcellsep 250 ?\tabcellsep 10 k?\\
Rc\tabcellsep 1 k?\tabcellsep 100 ?\tabcellsep 3 k?\\
Re\tabcellsep 400 ?\tabcellsep 100 ?\tabcellsep 1.5 k?\end{longtable} \par
 
\caption{\label{tab_6}Table 6 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{7} \par 
\begin{longtable}{P{0.28876712328767123\textwidth}P{0.09082191780821917\textwidth}P{0.12575342465753425\textwidth}P{0.1117808219178082\textwidth}P{0.11876712328767124\textwidth}P{0.11410958904109589\textwidth}}
Data type\tabcellsep Number of patterns\tabcellsep Upper base resistor Rb1(mse)\tabcellsep Lower base resistor Rb2(mse)\tabcellsep Collector resistor Rc(mse)\tabcellsep Emitter resistor Re(mse)\\
Training set\tabcellsep 5676\tabcellsep 0.704\tabcellsep 1.24\tabcellsep 0.031\tabcellsep 0.038\\
Test Set \#1\tabcellsep 6242\tabcellsep 0.656\tabcellsep 1.20\tabcellsep 0.030\tabcellsep 0.035\\
Test Set \#2\tabcellsep 8108\tabcellsep 0.603\tabcellsep 1.10\tabcellsep 0.027\tabcellsep 0.031\\
Test Set \#3\tabcellsep 9135\tabcellsep 0.560\tabcellsep 1.10\tabcellsep 0.281\tabcellsep 0.030\\
Test Set \#4\tabcellsep 10938\tabcellsep 0..627\tabcellsep 1.11\tabcellsep 0.029\tabcellsep 0.323\\
\multicolumn{6}{l}{The training set was included for comparative purposes.}\end{longtable} \par
 
\caption{\label{tab_7}Table 7 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{8} \par 
\begin{longtable}{P{0.16585365853658537\textwidth}P{0.33170731707317075\textwidth}P{0.3524390243902439\textwidth}}
Parameter\tabcellsep Minimum value\tabcellsep Maximum Value\\
Ro\tabcellsep 950 ?\tabcellsep 2.092 k?\\
Av\tabcellsep 93.9 V/V\tabcellsep 210 V/V\\
Vce\tabcellsep 2.02 V\tabcellsep 9.11 V\end{longtable} \par
 
\caption{\label{tab_8}Table 8 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{9} \par 
\begin{longtable}{P{0.15327868852459017\textwidth}P{0.15327868852459017\textwidth}P{0.10450819672131147\textwidth}P{0.3622950819672131\textwidth}P{0.020901639344262295\textwidth}P{0.05573770491803279\textwidth}}
Ro (k?)\tabcellsep \% error\tabcellsep Av V/V\tabcellsep \% error\tabcellsep Vce\tabcellsep \% error\\
1.305\tabcellsep 0.19\tabcellsep 206\tabcellsep \multicolumn{3}{l}{2.12 4.17 0.84}\\
1.748\tabcellsep -0.34\tabcellsep 202\tabcellsep \multicolumn{3}{l}{1.03 5.78 -0.76}\\
1.045\tabcellsep 0.012\tabcellsep 207\tabcellsep \multicolumn{3}{l}{2.51 3.24V 0.24}\end{longtable} \par
 
\caption{\label{tab_9}Table 9 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{10} \par 
\begin{longtable}{P{0.3333333333333333\textwidth}P{0.15\textwidth}P{0.14166666666666666\textwidth}P{0.225\textwidth}}
\multicolumn{3}{l}{Parameters Neural Network P-Spice}\tabcellsep \% difference\\
Ro\tabcellsep 1.045k?\tabcellsep 1.05k?\tabcellsep 0.478\\
Av\tabcellsep 207V/V\tabcellsep 211V/V\tabcellsep 1.93\\
Vce\tabcellsep 3.24V\tabcellsep 3.27V\tabcellsep 0.926\end{longtable} \par
 
\caption{\label{tab_10}Table 10 :}\end{figure}
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\end{document}
