Modeling and Simulation of Nonlinear Transmission Lines 

by Frank Crowne 

ARL-TR-5062 January 2010 

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Army Research Laboratory 

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ARL-TR-5062 January 2010 

Modeling and Simulation of Nonlinear Transmission Lines 

Frank Crowne 

Sensors and Electron Devices Directorate, ARL 

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Modeling and Simulation of Nonlinear Transmission Lines 

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Frank Crowne 

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14. ABSTRACT 

This technical report summarizes design work done for ARDEC Picatinny Arsenal from April to October 2009 as part of an 

effort to design and fabricate nonlinear transmission lines (NLTL) based on ferroelectric materials. These lines were intended 

to be key components in a compact source of high-power microwaves (HPM) that could be carried by artillery rounds and 

used to disrupt battlefield communications. The effort, which was begun in 2008, centered on implementing work previously 

done by General Atomics two decades ago. Although interrupted sporadically by changes in focus of the primary effort, the 

effort has progressed to a point where with proper resources it could be a valuable contribution to future HPM munitions. 

This TR incorporates five informal reports sent to ARDEC involving the modeling and simulation portion of the NLTL design 

effort, as requested by ARDEC. 

It will serve to document the resulting software products and to be a springboard for future design work. 

15. SUBJECT TERMS

Nonlinear, soliton, HPM, transmission line, modeling 

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Frank Crowne 

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iii 

Contents 

List of Figures iv

1. Introduction: The Nonlinear Transmission Line 1

2. Selection of Capacitance-Voltage Dependence for NLTL Circuit 2

2.1 Ikezi’s Model ...................................................................................................................3

2.2 Brown’s Model ................................................................................................................4

3. Nonlinear Circuit Software Development For Transmission Line 6

3.1 Brown and Ikezi Circuits .................................................................................................6

3.2 State Variables .................................................................................................................7

3.3 The Method of Partial Charges .......................................................................................9

4. Determining the Resistive Losses in the NLTL BASED on Loss Tangents 10

5. Realistic Antenna Coupling Via an RC Circuit 16

6. Description of Mathematica Code 18

7. Results of Preliminary Simulations 20

8. Movies of NLTL Interior 23

9. Systems Issues 24

10. Conclusions 26

11. References 27

Appendix. Brown-Ikezi Network Equations 29 

Distribution List 34 
iv 

List of Figures 

Figure 1. General discrete transmission-line circuit. ......................................................................2

Figure 2. NLTL structure used for fabrication at ARL. ..................................................................2

Figure 3. The function Q(V) evaluated from Brown’s fitting expression: red – multiplication, 

blue – integration. ......................................................................................................................5

Figure 4. Ikezi-Brown NLTL circuit. .............................................................................................6

Figure 5. Two-inductor circuit. .......................................................................................................8

Figure 6. Series LRC circuit. ..........................................................................................................8

Figure 7. Placement of resistors in Brown-Ikezi NLTL circuit. ...................................................11

Figure 8. Loss tangent of a BST varactor vs. frequency at different temperatures without (a) 

and under 20 V dc bias (b). The dotted line represents the tan δ ~ ω

1/3 power law. ...............15

Figure 9. NLTL with simple resistive load. ..................................................................................17

Figure 10. NLTL with RC circuit load to simulate antenna reactance. ........................................17

Figure 11. Ikezi normalized short test pulse shape; time scale in NLTL low-frequency 

single-stage periods. .................................................................................................................20

Figure 12. Low voltage response to figure 11 test pulse at end of NLTL. ...................................21

Figure 13. High voltage response to figure 11 test pulse at end of NLTL; note solitons. ............21

Figure 14. Ikezi normalized long test pulse shape. .......................................................................22

Figure 15. Low voltage response to figure 14 test pulse at end of NLTL. ...................................22

Figure 16. High voltage response to figure 14 test pulse at end of NLTL; note solitons. ............23

Figure A-1. Brown-Ikezi NLTL circuit. .......................................................................................29
1 

1. Introduction: The Nonlinear Transmission Line 

Transmission lines are ubiquitous in almost all high-frequency RF circuits, where they replace 

the standard wire connections used for assembling circuits at lower frequencies. The many forms 

they take, involving complex geometries and exotic materials, are conditioned by the task of 

delivering RF energy from one circuit location to another. A less common use of transmission 

lines is as actual RF devices – switches, phase shifters, filters, etc. – which exploit their ability to 

perform various impedance functions and versatility with regard to fabrication. For example, 

varying lengths of the same transmission line can be used as capacitors, inductors, or delay lines 

for matching circuit subunits such as amplifiers (1). When used in this way, the transmission line 

is usually a linear circuit component. A considerably less common use of nonlinear transmission 

lines (NLTL) is for large-signal device design – phase shifters come to mind (2) – although some 

early work on distributed amplifiers could qualify as such (3). 

The structure of a transmission line can be either continuous or discrete. Whereas continuous 

transmission lines are typically waveguide-based (3a), lines with discrete structure usually 

consist of repeated identical subcircuits made up of lumped elements. A useful definition of a 

NLTL is an ordinary transmission line whose electrical characteristics are functions of local 

voltages and currents. Thus, the prototypical discrete transmission line shown in figure 1 is made 

up of lumped elements – resistors, capacitors, inductors, etc. – any one of which can be voltage 

dependent. In the 1980’s, NLTL were fabricated by inserting shunt diodes periodically into a 

transmission line to create nonlinearity in its overall electrical response (3b). Although these 

lines had very useful properties, they were never capable of much power handling due to the 

semiconductor nature of the diodes used. At the end of the 1980’s, the Ikezi group at General 

Atomics (4) studied layered structures consisting of parallel-plate waveguides in which layers of 

ferroelectric material (specifically, barium strontium titanate, or BST) alternated with layers of 

simple dielectric (see figure 2); these structures now represent the canonical version of highvoltage NLTLs, and are often encountered in the NLTL literature to this day. Mark Litz and I 

were first made aware of the work of Ikezi’s group in 2007, as a possible way to address the need 

of ARDEC for a HPM source, and it has been the basis for our effort at ARL ever since. 

In this report I will describe my efforts to model the electromagnetic response of this type of 

transmission line numerically, as requested by ARDEC in the summer of 2009. 
2 

Figure 1. General discrete transmission-line circuit. 

Figure 2. NLTL structure used for fabrication at ARL. 

2. Selection of Capacitance-Voltage Dependence for NLTL Circuit 

Much of the NLTL design work published in the literature is numerical in nature. The need to 

use numerical modeling to design a useful NLTL is regrettable in many ways, the most important 

of which is that easy-to-use design rules are seldom derivable from simulations. After much 

labor, Ikezi et al., at General Atomics (5) published an example of one such design rule for 

finding the frequency at which the power from a NLTL was a maximum. At ARL we used this 

rule extensively in our previous effort. The design rule was formulated by performing a series of 

numerical simulations of their NLTL circuit model for various line parameters, and extracting 

the peak power frequency by Fourier-transforming the computed time dependence of the power 

out of the line and looking for its peak value. Their results were published in the form of a curve, 

which they fitted to the empirical function 

10

0

1.5 log 1.25

b

E

E

  

   

  

V0 V1 V2 V3

L L

C C C

i1 i2

L

i3 Rm

VN

L

iN

L

iL

VL

C

VN-1

r r r r

12 cm 

Epoxy Alumina BST Copper
3 

where  is the peak-power frequency of the output soliton pulse train, b is the width of the 

lowest NLTL pass band when treated as a linear (small-signal) transmission line, E is the electric 

field measured at the input of the NLTL, and E0 is a scaling field of order 10 kV. Because the 

relationship implied by this formula between the small-signal and large-signal quantities is 

obscure, its usefulness for evaluating other systems is problematic. Because we were unable to 

make any accurate power measurements, we were unable to check how good the equation was; 

nevertheless, we were able to use the dependence on b to design the NLTL at a prespecified 

frequency, using an additional equation taken from (6). 

Because the nonlinear response of the NLTL is based on the voltage dependence of BST 

capacitors, it is vital that these components be modeled as precisely as possible. At this time we 

have two literature-based modeling functions, one the original choice of Ikezi in 1988 (5) and the 

other that of Martin Brown in his 1997 thesis (7). In this section I will explain why the former 

was preferred over the latter as a basis for this modeling work, and propose a third alternative 

with features of both. 

2.1 Ikezi’s Model 

The model used by the Ikezi group for the voltage dependence of BST capacitors is based on a 

fundamental relationship between the electric displacement in BST and an external electric field, 

which they write as follows: 

3

33 2

0 0 0

D D E

E

 

  

The quantity D gives the charge per unit area induced on the metal walls of the parallel-plate 

transmission line, and hence, can be multiplied by the area of contact A between the metal and a 

given ferroelectric layer to extract the charge per ferroelectric layer induced on the walls: 

Q D Q AD A



while the electric field E equals the voltage V between the walls divided by the height of the 

guide: 

V E V Eh

h



Then together these quantities let us define a function V(Q) which can be used to specify the 

capacitance of the line: 
4 

 

   

3

0

00 00

3

2

0 0 3

00 00 00 0 0

2

0 00

1 2

0

0 0

1 3 1

1 3

1 3

Q Q V Eh

EA EA

d Q Q dQ Eh Eh Q dV E A E A E A dV E A

h Q dQ

A E A dV

dQ Q V A C V

dV E A h



   

     

     

     

         

         

   

     

       

    

             

Note that in order to compute the capacitance, we must invert the function V(Q) numerically. 

The form of this expression, which is a consequence of the crystal physics of BST (8), predicts a 

capacitance versus voltage with the following properties: 

1. At small voltages (small D) it predicts a constant capacitance of the usual form, i.e., 

0A C

h

  , 

2. The capacitance is independent of polarity, and 

3. For large voltages the capacitance decreases algebraically as V–2/3. 

It is worth noting that the expression used by Ikezi is clearly a truncation of an infinite series, 

and indeed, it is common practice with in most of the ferroelectrics literature to write 

3 5 E aD bD cD  

For our purposes, it is significant that this implies a falloff in the capacitance as V–4/5 with 

voltage. The large-V behavior is thus ambiguous, and can only be resolved by fitting. Note that 

the procedure for fitting experimental data to polynomials is especially simple, a further 

advantage to using Ikezi’s expressions. 

2.2 Brown’s Model 

In contrast to the Ikezi physics-based model, the model used by Brown seems to be entirely ad 

hoc, and violates both the material constraints and the standard laws of electrostatics, casting his 

entire simulation effort into doubt: 

(1) He assumes that the relationship between charge and voltage is 

Q CVV   
5 

whereas in fact the correct relationship is 

  

0

V

QV C d    

(2) He postulates a diodelike functional dependence for C(V) that depends on polarity: 

   /

0 1 CV C e     V VC   

In addition, this expression predicts a strong increase for negative V, which is unphysical. 

It is interesting to determine the difference between Brown’s expression and the correct 

expression for Q(V): his simple multiplication gives 

    /

0 1 Q V C V V CV e      V VC   

while the correct expression gives 

       / /

0 0

0

1 11 C C

V

VV VV Q V C e dV C V V e C

                  

The plot shown in figure 3 illustrates the difference: 

20 40 60 80 100

10

20

30

40

Figure 3. The function Q(V) evaluated from Brown’s fitting expression: red – multiplication, 

blue – integration. 
6 

Clearly the two functions are almost identical for small voltages, and differ only by an offset for 

large voltages; however, the severe change in slope at intermediate voltages will greatly distort 

the waveform of an ac signal as it passes through the intermediate voltage range.

These deficiencies persuade us to adopt the original Ikezi relations in our simulations. 

3. Nonlinear Circuit Software Development For Transmission Line 

Because of its flexibility and proven track record with regard to computational accuracy and 

speed, the mathematical package Mathematica has been chosen for software support of the 

NLTL design. In order to reduce the learning-curve burden for future users, code written for this 

package will be translated into MathCad in 2010 if funding permits; meanwhile, the basic 

analysis of the NLTL performance benefits greatly from the feasibility of writing transparent 

software that is easily changed and physically motivated, and is tailored to the specifics of the 

NLTL problem rather than relying on more broadly applicable circuit software such as PSPICE 

which cannot be “tuned” internally, i.e., lack of access to source code. 

The approach to circuit modeling adopted here takes advantage of the simplicity of the NLTL 

circuit model which in turn is a consequence of the underlying parallel-plate waveguide 

geometry of the latter. A more complex propagation structure would require much more 

powerful mathematical methods to treat correctly. The modeling will be described in the 

following sections. 

3.1 Brown and Ikezi Circuits 

Both Brown and Ikezi adopt a simple series-parallel circuit model for the NLTL (see figure 4). 

Figure 4. Ikezi-Brown NLTL circuit. 

A justification of this circuit structure will be given in a future report. The original circuit 

analyzed at General Atomics had no resistances; the various resistors shown here were added by 

Brown. The circuit is an oversimplification of the real NLTL; roughly speaking, the inductors 

represent the alumina layers, while the shunt lines represent the BST layers. This 

RL

V0 V1 V2 V3

L L

C C C

i1 i2

L

i3

VN

L

iN

L

iL

VL

C

VN-1

RIN r r r r

VS
7 

oversimplification will eventually be removed in our modeling effort. The capacitors C are taken 

to be nonlinear elements. 

3.2 State Variables 

Before the advent of circuit simulators, it was well understood that nonlinear circuits must be 

approached as individual problems. This is in contrast to linear circuits, which allow a universal 

approach to computations based on straightforward matrix computations. Textbooks on circuit 

theory routinely introduce the idea of the “state” of a circuit, by which they mean a list of all the 

currents flowing through the various circuit components and all the voltage drops across them. 

When these quantities depend on time, as in an ac circuit, their values are obtained from 

differential equations. For the special case of linear circuits driven by sinusoidal sources, these 

differential equations reduce to algebraic equations that can easily be solved by matrix methods. 

Transient excitation, e.g., by pulses, require the original differential equations and numerical 

methods, as do circuits with nonlinear elements. 

It is important for numerical reasons that these differential equations all be first-order in time, 

i.e., if we describe the circuit state using a vector of currents and voltages 

  11 2 2 33 x VIV I VI  ,, , , ,,   , the numerical problem must be of the form 

  dx F x

dt 

   (1) 

where F x    is a vector function of the vector variable x



. Finding this function can be very 

difficult for a complicated circuit topology, since the circuit equations usually take the form 

, 0 dxx

dt

      

    (2) 

Numerical simulators like PSPICE routinely use a numerical engine (a “root solver”) that 

converts equation 2 into equation 1, which greatly increases execution time since it must be done 

at each time step. 

It is usually found that not all the currents and voltages are needed to describe the circuit 

dynamics. Thus, for a simple ac circuit like the one in figure 5: only two differential equations 

are needed, the currents through the two inductors. These currents are said to be the state 

variables of the circuit. In general, state variables are defined as the smallest set of electrical 

quantities that can completely specify the circuit performance. 
8 

Figure 5. Two-inductor circuit. 

For more complex circuits there may be multiple choices for these state variables. While currents 

and voltages are appropriate for resistive elements, the natural variables for capacitors and 

inductors are actually charges and magnetic fluxes, and these quantities are often handy 

alternatives to currents and voltages, especially when the circuit does not immediately admit a 

description using first-order differential equations. 

A trivial example is a series LRC circuit (see figure 6). 

Figure 6. Series LRC circuit. 

This circuit is usually treated by writing a single equation 

VVVV 0  LRC   (3) 

V V0

L1 L2

r

RL I1 I2

V V0

L 

C 

R

I 
9 

where the voltage drops are given by 

 

0

1 

t

L RC

dI V L V RI V I d

dt C

    

This equation can be solved easily; however, from a numerical standpoint it has the problem that 

it has an integral in it, which means it is not a differential equation. Taking the derivative if 

equation 3 corrects this: 

2

0

2

dV d I dI 1 LR I

dt dt C dt

 

Although this is easily solved analytically, it is not first order and hence is problematic for 

numerical solution. A new choice of state variables fixes this: since the current is related to the 

charge on the capacitor by 

dQ I

dt 

we can write the following system of equations: 

0

1 1

dQ I

dt

dI R VI Q dt L L LC



 

We now have a state vector Qx

I

      

 to describe the circuit, and a vector function 

 

0

1 1

I

F x R VI Q

L L LC

        

 

which a simulator can accept. 

It is fortunate that this choice of state variables also works for the NLTL. The appendix contains 

the derivation of the set of network equations for the Brown-Ikezi circuit for an NLTL with an 

arbitrary number of stages. This set of equations has been coded into Mathematica, and seems to 

work well – circuits with over a thousand stages have been simulated without numerical 

problems. 

3.3 The Method of Partial Charges 

In writing the network equations for this circuit, it is useful to generalize the state variables 

discussed above. As discussed in the appendix, a natural way to treat the NLTL circuit is by loop 

currents, as shown in the circuit picture at the beginning of this report. Now, the discussion in the 
10 

previous section suggests that we introduce a “charge” qn associated with each loop current, 

which we define as 

n n

dq i

dt 

This identifies a set of state variables qq q q ii i i 1 2 12 , , , , and , , , ,   N L NL    for the 

circuit, which a differential-equation solver can handle easily. Once we have the solution to the 

network equations, we can get the node voltages from the obvious relation 

V V ri i n Cn n n    1 

where VCn is the voltage drop across the nonlinear capacitor. As discussed in Report #1, we write 

this voltage in the form 

1 3 V CQ Q Cn n n    

where the capacitor charge Qn = qn – qn+1 mimics the behavior of the shunt current. This is a true 

charge, arising from the flow of current into and out of the shunt branch. 

It is remarkable that when this decomposition of Qn into “partial charges” qn is made, 

preliminary simulation results reveal that the qn are unphysical even though their difference is 

not. When a NLTL is pulsed by a finite voltage signal, after a long time all the variables qn are 

nonzero and identical in value. This means that the currents and true charges on the capacitors 

must vanish, as they should in a dissipative network (i.e., one with resistors). This rather 

mysterious result has no impact on the modeling results, but it is an interesting consequence of 

the freedom to assign state variables to a nonlinear circuit. 

4. Experimental Fitting of Resistive Losses in the NLTL Based on Loss 

Tangent Data 

Accurate determination of losses in ferroelectric materials such as BST is extremely difficult, 

especially over a wide frequency range. Because such losses have a major impact on the 

performance of our NLTL, particular care was taken to identify a reasonable number for the 

single parameter we use to characterize these losses – the resistance r shown in the circuit (see 

figure 7). 
11 

Figure 7. Placement of resistors in Brown-Ikezi NLTL circuit. 

Neither Brown nor Ikezi provide any information about this number, although Brown suggests 

that he used loss-tangent data to obtain a value for r (note that he uses the circuit shown above, 

just as we do). Because loss-tangent data does not directly translate into r, a procedure was 

developed for connecting these numbers, which will be described. All the simulations performed 

will be based on numbers obtained from this procedure. 

Frequency dependent measurements of BST parameters usually are based on capacitance 

measurements made on fabricated capacitor structures. This procedure is typified by the work of 

Rimai et al., (9), who used capacitors to measure dielectric constant and loss tangent as a 

function of frequency. They fitted their data to the following functional form: 

 

2

00 0 2 2 1

t

j

j 

      

               

where the parameters t, 0,  and  are used to fit the data. The values of these parameters 

they found for their particular BST samples were: 

0

1020 GHz

1900

6

2.5

t

t



 

 

 

 

Now, the loss tangent is defined in most of the ferroelectric literature as follows: 

   

 

Im

tan

Re

 



 

RL

V0 V1 V2 V3

L L

C C C

i1 i2

L

i3

VN

L

iN

L

iL

VL

C

VN-1

RIN r r r r

VS
12 

so its values are obtained by finding 

      

   

   

2 2

0

0 2 2

2 2 22

0

0 2 2 2 22

2 2 2 2 22

0

0 2 2 2 22

1 1

Re Re

11 1

Re

11 1

t

t

t t

t

t t

t

j

j

j j



 

 

            

    

        

                             

           

                      

           

and likewise 

      

   

      

 

2 2

0

0 2 2

2 2 22

0

0 2 2 2 22

2 22 2

0

0 2 2 2 22

2

0

0

2 2

1 1

Im Im

11 1

Im

1 1 11

t

t

t t

t

t t

t

t

t

j

j

j j



 

 



            

    

        

                             

           

            

   

           

  

  

  

2 2 2    

Then the loss tangent is given by 

 

 

 

   

 

   

2

0

0 2 2 2 22

2 2 2 2 22

0

0 2 2 2 22

2

0

2 2 2 2 22

0

Im

tan

Re 11 1

11 1

t

t

t t

t

t

t t

 



 

  

 

           

                       

           

  

                
13 

The expressions above are valid from dc to near-IR frequencies, but for microwave applications 

they are useful primarily at low frequencies. It is therefore convenient to take the following limit: 

 

0

2

0

0 tan

1 t

        

  

The problem with this expression is that, unlike a true resistance, this quantity depends linearly 

on frequency. 

In order to make contact with our NLTL, we look at the capacitance of a BST layer with area A 

and thickness h and this kind of dielectric constant, and compute the corresponding impedance of 

the layer: 

      

    

2 2

0

0 2 2

2 2

2 2

0 0

1 1

1 1

1 1

t

t

t

t

j A A C

h h j

j h Z

j Cj A j





            

    

        

            

             

A low-frequency expansion of Z gives: 

 

 

   

2

2 2

2

0 0

2 2

0

2

2 2

00 0

0

2

00 0 00 0

1

1

1 1 1

1

1 1 1 1

1 1

1 1 1

1 11

t t

t t

t t

t t

j d Z

j A j

d j j A

d d j jA jA







       

    

                 

         

                                   

                                   

0

2 2

0

0

1

1

d

A

r

j C

 

 

 


14 

and thus it is clear that Z is the impedance of a capacitance C0 and a resistance r connected in 

series. This justifies our circuit topology, which is gratifying, but more importantly it gives us a 

formula for r: 

       

   

00 0

2 22 2

00 0 0 0 0

00 0

0

1

tan

1 1 1 1

1 1 lim tan 1 tan

1

tan

tt t

d r Zj A jC

r

C

 



            

          

     

   



 



This simple result almost gets us where we want to be. However, in order for r to be constant we 

need the linear frequency dependence of tan  alluded to above. The variation of tan  observed 

in experiment is not encouraging: the following plots (see figure 8) are given in the paper by 

Vorobiev and Gevorgian (10): 
15 

Figure 8. Loss tangent of a BST varactor vs. frequency at different 

temperatures without (a) and under 20 V dc bias (b). The 

dotted line represents the tan δ ~ ω1/3 power law. 
16 

These authors predict a peculiar dependence tan δ ~ ω

1/3 for the loss tangent, which is not very 

useful for our applications. Since the data used to make these plots was unavailable, the crude 

approach of connecting the points  = 0, tan  = 0, and  = 40 GHz, tan  = 0.07 with a straight 

line was used, and its slope, along with a value of C0 appropriate for our structure, was inserted 

in the formula 

0

tan

r

C

  

. Using the endpoint data tan = .07 at f = 40 GHz gives 

 

   

 

   

 

   

10 1 14 1

1

10 1 14 1 1

10 14 1

.07

2 4 10 s 8.85 10 f cm 5000

.07

1 f 1 Ohm -s

2 4 10 s 8.85 10 Ohm -s cm 5000

1 .07

 Ohm

cm 2 4 10 8.85 10 cm 5000

h

r

A

h

r

A

h

A

 



  

 

  

 

 

  

In all the simulations to date, Brown’s numbers have been used for the line dimensions. In his 

notation, A = wd = (2.5 cm)(.3cm) = .75 cm2

, h = .55 cm, which gives 

 

   

  

     

2 10 14 1

10 14

.55 cm .07

 Ohm

.75 cm 2 4 10 8.85 10 cm 5000

.07 .55

= Ohm .00046 Ohm

6.2832 .75 4 10 8.85 10 5000

r  



  

  

As better data is obtained, this approach can be refined. For now, however, the value obtained is 

suitable for simulations. 

5. Realistic Antenna Coupling Via an RC Circuit

The code written in Mathematica mimics that of both Brown and Ikezi, i.e., it is based on a 

simple series-parallel circuit model for the NLTL (see figure 9). 
17 

Figure 9. NLTL with simple resistive load. 

For the applications under study, this transmission line is excited by transient signals, i.e., pulses. 

Because pulse shapes are nonperiodic functions, there is a potential for a lossless transmission 

line to support signals with dc levels. However, a real antenna cannot pass a dc level, so that this 

problem is clearly mathematical rather than physical. To represent the antenna more realistically, 

a series capacitance can be included in the load, so the circuit becomes (see figure 10). 

Figure 10. NLTL with RC circuit load to simulate antenna reactance. 

The inductor voltage drops are related to the loop currents as stated previously. In particular, at 

the last stage of the transmission line we still have 

1

N

N N

di L VV

dt   

This last stage sees a voltage divider described by the equation 

   

     

1

1

1

1

1 1

1 11 3

1 11

N LN L N

N

N LN L N

N N N N LN L N

N L N N N N LN

V Ri C q

di L V Ri C q dt

q q r i i Ri C q

C q C C q q q ri r R i









 

 

 

 

 

     

       

RL

V0 V1 V2 V3

L L

C C C

i1 i2

L

i3

VN

L

iN

L

iL

VL

C

VN-1

RIN r r r r

VS

RL

V0 V1 V2 V3

L L

C C C

i1 i2

L

i3

VN-1

L

iN-1

L

iN

VN

C

VN-2

RIN r r r r

VS

CL

VL
18 

We can use these expressions to eliminate the voltages entirely, which gives us a set of network 

equations in terms of qn and in alone: 

     

    

    

    

  

1

1 1 2 12

1

21 1 2 1

1 11 1

1

1 1 2 12

1

21 1 2 1

2

2

2

2

S in

n n n n n n nn

n n n n n n nn

n nn n n n n n

N

N N NN N N

di L V t Ri q q ri i dt

di L q q q q ri i i dt

di L q q q q ri i i dt

di L q q q q ri i i dt

di L q q q q ri i i dt

    

  

    

    

     

      

      



      

      





 

      1 11 3

1 11

N

N

N L N N N N LN

di L C q C C q q q ri r R i dt

           

Then the set of network equations is completed by 

n n

N

N

dq i

dt

dq i

dt





and 

V Ri L LN 

It is no longer convenient to nondimensionalize these equations because there are two different 

capacitors in the system. 

6. Description of Mathematica Code

The circuit equations are easily converted to Mathematica form, which has a universal 

numerical differential equation solver called NDSolve. The only challenging problem about 

setting up these equations is to generate the code for an arbitrary number of layers. This can be 

done piecemeal by separating the equations into starting, intermediate, and ending segments. 

Each of these segments involves a set of equations, as shown above: 
19 

     

    

    

1

1 1 2 12

1

1

1

21 1 2 1

1

1

1 11 1

Starting segment: 

2

2

Intermediate segment: 

S in

n n n n n n nn

n n

n n n n n n nn

n n

di L V t Ri q q ri i dt

dq i

dt

di L q q q q ri i i dt

dq i

dt

di L q q q q ri i i dt

dq i

dt

    

 

  

       



  

      



      





    

     

1

1 1 2 12

1

1

1 11 3

1 11

2

Ending segment: 

n nn n n n n n

n n

N

N L N N N N LN

N

N

di L q q q q ri i i dt

dq i

dt

di L C q C C q q q ri r R i dt

dq i

dt

    

 

 

 





























        



 





         



  



The resulting code looks as follows: 

eqs=Table[{ cd[i]’[tau](1+LM (s[i1][tau]^2+s[i][tau]^2+s[i+1][tau]^2-s[i-1][tau]s[i][tau]-

s[i][tau]s[i+1][tau]-s[i-1][tau]s[i+1][tau]))(s[i-1][tau]-

2s[i][tau]+s[i+1][tau])+Qc (cd[i-1][tau]-

2cd[i][tau]+cd[i+1][tau]),s[i]’[tau]cd[i][tau]},{i,2,NN-1}]; 

eqst={cd[1]’[tau]U0[tau]-(s[1][tau]-s[2][tau])-LM (s[1][tau]-

s[2][tau])^3-Qc (cd[1][tau]-cd[2][tau]),s[1]’[tau]cd[1][tau]}; 

eqfn={cd[NN]’[tau](s[NN-1][tau]-s[NN][tau])+LM (s[NN-1][tau]-

s[NN][tau])^3-gamma s[NN][tau]+Qc cd[NN-1][tau]-

(Qc+QL)cd[NN][tau],s[NN]’[tau]cd[NN][tau]}; 

eqp=Prepend[eqs,eqst]; 

equ=Flatten[Append[eqp,eqfn]]; 

init=Flatten[Table[{s[i][0]0,cd[i][0]0},{i,1,NN}]]; 

NDSL=Flatten[Prepend[init,equ]]; 

vlist=Flatten[Table[{s[i],cd[i]},{i,NN}]]; 

sol=NDSolve[NDSL,vlist,{tau,0,500},MaxSteps->Infinity];
20 

In this code, the function Table generates an arbitrary number of intermediate equations, which 

describe the interior of the transmission line. This set of equations (the intermediate segment) is 

concatenated with the starting and ending segments, and the full set is solved by the numerical 

routine. The result is a list of currents and charges for the circuit elements, which can be 

examined and plotted. 

The translation of this mathematical routine into a MatLab code will be straightforward, 

provided that MatLab has utilities which can generate equation lists for arbitrary numbers of 

layers. Implementing this transfer will be an important deliverable when work on this project 

resumes. 

7. Results of Preliminary Simulations 

To date, the numerical code has been exercised in ways that bring out the physics of the 

frequency translation rather than parameter extraction; this focus lends credence to the 

correctness of the code results, which can be nonintuitive. Indeed, much of the value of 

numerical work lies in creating intuition where none existed before. 

Simulations were performed using a series of pulses based on Ikezi’s normalized trapezoidal 

pulse shape. I chose this pulse shape because it incorporates simply three numbers that 

characterize any pulse: a rise time, a pulse “width”, and a fall time. Thus, consider a pulse of 

almost triangular shape (see figure 11). 

Figure 11. Ikezi normalized short test pulse shape; time scale 

in NLTL low-frequency single-stage periods. 

5 10 15

0.2

0.4

0.6

0.8

1.0
21 

For a lossless NLTL, the time scale is measured by the period of the LC stages. In units of this 

period, the rise time of this pulse is 2 periods, its duration 1 period, and its fall time 5 periods. 

Let us choose L and C such that the period was 13 ns, and pulse the line with a 50 V signal, 

whose total pulse length is 104 ns. This low voltage should cause the line to respond like a linear 

filter. Plotting the time dependence of the voltage at the antenna end of the line gives (see figure 

12). 

Figure 12. Low voltage response to figure 11 test pulse at end of NLTL. 

The oscillatory tails following the pulse front exhibit typical dispersive behavior of the line. 

When the pulse voltage is increased to 6 kV, the following plot results (see figure 13), which 

reveals the evolution of solitons. 

Figure 13. High voltage response to figure 11 test pulse at end of NLTL; note solitons. 

500 1000 1500 2000

- 6000

- 4000

- 2000

2000

4000

6000

500 1000 1500 2000

- 40

- 20

20

40
22 

Increasing the 50 V pulse width to 10 periods (see figure 14). 

Figure 14. Ikezi normalized long test pulse shape. 

Gives a linear response of the form (see figure 15). 

Figure 15. Low voltage response to figure 14 test pulse at end of NLTL. 

The longer duration has separated the rise-time regions, producing spikes; these are not solitons, 

however, since the voltage is too low. Increasing the voltage to 5 kV shows multiple soliton 

generation. This suggests that long-duration pulses are better than short-duration ones for 

producing multiple solitons (see figure 16). 

500 1000 1500 2000

- 100

- 50

50

100

5 10 15 20 25 30

0.2

0.4

0.6

0.8

1.0
23 

Figure 16. High voltage response to figure 14 test pulse at end of NLTL; note solitons. 

8. Movies of NLTL Interior 

There are a number of features of these plots that require explanation. To this end, I have used a 

very handy feature of the Mathematica code to examine the interior of the NLTL as a pulse 

propagates within it. Because they exhibit only the output voltage, these plots treat the device as 

a “black box”, whose mysterious generation of alternating “bursts” of pulses is hard to 

understood. However, because the code simultaneously evaluates voltages and currents at every 

node, i.e., at every layer of the structure, these nodal quantities can be exhibited as “maps” of the 

voltage/current profile at each time step, which can be displayed as “movies” of the line interior. 

Multiple movies were made to examine the overall propagation of signals in the line. Two of 

these movies are included in this report, on a companion CDs for the hard copies; the data files 

they are based on are very large (~.5 Gb). Being able to examine the interior immediately reveals 

the origin of the bursts: the initial drive pulse propagates to the end of the line with a certain 

average velocity, disintegrating as it propagates. Once it reaches the end of the line, it is 

reflected. In the simulations I have made to date the antenna impedance is high, which makes the 

end of the line look like an open circuit. This causes the reflected signal to add to the incident 

signal, doubling the total amplitude. The first pulse burst marks the arrival of the distorted pulse 

at the end. The pulse then returns to the source, where the voltage is fixed, which makes the 

source look like a short circuit; this causes the reflected voltage to become negative. The 

negative reflected pulse then returns to the far end of the line, producing the negative-going 

second burst, which is once again doubled in magnitude. This cycle repeats until the pulse has 

given up all its energy to the load, in small increments because the load impedance is high. 

500 1000 1500 2000

- 10000

- 5000

5000

10000
24 

The first movie is of a long (120 ns) low-voltage (50 V) pulse, which changes shape as it 

propagates due purely to dispersion. Because the propagation velocity of a sinusoidal signal in a 

dispersive medium depends on frequency, the high-frequency components, which contribute to 

the rapid rise and fall times of the pulse, “dephase” during the propagation and produce the 

oscillations shown. Note that the frequency spectrum tends to redistribute energy to lower 

frequencies and not higher ones. The second movie is of a pulse with the same duration but much 

higher voltage (5 kV). The high-voltage pulse disintegrates into what appear to be solitons before 

reaching the end of the line. However, their shape is not stable – there is an unexpected 

oscillation in the soliton height. Compare these pulses with the superficially similar “ripples” in 

the low- voltage pulse, which are more or less stationary in shape. The origin of the high-voltage 

shape instability is unclear; while it is tempting to identify it with the “modulation instability” of 

soliton trains discussed in the literature, it could also be related to the discrete, i.e., “circuit”, 

nature of the model. Although this effect is small, and hence of no engineering importance, it is 

scientifically interesting, and cautionary given the tendency for any kind of pulse compression or 

amplification to be referred in the literature to as “soliton generation”. 

9. Systems Issues 

Although this modeling effort was preliminary, it is useful to bound the behavior of the NLTL 

based on optimal predictions of its performance. These predictions can only be based on the 

information available now, but they serve as a guide for future efforts. 

1. Maximum output power. Clearly we can use the breakdown field of the ferroelectric to 

estimate this quantity. In BST ceramics this field can approach 750 kV/cm. Assume that 

the line is a BST/alumina-filled parallel-plate waveguide, which propagates a TEM wave 

with electric and magnetic fields E and H  

 perpendicular and parallel to the conducting 

walls of the guide (top and bottom), respectively. At high power levels the dielectric 

constant of the BST saturates, dropping from 5500 for a very good sample to about 120. 

Then for a uniform line made of lossless material the characteristic impedance at large 

signal levels is 377 / 377 / 120 34.4 ohms ZC    . Since the power flux is given by 

the Poynting vector, the power flux at breakdown is 

   

2 12 5 1

2

ˆ 34.4 7.5 10 V/cm

16.3 GW/cm

n E H EH Z E       





 

Since the lines we fabricated had cross-sections of 2.5  .3 = .75 cm2

, one of our lines 

could handle ~12 GW. Note, however, that the inevitable impedance mismatch with the 

antenna due to the nonlinearity will reduce this, probably by about 50%, so ~6 GW is a 

more realistic number. 
25 

2. Maximum frequency: this is hard to specify without referring to geometric, i.e., 

manufacturing issues. The modeling we did was based on a formula Ikezi quotes in one of 

his General Atomics papers (Ikezi et al., Appl. Phys. Lett. 58, 986 (1991)), derived from 

simulation data. The structure he starts with is a parallel-plate waveguide with BST sheets 

in it, i.e., the one described above. His initial design numbers involve the linear modes of 

this structure, which is in essence a band-stop filter. As such, it allows frequencies to 

propagate that are below a certain maximum frequency – he calls it b – and blocks signals 

with higher frequencies within a “stop band”. His nonlinear simulations reveal that this 

maximum frequency also determines the oscillation frequency osc from the empirical 

formula 

osc 10

0

1.25 1.5log

b

E

E

  

   

  

where the field E is the maximum field in the line, and E0 is a material-dependent constant 

with the dimensions of field; for BST this number is around 10 kV/cm. This frequency b

depends mostly on the widths of the ferroelectric and dielectric slabs, and their linear 

dielectric constants. If we assume the ferroelectric is BST6-4, i.e., with 60% barium and 

40% strontium,  for his material is 5500 at room temperature, i.e., 20 °C (this is an 

optimistic number); at lower temperatures it is higher (the Curie temperature is 0 °C), but 

clearly room temperature is appropriate for our application. Given the volume constraints 

imposed by the latter, the optimal transmission line length is probably about 15 cm. To 

increase b, we want to make the layers as thin as possible. However, we found that layers 

thinner than .1 mm are too fragile to handle, so let us assume the ferroelectric layers are .1 

mm thick. If we have 15 cm to fit the transmission line into the weapon, and we want 100 

layers of each kind of material, this implies a total thickness of ferroelectric of 1 cm, 

leaving 14 cm for dielectric, or 1.4 mm per layer. Then using the formula above along with 

Ikezi’s linear-design formula (equation 3 of the Applied Physics paper mentioned above), 

we find the resulting bandstop filter corresponds to an osc of 3.4 GHz. 

3. Pulse repetition frequency. This can be estimated based on the observation that in this type 

of line, impedance mismatch with the antenna is inevitable due to the nonlinearity. This in 

turn gives rise to reflections, which return to the source and interfere with subsequent 

pulses. The safe thing to do is to wait until all these have died away. If we define “died 

away” to mean that only 1% of the original power is left after N reflections, and the 

reflected power is 50% = .5 of the incident power for each reflection, we are saying that 

(.5)N

 = .01, or N = between 6 and 7. Now, the maximum propagation velocity of a pulse in 

the transmission line is the velocity of light for the reduced (saturated) high-field dielectric 

constant, which is 120 for BST at high fields. This velocity is 2.7  109

 cm/s, so that for a 

15 cm line we need 7 round trips or a distance of 210 cm, which takes 7.7  10–8 s = 77 ns. 

Then the PRF is the reciprocal of this, or 13 MHz. 
26 

Now, suppose we have a train of 30 pulses coming at a rate of 3.4 GHz. Then the time 

between pulses is .29 ns and hence, the total duration of the pulse train is 30  .29 = 8.8 ns. 

This implies a duty cycle of 11%. Then the time it takes for the first pulse to hit the antenna 

is 5.6 ns, and the time to return to the source is 11.1 ns. Since this is greater than 8.8 ns, all 

the pulses in the train will be transmitted to the antenna before the first reflected pulse 

returns to the source. Since the power is 6 GW, over a period of 8.8 ns the pulse train 

delivers 53 J to be radiated by the antenna. If the pulser can supply a kilojoule of energy, 

19 of these pulse trains can be generated for a total dwell time of 1.46 s. 

Summary: 

Max power: 6 GW 

Max frequency 3.4 GHz 

PRF 13 MHz 

Time between pulses 290 ps 

No. of cycles of 3.4 GHz/train 30 

Duration of pulse train 8.8 ns 

Energy/pulse 53 J 

No. of pulse trains in dwell 19 

Dwell time 1.46 s 

10. Conclusions 

The results of this work validate the previous work of Ikezi and, to a lesser extent, Brown, and 

show that we now have a useful numerical tool for studying NLTL and guiding efforts to 

fabricate them. Future work will center on parameter studies for designing and optimizing the 

performance of these devices as HPM sources. 
27 

11. References 

1. Gonzalez, G., Microwave Transistor Amplifiers; Prentice Hall Inc.: Englewood Cliffs, NJ, 

1984. 

2. Qiu, J. X.; Judy, D.; Pulskamp, J. S.; Polcawich, R. G.; Kaul, R.; Crowne, F., 

Characterization of Nonlinear Behavior in a Tunable Phase Shifter Using Ferroelectric PZT 

Thin-Film Capacitors and Its Effect on System Performance. Microwave Symposium Digest, 

2009, MTT ’09. IEEE MTT-S International 7-12 June 2009, 341. 

3. Landauer, R. IBM Journal 1960, 4, 391. 

3a. An early description of such a transmission line can be found in R. V. Khokhlov, 

Radiotekhnika i Elektronika 1961, 6, 917. 

3b. Champlin, K. S.; Singh, D. R. IEEE Trans. Microwave Theory and Techniques 1986, MTT34, 351; Rodwell, M.J.W.; Bloom, S. M.; Auld, B. A. Electronics Letters 1987, 23, 110; 

Rodwell, M.J.W.; Madden, C. J.; Khuri-Yakub, B. T.; Bloom, D. M.; Pao, Y. C.; Gabriel, N. 

S.; Swierkowski, S. P. Electronics Letters 1988, 24, 101; Madden, C. J.; Rodwell, M.J.W.; 

Marsland, R. A.; Bloom, D. M. IEEE Electron Device Letters 1988, 9, 303; Yu, R. Y.; Case, 

M.; Kamegawa, M.; Sundaram, M.; Rodwell, M.J.W.; Gossard, A. W. Electronics Letters

1990, 26, 951; Marsland, R. A.; Sharouri, M. S.; Bloom, D. M. Electronics Letters 1990, 26, 

1237; Carman, E.; Giboney, K.; Case, M.; Kamegawa, M.; Yu, Ruai; Abe, K.; Rodwell, 

M.J.W.; Franklin, J. IEEE Microwave and Guided Wave Letters 1991, 1, 28. 

4. Ikezi, H. J. Appl. Phys. 1988, 64, 3273; Ikezi, H.; Wojtowicz, S. S.; Waltz, R. E.; DeGrassie, 

J. S.; Baker, D. R. J. Appl. Phys. 1988, 64, 3277; Ikezi, H.; Lin-Liu, Y. R.; Ohkawa, T.; 

DeGrassie, J. S. J. Appl. Phys. 1988, 64, 4717; Ikezi, H.; Wojtowicz, S. S.; Waltz, R. E.; 

Baker, D. R. J. Appl. Phys. 1988, 64, 6836. 

5. DeGrassie, J. Soliton Microwave Generator Final Technical report. prepared for the U. S. 

Department of Energy under Contract DE-AC03-87SF17119, New and Innovative Concepts 

Program in Support of the Strategic Defense Initiative, Sept. 1987 – Nov. 1989. 

6. Ikezi, H.; DeGrassie, J. S.; Drake, J. Appl. Phys. Lett. 1991, 58, 986. 

7. Brown, M. High Voltage Soliton Production in Nonlinear Transmission Lines and Other 

Pulsed Power Applications. Thesis, Oriel College, Oxford Univ., England, Sept. 1997. 

8. Devonshire, A. F. Adv. Phys. 1954, 3, 85. 

9. Rimai, L.; Parsons, J. L.; Hickmott, J. T.; Nakamura, T. Phys. Rev. 1968, 168, 623. 
28 

10. Vorobiev, A.; Gevorgian, S. Microwave Performance of Thin Film Ferroelectric Varactors 

in the Wide Temperature Range of −223 °C to +227 °C. Journal of the European Ceramic 

Society 2007, 27 (13–15), 3847–3850. 
29 

Appendix. Brown-Ikezi Network Equations 

The circuit is defined using Kirchhoff’s law in the usual way, based on loop currents. Source and 

load resistances are introduced to simulate coupling to the PFN and antenna (see figure A-1). 

Figure A-1. Brown-Ikezi NLTL circuit. 

The inductor voltage drops are related to the loop currents as follows: 

  1

1 1

1

2 1

1

1

1

S in

n n n

n n n

n n n

L

N L

di L V t Ri V

dt

di L VV

dt

di L VV

dt

di L VV

dt

di L VV

dt

  



 

  

 

 

 

 





For the shunt branches we introduce the nonlinear function   1 3 Q CQ Q     , where C is the 

linear term in the capacitance and  measures the circuit nonlinearity. As discussed in the text, 

we introduce “partial charges” qn related to the physical currents by n n

dq i

dt  . This gives the 

following expressions for the node voltages: 

RL

V0 V1 V2 V3

L L

C C C

i1 i2

L

i3

VN

L

iN

L

iL

VL

C

VN-1

RIN r r r r

VS
30 

 

   

   

   

   

   

0 1

1 1 2 12

11 1

1 1

1 1 2 12

 

 

S in

n n n nn

n n n nn

n n n nn

N N L NL

L LL

V V t Ri

V q q ri i

V q q ri i

V q q ri i

V q q ri i

V q q ri i

V Ri

 

 

   

 

   

   

   

   

   







We can use these expressions to eliminate the voltages entirely, which gives us a set of network 

equations in terms of qn and in alone: 

     

    

    

    

    

1

1 1 2 12

1

21 1 2 1

1 11 1

1

1 1 12

1 1

2

2

2

2

S in

n n n n n n nn

n n n n n n nn

n nn n L n n n

N

N N N L N NL

L

di L V t Ri q q ri i dt

di L q q q q ri i i dt

di L q q q q ri i i dt

di L q q q q ri i i dt

di L q q q q ri i i dt

di L q dt

    

  

   

 

     

      

      



      

      

 





  NL N L L   q ri R r i  

It is useful to further reduce the terms involving the function Q as follows: 

   

 

 

1 31 3

1 11 1 1 1

1 33

1 11 1

12 2

1 1 1 11 1

1 22 2

1 1

{ }{ } { }{ }

2 { }{ }

({ } { }{ } { } ) 2

{

n n nn n n n nn nn

n nn n n nn

n n n nnn nn n nn

n nn

q q qq Cq q q q Cqq qq n

Cq qq q q qq

C q q q qqq qq q qq

C q qq

 

 





 

    



  



    



 

           

         

            

  



q q qq q q q q q n n nn n n n n n      1 1 11 1 1 } 2          
31 

which we rewrite as follows: 

   

   

      

22 2

1 1 1 1 1 1 11

2

01 1

1

1 1 1 11 1

,, 1

, 1

,, 2

n n n n n n n n nn n n

nn n n

n n n n n nn n n n

q q q C q q q q q qq q q

q q Cq q

q q q q C q qq q q q

      

 



    

       

   

        

Then the final set of network equations is 

n n

dq i

dt 

and 

      

    

    

   

1 1

1 012 1 2 1 2

1 1

21 2 1 2 1

1

1 11 1 1 1

1 1

12 1 2 1

,

,, 2 2

,, 2 2

,, 2 2

S in

n n nn n n n n n n

n n nn n n n n n n

n nn n n n n n n n

di L V t Ri C q q q q ri i dt

di L C q q q q q q ri i i dt

di L C q q q q q q ri i i dt

di L C q q q q q q ri i i dt



 

    



    

 

    

    

    

    

     











 

    

     

2

1

11 1

0

,, 2 2

,

N

N NL N N L N N L

L

NL N L N L L

di L C q q q q q q ri i i dt

di L q q q q ri R r i dt

      

    







where the state variables are qq q q ii i i 1 2 12 , , , , and , , , ,   N L NL   
32 

Dimensionless form: let 2

0

1 , , , and C in in L L rC R C R C

LC

      . Then 

      

    

    

   

2 1

0 1 012 1 2 1 2

2 1

0 21 2 1 2 1

2

0 1 11 1 1 1

2 1

0 12 1 2

 ,

,, 2 2

,, 2 2

,, 2 2

S in C

n n n n n n n Cn n n

n n nn n n n Cn n n

n n n n n n n Cn

di CV t R C i q q q q i i dt

di

q qq q q q i ii

dt

di q qq q q q i i i dt

di qq q q q q i i dt



      



    

    

      

      

      

     











 

    

    

1 2

2

0 11 1

2

0 0

,, 2 2 

,

n n

N

N N L N N L CN N L

L

N L N L CN L C L

i

di q qq q q q i i i dt

di qq q q i i dt

 



 





      

      







Setting 00 0 0 0 0 0 , , , C C in in in L L L              t rC Q R C Q R C Q gives 

       

    

    

    

1

0 012 0 1 2 1 2

1

21 0 2 1 2 1

1 10 1 1 1 1

1

120 1 2 1 2

,

,, 2 2

,, 2 2

,, 2 2

S in C

n n n n n n n Cn n n

n n nn n n n Cn n n

n n n n n n n Cn n n

di CV q q q q Q Q i Q i C d

di

q qq q q q Qi i i d

di q qq q q q Q i i i d

di qq q q q q Q i i i d

     

    

     

       



      



      



      













    

    

1 01 1

0 0

,, 2 2

,

N

N NL N N L N N L

L

N L N L CN L C L

di q qq q q q Q i i i C d

di q q q q Qi Q Q i d

         

     









Finally, setting V t V q t CV i t CV S m n mn n m n       , ,         0   gives 
33 

       

    

    

1

0 0 012 0 1 2 0 2 0 1

1

0 21 0 2 1 0 2 1

0 1 10 1 1 0 1 1

0

,

,, 2 2

,, 2 2

m m m in m in C m

n m n n n mn n n C m n n n

m n nn m n n n C m n n n

m

d CV CV t CV Q CV Q Q CV d

d CV CV Q CV d

d

n CV CV Q CV d

d CV

     

    

                



                



                



 









    

    

    

1

120 1 2 0 1 2

0 1 01 0 1

0 00 0 0

,, 2 2

,, 2 2

,

n nn n m n n n C m n n n

N

m N NL mN N L C m N N L

L

m N L m N L C mN L C mL

CV Q CV d

d CV CV Q CV d

d CV CV Q CV Q Q CV d

     

 

              



                



              











or 

       

    

    

    

1

012 1 2 2 1

1

21 2 1 2 1

1 11 1 1 1

1

12 1 2 1 2

,

,, 2 2

,, 2 2

,, 2 2

in in C

n n n n n n n Cn n n

n n nn n n n C n n n

n nn n n n n C n n n

N

N

d Q QQ d

d Q d

d Q d

d Q d

d

d

     

    

     



            



             



             



             



  















    

    

11 1

0

,, 2 2

,

NL N N L C N N L

L

N L N L CN L C L

Q

d Q QQ d

            

          







where 

   

   

22 2

1 1 1 1 1 1 11

2

01 1

,, 1

, 1

n n n n n n n n nn n n

nn n n q q

      

 

              

    

and 3 2 C VS measures the system nonlinearity. 
34 

NO. OF 

COPIES ORGANIZATION 

 1 ADMNSTR 

 ELEC DEFNS TECHL INFO CTR 

 ATTN DTIC OCP 

 8725 JOHN J KINGMAN RD STE 0944 

 FT BELVOIR VA 22060-6218 

 1 HC DARPA 

 ATTN IXO S WELBY 

 3701 N FAIRFAX DR 

 ARLINGTON VA 22203-1714 

 1 CD OFC OF THE SECY OF DEFNS 

 ATTN ODDRE (R&AT) 

 THE PENTAGON 

 WASHINGTON DC 20301-3080 

 1 HC US ARMY RSRCH DEV AND ENGRG 

 CMND 

 ARMAMENT RSRCH DEV & ENGRG 

 CTR ARMAMENT ENGRG & 

 TECHNLGY CTR 

 ATTN AMSRD AAR AEF T J MATTS 

 BLDG 305 

 ABERDEEN PROVING GROUND MD 

 21005-5001 

 1 HC PM TIMS, PROFILER (MMS-P) 

 AN/TMQ-52 

 ATTN B GRIFFIES 

 BUILDING 563 

 FT MONMOUTH NJ 07703 

 1 HC US ARMY ARDEC 

 ATTN AMSRD AAR AEE P 

 C HAINES 

 BLDG 25 

 PICATINNY ARSENAL NJ 07806-5000 

 1 HC US ARMY ARDEC 

 ATTN AMSRD AAR AEM C 

 A MARSTON 

 BLDG 61S 

 PICATINNY ARSENAL NJ 07806-5000 

 3 HCS US ARMY ARDEC 

 ATTN AMSRD AAR AEM L 

 S GILMAN 

 ATTN AMSRD AAR AEP F 

 O NGUYEN 

 ATTN AMSRD AAR AEP F S SHRI 

 BLDG 65S 

 PICATINNY ARSENAL NJ 07806-5000 

 1 HC US ARMY INFO SYS ENGRG CMND 

 ATTN AMSEL IE TD A RIVERA 

 FT HUACHUCA AZ 85613-5300 

NO. OF 

COPIES ORGANIZATION 

 1 HC COMMANDER 

 US ARMY RDECOM 

 ATTN AMSRD AMR 

 W C MCCORKLE 

 5400 FOWLER RD 

 REDSTONE ARSENAL AL 35898-5000 

 1 HC US GOVERNMENT PRINT OFF 

 DEPOSITORY RECEIVING SECTION 

 ATTN MAIL STOP IDAD J TATE 

 732 NORTH CAPITOL ST NW 

 WASHINGTON DC 20402 

 1 HC DEPT OF PHYSICS AND GEOLOGY 

 UNIV OF TEXAS - PAN AMERICAN 

 ATTN S TIDROW 

 1201 W UNIVERSITY DR 

 EDINBURG TX 78541 

 1 HC US ARMY RSRCH LAB 

 ATTN RDRL CIM G T LANDFRIED 

 BLDG 4600 

 ABERDEEN PROVING GROUND MD 

 21005-5066 

25 HCS US ARMY RSRCH LAB 

 ATTN IMNE ALC HRR 

 MAIL & RECORDS MGMT 

 ATTN RDRL CIM L TECHL LIB 

 ATTN RDRL CIM P TECHL PUB 

 ATTN RDRL SE J PELLEGRINO 

 ATTN RDRL SED E D KATSIS 

 ATTN RDRL SED E S HENRIQUEZ 

 ATTN RDRL SED M J TATUM 

 ATTN RDRL SED P K JONES 

 ATTN RDRL SEE I S SVENSSON 

 ATTN RDRL SEE O W M GOLDING 

 ATTN RDRL SER E E VIVEIROS 

 ATTN RDRL SER E F CROWNE 

 (5 COPIES) 

 ATTN RDRL SER E R DEL ROSARIO 

 ATTN RDRL SER J MAIT 

 ATTN RDRL SER L D POTREPKA 

 ATTN RDRL SER L J PULSKAMP 

 ATTN RDRL SER L M DUBEY 

 ATTN RDRL SER L R POLCAWICH 

 ATTN RDRL SER P AMIRTHARAJ 

 ATTN RDRL SER U C FAZI 

 ATTN RDRL SES M PATTERSON 

 ADELPHI MD 20783-1197 

TOTAL: 40 (1 ELEC, 1 CD, 38 HCS)