generalized model for magnetically insulated transmission line flow

2708 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 36, NO. 5, OCTOBER 2008

Generalized Model for MagneticallyInsulated Transmission Line Flow

Paul F. Ottinger, Senior Member, IEEE, Joseph W. Schumer, Member, IEEE,David D. Hinshelwood, and Raymond J. Allen, Member, IEEE

Abstract—A generalized fluid model is developed for electronflow in a magnetically insulated transmission line (MITL). By in-cluding electron pressure in the fluid model and allowing nonzerovalues of the electric field at the cathode, the model can treatboth emission and retrapping of flow electrons. For the first time,a direct derivation of the space-charge correction term in theflow equations is also obtained by identifying a new condition atthe boundary of the electron layer. Also, a free parameter in themodel is chosen so that previously derived MITL flow equationsare recovered when the electric field at the cathode is taken tozero; consequently, recent equilibrium MITL rescaling results stillapply. Generalized MITL flow equations are derived from themodel and solutions presented. These new equations form the basisfor a description of the dynamic MITL flow.

Index Terms—Electron emission, electron pressure, flow im-pedance, fluid model, magnetically insulated transmission line(MITL), power flow.

I. INTRODUCTION

MANY modern pulsed power generators use magneti-cally insulated transmission lines (MITLs) to couple

the power between the driver and the load [1]. In an MITL,the electric field stress on the cathode exceeds the vacuumexplosive-emission threshold, and electron emission occurs.For sufficiently high current, emitted electrons are magneticallyinsulated from crossing the anode–cathode gap and flow axiallydownstream in the direction of the power flow, as illustratedin Fig. 1. The return current from total anode current Ia isdivided between current Ic flowing in the cathode and currentIf flowing in the vacuum electron layer, i.e., If = Ia − Ic. As aresult of the electron flow in vacuum between the electrodes, theimpedance of the MITL is altered, and, thus, the power couplingbetween the machine and the load changes. For equilibriumflow, it has been shown that the effective impedance of theMITL is best described by the flow impedance Zf [2], [3]. In adynamic system, where the voltage and the current are changingin time (e.g., due to a finite duration power pulse and/or a time-varying load impedance), the flow impedance also varies intime along the line [4].

Manuscript received September 28, 2007; revised March 14, 2008. Currentversion published November 14, 2008. This work was supported by the U.S.Department of Energy through Sandia National Laboratories under ContractDE-AC04-94AL-85000.

The authors are with the Plasma Physics Division, Naval ResearchLaboratory, Washington, DC 20375 USA (e-mail: [email protected];[email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPS.2008.2004221

Fig. 1. Schematic of MITL flow in negative polarity illustrated here inplanar geometry for a transmission line element of length �, width w, andanode–cathode gap d. The equivalent anode and cathode radii in cylindricalgeometry would be ra and rc, respectively. The voltage across the gap is V ,and the anode and cathode currents are Ia and Ic, respectively. Vs is the voltageat the edge of the electron flow layer whose thickness is ds. Not shown, butimplied by this illustration, are the generator to the left and the load to the right.

In this paper, a new generalized model for MITL flow isdeveloped for incorporation into the transmission line code(TLC) Bertha [5] to treat dynamic MITL problems. The modeldescribes the self-limited flow, as the pulse initially propagatesdown the MITL toward the load, and the subsequent electronpower flow along the MITL after the pulse encounters the load.To accomplish this, the model must treat electron emission atthe pulse front and at impedance transitions along the MITLwhere required. For low impedance loads, this description mustalso include electron retrapping [6], as the flow is modifiedby the wave reflection off the load, and the percentage of thereturn current in vacuum electron flow decreases. The goalis to develop an analytic model that provides a reasonablerepresentation of what is observed in particle-in-cell (PIC)simulations of MITL flow. Simulation features of interest thatare represented in the model include a finite thickness flowlayer that extends from the cathode to the electron layer edge,as well as a distribution of various electron orbits within thatlayer, which is indicative of transverse temperature or pressurein the center of the flow layer. The ultimate objective of thispaper is to use this model in a TLC code or a circuit code toefficiently and accurately simulate power MITL flow, so thata more computationally intensive PIC code treatment is notrequired.

To this end, a new fluid model for the electron flow layerincluding pressure is developed to extend the basic MITL flowmodel, allowing solutions with nonzero electric field at thecathode (i.e., Ec �= 0) and, therefore, the treatment of electronemission and retrapping. The basic MITL flow model refers tothe case with Ec = 0 [2], whereas the model developed here re-moving this restriction will be called the generalized MITL flowmodel. Three additional issues are also addressed here. First,the space-charge term in the previous basic models results fromchoosing a scaling for the plasma density rather than deriving

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OTTINGER et al.: GENERALIZED MODEL FOR MAGNETICALLY INSULATED TRANSMISSION LINE FLOW 2709

one. By introducing an additional new condition that the flowvelocity at the electron layer edge can be related to the voltagethere, the space-charge term in the MITL flow equations is nowderived directly. Second, the predictive capability of the voltageequation from the basic model is not accurate in the region ofself-limited and saturated flow. This issue has been recentlyresolved for MITL flow with Ec = 0 by rescaling the basicMITL model [3]. By appropriately choosing the magnitude ofthe pressure term in this generalized MITL model, the resultsof the newly rescaled basic MITL theory can be applied here aswell, and the accuracy of the voltage predictions is preserved.In fact, the new generalized model presented here may be bestunderstood after reviewing [3]. Third, the impedance that bestdescribes the MITL in the generalized model, where Ec is notzero, is investigated. The basic model introduces the electricflow impedance Zf and the magnetic flow impedance Zm asmeasures of the distances of the centroid of the charge andthe centroid of the current in the electron flow layer from theanode, respectively [2]. The definitions of the centroid of thecharge and the centroid of the current are discussed below.By assuming in the fluid model developed here that the elec-tron charge density is uniform across the electron flow layer,and that the pressure has a parabolic profile, these two flowimpedances can be calculated and compared quantitatively.Although it is found that Zm < Zf , their difference is usuallysmall. However, it is shown that the capacitive impedance Zcap

and the inductive impedance Zind described herein are moreappropriate than Zf and Zm for describing the MITL in thegeneralized model with Ec �= 0. When Ec = 0, Zcap = Zf ,and the previous result from the basic model is recovered. In thegeneral case, Zind > Zcap, but, again, their difference is small,and Zcap best describes the MITL impedance.

To build a dynamic model for MITL flow in a TLC, thisnew generalized MITL flow model can be combined with time-dependent field equations [4]. However, to complete the TLCmodel, a robust numerical technique must be constructed forsolving these new MITL flow equations, and techniques mustbe developed to treat the emission front, adders, load coupling,nonemitting regions, etc. In this paper, the new generalizedMITL flow model is presented in Section II, and its connec-tion to the previous basic MITL flow model is discussed inSection III. Example solutions of these generalized MITL flowequations are then given in Section IV, and the conclusions thatcan be drawn from this paper are given in Section V.

II. GENERALIZED MITL FLOW MODEL

Negative polarity MITL flow is illustrated in Fig. 1, where ra

and rc denote the anode and cathode radii in cylindrical geom-etry. For simplicity, the model is developed in planar geometryconsidering a transmission line element with anode–cathodegap d, length �, and width w. As shown in [2], these results canbe easily extended to cylindrical geometry. In what follows, thesubscripts c, s, and a will denote quantities that are evaluatedon the cathode at x = 0, the edge of the electron flow layerat x = ds, and the anode at x = d, respectively. The electronflow layer extends from the cathode to its edge at ds andis described by a single-fluid model. The three fundamental

equations for this 1-D model are the momentum transportequation, Poisson’s equation, and Ampere’s law, which aregiven in meter–kilogram–second units by

∂T

∂x= ρ(E − vzB) (1)

∂E

∂x=

ρ

ε0(2)

∂B

∂x=μ0ρvz (3)

and only variations in x across the layer are considered. Here,T is the xx component of the electron pressure tensor, ρ isthe electron charge density, vz is the electron fluid velocityin the z-direction, E is the electric field in the x-direction, Bis the magnetic field in the y-direction, ε0 is the permittivity offree space, and μ0 is the permeability of free space. Whereasthe general directions of �E, �B, and �vz are shown in Fig. 1,as expressed in (1)–(3), the quantities ρ, vz , T , E, and Bare generally all positive (i.e., E = −Ex, B = −By , and ρ =ene > 0, where e is the electron charge, and ne is the electrondensity). Note, however, that the freedom for E to becomenegative near the cathode is retained in the model, and vz canalso be negative near the cathode. The vacuum impedance of theline is given by Z0 = cμ0d/w, where c = 1/(ε0μ0)1/2 is thespeed of light. Also, Ea,c = Z0cQa,c/d and Ba,c = Z0Ia,c/cd,where Qa,c and Ia,c are the charge per unit length and thecurrent (on the anode and the cathode), respectively.

Because the time derivatives have been assumed negligiblein (1)–(3), this is a quasi-equilibrium model. It is assumed thatthe electrons quickly react and come to equilibrium on a fasttime scale, so that τωpe � 1 and τωce � 1, where τ is the timescale for voltage and current variations, and ωpe and ωce are theelectron plasma and cyclotron frequencies, respectively. Thus,as a dynamic model is developed for MITL flow, the voltageand the current are dynamically treated, whereas the electronsare assumed to be in quasi-equilibrium at all times. In the basicMITL flow model, it is assumed that Ec = 0 and, althoughgenerally not true, that this implies no emission (or retrapping)occurs. In the generalized MITL flow model developed here,Ec is allowed to be nonzero, so that emission and retrappingcan be treated in a dynamic MITL model.

Two additional quantities need to be defined. The voltage inthe gap is given by

V (x) =

x∫0

E(x′)dx′ (4)

where Vc = V (0) = 0, and the full voltage across the gap isV = V (d) (with the subscript a suppressed for simplicity).Similarly, the axial component of the magnetic vector potentialin the gap is given by

A(x) =

x∫0

B(x′)dx′ (5)

where Ac = A(0) = 0, and the vector potential at the anode isA = A(d) (again, with the subscript a suppressed). Although Awas not required for the basic MITL flow model, V and A are


necessary to complete the description of the generalized MITLflow model for application in a dynamic MITL model.

Solving (1) for the axial fluid velocity yields

vz =E

B− ∂T/∂x

ρB(6)

which shows that vz is equal to the E × B drift velocity whenthe pressure gradient is zero. Substituting this expression for vz

into (3) and using (2) to substitute for ρ give

∂(B2)∂x

=1c2

∂(E2)∂x

− 2μ0∂T

∂x(7)

which can be integrated to obtain the generalized pressurebalance equation

B2(x) = B2c +

E2(x) − E2c

c2− 2μ0 (T (x) − Tc) . (8)

Because electron space charge and current are distributed inthe line, there is not, in general, a well-defined wave impedance.The electric flow impedance Zf and the magnetic flow im-pedance Zm have been defined by the distances of the centroidof the charge and the centroid of the current from the anode,respectively [2]. Thus, they depend on the functional forms ofρ(x) and vz(x) and will be formally defined in (31) and (32).By assuming in the model presented here that the charge densityis uniform in the flow layer, the electric flow impedance Zf iseasily calculated [see the definition of Zf given in (31)], withthe centroid of the charge located at a distance of d − ds/2 fromthe anode and

Zf = Z0

(1 − ds

2d

). (9)

Therefore, Z0 ≥ Zf ≥ Z0/2 for electron flow layer thicknessds that ranges from 0 to d. Magnetic flow impedance Zm willalso be calculated below and shown to be smaller than Zf .

The electron pressure is due to the motion that is perpen-dicular to the axial flow, which results from the distributionof electrons with various orbit types [7], as observed in PICsimulations of MITL flow. Hence, this generalized MITL flowmodel significantly differs from the parapotential flow model[8], where all electrons axially move in straight-line orbits and,consequently, T (x) = 0. For the model here, the pressure isassumed to have a parabolic profile with

T (x) =

⎧⎨⎩ 4Tm

(xds

− x2

d2s

)for 0 ≤ x ≤ ds

0 for ds ≤ x ≤ d

(10)

where the pressure has a maximum of Tm at x = ds/2, andT (0) = T (ds) = 0. The assumption that T (0) = 0 is reason-able because electrons are born on the cathode with zerokinetic energy, and the assumption that T (ds) = 0 is reasonablebecause, by definition, electrons at the layer edge must havezero velocity in the x-direction. Also, the pressure is maximumnear midgap, where electrons of various orbit types cross, halfmoving outward and half moving inward. Because the gradientin the pressure is positive near the cathode and negative near

the electron layer edge, (6) shows that it subtracts from theE × B drift near the cathode and adds to the E × B drift nearthe electron layer edge. Thus, under some circumstances (i.e.,for small or negative Ec), electron current density ρvz near thecathode can be negative, and the magnetic field can decreasebefore increasing further out into the layer.

The assumption of uniform ρ makes the integration of (2)straightforward, yielding

E(x) ={

Ec + ρxε0

for 0 ≤ x ≤ ds

Ea for ds < x ≤ d.(11)

Substituting (10) and (11) into (8) gives

B(x)=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

[B2

c +(

2Ecρds

ε0c2 − 8μ0Tm

)xds

+(

ρ2d2s

ε20c

2 + 8μ0Tm

)x2

d2s

]1/2

for 0 ≤ x ≤ ds

Ba for ds < x ≤ d.(12)

In this planar geometry, E(x) and B(x) are constant in theregion between the edge of the electron flow layer and theanode. Demanding that the electric and magnetic fields arecontinuous at the edge of the electron flow layer at x = ds

implies that

Ea = Ec +ρds

ε0(13)

and consequently

Ba =(

B2c − E2

c

c2+

E2a

c2

)1/2

. (14)

This last expression is the pressure balance equation and couldhave been directly obtained by evaluating (8) at x = ds withTc = T (0) = 0 and T (ds) = 0.

The voltage is obtained by integrating (4) from x = 0 to x =d and using the boundary conditions for V (x) defined above,yielding

V = Ecd + (Ea − Ec)(

d − ds

2

)(15)

where (13) has been used to substitute for ρds. This result willbe used later to show that the relationship between ds and flowimpedance Zf given in (9) is valid for this model.

Expressions for vz(x) and flow current density Jf (x) =ρvz(x) in the electron layer are obtained by substituting (11)and the derivative of (10) into (6), yielding

vz(x) =[Ec +

ρx

ε0− 4Tm

ρds

(1 − 2x

ds

)]/B(x) = Jf (x)/μ0ρ

(16)

where B(x) is given by (12). Note that at the cathode

vzc =(

Ec −4Tm

ρds

)/Bc = Jfc/μ0ρ (17)


can be either positive or negative depending on the values ofEc and Tm, whereas at the electron layer edge, vzs is alwayspositive with

vzs =(

Ea +4Tm

ρds

)/Ba = Jfs/μ0ρ. (18)

Because vz(x) monotonically increases with x and the chargedensity ρ is uniform, the centroid of the current will be foundlater to be closer to the anode than the centroid of the charge.Thus, the magnetic flow impedance Zm is smaller than theelectric flow impedance Zf .

There are a total of 14 variables in the model, namely, Ea,Es, Ec, Ba, Bs, Bc, Zf , V , A, ds, ρ, vzc, vzs, and Tm.Magnetic flow impedance Zm will also be calculated below,but is not part of the model. As will be discussed below, thesolutions will be parameterized by three of these variables,which will be determined by the TLC. However, there are onlynine independent relationships, namely, (11) and (12) evaluatedat x = ds for Es and Bs, (13) and (14) for Ea and Ba, (4) [or(15)] and (5) evaluated at x = d for V and A, (9) for Zf , and(17) and (18) for vzc and vzs. Thus, two conditions are requiredto close the solution.

First, a new condition is applied at the electron layer edge.It is assumed that the axial flow velocity of the electrons at thelayer edge can be related to the voltage there. Consequently, thisnew condition also assumes that there is negligible motion inthe y-direction (or azimuthal direction in cylindrical geometry)at the layer edge, there is negligible spread in the axial velocityof the electrons at the layer edge, and electrons at the layer edgeare all born on the cathode with negligible kinetic energy.

Second, the value of Tm will be chosen to fit the PICsimulation results presented in [3] for the case where Ec = 0. Inthis case, Tm can be related to the scaling factor g used in [3]. Itis important to note that although Tm will be fit to equilibriumsimulation results, which are associated with the basic MITLmodel [3], the results are assumed to extend to the generalizedmodel developed here. The validity of this assumption will needto be tested in the future by comparing predictions from thisgeneralized model with PIC simulations of dynamic problems.

For ease of comparison with previous work [3], [4], themodel will be expressed in terms of the variables Z0Ia, Z0Ic,Z0cQa, Z0cQc, Zf/Z0, and cA rather than Ba, Bc, Ea, Ec, Zf ,and A. The transformation to these variables uses the definitionsin the text following (3). With the exception of Zf/Z0, whichis dimensionless, each new variable has units of megavolts asdoes the variable V.

The first condition described above is applied by equating theaxial flow velocity of the electrons at the layer edge to the axialvelocity derived with the relativistic gamma factor associatedwith the voltage at the edge of the electron layer, yielding

vzs

c=(

1 − 1γ2

s

)1/2

. (19)

Here, γs = 1 + eVs/mc2 is the relativistic gamma factor, Vs =V (ds) is the voltage at the electron layer edge, and m is the

electron mass. Integrating (4) from the cathode to the electronlayer edge and using (13) to substitute for ρ give

Vs = (Z0cQa + Z0cQc)ds

2d. (20)

Substituting (18) and (20) into (19) and solving for ds yield

ds

d=

2mc2

e

×Z0Ia−

[Z2

0I2a−(Z0cQa+ 4Tmd2/ε0

(Z0cQa−Z0cQc)

)2]1/2

(Z0cQa+Z0cQc)

[Z2

0I2a−(Z0cQa+ 4Tmd2/ε0

(Z0cQa−Z0cQc)

)2]1/2

(21)

where ds varies from 0 (superinsulated flow) to d (saturatedflow).

The four generalized MITL flow equations are now given by(22)–(25), shown at the bottom of the next page, where a1, a2,and a3 are given by (26)–(28), shown at the bottom of thenext page.

Equation (22) comes directly from (14), (23) comes from(9) and (15), (24) is derived from (9) using (21) for ds, and,finally, (25) is obtained by integrating (5) from x = 0 to x = dand using the boundary conditions for cA(x) defined above. Aswill be discussed in Section III, basic MITL flow equations areobtained from (22)–(25) when Z0cQc = 0. Although (24) and(25) are algebraically cumbersome, the numerical solution ofthese four generalized MITL flow equations is straightforward,as will be discussed in Section IV.

Typically, (22) is referred to as the pressure balance equa-tion. Furthermore, (23) and (24) can be combined to obtain

V = Z0cQa − mc2

e

(Z0cQa − Z0cQc

Z0cQa + Z0cQc

)

×Z0Ia −

[Z2

0I2a −(

Z0cQa +4Tmd2/ε0

(Z0cQa − Z0cQc)

)2]1/2

[Z2

0I2a −(

Z0cQa +4Tmd2/ε0

(Z0cQa − Z0cQc)

)2]1/2

(29)

which is referred to as the voltage equation. In terms of thecomplete set of four generalized MITL flow equations, (24) and(29) are interchangeable. Additionally, the second terms, whichare proportional to m, in (24) and (29) are referred to as thespace-charge correction terms.

Regrouping the terms in (23) as

V = Z0cQc + Zf (cQa − cQc) (30)

provides insight into the motivation for the definition of theelectric flow impedance Zf . The first term on the right side of


Fig. 2. Plots of the enclosed charge per unit length in flow layer Q(x)−Qc

as a function of position x across the AK gap. This plot illustrates the definitionof electric flow impedance Zf . The solid curve shows Q(x)−Qc for a samplecase, and the dashed curve shows the square profile generated by placing thetotal charge per unit length in electron layer Qa−Qc for that sample case inan infinitely thin layer at x = df = d − xf , so that the integral from x = 0 tod under the two curves are equal. This requires that the area of the two shadedregions must be equal. Location x = df is defined as the centroid of the charge.

this equation represents the contribution to the voltage associ-ated with the charge per unit length Qc on the cathode locatedat a distance d from the anode or at vacuum impedance Z0 =(cμ0/w)d. The second term represents the contribution fromthe charge per unit length Qa − Qc in the electron layer, whichwas found in (15) to be located at a representative distancexf = (d − ds/2) from the anode or at flow impedance Zf =(cμ0/w)xf = Z0(1 − ds/2d). This representative distance xf

from the anode is the distance that the total charge per unitlength in the electron layer (i.e., Qa − Qc) would need to beplaced from the anode in an infinitely thin layer, so that theintegral from x = 0 to x = d of the enclosed charge per unitlength is preserved. This equivalence is illustrated in Fig. 2for an arbitrary charge density profile in the electron layer. Forpurposes here, the location x = df = d − xf will be defined as

the centroid of the charge. Solving (30) for Zf and using (4)evaluated at the anode to substitute for V provide

Zf =

∫ d

0 E(x)dx − Z0cQc

cQa − cQc=

(Z0/d)∫ d

0 cQ(x)dx − Z0cQc

cQa − cQc(31)

where Q(x) = E(x)d/cZ0 is the charge per unit length en-closed between the cathode (at x = 0) and a distance x fromthe cathode, including the charge per unit length Q(0) = Qc onthe cathode. This definition of electric flow impedance Zf isdescribed in [2].

Similarly, the magnetic flow impedance is defined in [2] as

Zm =

∫ d

0 cB(x)dx − Z0Ic

Ia − Ic=

(Z0/d)∫ d

0 I(x)dx − Z0Ic

Ia − Ic(32)

where I(x) = B(x)cd/Z0 is the current enclosed between thecathode (at x = 0) and a distance x from the cathode, includingcurrent I(0) = Ic on the cathode. Using (5) and this definitionyields

cA = Z0Ic + Zm(Ia − Ic). (33)

This equation then provides insight into the motivation forthe definition of magnetic flow impedance Zm. The first termon the right side of (33) represents the contribution to themagnetic flux associated with current Ic on the cathode locatedat a distance d from the anode or at vacuum impedance Z0 =(cμ0/w)d. The second term represents the contribution to themagnetic flux from flow current Ia − Ic, which is distributedacross the electron flow layer with a current density givenby Jf (x) in (16). This contribution to the flux can be de-scribed through (32) as located at a representative distance

Z20I2

a − Z20I2

c = Z20c2Q2

a − Z20c2Q2

c (22)

V =(

Zf

Z0

)Z0cQa +

(1 − Zf

Z0

)Z0cQc (23)

Zf

Z0= 1 − mc2

e

Z0Ia −[Z2

0I2a −(

Z0cQa +4Tmd2/ε0

(Z0cQa − Z0cQc)

)2]1/2

(Z0cQa + Z0cQc)

[Z2

0I2a −(

Z0cQa +4Tmd2/ε0

(Z0cQa − Z0cQc)

)2]1/2

(24)

cA = a1 + a2 ln(a3) (25)

a1 =Zf

Z0Z0Ia +

(1 − Zf

Z0

)(Z0Ia − Z0Ic)

Z0cQc(Z0cQa − Z0cQc) − 4Tmd2/ε0

(Z0cQa − Z0cQc)2 + 8Tmd2/ε0(26)

a2 =(

1 − Zf

Z0

)Z2

0I2c

[(Z0cQa − Z0cQc)2 + 8Tmd2/ε0

]−[Z0cQc(Z0cQa − Z0cQc) − 4Tmd2/ε0

]2[(Z0cQa − Z0cQc)2 + 8Tmd2/ε0]

3/2(27)

a3 =Z0Ia

[(Z0cQa − Z0cQc)2 + 8Tmd2/ε0

]1/2 + Z0cQa(Z0cQa − Z0cQc) + 4Tmd2/ε0

Z0Ic [(Z0cQa − Z0cQc)2 + 8Tmd2/ε0]1/2 + Z0cQc(Z0cQa − Z0cQc) − 4Tmd2/ε0

(28)


xm from the anode or at flow impedance Zm = (cμ0/w)xm =Z0(1 − dm/d), where dm = d − xm. As for the electric flowimpedance, this representative distance xm from the anode isthe distance that the total current in the electron layer (i.e.,Ia − Ic) would need to be placed from the anode in an infinitelythin layer, so that the integral from x = 0 to x = d of theenclosed current was preserved. For purposes here, the locationx=dm =d−xm will be defined as the centroid of the current.Using (25) for cA and this relationship between Zm and dm,an expression for dm can be directly calculated from (32) [or(33)]. As discussed above, dm >ds/2, so that Zm <Zf .

For completeness, (21) can be used to express ρ, vzc, and vzs

in terms of the model variables. Substituting for ds/d into (13),(17), and (18), these quantities become

ρ=eε0

(Z2

0c2Q2a − Z2

0c2Q2c

)2mc2d2

×

[Z2

0I2a−(Z0cQa+

4Tmd2/ε0

(Z0cQa−Z0cQc)

)2]1/2

Z0Ia−[Z2

0I2a−(Z0cQa+

4Tmd2/ε0

(Z0cQa − Z0cQc)

)2]1/2

(34)

vzc

c=

Z0cQc−4Tmd2/ε0

(Z0cQa − Z0cQc)Z0Ic

(35)

vzs

c=

Z0cQa+4Tmd2/ε0

(Z0cQa−Z0cQc)Z0Ia

. (36)

Potential Vs and magnetic vector potential cAs at the edge ofthe electron layer can also be expressed in terms of the modelvariables. Using (9) and (20), the potential at the edge of theelectron layer becomes

Vs = (Z0cQa + Z0cQc)(

1 − Zf

Z0

). (37)

Similarly, the magnetic vector potential at the edge of theelectron layer is obtained by integrating (5) from x = 0 tox = ds, yielding

cAs = cA − Z0Ia

(2Zf

Z0− 1)

(38)

where cA is given in (25), and Zf/Z0 is given in (24).To incorporate this generalized MITL model into a TLC

for treating dynamic problems, a prescription for assigning theimpedance of the transmission line elements in the presenceof the MITL flow must be devised. In terms of capacitanceC of a given element, the impedance of the element is givenby Zcap = τ/C, and the charge on the element is given by�Qa = CV . Noting that, by definition, �/τ = c and using (23)to substitute for voltage V , this capacitive impedance Zcap canbe written as

Zcap =Z0cQc + Zf (cQa − cQc)

cQa. (39)

For the special case where cQc = 0 (i.e., for the basic MITLflow model), Zcap = Zf , and the electric flow impedance be-comes the capacitive impedance of the line [2]. However, forthe general case, (39) applies.

Similarly, in terms of inductance L of a given element, theimpedance of the element is given by Zind = L/τ , and thecurrent carried in the element is given by Ia = �A/L. Using(33) to substitute for magnetic flux A, inductive impedanceZind can be written as

Zind =Z0Ic + Zm(Ia − Ic)

Ia. (40)

It will be shown below that Zcap < Zind, but, generally, theirdifference is small, so that Zcap can be used to describe the im-pedance of the line with reasonable accuracy. The range in theparameter space over which this approximation is reasonablewill also be assessed below.

The dynamic MITL model also requires a TLC to advanceV , Z0Ia, and cA along the transmission line for each time step.The TLC naturally advances V and Z0Ia, but advancing cArequires an additional step. Within a fixed vacuum impedancetransmission line element, the magnetic vector potential cA isadvanced using

∂(cA)∂t

= −c∂V

∂z(41)

which is derived from integrating the y component (azimuthalcomponent in cylindrical geometry) of Faraday’s law from x =0 to x = d and applying Leibnitz formula [9] at the electronlayer edge and boundaries. Because B(x) is continuous at theelectron layer edge, contributions to (41) from the motion ofthe layer edge cancel. It is also assumed that the walls of thetransmission line are fixed, so that dZ0/dt = 0. Using (41) toadvance cA in time, values of the parameters V , Z0Ia, and cAare applied to (22)–(25) to solve for Z0cQa, Z0cQc, Z0Ic, andZf/Z0, determining the flow conditions along the MITL forthe subsequent time step. However, to complete the model, aprescription for determining any loss current that is shuntedacross the MITL is required.

The x component of Ampere’s law evaluated at the anode

Z0Ja = −∂(Z0cQa)c∂t

− ∂(Z0Ia)∂z

(42)

is used to calculate the current lost to the anode (when Ja > 0),where Ja is the linear current density of the electrons and isdefined as positive when electrons flow in the positive x (radial)direction (i.e., when electrons flow toward the anode). The losscurrent for a TLC element of length � = cτ is Iloss = �Ja,where τ is the element’s length in seconds. This is accom-plished by using a shunt resistor with resistance Rloss = V/Iloss

to shunt the loss current for that TLC element. It is assumed thatthe anode is not turned on to electron emission, so that, whenJa ≤ 0, there is no current loss, and Iloss is set to zero (i.e.,Rloss = ∞).


When evaluated at the cathode, the x component of Ampere’slaw yields

Z0Jc = −∂(Z0cQc)c∂t

− ∂(Z0Ic)∂z

. (43)

This equation can be used to calculate current �Jc for a TLCelement that is emitted from the cathode when Jc > 0 orretrapped on the cathode when Jc < 0. Although instructive,this information is not needed for the model.

It has been observed in PIC simulations that when retrappingoccurs, the transition that propagates back up the MITL alongthe line moves at a speed that can be significantly less than thevacuum wave speed (i.e., the speed of light given by c). In fact,this “retrapping” speed has been observed to vary dependingon the difference between the low impedance of the load thatcaused the retrapping wave and the self-limited impedanceof the line [6]. It is expected here that the dwell time at aspecific location along the line required to transition the flowimpedance and to retrap the excess charge in the flow layerwill determine this “retrapping” speed. Thus, this phenomenonshould be naturally reproduced by applying the generalizedMITL model.

III. IMPLICATIONS FOR THE BASIC MITL FLOW MODEL

The basic MITL flow model for this fluid treatment withpressure is obtained from the generalized MITL flow modelby taking the electric field at the cathode to zero. In this case,Z0cQc = 0, and (22)–(25) become

Z20I2

a − Z20I2

c = Z20c2Q2

a (44)

V =Zf

Z0Z0cQa (45)

Zf

Z0= 1− mc2/e

Z0cQa

Z0Ia−[Z2

0I2a−(Z0cQa+

4Tmd2/ε0

Z0cQa

)2]1/2

[Z2

0I2a−(Z0cQa+

4Tmd2/ε0

Z0cQa

)2]1/2

(46)

cA= a10+a20 ln(a30) (47)

where

a10 =Zf

Z0Z0Ia−

(1−Zf

Z0

)(Z0Ia−Z0Ic)

4Tmd2/ε0

Z20c2Q2

a+8Tmd2/ε0

(48)

a20 =(1−Zf

Z0

)Z2

0I2c

[Z2

0c2Q2a+8Tmd2/ε0

]−[4Tmd2/ε0

]2[Z2

0c2Q2a+8Tmd2/ε0]

3/2

(49)

a30 =Z0Ia

[Z2

0c2Q2a+8Tmd2/ε0

]1/2+Z20c2Q2

a+4Tmd2/ε0

Z0Ic [Z20c2Q2

a+8Tmd2/ε0]1/2−4Tmd2/ε0

.

(50)

The voltage equation for this basic MITL flow model becomes

V = Z0cQa − mc2

e

×Z0Ia −

[Z2

0I2a −(

Z0cQa + 4Tmd2/ε0

Z0cQa

)2]1/2

[Z2

0I2a −(

Z0cQa +4Tmd2/ε0

Z0cQa

)2]1/2

(51)

and is interchangeable with (46) when applying this basicmodel. Note that the pressure balance equation given in (44)can be used in (45)–(51) to eliminate Z0cQa in favor of writingthe equations in terms of the current Z0Ia and Z0Ic.

By assuming that Tm = 0, the model can be further simpli-fied. In this case, (44) and (45) are unchanged, and (46) and(47) become

Zf

Z0= 1 − mc2/e

Z0Ic

(Z0Ia − Z0Ic

Z0Ia + Z0Ic

)1/2

(52)

cA =Zf

Z0Z0Ia +

(1 − Zf

Z0

)Z2

0I2c

(Z20I2

a − Z20I2

c )1/2

× ln

[Z0Ia +

(Z2

0I2a − Z2

0I2c

)1/2

Z0Ic

]. (53)

Similarly, the voltage equation becomes

V =(Z2

0I2a − Z2

0I2c

)1/2 − mc2

e

Z0Ia − Z0Ic

Z0Ic. (54)

This equation is similar but not identical to the voltage equa-tion [2], [3] that has been extensively used in experiments tocalculate the voltage from measurements of Z0Ia and Z0Ic.

Defining the parameter G ≡ 4Tmd2/ε0 (in units of squaremegavolts) as a measure of the electron pressure, the effectof pressure on the solution can be investigated. As shownin Fig. 3, this model with V = 6 MV and G = 0 (i.e., nopressure), in fact, does not accurately predict the results foundin previous equilibrium PIC simulations of MITL flow [3]. Thesimulation data are plotted as filled circles. In particular, thesimulation data point for self-limited flow on the saturated flow(left) side of the curve is well off the G = 0 curve. However,G can be adjusted in (51) to fit these simulation results. The fitis made at the self-limited flow point. Results in Fig. 3 for thecase with V = 6 MV and G = 1.63 show a good fit to the data.Using the fit

G(V ) ≡ 4Tmd2

ε0= 0.1316V + 0.0234V 2 (55)

the basic model derived above accurately predicts equilibriumsimulation results over the voltage range of 2–7 MV from[3], as shown in Fig. 4. The data points in Fig. 4 representthe simulation data, and the curves are derived from the basicmodel presented here with G(V ) given by (55). In fitting thesimulation data, it was assumed that G(V ) was a second-orderpolynomial and that G = 0 at V = 0. Recall, however, thatsolutions of the basic model with Z0cQc = 0 cannot representcases where there is emission or retrapping of electron flow. It


Fig. 3. Plot of Z0Ic as a function of Z0Ia for solutions of the basic MITLflow model with Z0cQc = 0 for V = 6 MV. Note that the independentvariable is displayed as the vertical axis. Cases for G = 0 (i.e., no pressure)and G = 1.63 are shown. PIC simulation data is shown as filled circles. Thelocations of minimum-current flow (open diamond), self-limited flow (left-most filled circle) and saturated flow (open square) are indicated for theG = 1.63 case.

Fig. 4. Plot of Z0Ic as a function of Z0Ia for solutions for MITL flow fromthe basic MITL flow model with Z0cQc = 0 and G(V ) given by (55). Notethat the independent variable is displayed as the vertical axis. Each curve is for adifferent voltage over the range of 2–7 MV. The data points are equilibrium flowsolutions from PIC simulations presented in [3]. The dashed curves indicatesolutions for minimum-current flow and self-limited flow.

is assumed here that the validity of the fit for G(V ) provided in(55) for Z0cQc = 0 solutions can be extended to the solutionsof the generalized MITL flow model with Z0cQc �= 0.

There are two differences between the basic model as derivedhere (by setting Z0cQc = 0) and the basic model derivedpreviously [2], [3]. First, the electron fluid has a perpendicularpressure given by T (x) in the model presented here, whereasthe electron pressure is assumed negligibly small in the pre-vious model. Second, the axial velocity of the electron flowlayer is calculated here, whereas, previously, only average flowvelocity was found (e.g., see [3, eq. (A1)]). More importantly,here, the value of the flow velocity at the edge of the electronlayer is related to the relativistic gamma factor from the voltageat the layer edge to close the set of equations. Additionally, themagnetic vector potential A and the magnetic flow impedanceZm are usually not considered in the previous basic MITL flowmodeling, whereas, here, cA is explicitly included in the modelas expressed in (47) because it is needed for the generalizedmodel, and Zm is calculated and compared to Zf to explicitlyshow that they are close in value.

Fig. 5. Plot of cA as a function of Z0Ia for solutions of the basic MITL flowmodel with Z0cQc = 0 for V = 6 MV. Note that the independent variableis displayed as the vertical axis. Cases for G = 0 (i.e., no pressure) and G =1.63 are shown. The locations of minimum-current flow (open diamond), self-limited flow (filled circle) and saturated flow (open square) are indicated for theG = 1.63 case.

To show the range of values for cA corresponding to theseZ0cQc = 0 solutions, (47) is used to calculate cA as a functionof Z0Ia, and the results are plotted in Fig. 5. Note that, althoughZ0Ic in Fig. 3 and cA in Fig. 5 are plotted as a function of Z0Ia,the axes have been oriented with independent variable Z0Ia onthe vertical axis to display the results in the traditional manner(i.e., Z0Ia versus Z0Ic). There is a one-to-one correspondenceof MITL flow solutions along these Z0Ic and cA curves.

Over the range of Z0Ic and cA, only one (Z0cQc = 0) solu-tion exists for each value of Z0Ic or cA. These curves terminateon the left-hand side at saturated flow, where Zf/Z0 = 1/2, andthe electron layer extends all the way to the anode (i.e., ds = d).The saturated flow points along these curves with Z0cQc = 0are indicated by the open squares in Figs. 3 and 5. EquatingZf/Z0 = 1/2 in (46) and applying (44) provide the followingexpressions for saturated flow:

Z0Isat0a =

2V γ(V )(

1 +G(V )4V 2

)[γ2(V ) − 1]1/2

(56)

Z0Isat0c =

2V

{γ2(V )

[(1 +

G(V )4V 2

)2

− 1

]+ 1

}1/2

[γ2(V ) − 1]1/2

(57)

and cAsat0 can be obtained by substituting (56) and (57) into(47) and applying (44). Here, γ(V ) = 1 + eV/mc2.

Solutions to the left of the minimum of the Z0Ia versusZ0Ic curve have traditionally been referred to as being onthe saturated-flow side of this curve (with ds approaching d);solutions to the right of the minimum are referred to as beingon the superinsulated side of the curve (with ds approaching 0).Minimum current flow refers to the solution at the minimum inZ0Ia along the curve and is indicated by the open diamonds inFigs. 3 and 5 (and the dashed line in Fig. 4). Self-limited flow isobserved when a pulse is propagating down a transmission linebefore reaching the load and continues after the pulse reachesthe load if the load impedance is larger than the self-limited


impedance of the line. Self-limited flow occurs on the saturated-flow side of the curve between minimum-current flow andsaturated flow and is indicated by the furthest left simulationdata point (filled circle) in Fig. 3 and by the filled circle inFig. 5. In Fig. 4, self-limited flow is indicated by the dashedline that passes through the leftmost simulation data pointson each curve. Equivalence of the basic model used in [3]with the basic model derived here is obtained by relating thescaling factor g used in [3] with the pressure [or G(V )] in thismodel. This equivalence is obtained by equating [3, eq. (36)]with (51) and using (44) to write Z0cQa in terms of Z0Ia andZ0Ic. The result is given in (58), shown at the bottom of thepage, where the fits are matched for the basic models at self-limited flow, so that (58) is evaluated at Z0Ia = Z0I

SLa (V ) and

Z0Ic = Z0ISLc (V ). Thus, g is only a function of V . Because

the simulation data are being fit, the simulation data at self-limited flow (leftmost data point for each voltage) can be usedfor this purpose, or the expressions given in [3, eqs. (49) and(50)] for Z0I

SLa (V ) and Z0I

SLc (V ) can be applied. Although it

was assumed in [3] that g was given by a simple function of V ,this equivalence provides more complex scaling. However, bothapproaches provide excellent fits to the simulation data withinthe range of their applicability (i.e., V = 2−7 MV). This rangecan easily be expanded by fitting to new simulation data outsidethis range.

The electric and magnetic flow impedances are compared inFig. 6 for the two cases for V = 6 MV, with G = 0 in Fig. 6(a)and G(6) = 1.63 in Fig. 6(b). As expected, all solutions haveZm < Zf , with the largest difference occurring at saturatedflow (i.e., the solution at the left end of the curves). Thedifference is about 14% at saturated flow for the case withG(6) = 1.63 and about 7% for the case with G = 0. At self-limited flow, the difference is only about 5% for the casewith G(6) = 1.63. Capacitive impedance Zcap is compared toinductive impedance Zind in Fig. 6(c), where, in this case,with Z0cQc = 0, it was shown above that Zcap = Zf . Here,Zind > Zcap over the entire range of solutions; however, theyare not very different. For the solutions shown in Fig. 6(c), Zind

is never more than 6% larger than Zcap. Thus, it is reasonableto use Zcap to describe the effective impedance of the MITL,keeping in mind that this inaccuracy in the model should bequantified and assessed as the model is applied. This is thesame assumption previously used [2] for the basic model (withZ0cQc = 0), where flow impedance Zf (= Zcap for Z0cQc =0) was used as the best description of the line impedance.

IV. SOLUTIONS OF THE GENERALIZED EQUATIONS

The TLC model for MITL flow depends on a robust nu-merical technique for solving (22)–(25), generalized to allow

Fig. 6. Plots of electric flow impedance Zf /Z0 (solid line) and magnetic flowimpedance Zm/Z0 (dash-dot line) as functions of Z0Ia for solutions of thebasic MITL flow model with Z0cQc = 0 and V = 6 MV for (a) G = 0 (i.e.,no pressure) and (b) G = 1.63. Note that the independent variable is displayedas the vertical axis. The locations of minimum-current flow (open diamond),self-limited flow (filled circle) and saturated flow (open square) are indicated onthe Zf /Z0 curve for the G = 1.63 case in (b). Also shown in (c) are plots ofcapacitive impedance Zcap/Z0 (solid line) and inductive impedance Zind/Z0

(dash-dot line) as functions of Z0Ia for the same solutions as in (b).

solutions with Z0cQc �= 0 and, thus, dynamic phenomena suchas electron emission and retrapping. For the particular TLCelement of vacuum impedance Z0, where the solution is sought,Z0cQa, Z0cQc, Z0Ic, and Zf/Z0 are the unknown variables,and V , Z0Ia, and cA are the known parameters. These knownparameters are determined at each time step by advancingthe solutions for each element in the TLC, as described at

g(V ) =2Z2

0I2c

Z20I2

a − Z20I2

c

Z0Ia

(Z2

0I2a − Z2

0I2c

)1/2 −[Z2

0I2a

(Z2

0I2a − Z2

0I2c

)−(Z2

0I2a − Z2

0I2c + G(V )

)2]1/2

[Z2

0I2a (Z2

0I2a − Z2

0I2c ) − (Z2

0I2a − Z2

0I2c + G(V ))2

]1/2(58)


the end of Section II. The numerical solution of (22)–(25) isstraightforward.

One special solution of the generalized equations can besolved immediately. For saturated flow, Zf/Z0 = 1/2, ds/d =1, and Vs = V . In this case, (23) [or (37)] becomes V =(Z0cQa + Z0cQc)/2. Substituting this result into (21) evalu-ated at ds/d = 1 provides the solutions

Z0cQsata =

V

2+

Z0Ia

2

⎡⎢⎢⎢⎣1−

(mc2

eV

)2

γ2(V )

⎤⎥⎥⎥⎦

1/2

± 12

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

⎛⎜⎜⎜⎜⎝V −Z0Ia

⎡⎢⎢⎢⎣1−

(mc2

eV

)2

γ2(V )

⎤⎥⎥⎥⎦

1/2⎞⎟⎟⎟⎟⎠

2

−2G(V )

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

1/2

(59)Z0cQ

satc =2V −Z0cQ

sata (60)

Z0Isatc =

[(Z0Ia)2−

(Z0cQ

sata

)2+(Z0cQ

satc

)2]1/2

(61)

cAsat =cA(Z0Ia, Z0I

satc , Z0cQ

sata , Z0cQ

satc

)(62)

where (25) with Zf/Z0 = 1/2 and 4Tmd2/ε0 = G(V ) is usedto evaluate (62). These expressions are obtained for the givenvalues of Z0Ia, V , and G(V ) [see (55)]. Because this isthe special case for saturated flow (i.e., Zf/Z0 is a knownparameter rather than cA), cA is not a free parameter, but isspecified by (62). Alternately, cA, V , and G(V ) could havebeen chosen for saturated flow, in which case Z0I

sata would be

specified. Note that, because of the “±” sign in (59), there aretwo saturated flow solutions for each set of these values. This isillustrated in Fig. 10.

Aside from this one special case, generalized solutions (i.e.,with Z0cQc not necessarily equal to zero) are found numeri-cally by fixing V and cA and solving (22)–(25) as functions ofZ0Ia. Solutions for Z0Ic are shown as the solid curves in Fig. 7for V = 6 MV and G = 1.63, with cA = 7 MV in Fig. 7(a),with cA = 8.5 MV in Fig. 7(b), and with cA = 10 MV inFig. 7(c). Again, the axes are oriented to show the familiarZ0Ia versus Z0Ic curve. The basic Z0cQc = 0 solutions forV = 6 MV are indicated by the dash-dot curves. The dashedlines indicate the boundaries specified by physical constraintson the solutions; meaningful solutions for these values of Vand cA fall within these boundaries in the unshaded region.The horizontal boundary is determined by Z0Ia ≥ cA. BecausecA is reduced when the current is distributed in the gap, cA isa maximum for superinsulated flow, where Z0Ic = Z0Ia andcA = Z0Ia [from the definition of cA in (5)]. The diagonalboundary is simply Z0Ic ≤ Z0Ia. Note that the generalizedsolutions (solid curves) in Fig. 7 shift toward superinsulatedflow (Z0Ic approaches Z0Ia) as cA increases from 7 MV inFig. 7(a) to 10 MV in Fig. 7(c); conversely, as cA decreases, thegeneralized solutions move toward saturated flow. This trend isseen in Fig. 5, with cA approaching V at saturated flow. Also,the ends of the curves that fall within the range that is plottedoccur at saturated flow, as is clear from examining Fig. 10. The

Fig. 7. Plots of Z0Ic as a function of Z0Ia for solutions (solid curve) of thegeneralized MITL flow model with V = 6 MV (and G = 1.63) and (a) cA =7 MV, (b) cA = 8.5 MV, and (c) cA = 10 MV. Note that the independentvariable is displayed as the vertical axis. Physical constraints restrict validsolutions to the unshaded region bounded by the dashed lines. The dash-dotcurve shows the solutions of the basic MITL flow model with Z0cQc = 0 (withcA varying along the curve). The open circle indicates where the generalizedsolutions pass through Z0cQc = 0 (for the fixed value of cA of each plot) andcoincide with the solutions of the basic model. The open squares correspond tosaturated flow solutions.

locations of the saturated flow solutions are indicated by theopen squares.

Imagining that there is a Z0cQc axis projecting in and outof the paper in Fig. 7, and that solutions with different valuesof Z0cQc are projected onto the plane of the paper, the onlyZ0cQc = 0 solution occurs at the location that is indicated bythe open circle. As expected, these generalized solutions withZ0cQc = 0 (indicated by the open circles) fall on the dash-dotcurves corresponding to the solutions for V = 6 MV from thebasic model. Other points along the solid curves that seem tocross the dash-dot Z0cQc = 0 curves (not marked with opencircles) actually lie off the plane of the paper with Z0cQc �= 0.An alternate way of viewing this is that cA is not fixed alongthe dash-dot Z0cQc = 0 curve and only acquires the cA valuesthat are appropriate for the specific plot at the location of theopen circles.


Fig. 8. Plots of Z0cQc as a function of Z0Ia for solutions (solid curve)of the generalized MITL flow model with V = 6 MV (and G = 1.63) and(a) cA = 7 MV, (b) cA = 8.5 MV, and (c) cA = 10 MV. Note that theindependent variable is displayed as the vertical axis. Physical constraintsrestrict valid solutions to the unshaded region bounded by the dashed lines.The dash-dot curve shows the solutions of the basic MITL flow model withZ0cQc = 0 (with cA varying along the curve). The open circle indicates wherethe generalized solutions pass through Z0cQc = 0 (for the fixed value of cA ofeach plot) and coincide with the solutions of the basic model. The open squarescorrespond to saturated flow solutions.

Solutions for Z0cQc, Z0cQa, Zf/Z0, and Zcap/Z0 as func-tions of Z0Ia for the same cases as in Fig. 7 are shown assolid curves in Figs. 8–11, respectively. For ease of compari-son, the axes are also oriented in the same fashion as Fig. 3(and Fig. 7). As in Fig. 7, the locations of the Z0cQc = 0solutions from the basic model for V = 6 MV are shown asthe dash-dot lines. The bound Z0Ia ≥ cA applies in all fourof these plots and is shown as the horizontal dashed line. Theother constraints are derived from the following. The pressure

Fig. 9. Plots of Z0cQa as a function of Z0Ia for solutions (solid curve)of the generalized MITL flow model with V = 6 MV (and G = 1.63) and(a) cA = 7 MV, (b) cA = 8.5 MV, and (c) cA = 10 MV. Note that theindependent variable is displayed as the vertical axis. Physical constraintsrestrict valid solutions to the unshaded region bounded by the dashed lines.The dash-dot curve shows the solutions of the basic MITL flow model withZ0cQc = 0 (with cA varying along the curve). The open circle indicates wherethe generalized solutions pass through Z0cQc = 0 (for the fixed value of cA ofeach plot) and coincide with the solutions of the basic model. The open squarescorrespond to saturated flow solutions.

balance equation [see (22)] and the constraint Z0Ia ≥ Z0Ic

provide the bound Z0cQa ≥ Z0cQc ≥ −Z0cQa, whereas theconstraint 1 ≥ Zf/Z0 ≥ 1/2 and (23) provide the two boundsZ0cQa ≥ V and 2V ≥ Z0cQa + Z0cQc. These three boundscan be combined as Z0cQa ≥ V ≥ Z0cQc ≥ −Z0cQa. Theright and left bounds in Fig. 8 are given by the middle andthe last of these inequalities, whereas the left bound in Fig. 9is given by the first of these inequalities. The right-hand boundin Fig. 9 comes from pressure balance. To see this, considerZ0cQc = 0 solutions in (22). Because (Z0Ic)2 > 0 and bothZ0Ia and Z0cQa are positive quantities, Z0Ia ≥ Z0cQa. Fi-nally, the vertical dashed line in Fig. 10 shows the bounds setby 1 ≥ Zf/Z0 ≥ 1/2, and the vertical dashed line in Fig. 11shows the bound Zcap/Z0 ≤ 1.


Fig. 10. Plots of Zf /Z0 as a function of Z0Ia for solutions (solid curve)of the generalized MITL flow model with V = 6 MV (and G = 1.63) and(a) cA = 7 MV, (b) cA = 8.5 MV, and (c) cA = 10 MV. Note that theindependent variable is displayed as the vertical axis. Physical constraintsrestrict valid solutions to the unshaded region bounded by the dashed lines.The dash-dot curve shows the solutions of the basic MITL flow model withZ0cQc = 0 (with cA varying along the curve). The open circle indicates wherethe generalized solutions pass through Z0cQc = 0 (for the fixed value of cA ofeach plot) and coincide with the solutions of the basic model. The open squarescorrespond to saturated flow solutions.

As in Fig. 7, the generalized MITL flow solutions withZ0cQc = 0 in Figs. 8–11 are indicated by the open circles andoccur at locations where the solid curves for the generalizedsolutions cross the dash-dot curves for the basic solutions.Other locations (without the open circles) where the solidcurves cross the dash-dot curves do not have Z0cQc = 0 forthe generalized solution. Fig. 8 shows that the generalizedsolutions occur over a broad range of positive and negativevalues of Z0cQc. Recall that in the generalized MITL flowmodel developed here, Z0cQc is allowed to be nonzero, so thatemission and retrapping can be treated in the dynamic model.The solutions of the generalized MITL flow model for saturatedflow described by (59)–(62) are reproduced by this numericaltechnique and, by definition, appear at Zf/Z0 = 1/2. These

Fig. 11. Plots of Zcap/Z0 as a function of Z0Ia for solutions (solid curve)of the generalized MITL flow model with V = 6 MV (and G = 1.63) and(a) cA = 7 MV, (b) cA = 8.5 MV, and (c) cA = 10 MV. Note that theindependent variable is displayed as the vertical axis. Physical constraintsrestrict valid solutions to the unshaded region bounded by the dashed lines.The dash-dot curve shows the solutions of the basic MITL flow model withZ0cQc = 0 (with cA varying along the curve and Zcap = Zf ). The opencircle indicates where the generalized solutions pass through Z0cQc = 0 (forthe fixed value of cA of each plot) and coincide with the solutions of the basicmodel. The open squares correspond to saturated flow solutions.

saturated flow solutions are indicated by the open squares.In particular, Fig. 10(a) shows that these two saturated flowsolutions lie within the range that is plotted with the ends ofthe generalized solution curves occurring at Zf/Z0 = 1/2, asnoted above in the discussion of Fig. 7. In Fig. 9, showingthe Z0cQa versus Z0Ia plots, the generalized solution curvesdescend down from the upper right and, after reaching a min-imum, begin to rise again for a short length with the solutionslying slightly below the descending part of the curve. Becauseof this behavior, the rising parts of the curves in Fig. 9(a)–(c)are difficult to see, and the open squares, which indicate the endof the curves at saturated flow, obscure the open circles, whichindicate the location of the Z0cQc = 0 solutions. ComparingFigs. 10 and 11, it is clear that Zcap can significantly differ from


Fig. 12. Plot of Zcap/Zind as a function of Z0cQc for the solutions of thegeneralized MITL flow model with V = 6 MV (and G = 1.63) for (solidcurve) cA = 7 MV, (dashed curve) cA = 8.5 MV, and (dot-dash curve) cA =10 MV. The horizontal dashed line indicates the 0.9 level for Zcap/Zind, andthe vertical drop-down dashed line shows that Zcap/Zind ≥ 0.9 for Z0cQcgreater than about −2.5 MV for all these solutions.

Zf as Qc moves away from zero (at the open circles). As inFig. 9, the solutions for Zcap/Z0 versus Z0Ia in Fig. 11 descenddown from the upper left and, after reaching a minimum, beginto rise again for a short length; however, unlike in Fig. 9, thesesolutions in Fig. 11 lie slightly above the descending part of thecurve.

In Fig. 12, Zcap/Zind is plotted as a function of Z0cQc forthe solutions of the generalized MITL flow model with V =6 MV (and G = 1.63) for the same three values of cA = 7,8.5, and 10 MV as plotted in Figs. 7–11. These data show thatthere is a significant range of Z0cQc over which Zcap is close invalue to Zind. The horizontal dashed line indicates the 0.9 levelfor Zcap/Zind, and the vertical drop-down dashed line showsthat Zcap/Zind ≥ 0.9 for Z0cQc greater than about −2.5 MVfor all these solutions. These results show that there is areasonable range of Z0cQc about the Z0cQc = 0 point overwhich the difference between Zcap and Zind is small. Thus,Zcap provides a reasonable representation of the impedanceof the MITL for most Z0cQc values of interest. Because it isexpected that both retrapping and emission will push the flowback toward equilibrium (i.e., toward Z0cQc = 0), |Z0cQc|should not become large. In any case, Zcap/Zind should betracked to monitor the accuracy of using Zcap to describethe MITL impedance. If necessary, better accuracy may beobtained by using another value such as the average impedance(Zcap + Zind)/2 to describe the MITL.

In Fig. 13, Zcap/Zind and Zm/Zf are plotted as functions ofZ0cQc for saturated flow solutions of the generalized MITLflow model with V = 6 MV (and G = 1.63). Note that cAvaries along these curves. The horizontal dashed line indicatesthe 0.9 level for the ratio of impedances, and the verticaldrop-down dashed line illustrates that Zcap/Zind ≥ 0.9 forZ0cQc greater than about −2 MV. Also, these results showthat Zcap/Zind > Zm/Zf over the entire range of solutions forsaturated flow.

V. CONCLUSION

A new fluid model including pressure has been introducedhere to treat electron flow in an MITL. The goal of this paper is

Fig. 13. Plots of Zcap/Zind and Zm/Zf as functions of Z0cQc for saturatedflow solutions of the generalized MITL flow model with V = 6 MV (andG = 1.63). Note that cA varies along these curves. The horizontal dashed lineindicates the 0.9 level, and the vertical drop-down dashed line illustrates thatZcap/Zind ≥ 0.9 for Z0cQc greater than about −2 MV. Also, Zcap/Zind >Zm/Zf over the entire range of solutions for saturated flow.

to develop a model that provides a reasonable representationof what is observed in PIC simulations of MITL flow. Thisincludes a finite thickness flow layer that extends from thecathode to the electron layer edge, as well as a distributionof various electron orbits within that layer, which is indicativeof transverse temperature or pressure in the center of the flowlayer. Equally important, the model allows for nonzero electricfield at the cathode and, therefore, is capable of treating electronemission and retrapping in dynamic MITL modeling. A pre-scription has also been presented for calculating any necessaryelectron loss current along the line.

Other important features have been also included in the newmodel. By introducing an additional new condition that relatesthe flow velocity at the electron layer edge to the voltage there,the space-charge term in the MITL flow equations has beenderived directly. Also, by appropriately choosing the magnitudeof the pressure term in this generalized MITL model, the resultsof the recent rescaled MITL theory presented in [3] can beapplied here, so that the improved accuracy of the voltagepredictions developed there is preserved. Solutions of the gen-eralized MITL model have been presented, which illustrate thefeatures of the model, where Z0cQc is allowed to be nonzero,so that emission and retrapping can be treated in dynamic MITLmodeling. Solutions of the basic MITL model are recoveredwhen Z0cQc = 0.

Because electron space charge and current are distributed inthe line, there is not, in general, a well-defined single waveimpedance that describes an MITL. Electric flow impedanceZf and magnetic flow impedance Zm are defined by the dis-tances of the centroid of the charge and the centroid of thecurrent from the anode, respectively. Additionally, capacitiveimpedance Zcap and inductive impedance Zind have been in-troduced and related to Zf and Zm to describe the electricalproperties of the MITL. In fact, Zcap = Zf for the basic model,where Z0cQc = 0. The difference between Zcap and Zind isusually small, so that Zcap provides a reasonable approximationof the MITL flow impedance under most circumstances of


interest. For purposes of incorporating the generalized MITLflow model into a TLC, where a single impedance is needed tocharacterize a transmission line element, Zcap has been usedto describe the MITL impedance. Alternately, another valuesuch as the average value (Zcap + Zind)/2 could be used todescribe the MITL impedance. For incorporating the model intoa circuit code, where the MITL is modeled by a sequence ofseries inductors and parallel capacitors, Zcap and Zind couldbe used to describe the capacitance and the inductance ofthese circuit elements, respectively, to provide better simulationfidelity.

Because it is assumed that electrons instantaneously react tothe time-dependent fields, the generalized MITL flow modelis a quasi-equilibrium model. However, when combined withtime-dependent circuit equations for evolving V , Ia, and Ain time, this quasi-equilibrium model can be used to build adynamic model for MITL flow, for example, in a TLC. Futurework is required to develop a robust numerical technique forsolving the new MITL flow equations and to develop techniquesto treat the emission front, adders, load coupling, nonemittingregions, etc., which are needed to implement the dynamicmodel in a useful TLC. Ultimately, this will allow efficient andaccurate modeling of MITL flow in a fast TLC to replace themore computationally intensive PIC code treatment.

ACKNOWLEDGMENT

The authors would like to thank the support, encouragement,and leadership of Dr. J. E. Maenchen and Dr. B. V. Oliver forthis paper. Also, they would like to acknowledge useful discus-sions with Dr. C. Mendel, Dr. V. Bailey, and Dr. S. Rosenthal.Additionally, the referees’ comments are greatly appreciatedand have significantly helped in improving this paper.

REFERENCES

[1] J. P. VanDevender, J. T. Crow, B. G. Epstein, D. H. McDaniel,C. W. Mendel, E. L. Neau, J. W. Poukey, J. P. Quintenz, D. B. Seidel,and R. W. Stinnett, “Self-magnetically insulated electron flow in vacuumtransmission lines,” Physica, vol. 104C, no. 1/2, pp. 167–182, 1981.

[2] C. W. Mendel and S. E. Rosenthal, “Modeling magnetically insulateddevices using flow impedance,” Phys. Plasmas, vol. 2, no. 4, pp. 1332–1342, Apr. 1995.

[3] P. F. Ottinger and J. W. Schumer, “Rescaling of equilibrium magneticallyinsulated flow theory based on particle-in-cell simulations,” Phys. Plasmas,vol. 13, no. 6, pp. 063 109-1–063 109-17, Jun. 2006.

[4] C. W. Mendel and S. E. Rosenthal, “Dynamic modeling of magnetically in-sulated transmission line systems,” Phys. Plasmas, vol. 3, no. 11, pp. 4207–4219, Nov. 1996.

[5] D. D. Hinshelwood, “Bertha—A versatile transmission line and circuitcode,” Naval Res. Lab., Washington, DC, NRL Memorandum Report 5185,Nov. 21, 1983.

[6] V. L. Bailey, P. A. Corcoran, D. L. Johnson, I. D. Smith, J. E. Maenchen,K. D. Hahn, I. Molina, D. C. Rovang, S. Portillo, E. A. Puetz, B. V. Oliver,D. V. Rose, D. R. Welch, D. W. Droemer, and T. L. Guy, “Re-trapping ofvacuum electron current in magnetically insulated transmission lines,” inProc. 15th Int. Conf. High-Power Particle Beams, G. Mesyats, V. Smirvov,and V. Engelko, Eds., St. Petersburg, Russia, Jul. 2004, pp. 247–250.

[7] C. W. Mendel, D. B. Seidel, and S. A. Slutz, “A general theory of magneti-cally insulated electron flow,” Phys. Fluids, vol. 26, no. 12, pp. 3628–3635,Dec. 1983.

[8] J. M. Creedon, “Relativistic Brillouin flow in the high ν/γ diode,” Appl.Phys., vol. 46, no. 7, pp. 2946–2955, Jul. 1975.

[9] G. Arfken, Mathematical Methods for Physicists, 2nd ed. New York:Academic, 1970, p. 410.

Paul F. Ottinger (M’90–SM’01) was born inPhiladelphia, PA, in 1948. He received the B.A. de-gree in physics from the University of Pennsylvania,Philadelphia, in 1970 and the M.S. and Ph.D. de-grees in theoretical plasma physics from the Univer-sity of Maryland, College Park, in 1974 and 1977,respectively.

He was a Senior Research Scientist in the privateindustry with JAYCOR, Inc. He was a National Re-search Council Associate with the Naval ResearchLaboratory (NRL), Washington, DC, where, since

1985, he has been the Head of the Theory and Analysis Section in the PulsedPower Physics Branch. During his career, he has developed expertise in thespecialized areas of intense particle beams, fast opening plasma switches, diodephysics, pulsed power physics, and plasma radiation sources. In particular, hisresearch has concentrated on the areas of ion beam generation, transport, andstability.

Dr. Ottinger is a Fellow of the American Physical Society and a SeniorMember of the IEEE Nuclear and Plasma Science Society. He was a recipientof the 1992 IEEE Plasma Science and Applications Award.

Joseph W. Schumer (M’99) was born in CapeGirardeau, MO, on February 9, 1969. He receivedthe B.S. degree in nuclear engineering from theUniversity of Missouri-Rolla, Rolla, in 1992 and theM.S. and Ph.D. degrees in nuclear engineering fromthe University of Michigan, Ann Arbor, in 1994 and1997, respectively, under the Department of EnergyMagnetic Fusion Science Fellowship.

He was with the Idaho National Engineering Lab-oratory in 1989, the Oak Ridge National Laboratoryin 1990 and 1991, and the Institute for Fusion Stud-

ies, Austin, TX, in 1993, where his research interests evolved. Since 1997,he has been with the Naval Research Laboratory (NRL), Washington, DC, asa National Research Council Associate in 1997–1999, a Research Scientistwith JAYCOR, Inc., in 2000, and, currently, a Research Physicist with thePulsed Power Physics Branch, Plasma Physics Division (since August 2000),primarily involved with the development of numerical models for collisionlessand collisional plasmas, study of advanced energetics using nuclear isotopes,improving the design of power conditioning systems, and study of high-energyelectron and ion beam transport physics for radiography and nuclear weaponseffects simulation. He also serves as a Contracts Monitor for various projectswith the NRL.

Dr. Schumer served as the Vice-Chair for the IEEE PSAC in 2002–2005 andas the Local Chair for the ICOPS 2004 meeting (Baltimore, MD).

David D. Hinshelwood was born in Bethesda, MD.He received the B.S. degree in physics and math andthe Ph.D. degree in physics from the MassachusettsInstitute of Technology, Cambridge, in 1977 and1984, respectively.

From 1984 to 1995, he was with JAYCOR as aResearch Physicist with the Naval Research Labora-tory (NRL), Washington, DC, working on the plasmaphysics of opening switches, pulsed power, and in-tense electron and ion beams. Since 1995, he hasbeen with the Pulsed Power Physics Branch, Plasma

Physics Division, NRL. His interests include electron-, ion-, and plasma-filleddiodes, plasma opening switches, plasma radiation sources, pulsed processingplasmas, code development, and advanced pulsed power development.

Raymond J. Allen (M’96) was born in Oakland,CA. He received the B.S., M.S., and Ph.D. degreesfrom Old Dominion University, Norfolk, VA, in1992, 1994, and 1998, respectively, all in electricalengineering.

Since 2000, he has been with the Naval ResearchLaboratory, Washington, DC, working on pulsedpower issues.

generalized model for magnetically insulated transmission line flow

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