A Modified Genetic Algorithm for Design Peak Pulse Power  
in Nonlinear Transmission Lines 
 
by 
 
Hai Lu 
 
 
 
 
A thesis submitted to the Graduate Faculty of 
Auburn University 
in partial fulfillment of the 
requirements for the Degree of 
Master of Science 
 
Auburn, Alabama 
May 9, 2011 
 
 
 
 
Keywords: Nonlinear transmission lines; peak pulse power;  
Genetic Algorithm; interface; circuit optimization; 
 
 
Copyright 2011 by Hai Lu 
 
 
Approved by 
 
Michael Baginski, Chair, Associate Professor of Electrical and Computer 
Engineering 
Hulya Kirkici, Associate Professor of Electrical and Computer Engineering 
Stuart Wentworth, Associate Professor of Electrical and Computer Engineering 
ii 
 
 
 
 
Abstract 
 
 Non-linear transmission lines (NLTL) have become increasingly popular 
as a way to develop very high frequency, wide band waveforms that offer 
extremely fast rise times usually unobtainable by any other method. The 
applications of non-linear transmission lines range from very wideband 
telecommunications systems to high energy physics. An integrated optimization 
algorithm was built using between Matlab and LTspice to optimize the 
performance of the NLTL. A bidirectional interface was also created that allowed 
data to be easily transferred between the circuit simulator and the optimization 
algorithm batch command. A genetic algorithm was used for the optimization and 
designed for relative rapid convergence. Using this integrated optimization 
method, the NLTL circuit was designed that showed a peak pulse power 
enhancement of 570 % without the tedious trial and error process usually required. 
 
 
 
 
 
 
 
 
 
 
 
iii 
 
 
 
 
 
 
 
 
Acknowledgments 
 
 I would like to thank my advisor Dr. Michael Baginski for his help and 
inspiration over the past two years of this project. It was he who helped me 
overcome several obstacles during the research process. Also, I would like to 
thank my committee members Dr. Kirkici and Dr. Wentworth for their help on the 
theoretical problem description and experimental evidence. I would especially 
like thank my parents for their tremendous support during my entire life. 
 
 
  
iv 
 
 
 
 
 
 
Table of Contents 
 
Abstract ............................................................................................................... ii 
Acknowledgments .............................................................................................. iii  
List of Figures.................................................................................................... vii 
List of Abbreviations ........................................................................................ viii 
1.   Introduction ................................................................................................... 1                                                                                                                        
      1.1. Overview ........................................................................................... 1 
      1.2. Soliton Theory ................................................................................... 2 
      1.3. Magnetic Material ............................................................................. 3 
      1.4. New Techniques ................................................................................ 4 
             1.4.1. Hybrid NLTL ........................................................................... 4 
             1.4.2. Gradually Scaled NLTL ........................................................... 5 
 1.4.3. Optimal Design of NLTL ......................................................... 6 
      1.5. Benefit of the Method ........................................................................ 7 
2.   Theory ........................................................................................................... 8 
      2.1. The NLTL Implementation in LTspice .............................................. 8 
 2.1.1. The Basic Circuit for NLTL ..................................................... 8 
             2.1.2. Nonlinear Inductor ................................................................... 9 
             2.1.3. Nonlinear Capacitance ........................................................... 13 
                               2.1.4. Simulation Time Step............................................................. 14 
                               2.1.5. Voltage Source ...................................................................... 14 
v 
 
                               2.1.6. Matched Load ........................................................................ 15 
2.2. Function of NLTL ........................................................................... 15 
                               2.2.1. Pulse Narrowing .................................................................... 15 
                        2.3. GA .................................................................................................. 17 
                               2.3.1. Basic Theory and Implementation .......................................... 17 
                               2.3.2. Flowchart of GA .................................................................... 23 
                               2.3.3. Parameters of GA................................................................... 24 
                        2.4. Software Used ................................................................................. 25 
                        2.5. Interface Implementation ................................................................. 25 
                               2.5.1. Function of the Interface ........................................................ 25 
                               2.5.2. LTspcie Batch Command ....................................................... 26 
                               2.5.3. LTspice to Matlab .................................................................. 27 
                               2.5.4. Matlab to LTspice .................................................................. 28 
                               2.5.5. Packet the Interface ................................................................ 33 
                               2.5.6. GA and interface operation .................................................... 34 
                               2.5.7. Computation Complexity and Iteration Time ......................... 34 
                               2.5.8. Optimization System .............................................................. 35 
3.   Results ......................................................................................................... 37 
3.1. Simulation ....................................................................................... 37 
                               3.1.1. Simulation Settings ................................................................ 38 
                               3.1.2. Components Optimized .......................................................... 39 
                               3.1.3. Initial Conditions ................................................................... 39 
                               3.1.4. Matlab GUI ............................................................................ 40 
vi 
 
3.2. Simulation Results ........................................................................... 41 
                        3.3. Method Confirmation ...................................................................... 43 
4.   Conclusions ................................................................................................. 44 
                        4.1. Suggestion for Future Research ....................................................... 45 
References   ....................................................................................................... 46 
 
 
  
vii 
 
 
 
 
List of Figures 
 
2.1      Basic NLTL circuit ................................................................................... 8 
2.2      Equivalent circuit of NLTL  ...................................................................... 9 
2.3      Inductance versus current curve............................................................... 10 
2.4      The inductance versus current curve of nonlinear inductor  ..................... 11 
2.5      Diode capacitance versus reverse voltage curve  ...................................... 13 
2.6      Improvement of the pulse through the NLTL........................................... 17 
2.7      Flowchart of the GA used in this thesis ................................................... 23 
2.8      Key words in Matlab codes building the interface ................................... 27 
2.9      Data imported into Matlab ....................................................................... 28 
2.10    Screen shoot on the dos batch file ............................................................ 32 
2.11    Flowchart of the optimization system ...................................................... 35 
3.1      Schematic of the main NLTL circuit........................................................ 37 
3.2      LTspice simulation command setting ...................................................... 38 
3.3      Layout of the Matlab GUI ....................................................................... 40 
3.4      Simulation results showed in Matlab GUI ............................................... 41 
3.5      Confirmation of LTspice ......................................................................... 43 
 
 
 
 
 
 
viii 
 
 
 
 
 
 
List of Abbreviations 
 
NLTL             Nonlinear Transmission Lines 
GA                  Genetic Algorithm 
GUI                 Graphical User Interface 
1 
 
 
 
 
1. Introduction 
1.1. Overview 
Since 1940s non-linear materials have been used in transmission lines to 
generate wide-band signals having an extraordinary peak pulse powers. These 
types of waveforms are made possible by taking advantage of the nonlinear 
material properties to create solitons that propagate down the periodic structure. 
Over the last six decades a significant amount of research has focused on 
developing applicable, non-linear materials as well as designing novel 
transmission line housings.  In 1950?s linear capacitors were combined with 
nonlinear inductors as the basis for lumped-element Non-Linear Transmission 
Line?s (NLTL?s) that produced sub-us pulses. By the 1960?s high resistivity 
ferrite cores were added to NLTL?s that allowed even higher frequency pulses 
(pulse pumps). The Star Wars program of the 1980?s created a need for extremely 
high power ? short duration sources. This need lead to further advancements in 
NLTL designs that included the addition of periodically separated blocks of 
nonlinear ceramics in NLTL?s or ?shocklines?.  By the 1990?s, fast responding 
Schottky diodes were used as nonlinear capacitance elements increasing the 
NLTL bandwidth further.   
 
 
   
2 
 
 
1.2. Soliton Theory 
A soliton is defined as a wave that preserves its shape and propagates at a 
constant velocity. The technology of generating solitons [1] using nonlinear 
materials has been researched since 1940s and is widely used for many industrial 
applications. For instances, ultra-short duration pulse lasers rely on optical 
solitons to deliver the repetitive, ultra short duration pulses. This was first 
observed when researchers showed that the nonlinear and dispersive nature of 
nonlinear optical materials could reshape a series of relatively long duration 
pulses into a series of much shorter duration optical solitons [1]. Using the same 
concept, research in the microwave area has investigated different types of 
dielectric and magnetic materials to generate similar solitons.  
  A NLTL creates solitons or quasi-solitons by using a high voltage pulse 
pump to generate a series of pulses. The nonlinearity of the periodical lumped 
materials in the circuit will sharpen the rising and falling edges of the pulse. At 
the same time, the dispersive and nonlinear nature of the nonlinear transmission 
will break the relative long input pulse into an array of short duration solitons as 
the pulse is propagating through the transmission line. 
The fundamental reason nonlinear materials allow the creation of solitons 
is that the wave velocity is a function of the electric and magnetic field strength. 
Specifically, the wave velocity in a transmission line is equal to 1v
???
 , where 
? is the permittivity and ? the permeability of the material. Both the permeability 
and permittivity decrease with increasing field strengths. Therefore the portion of 
3 
 
 
the wave having a relatively large field strength propagates rapidly and the lower 
field strength portion at a reduced velocity thereby sharpening broad pulses into 
shorter duration pulses that usually having a larger peak value.   
This phenomenon can be observed by analyzing a simple NLTL. The 
NLTL can be modeled as a typical L-C ladder network [1] where the series 
inductors are connected to ground by a shunt capacitor. Both the capacitors and 
inductors are made from nonlinear materials. The nonlinear capacitors are simply 
reverse biased diodes and the capacitance will decrease with increasing voltage, 
so that the crest of the pulse has a larger voltage value and will travel faster than 
lower voltage portions.  In this way, the peak of the pulse will overtake the rising 
edge and a very short duration shockwave is generated. The same type of pulse 
compression is observed in nonlinear inductors. The inductance values will 
decrease as current increases causing the peak of the current pulse to travel faster 
thereby sharpening the wave. 
Selection of the nonlinear materials for each of the respective components 
for pulse sharpening is critical in the design of NLTL?s. Typically reverse biased 
diodes are used for the nonlinear capacitors and nonlinear magnetic materials 
receive the most consideration for nonlinear inductors. 
 
1.3. Magnetic Materials 
Many types of magnetic materials have been characterized and considered 
for their ability to sharpen a propagating voltage pulse [1, 2]. The most promising 
material to date was discovered by BAE and is a patented highly resistive ferrite. 
4 
 
 
Generally, most nonlinear magnetic materials used for NLTL?s or shocklines 
operate near the Curie temperature. They behave as a ferromagnetic material 
below the Curie temperature and a paramagnetic material above the Curie 
temperature. Since operation in the ferromagnetic region increases losses due to 
the magnetic hysteresis, NLTL?s are normally operated above the Curie 
temperature, where the ferromagnetic material becomes paramagnetic. Eddy 
current losses are also of concern and minimized by laminating the magnetic 
material [1] and adding insulation layers. 
A majority of nonlinear magnetic materials used today are a type of soft 
ferrites which are typically characterized at low frequencies (f ~ 10 KHz). They 
have a wide range of permeability; saturation flux and resistivity values and are 
relatively inexpensive [1].  This is a major problem when someone attempts to use 
Commercially-Off-The-Shelf (COTS) ferrites in NLTL design. It is therefore 
important that the constitutive parameters are measured over the frequency range 
of interest.  
 
1.4. New Techniques 
1.4.1. Hybrid NLTL 
A majority of the NLTL?s have been designed with invariant capacitance 
and inductance values along the line that are based on two separate sets of 
inductors and capacitors. These lines have the disadvantage of the characteristic 
impedance having discrete variations along the line. This can be avoided by 
integrating all materials in a single housing structure. A transmission line of this 
5 
 
 
type is termed a hybrid line. Research by Smith investigated a type of hybrid line 
created by filling a planar conducting line with alternating ceramic and ferrite 
sections [3]. The nonlinear dielectric and ferrite materials tend to rapidly sharpen 
the pulse and create solitons. This type of structure has impedance that is invariant 
along the line length but operates near the relaxation time of the nonlinear 
material dielectric. 
 
1.4.2. Gradually Scaled NLTL 
A major problem that arises when a relatively wide voltage pulse is 
compressed into sharper and narrower pulse is that it tends to degenerate into 
multiple pulses. A technique that has been shown to reduce the likelihood of this 
occurring is to gradually scale the lumped parameter inductance and capacitance 
values along the direction of signal propagation. This technique has been shown 
to sharpen both the leading and trailing edges of the pulse [4]. Generally, it is 
difficult to reduce the pulse width when the parameters remain invariant in the 
NTLT. What may happen is that the input voltage pulse, having a relative long 
time duration, will be broken into a pulse sequence of shorter duration solitons. 
The problem of a wide pulse degenerating into multiple pulses rather than a single 
pulse is solved by using a gradually scaled NLTL. By gradual changing these 
parameters it is possible to achieve a series of very narrow pulses without pulse 
degeneration.   
 
 
6 
 
 
1.4.3. Optimal Design of NLTL 
 All the aforementioned NLTL designs were created by designers using a 
common sense approach based on trial and error.  The goal of this research is to 
optimally design a NLTL using a genetic algorithm (GA). GAs are a type of 
global optimization that converge to a global minimum in a weakly probabilistic 
sense. They have been used in the design of many highly complicated systems 
with a remarkable degree of success.  
 In the research presented here, the controlling GA was coded in MATLAB 
and interfaced with the software LTspice to exchange information and run 
LTspice. The fitness function required that the circuit maximize the peak output 
pulse power. Generally this type of design is quite complicated and the circuit 
structure and types of materials used could span a large range of values. Therefore 
care must be taken when selecting materials knowing that they?ll operate on the 
cusp of the ferrite B-H and dielectric P-E nonlinear saturation curve.  
 The GA algorithm emulates the natural evolutionary process by creating 
chromosomes, mutation rates, population sizes, etc. that are related to 
fundamental electrical parameters of the circuit. The parameter values in the 
circuit gradually evolve to the best solution of the fitness function without being 
limited by local minimum trapping.  Each of the variable components? values 
(capacitors and inductors) are converted to a floating point number or ?string? and 
used as the chromosomes of each child.  The total number of chromosomes 
determines the genetic sequence of each child and the total number of children is 
the population size. Each child is simulated in LTspice and the results evaluated 
7 
 
 
and ranked by the fitness function.  The next step is the defining phase of the GA 
that allows a new generation to be created by crossover and mutation and the 
subsequent solutions ranked and the process repeated.  
 
1.5.  Benefit of the Method 
Since the NLTL has been widely employed in many different areas 
including radar signal compression and high power microwave generation, the 
requirements for the NLTL will vary a great deal. After determining the system?s 
operating parameters, including the generator frequency and nonlinear component 
characteristics, the task of optimizing of the NLTL to achieve a maximum output 
pulse power can be done using the integrated optimization technique described 
earlier. This will circumvent the usual trial and error approach that relies on 
human insight into the problem and directly solves for the best set of parameter 
values based on the system constraints.  
 
 
 
 
 
 
 
8 
 
 
 
 
 
2. Theory 
2.1. The NLTL Implementation in LTspice 
2.1.1. The Basic Circuit for NLTL 
The basic building blocks of the NLTL are L-C sections comprised of 
nonlinear inductors and reverse biased diodes acting as nonlinear capacitors. The 
L-C sections act as a low-pass filter below frequencies specified by the Bragg 
frequency ( 1
Bf LC??
) [5]. Figure 2.2 shows the equivalent lumped parameter 
model of the NTLT, where dC  and dR  represent the nonlinear capacitance and 
parasitic resistance of the reverse biased diode and L and C represent the 
nonlinear inductance and linear capacitance of the nonlinear transmission line 
respectively.  
 
Figure 2.1:  Basic NLTL circuit 
9 
 
 
 
Figure 2.2: Equivalent circuit of NLTL 
 
2.1.2. Nonlinear Inductor 
Since the permeability for any magnetic material is realistically a function 
of both the magnetic field intensity and, to a lesser extent, the operational 
frequency, an empirical model must approximate the B-H curve of realistic 
ferrites in the frequency domain of interest. Also, the inductor?s L-I curve must be 
directly proportional to the ferrite?s B-H curve and since the magnetic field will 
be proportional to I. The inductor?s L-I curve was designed to be representative of 
many commonly used nonlinear magnetic materials and based on the 
measurements taken at a frequency of 77 kHz shown below in Figure 2.3. The 
mathematically model used to approximate this behavior is shown in equation 2.1 
and the corresponding curve in Figure 2.4. Several different nonlinear models 
have been presented in the research but were not used due to computational 
difficulties. The parameter values for max 0,LLand mag were found from a least 
squares curve fit of the measured data and modeled in LTspice. It should be noted 
10 
 
 
that this model may easily be adapted to describe other nonlinear inductor 
behavior.  
 
Figure 2.3 Inductance versus current curve  
The nonlinear behavior of the inductor is modeled in this research using 
hyperbolic tangent function as shown below 
m a x 0()L = L 1 t a n h s a ta b s I I Lm a g???????????????                              (2.1) 
Equation 2.1 is plotted in Fig 2.4 with the parameters specified as
m a x 07  , 1  , 1  , 1 3satL nH I m A L nH m ag e? ? ? ? ?. 
11 
 
 
 
Figure 2.4: The inductance versus current curve for the nonlinear inductor 
The voltage across the non-linear inductor is given by equation 2.2 and 
dLdI given in equations 2.2 and 2.3 respectively.  
    ()d I d L d I d L d Iv L Ld t d I d t d I d t? ? ? ?                               (2.2) 
2m a xL ( )1 t a n h() s a ta b s I Id L Id I m a g a b s I m a g?????? ? ?????????
                      (2.3) 
 
By inserting equation 2.1 and 2.3 into equation 2.2 the final form of the 
equation used in LTspice is shown in equation 2.4.  
0 1 2 3 4 5 6 7
x  1 0
-3
0
0 . 2
0 . 4
0 . 6
0 . 8
1
1 . 2
1 . 4
x  1 0
-8
C u r r e n t ( A )
I
n
d
u
ct
a
n
ce
(
H
)
N o n l i n e a r  I n d u ct a n ce
12 
 
 
2m a xm a x 0( ) L ( )L 1 ta nh 1 ta nh()sat satabs I I abs I II dIvL m ag m ag abs I m ag dt??? ? ? ?? ? ? ?????? ? ? ? ???? ? ? ?? ? ? ?? ? ? ?? ? ? ?
??
(2.4) 
The nonlinear inductor behavior is described first using a subcircuit in 
LTspice and the netlist shown below.  
.subckt nonind n2 n3 
B1 n1 n3 V=(({Lmax}*(1-tanh((abs(I(B1)-{Isat})/{mag})))+{L0}-
{Lmax}*(1-pow(tanh((abs(I(B1)-
{Isat})/{mag})),2))*(I(B1)/({mag}*abs(I(B1)))))*ddt(I(B1))) 
R2 n2 n1 .0001 
.param Lmax=300n Isat=2.5 L0=1n mag=3 
.ends 
The netlist calls the subcircuit definition according to the requirement of 
LTspice. The subcircuit parameter Lmax defines the maximum value of the 
nonlinear inductance, Isat the current where the magnetic material begins to 
saturate, and mag is the variance. The resistance accounts for the magnetic 
material?s parasitic losses. 
 
 
 
13 
 
 
2.1.3. Nonlinear Capacitance 
The value of the NLTL capacitance selected must have a relatively large 
breakdown voltage and operate with nonlinear behavior for range of voltages 
developed in the line. After some experimentation, a diode was selected that had a 
reverse breakdown voltage of several hundred volts and labeled MMSD4148 [6]. 
Figure 2.5 depicts the capacitance versus reverse voltage curve at the lower end of 
the reverse voltage range. 
 
Figure 2.5: Diode capacitance versus reverse voltage curve 
 
 
 
14 
 
 
2.1.4. Simulation Time Step 
For transmission line analysis, the time step is required to be equal to or 
smaller than the period associated with the Bragg frequency [2], which  is defined 
in equation 2.5. Since the value of the per unit length capacitance and inductance 
vary, an effective capacitance and inductance must first be calculated according to 
equations 2.6-7. 
1Bf LC??                                                 (2.5) 
m a x
m in_ m a x m in
1 ()VD a v e D
VC C v d vVV? ? ?                                    (2.6) 
          m a x
m inm a x m in
1 ()Ia v e
IL L i d iII? ? ?
        (2.7)
 
 
The Bragg cutoff frequency for the components used is 82.9300 10? Hz 
and associated time step 91 3 .4 1 3 0 e 1 0
Bt f
?? ? ?seconds. 
 
2.1.5. Voltage Source 
Another parameter that determines the NLTL design is the source voltage 
magnitude and frequency. The voltage level was arbitrarily set to values of ?50 
volts and at a pulse period of 15 ns. These parameters are typical for pulsed power 
sources and the entire pulse described below [3,4]. 
15 
 
 
 Voltage:                     ?50 V 
 Rise time:                1.5ns 
 Fall time:                1.5ns 
 Pulse width:                5ns 
 Pulse period:               15ns 
 
2.1.6. Matched Load 
The load impedance and source resistance was set to 50 ohms and the 
peak power enhancement that occurs at the load is calculated according equation 
2.8. The denominator in this equation is equal to average power supplied to the 
source without the NLTL present and attaches a figure of merit to the pulse power 
enhancement.  
2
0
2
0
in
( ) /
/2
w he r e :
V ( t)  = ins ta nta ne ou s ou tput v ol ta ge
V  = p e a k inp ut v ol ta ge
 
L
in S
V t RP owe rR atio
VR
?
                                        (2.8) 
 
2.2. Function of NLTL 
2.2.1. Pulse Narrowing 
The specific nonlinear and dispersive qualities of the NLTL are the key to 
the pulse sharpening that occurs. The NLTL allows velocity of the frequency 
components of the pulse to be proportional to pulse voltage and current. This is 
16 
 
 
due to the decrease in capacitance and inductance with increasing pulse voltage 
and current as shown in equation 2.9 below. 
   
1
( ) ( )
w he r e :
ph a se  v e l oc it y
( )  indu c ta nc e  pe r  un it  l e ng th 
( )  c a pa c it a nc e  pe r  un it  l e ng th
p
p
v
L I C V
v
LI
CV
?
?
?
?                                            (2.9)
 
 
 Thus the leading edge of the voltage pulse will travel slower than the crest 
of the pulse voltage and pulse magnitude will increase and width of the pulse 
decrease. Also, dispersive effects of the NLTL will be effectively canceled out if 
the NLTL is designed properly. Ideally, if the NLTL is designed or ?tuned? 
correctly, controllable solitons and quaisi-soliton pulses will be produced as 
shown in Figure 2.6 and the pulse will gradually sharpen as it propagates down 
the line.  
 
17 
 
 
 
Figure 2.6: Improvement of the pulse through the NLTL 
 
2.3. GA 
2.3.1. Basic Theory and Implementation 
Genetic algorithms (GAs) are one of the most popular global optimization 
methods used today for solving the complex optimization problems. They are 
based on the biological sciences and have been used very effectively on many 
?noisy? or highly nonlinear problems not well suited for gradient based solvers 
(Quasi-Newton or conjugant gradient) [7]. Since NLTL are inherently nonlinear, 
the GA is ideal for optimizing peak pulse output power for any combination of 
material properties and input voltage characteristics [7]. 
18 
 
 
GAs are relatively easy to understand since they attempt to replicate the 
Darwnian concepts of natural selection and evolution. The parameters solved for 
in the research are encoded as floating point numbers and become the 
chromosome of each child of a single generation represented as a [1?18] vector. 
Each of the children represents a specific set of capacitance and inductance values 
for a NLTL and each child is ranked based on its performance by evaluating its 
strength using a fitness function. The fitness function determines a numerical 
figure indicative of the relative ability of each child to obtain a desired solution. 
After the children are ranked, the elite children are copied into the next generation 
and the remaining population is used in a roulette wheel selection process for 
crossover and mutation. The crossover process is done by taking a portion of the 
chromosomes from two children and combining them to form two new children. 
The rate that this takes place is called the crossover rate.  Mutation also takes 
place where a certain number of children have a small percentage of their 
chromosomes changed in a random way.  The crossover and mutation process is 
used to create the next generation while holding the total population of each 
generation fixed. This process will create the new generation and will continue 
until either a certain value of the fitness function is attained or a specified number 
of generations completed.  
When designing a GA to solve any problem there are a number of initial 
parameters that must be set. The first parameter is an initial guess at the starting 
values for all unknowns. In our case we begin by initializing the values of the 
linear capacitance and inductance by assigning a random value within a specified 
19 
 
 
range to all linear capacitive and nonlinear inductive elements. The total time 
duration and period of the initial pulse voltage also needs to be determined. This 
is somewhat arbitrary and was found to be highly dependent on the number of 
sections used.   
a. Chromosome encoding 
As mentioned earlier, either binary coding or floating point coding could 
be used for the chromosomes.  It was found that using floating point numbers 
allowed a more rapid convergence to a solution and thus used. Also, converting 
the parameter values to binary numbers and then back to floating point numbers 
would add an additional processing step.  
b. Population size 
The population size needs to be determined before proceeding. There is 
however a tradeoff between the population size and rate of new generation 
creation. A larger population will likely yield a larger number of solutions with 
relatively ?good? fitness values but will slow the overall creation rate of the new 
generation. If the population is too small many ?good? solutions may be 
overlooked. Therefore, it is recommended that the population be of medium size 
suitable for the specific NLTL problem. In this work the population size was set 
to 200. 
c. Selection scheme  
All the children have their relative fitness evaluated and ranked. The best 
12.5% of the children (largest fitness value) will be copied directly into the next 
20 
 
 
generation to protect good chromosomes (solutions). A Roulette Wheel selection 
process is then used on the remaining 87.5% population to determine the children 
that will be used to create the remainder of the next generation. The probability of 
being selected for crossover is calculated using equation 2.10: 
1
i th
i th
w h e r e
P = p o s s ib il ty o f b e in g s e l e c te d f o r i c h il d
F = n o r m a l iz e d f itn e s s v a l u e o f i c h il d
n = n u m b e r o f c h il d r e n
i
i n
i
i
F
P
F
?
?
?
                 (2.10)
 
The value calculated by equation 2.10 is assigned to each chromosome and then 
selection takes place in the following manner. A uniformly distributed random 
number between 0 and 1 is generated and designated a. Then selection 
(selected_child) takes place according to the following pseudo code: 
 
   
 
 
  
(2.11) 
 
 1 :
       ;
       0 ; 1 ;
        
                ;
                1 ;
       
_ ( ) _  ( 1 ) ;
w he r e   is the  s e l e c ti on  p r ob a bil ty  o f
j
j
for i n
a ran d
pj
while p a
p p P
jj
e nd
c hild se le c te d i c hild pre v ious ge ne ration j
e nd
P
?
?
??
?
??
??
??
th
 the  j c hil d
 is the  n um be r  o f  the  c hil dr e n ( 17 5)
 is the  r a nd om  n um be r  g e ne r a te d f or  s e l e c ti on  p r oc e ss
n
a
21 
 
 
The greater the value of the fitness function the greater the possibility of 
being selected for crossover. 
d. Mutation and crossover 
The last 5% of the children selected experience only mutation and are 
placed in the new generation.  Mutation takes place based by adding a small 
random number as follows: 
child mutated = original child?s chromosome + random value 
and the random value given by equation 2.12. 
 
(2.12) 
 
 
The function erfinv creates random values that will be added to each entry of the 
chromosome. These values are limited to the range of each component as shown 
below: 
 
 
           
                          (2.13) 
 
  
 
 
 [ ( 2 ) - 1 ]
whe r e :
i s a r a nd om nu m be r  betwe e n 0  a nd  1
 i s t he  i nv e r se e r r or f un c t i on
ra nd om v al ue e rf i nv ra nd
ra nd
e rf i nv
??
 
th
 1 :
        _
                 _ ;
        _
                 _ ;
       
       
whe r e  
 i s  t he v a l ue of  t he i
i
i
i
i
i
f or i n
i f c hi l d l ow e r bo un d
c hi l d l ow e r bo un d
e l s e i f c hi l d up per bo un d
c hi l d up per bo un d
e nd
e nd
e nd
c hi l d
?
?
?
?
?
 
c hi l d i n m ut a t i on  
 i s  t he num ber  of th e  c hi l dre n i n t he m u t a t i on  n
 
22 
 
 
The remaining 82.5% of selected children are divided into two equally 
sized subgroups. Each of the parents of the first subgroup experience crossover 
with members of the second subgroup and create two new children according to 
equation 2.14. The technique used for the crossover process uses the first and last 
parents of each group to create the first two new children, then the second and 
second to last parents are used to create the next two children and so on. The 
entire process is repeated until the new generation is complete.  
1 1 ( 1 ) 2
2 2 ( 1 ) 1
w he r e r a nd is a r a nd om nu m be r be tw e e n 0 a nd 1
c hild ran d pa re nt ran d pa re nt
c hild ran d pa re nt ran d pa re nt
? ? ? ? ?
? ? ? ? ?           (2.14) 
e. Fitness Evaluation 
The fitness value assigned to each chromosome was set equal to the peak 
output voltage generated by the chromosome. This condition enforces the desire 
to achieve a peak pulse output power since the output resistance was fixed. If a 
certain minimum pulse width is required the fitness function could be modified to 
include the any width speciation.  
 
 
 
 
 
23 
 
 
2.3.2.  GA Flowchart 
 
Figure 2.7: Flowchart of the GA used in this thesis 
The flowchart shown in Figure 2.7 indicates the order that each operation 
occurs in the GA. The first step is the creation of initial values for the inductances 
and capacitances using a random number generator in MATLAB. These numbers 
24 
 
 
are then used to create a netlist in LTspice and the circuit?s output voltage 
calculated. This is followed by determining the fitness function each child. The 
top 12.5% of the chromosomes will be directly copied to the next generation. The 
remaining children will be selected using a roulette wheel selection process and 
then used for crossover and mutation processes. The crossover and mutation 
processes are repeated until the new population is complete. This process of 
selection, crossover and mutation will be repeated until the termination criteria is 
met. 
 
2.3.3. Parameters of the GA 
The goal of this research is to optimize the output voltage of a NLTL 
using a GA and thus provide proof of concept. To limit the amount of computer 
usage, only six L-C sections will be cascaded together for the design. This could 
be extended to any number of sections depending on the specific design 
requirements. The parameters selected for use in the GA include population and 
generation size, crossover and mutation rates. All these parameters were adjusted 
as necessary to increase the algorithm?s performance.  The specific NLTL design 
undertaken and has the parameters listed below: 
a. 18 component values are optimized  
b. Selection scheme: roulette wheel 
c. Population size:200 
d. Mutation rate:0.05 
e. Number of generations:170 
25 
 
 
2.4. Involved Software 
After choosing the GA as the optimization algorithm for NLTL, it?s 
important to decide what software is most appropriate to realize the optimization 
process. The software must be efficient and able to provide accurate nonlinear 
circuit simulations. LTspice was chosen since it is both efficient and contains an 
extensive library of standard electronic components in addition to being free to 
use. The GA and the bi-directional interface to LTspice was written in Matlab. 
The main function of the interface is to forward the LTspice simulation data to 
Matlab for later processing and transfer the updated circuit netlist to LTspice 
without the need for any mouse and keyboard operations. 
 
2.5. Interface Implementation 
2.5.1. Function of the Interface 
The interface between Matlab and LTspice is critical to proper operation 
of the algorithm and must be the compatible with both Matlab and LTspice. Since 
this is a nonlinear optimization problem, there is the possibility of non realizable 
or artifact solutions occurring. Therefore, appropriate measures must be 
incorporated into the interface to prevent this from occurring. The interface 
should also be efficient to minimize the iteration time of the optimization process.  
Finally, the interface needs to allow information to flow bi-directionally between 
Matlab and LTspice in a closed loop so that the optimization will be continue 
automatically without any further commands.  The interface was written in 
Matlab and performed the following tasks: 
26 
 
 
a. Create netlist for LTspice with component values using Matlab 
b. Access LTspice using Matlab and simulate circuit behavior 
c. Read output voltage from LTspice and calculate fitness function 
d. Use GA to calculate new component values and repeat from step  a. 
 
2.5.2. LTspice Batch Command 
The biggest problem experienced when creating the interface was finding 
a method to automatically start the simulation in LTspice from Matlab. It?s not 
possible to directly access the GUI interface radio button on the LTspice control 
panel using Matlab. However, every time a new combination of components? 
values is obtained from the GA, LTspice must be rerun. Fortunately LTspice 
provides batch command for circuit simulation. This is done using the batch 
command ? ?b?. This command will run and leave the results in a file named 
?.raw?. It?s important that the windows batch file be written and saved in the 
same directory as LTspice to allow easy access by Matlab. The batch command 
used is shown below [8]: 
C:\batch\LTC\LTspiceIV\scad3.exe-b C:\batch\LTC\LTspiceIV\nltl.net 
The first half of the command specifies the location of the executable file, 
command ?-b? for automatic simulation and data storing, and the second half 
refers to the location of the netlist file of the NLTL that is optimized. Also, the 
batch command should be saved as a windows executable batch file. The 
27 
 
 
commands used to access the interface are given below that will allow the batch 
job to be executed in Matlab.  
 
Figure 2.8: Key words in Matlab codes building the interface 
 
2.5.3. LTspice to Matlab 
The output of LTspice in the ?.raw? file is imported into a Matlab as a data 
structure that contains the variable components? name, values and the time axis 
vector. By processing these results the peak voltage is found and set equal to the 
fitness value for the specific parameters simulated. Figure 2.9 is a plot of the 
output voltage versus time for an example LTspice simulation.  
28 
 
 
 
Figure 2.9: Data imported into Matlab 
 
2.5.4. Matlab to LTspice 
a. Netlist creation 
Each new generation created by the GA in MATLAB generates a new 
netlist file that must be evaluated by LTspice.  The following set of Matlab 
commands is used for the process.  
i. fid = fopen(?filename?) 
fopen will open a file and requires the filename argument to be in 
single quotation marks 
ii. count = fwrite (fid, A) 
0 1000 2000 3000 4000 5000 6000 7000 8000
- 2 0
0
20
40
60
80
100
120
140
160
180
T i m e  S t e p  1 e - 1 2
P
e
a
k 
V
o
l
t
a
g
e
29 
 
 
After obtaining the identification of the target file, fwrite is used to 
write the matrix of input information into the file.  
iii. Status = fclose(fid) 
The fclose command will close the file specified by the argument 
and return the status after closing. 
b. Sub-circuit and main circuit netlist 
It is important that the correct data format is used by Matlab to create the 
netlist used by LTspice or an error will occur. The Matlab executable file used to 
generate the subcircuit netlist for the nonlinear inductors is shown below.  
line{01} = ['.subckt nonind n2 n3']; 
line{02} = ['B1 n1 n3 V=(({Lmax}*(1-tanh((abs(I(B1)-
{Isat})/{mag})))+{L0}-{Lmax}*(1-pow(tanh((abs(I(B1)-
{Isat})/{mag})),2))*(I(B1)/({mag}*abs(I(B1)))))*ddt(I(B1)))']; 
line{03} = ['R2 n2 n1 .0001']; 
line{04} = ['.param Lmax=' num2str(x(1)) 'n Isat=2.5 L0=' 
num2str(x(7)) 'n mag=3']; 
line{05} = ['.ends']; 
fid = fopen('C:\batch\LTC\LTspiceIV\nonind.lib', 'wb'); 
for i = 1:length(line) 
    fwrite(fid, [line{i} char(13) char(10)], 'char'); 
end 
fid = fclose(fid); 
30 
 
 
There are six sections in the NLTL simulated and the main program netlist 
is given below.  
Main circuit for NLTL 
line{01} = ['* C:\batch\LTC\LTspiceIV\nltl.asc']; 
line{02} = ['XU1 N004 N003 nonind']; 
line{03} = ['V1 N001 0 PULSE(-50 50 0 1.5n 1.5n 5n 13n)']; 
line{04} = ['R1 N002 N001 50']; 
line{05} = ['D1 0 N004 MMSD4148']; 
line{06} = ['R2 N014 0 50']; 
line{07} = ['C1 N004 0 ' num2str(x(13)) 'f']; 
line{08} = ['D2 0 N006 MMSD4148']; 
line{09} = ['C2 N006 0 ' num2str(x(14)) 'f']; 
line{10} = ['D3 0 N008 MMSD4148']; 
line{11} = ['C3 N008 0 ' num2str(x(15)) 'f']; 
line{12} = ['D4 0 N010 MMSD4148']; 
line{13} = ['C4 N010 0 ' num2str(x(16)) 'f']; 
line{14} = ['XU3 N008 N007 nonind3']; 
line{15} = ['XU2 N006 N005 nonind2']; 
line{16} = ['XU4 N010 N009 nonind4']; 
line{17} = ['R3 N004 N005 100?']; 
line{18} = ['R4 N007 N006 100?']; 
line{19} = ['R5 N003 N002 100?']; 
line{20} = ['R6 N009 N008 100?']; 
31 
 
 
line{21} = ['D5 0 N012 MMSD4148']; 
line{22} = ['C5 N012 0 ' num2str(x(17)) 'f']; 
line{23} = ['D6 0 N014 MMSD4148']; 
line{24} = ['C6 N014 0 ' num2str(x(18)) 'f']; 
line{25} = ['R7 N011 N010 10?']; 
line{26} = ['R8 N013 N012 10?']; 
line{27} = ['XU5 N012 N011 nonind5']; 
line{28} = ['XU6 N014 N013 nonind6']; 
line{29} = ['.model D D']; 
line{30} = ['.lib C:\batch\LTC\LTspiceIV\lib\cmp\standard.dio']; 
line{31} = ['.tran 0 100n 0 1e-11']; 
line{32} = ['.lib C:\batch\LTC\LTspiceIV\nonind4.lib']; 
line{33} = ['.lib C:\batch\LTC\LTspiceIV\nonind3.lib']; 
line{34} = ['.lib C:\batch\LTC\LTspiceIV\nonind2.lib']; 
line{35} = ['.lib C:\batch\LTC\LTspiceIV\nonind.lib']; 
line{36} = ['.lib C:\batch\LTC\LTspiceIV\nonind6.lib']; 
line{37} = ['.lib C:\batch\LTC\LTspiceIV\nonind5.lib']; 
line{38} = ['.backanno']; 
line{39} = ['.end'];  
fid = fopen('C:\batch\LTC\LTspiceIV\nltl.net', 'wb'); 
for i = 1:length(line) 
    fwrite(fid, [line{i} char(13) char(10)], 'char'); 
end 
32 
 
 
fid = fclose(fid); 
The list includes six nonlinear inductors each having separate 
max 0andLL values and a six capacitance that are optimized. Each of the nodes 
for the nonlinear inductor must be defined six times in the netlist (a simple call 
statement is insufficient). The vector x[n] called above represents the component 
values for the inductance and capacitance in the netlist file created by Matlab and 
the num2str command is used to convert the floating point value into a string 
which is used by Matlab.  
c. Call batch file to simulate the circuit 
After creating the netlist described above, the batch file is executed in 
LTspice from Matlab by calling the DOS shell script and running it in Microsoft 
Windows. The Matlab command used for this is shown below along with the 
corresponding DOS batch file [8]. 
dos('C:\batch\LTC\LTspiceIV\lt2mat.bat'); 
 
Figure 2.10: Screen shoot on the dos batch file 
 
33 
 
 
Every time this command is called the batch file will be displayed in the 
Matlab command window and a completion status will be returned. 
d. Obtain the peak voltage 
The process of obtaining the maximum output voltage for a given ?.raw? 
file was done by sorting the time dependant voltage data in descending order and 
keeping the maximum value.  
e. Artificial situation handling 
Since the inductance and capacitance used are nonlinear, LTspice 
sometimes fails to converge and gives non-realizable results which cause an error 
to occur when Matlab attempts to accesses the peak voltage. A Matlab command 
was added to the interface to prevent this error from occurring. When an 
artificially large voltage occurs, the unreasonable result is ignored by Matlab and 
the component values causing this behavior returned to their pre-defined settings 
that would result in a relatively low output voltage occurring. This will allow the 
combination of the components? values to be treated as a child with low fitness 
value and will not affect the optimization process negatively. 
 
2.5.5. Packet the Interface 
The entire bi-directional interface between Matlab and LTspice was 
written as a single Matlab function called lt2mat that was called by the GA. The 
inputs of the function are the component values obtained from the GA that are 
34 
 
 
used to create the LTspice netlist file. The return value of this function is the peak 
voltage of the single circuit simulation which will be treated as fitness evaluation 
in the genetic algorithm. 
 
2.5.6. GA and interface operation 
The entire optimization scheme needs to operate efficiently and the 
specific working of all the operators is described below. Following the creation of 
a new generation, each child?s fitness must be evaluated. This is done by calling 
the interface described previously where the child?s chromosome was used to 
create the netlist. LTspice will then be executed and the resulting peak output 
voltage of the circuit returned to the GA using the interface previously described. 
The peak value of the output voltage will be assigned as the child?s fitness value.  
 
2.5.7. Computation complexity and iteration time 
The population size, cross over rate, and mutation rates all affect the 
performance of the algorithm. Following a thorough literature review that 
provided many helpful suggestions, a trial and error process was used to 
determine the parameter combination that would provide the best overall 
algorithm performance.  The parameters selected provided a sufficient 
convergence rate while still yielding optimal results in real-time. It should be 
noted all Matlab for loops were vectorized to reduce runtime when possible. 
35 
 
 
Besides adjusting parameters used in the Matlab GA, a minimal number of 
sections used for the simulation needed to be set.  The number of sections was 
required to be sufficient to allow true transmission line behavior to occur. It could 
not be made too large or the simulations would not be completed in a reasonable 
time. This is somewhat dependant on the computer resources available and after 
some experimentation the final number of sections was set to six and the results 
found in a reasonable time (approximately 1 day to complete a single simulation).  
 
2.5.8. Optimization System 
 
Figure 2.11: Flowchart of the optimization system 
 
36 
 
 
Figure 2.11 contains a flowchart of the entire optimization process.  The 
process can basically be divided into two parts: The calculations done in Matlab 
and the simulations done by LTspice.  Matlab is responsible for all the GA 
calculations and the data interface while LTspice is used to generate the transient 
output voltage waveform. To summarize, a single iteration starting at the LTspice 
output file is described: the ?.raw? file is imported by Matlab and the peak output 
voltage is obtained. This value is treated as the chromosome?s fitness value and a 
new combination of all the component values is obtained in the GA and written 
into the netlist file. Finally, the batch file is called (run) again, circuit simulated in 
LTspice and results saved in the ?.raw? file.  
 
 
 
 
 
 
 
 
 
 
 
 
 
37 
 
 
 
 
 
 
3. Results 
3.1. Simulation 
The NLTL circuit design contains six L-C sections. In each section, the 
nonlinear inductor is defined using a subcircuit that mathematically defines its 
nonlinear behavior and the nonlinear capacitance is realized using a linear 
capacitor in parallel with reverse-biased diode. The pulse voltage source was 
based on typical parameters found in the literature [9] and the load and source 
impedance arbitrarily set to 50 ?.  Figure 3.1 illustrates the structure of the final 
NLTL circuit optimized by the simulations. Figure 3.1 is a screen capture image 
of the actual circuit as it appears in the LTspice Graphical User Interface (GUI). 
The ?lib.? files define each nonlinear inductor and need to be included as part of 
the netlist that describes the main circuit. 
 
Figure 3.1: Schematic of the main NLTL circuit 
38 
 
 
3.1.1. Simulation Settings 
 
Figure 3.2: LTspice simulation command setting 
Since the NLTL circuit contains many nonlinear components (nonlinear 
inductors and capacitors), the equivalent set of nonlinear differential equations 
must be solved using a relatively small time step to achieve an accurate solution. 
The initial time step was set to equal to the inverse of the Bragg frequency and 
steadily reduced until the resulting simulations were insensitive to further time 
step reductions. The final time step was equal to 1 ps. The LTspice simulation 
duration was set to 140 ns since output voltage waveform became approximately 
steady state. 
 
 
 
39 
 
 
3.1.2. Components optimized 
When the simulations were completed the values for the nonlinear 
inductor and capacitors were determined. These include six values of 
max 0andLL and six capacitance values.  
 
3.1.3. Initial conditions 
The initial values for maxL  were determined using the random number 
generator in Matlab that ranged from 10 to 500 nH. All capacitor values were 
initial based on the inductance range and the source voltage period and the 
estimated range was found using the formula shown in equation 3.1 
2
2(2 )TC L??
                                                  ( 3.1)
 
The value of the total capacitance of each section is ~ 6.04pF and the 
average value of the nonlinear inductors is ~ 268.23 nH . Since the parasitic 
capacitance is in parallel with the diodes, the estimated value of the parasitic 
capacitance is ~ 1.64pF .  
The initial value of the capacitance was arbitrarily set using a random 
number generator in Matlab that ranged from 1 fF  to 600 fF  and the final 
values calculated in the optimization. 
 
 
40 
 
 
3.1.4. Matlab GUI 
Because it was impossible to predict the behavior of the GA, the process 
was monitored and the parameters adjusted if the rate of convergence was too 
slow.  A GUI was developed in MATLAB to monitor the maximum voltage 
output and power ratio versus number of generations and output voltage 
waveform. It also allowed the population size, number of generations, and number 
of elite children to be set in the GUI as shown in Figure 3.3. 
 
Figure 3.3: Layout of the Matlab GUI 
If the process does not proceed at a reasonable rate the optimization 
process was terminated and the parameters modified.  
 
G A  f o r  N L T L
T i m e  S t e p  1 e - 1 2
N u m b e r  o f  G e n e r a t i o n s
N u m b e r  o f  G e n e r a t i o n s
P e a k  O u t p u t  V o l t a g e  ( V )
P o w e r  R a t i o
In s t a n e o u s  O u t p u t  W a v e f o r m
0 20 40 60 80 100 120 140 160 180
50
100
150
200
P e r f o r m a n c e  o f  G A
0 1000 2000 3000 4000 5000 6000 7000 8000
- 5 0
0
50
100
150
200
C u r r e n t  P e a k :
1 6 5 . 9 2 2 9
G e n e r a t i o n  P e a k :
1 6 7 . 2 4 4 4
0 20 40 60 80 100 120 140 160 180
0
1
2
3
4
5
6
C u r r e n t  C h i l d :
3 6 1 . 3 2 1       9 1 . 5 7 5 6 3       5 0 . 5 5 6 8 6        1 3 . 1 1 4 5       1 2 . 9 3 7 8 1       1 0 . 0 0 1 5 2
1 1 . 6 6 5 9       1 7 . 3 0 2 3       4 . 1 7 4 5 2       6 . 8 7 3 6 8      0 . 6 3 2 0 2 5     0 . 0 1 1 4 5 8 9
2 . 1 6 1 1       1 . 6 9 6 4        2 . 3 7 1       1 . 8 1 8 7       3 . 8 8 5 4       2 . 8 4 5 1
Peak output voltage 
Power ratio 
Instantaneous output waveform 
GA?s parameters 
Simulation results 
41 
 
 
3.2. Simulation Results 
The interface was specifically designed to be transferable across Windows 
platforms providing that MATLAB and LTspice are available.  In this research, 
the entire simulation process was done on a HP Pavilion dv7-4070 us laptop with 
a Windows 7 operation system for testing purpose. 
 
Figure 3.4: Simulation results showed in Matlab GUI 
The simulation resulted in the peak voltage being increased from?  50 V 
to 165.9229 V and the width of the pulse being narrowed appreciable to ~ 0.17 ns 
compared to the input pulse width of 8 ns. The power ratio, which indicates the 
degree of peak pulse power enhancement, was increased to a value of 5.7.  
The final component values will appear in the Matlab GUI after the 
termination criteria has been met. It is obvious that the number of generations 
G A  f o r  N L T L
T i m e  S t e p  1 e - 1 2
N u m b e r  o f  G e n e r a t i o n s
N u m b e r  o f  G e n e r a t i o n s
P e a k  O u t p u t  V o l t a g e  ( V )
P o w e r  R a t i o
In s t a n e o u s  O u t p u t  W a v e f o r m
0 20 40 60 80 100 120 140 160 180
50
100
150
200
P e r f o r m a n c e  o f  G A
0 1000 2000 3000 4000 5000 6000 7000 8000
- 5 0
0
50
100
150
200
C u r r e n t  P e a k :
1 6 5 . 9 2 2 9
G e n e r a t i o n  P e a k :
1 6 7 . 2 4 4 4
0 20 40 60 80 100 120 140 160 180
0
1
2
3
4
5
6
C u r r e n t  C h i l d :
3 6 1 . 3 2 1       9 1 . 5 7 5 6 3       5 0 . 5 5 6 8 6        1 3 . 1 1 4 5       1 2 . 9 3 7 8 1       1 0 . 0 0 1 5 2
1 1 . 6 6 5 9       1 7 . 3 0 2 3       4 . 1 7 4 5 2       6 . 8 7 3 6 8      0 . 6 3 2 0 2 5     0 . 0 1 1 4 5 8 9
2 . 1 6 1 1       1 . 6 9 6 4        2 . 3 7 1       1 . 8 1 8 7       3 . 8 8 5 4       2 . 8 4 5 1
42 
 
 
could be increased to attain an even greater output voltage. However, the goal of 
this research was to show the feasibility of using a GA to optimally design a 
NLTL. The final parameter values found are shown below. (The max 0andLL
values of each nonlinear inductor and capacitance value of each parasitic 
capacitance are given) 
The maximum values for the nonlinear inductors were found to be 
m a x _ 1 m a x _ 2 m a x _ 3
m a x _ 4 m a x _ 5 m a x _ 6
3 6 1 . 3 2 1  9 1 . 5 7 6  5 0 . 5 5 7  1 3 . 1 1 5 1 2 . 9 3 8 1 0 . 0 0 2  L n H L n H L n HL n H L n H L n H? ? ?? ? ? 
and 0L  values for the nonlinear inductors are 
0 _ 1 0 _ 2 0 _ 3
0 _ 4 0 _ 5 0 _ 6
1 1 . 6 6 6 1 7 . 3 0 2  4 . 1 7 5 6 . 8 7 4  0 . 6 3 2  0 . 0 1 2  L n H L n H L n HL n H L n H L n H? ? ?? ? ?
 
The component values for the parasitic capacitors are 
1 2 3
4 5 6
2 . 1 6 1  1 . 6 9 6  2 . 3 7 11 . 8 1 9 3 . 8 8 5 2 . 8 4 5 C fF C fF C fFC fF C fF C fF? ? ?? ? ? 
The resulting output voltage waveform is shown below 
43 
 
 
 
Figure 3.5: Confirmation of LTspice 
 
3.3. Method Confirmation 
The component values of the nonlinear inductors obtained from the 
optimization gradually decrease from section to section. This was also reported by 
Afshari [4] as a method of enhancing the NLTL performance. It is also 
noteworthy that by gradually scaling, the NLTL soliton degeneration will be 
minimized [4].  
 
 
 
 
 
44 
 
 
 
 
 
 
 
4. Conclusions 
In this research, a novel integrated optimization method was presented that 
maximized the peak output voltage and power of a six section NLTL. A GA was 
chosen for the optimization method and the software code written in Matlab 
which created component values that were then used to simulate the circuit?s 
behavior in LTspice. The results show that the peak output voltage steadily 
increased for each new generation proving that code was functioning correctly. 
 A special emphasis was placed on trying to replicate NLTLs currently 
used. A standard NLTL circuit was modeled using a L-C ladder network. Each of 
the nonlinear inductor?s L-I curves was created by scaling the permeability curve 
of realistic magnetic material to the current range of the inductor.  A typical  ?  
50 V pulsed voltage with 50 ? of internal resistance was used to simulate a 
realistic input pulse source.  
The final optimization results showed that the width of input impulse was 
narrowed and the peak voltage had been increased. The power ratio indicated that 
the peak output peak power had been enhanced by 5.7 times the maximum 
possible peak power that would have occurred without a NLTL. Also, a somewhat 
gradually scaling of the nonlinear inductor?s value from the source to load end 
was observed. The actual degree of scaling had similarities to the gradually scaled 
NTLT design currently used for enhancing the NLTL?s performance. 
45 
 
 
It is also noteworthy to mention that the interface between Matlab and 
LTspice was reliable and worked very efficiently. Finally, the complicated 
problem of designing NTLT was simulated in a 24 hour time frame and clearly 
showed steady improvement.  If a much faster computer was available many more 
sections could be added and the optimization allowed to run until the maximum 
output voltage settled to a constant value.  
 
4.1. Suggestions for Future Research 
The ultimate goal of this thesis is to demonstrate the feasibility of 
employing a modified GA in the optimization of the NLTL?s peak pulse power.  
If a more complicated NLTL with known adjustable nonlinear inductor and 
capacitor values was optimized, it would be reasonable to expect that significantly 
better performance could be achieved using an extension of this method. 
Additionally, further research on improvements to the optimization speed should 
be considered.  For instance, the entire system of nonlinear equations could be 
written directly in either FORTRAN or C and solved thus eliminating the need for 
an interface to the circuit simulation software. Also, parallel computing is another 
feasible method to speed up the optimization. A very large population could be 
used in the GA and shared by several computers. Parallel processing has been 
used to achieve remarkable speed improvements in many optimization processes.  
 
 
 
46 
 
 
 
 
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