Time-Varying Electrostatic Modeling Techniques 

David M. Hull 

U.S. Army Research Laboratory 

Adelphi, MD 20783-1 197 

Phone: 301 -394-3140 

email: hull @arl.mil 

Abstract 

ARL developed computer models and modeling techniques based on the Method of Moments, and has used 

them for some time to study electrostatic fields 

associated with targets, terrain clutter, and sensors of 

interest. Recent extensions to these unique ARL capabilities allow some dynamic conditions to be modeled as 

a time series of quasi-static models. These new 

techniques have enabled us to study the extremely 

low-frequency (ELF) effects of rotating helicopter 

blades on both airborne and remote sensors. Examples 

show how a dynamic helicopter model can be used to 

compute time-varying airbome fieldmeter calibration 

factors for aircraft charge and atmospheric electric field 

measurements, and remote ELF electric fields which 

might be detected by passive surveillance sensors. 

Introduction 

Numerical analysis techniques are widely used today to 

solve electromagnetic field problems that are too 

complex to be solved by analytical methods alone. We 

chose to develop a significant electrostatic modeling and 

analysis capability based on the Method ($Moments, as 

described by Harrington [I]; this approach is sometimes 

called the Method c?f' Subsections [2] or the Moment 

Method [3]. Unlike the more popular finite-element 

approach, one does not solve for fields directly, but 

rather for the field sources (in our case, the charge 

distribution). Once the field sources have been determined, the fields can be readily computed as needed, 

using the Principle qj' Superposition [4]. 

To compute the field sources, the governing differential 

equations (in our case, the Laplace or Poisson equations) 

are transformed into integral identities, which are 

applied to a finite number of elements that form the 

boundary surfaces of the field problem. A system of 

linear equations is defined that satisfy the boundary 

conditions at each element. In general, the coefficient 

matrix associated with the resulting system of equations 

is dense, non-symmetric, and not diagonally dominant, 

so direct linear system solvers must be used. We have 

successfully developed models with over 5000 elements. 

As applied to electrostatic boundary-value problems, 

this method is summarized as follows. In a region of 

constant permittivity E and a volume charge distribution 

p(x, y, z), the electrostatic potential $(x, y, z) satisfies the 

Poisson equation: 

Dirichlet boundary conditions are those conditions for 

which the scalar potential is specified at every point on 

the conductor-dielectric boundary. Under these conditions, the unique solution to this problem is: 

where r is the distance between the source point (x, y, z) 

and the field point (x', y', z'): 

In general, we cannot solve equation (2) at every point 

on the boundary surfaces. However, we can divide the 

boundaries up into N elements. For each element, we 

define a linear equation: 

In this way, we have reduced our problem to that of 

solving the matrix equation: 

The coefficient matrix A can be thought of as an "inverse 

capacitance density" matrix. The coefficients describe 

the mutual coupling between the elements, and are 

defined in terms of the size and orientation of each 

element relative to the others: 

where the field point (x', y ', z') in the fimction r is chosen 

to be the centroid of element n, and the integration is 

carried out over element m. The potential vector V 

specifies the Dirichlet boundary conditions over the 

boundary surfaces. The elements of the solution vector 

p,, are the modeled charge densities for each element. 

Generally, we model charged objects as equipotential 

surfaces. This does not mean that these objects are 

"good" conductors, only that the charge relaxation time 

(i.e., the time needed to redistribute any charge to the 

steady-state conditions) is small, compared to the time 

associated with any changes in the specified boundary 

conditions. To be more precise, we should say we are 

solving quasi-static boundary-value problems. Timevarying boundary conditions can be modeled with a 

time-series of quasi-static models, up to the frequency 

at which these assumptions are no longer valid. This 

upper-frequency limit is dependent on the size and 

conductivity of the objects being modeled, and on the 

rate of change of the boundary conditions. The series of 

quasi-static models is called a dynamic model; each 

quasi-static model in a series is called a frame. 
Dynamic Hind-D Helicopter Model Table 1. First-order helicopter charge statistics. 

To construct the first frame, the surface of the Hind-D 

(shown in figure I), was divided into small trapezoids 

(often rectangles). Each of these 1167 areas is represented as a line segment, as shown in figure 2. These 

elements are normally positioned in the middle of each 

area, although they can also be placed on the edges to 

better model the rapid increases in surface charge 

densities in those areas. Each element is assigned a 

constant (linear) charge density using the Method of 

Moments; the modeled surface charge distribution can 

vary with the inter-element spacing. The dynamic 

model is constructed with the five-blade main rotor 

turning at 240 rpm (4 rps), and the three-blade tail rotor 

turning at 1200 rpm (20 rps). In this way, we 

constructed a self-looping high-resolution mode1 (with 

frames every 2" of main rotor travel), using 36 individual 

quasi-static models. 

Figure 1. Surface model of Hind-D helicopter. 

Figure 2. Line-segment model of Hind-D helicopter. 

Each frame in the dynamic charge model is the solution 

to the Laplace equation subject to modified Dirichlet 

boundary conditions. Specifically, at each time instant, 

we assumed that the potential was constant over the 

entire surface of the aircraft; however, we also assumed 

that this potential "floated" about some mean in such a 

way that the total aircraft charge remained constant 

throughout the time series. Since Q = CV, the modeled 

potential varies inversely with the capacitance. 

Summary information is shown in table 1 for every third 

quasi-static model; the curves in later figures use data 

from all 36 frames. 

For the Hind-D with a 5-blade main rotor, the capacitance varies by less than 0.5 percent about the mean, as 

a function of blade phase. Other researchers have 

reported changes in capacitance of more than +3 percent 

for a similar-sized Seahawk helicopter, which has only 

four main rotor blades [5]. It is reasonable to assume 

that this effect would be even greater for a UH- 1 (Huey) 

or an AH-1 (Cobra), since these older U.S. helicopters 

have only two main rotor blades. 

Time 

(ms) 

0.00 

4.17 

8.33 

12.50 

16.67 

20.83 

25.00 

29.17 

33.33 

37.50 

41.67 

45.83 

Rotor (deg) Charge 

Q (PC) 

Cap. 

c (PF) 

Voltage 

v (V) 

It is obvious that the charge on the blades must move 

with the blades. Perhaps not so obvious is the fact that, 

as the blades turn, charge moves on and off the blades 

in concert with the changing boundasy conditions. 

Indeed. the total available charge is continually redistributed over the entire surface orthe aircraft as the main 

and tail rotors turn. An animated "movie" which shows 

the movement of charge over time with false color is 

accessible via the World Wide Web [6]. 

Figure 3 shows the movement of the electrostatic charge 

centroid about the mean, as a function of main rotor 

blade phase in each of the three Cartesian directions. 

Since there are five main rotor blades, there are five 

main "beats" in these functions for each complete 

revolution of the main rotor. As expected, the tail rotor 

effects are at a frequency three times that of the main 

rotor, and are most noticeable in the Z (up-down) 

direction. 

Charge Centroid (m) 

0.08 I I. I I I .I 

X (f_Op?:aft) Y (side-side) Z (uewn) - 11 

I1 , 

I 1 I I 

I I 11' 1 

I I 

-0.08 ''''''"'''''' 

0 72 144 21 6 288 360 

Main Rotor Blade Phase, degrees (0 =forward) 

Figure 3. Hind-D charge centroid. 

Focus on the Rotor Blades. 

Figure 4 shows the percent of total helicopter charge that 

is present at any given time on two selected rotor blades. 

The main and tail rotor blades chosen are defined to face 

forward at zero phase. The other four main rotor blades 

carry the same charge as the one shown, but staggered 

in time by 72" increments. Similarly, the charge on the 

other two tail rotor blades is offset by increments of 24" 

(or 120" of tail-rotor travel). From the figure, it can be 

seen that, at any given point in time, over half of the total 

helicopter charge resides on the blades. 
Percent of total charge 

12 

Q I I I I Ell = kvt, - C + kX,lEX + ky,,E; + k,,E:, (7) I I I I 

0 72 144 21 6 288 360 

Main Rotor Blade Phase (0 degrees = forward) 

where the four k terms are called theafield enhancement 

fuctors at each fieldmeter location n. These factors vary 

as a function of the size and shape of the aircraft, and of 

the position of the sensor on the aircraft. Since the 

Figure 4. Charge on selected rotor blades. 

6------1------1------L-----1------ I I I I 

I I I I 

I I I I 4------ ------------ r-----7------ Tail rotor blade #I13 1 I I 

We can also use the dynamic model to estimate various 

currents that are flowing over the airframe. These 

currents are directly proportional to the airframe charge, 

which we assume to be 5.1 PC. In the case of the main 

rotor, we have 32 percent of 5.1 /.LC (moving on and off 

each blade) at a fundamental frequency of 8 Hz (there 

arf two minima and two maxima per revolution). If we 

treat the oscillating charge as a simple sinusoid, and 

differentiate with respect to time, we expect a peak 

current flowing between the rotor hub and each blade of 

about 5 PA. For a more precise estimate, we can 

numerically differentiate the computed charge functions 

shown in figure 4 with respect to time. These currents 

are shown in figure 5. 

enhancement factors apply to fixed fieldmeter locations, 

they are usually considered constants. The calibration 

problem becomes one of precomputing the enhancement 

factors. Once these factors are known, the measured 

Current onloff rotor blades (uA) 

12 - I I I I 

Main rot~blsde #I15 Tail rotoede #I13 

8 ----- 

fields can be related to the actual quantities of interest: 

the aircraft charge and/or the ambient electric field. 

-8 I I I I I J 

0 50 100 150 200 250 

Time (ms) 

Figure 5. Current at roots of selected rotor blades. 

Application of the Model to the Airborne Fieldmeter Calibration Problem 

Continuous measurements of aircraft charge and/or 

atmospheric electric fields (E-fields) can be made during 

flight operations using one or more jieldmeters mounted 

on the surface of a test aircraft. These instruments use 

a rotor to alternately shield and unshield stationary 

sensing elements; this "chopping" technique allows DC 

(and extremely low-frequency AC) E-fields to be 

detected. In practical measurement scenarios, the 

measured electric field at any given point on the aircraft 

(E,,) is equal to a weighted sum of the aircraft charge (Q) 

and the ambient atmospheric field (Ex, E,, E,): 

If four fieldmeters are used, four equations can be solved 

for the four unknowns (Q, Ex, Ex, and E,) at any point in 

time. It has been shown that estimates of aircraft charge 

and ambient atmospheric fields are sensitive to errors in 

calibration factors [7]. Moreover, the degree of sensitivity is related to the choice of fieldmeter locations on 

the surface of the aircraft. 

If helicopters are used as the test aircraft, the size and 

shape of the overall airframe is modulated by the rotor 

blades, and the enhancement factors can no longer be 

considered to be constant (in general). For the present 

discussion, we will limit ourselves to examination of 

variations in the k,, terms, but we note that similar 

effects are expected for the other factors. 

The amplitudes and phases of the AC components of the 

enhancement factors are due to the position of the 

fieldmeters relative to the rotor blades. Computed 

factors at four selected locations on the fuselage of the 

Hind-D are shown in figure 6. The model shows that 

these enhancement factors can vary from less than 1 

percent about their mean (as in the "bottom" location) to 

more than k20 percent (as in the "top" location). 

OL ' I I I I I 

0 50 100 150 200 250 

Time (ms) 

k""(1 lm) 

Figure 6. Selected electric field enhancement factors. 

The "top" and "bottom" locations on the Hind-D model 

were chosen to correspond to the fieldmeter positions 

used in earlier UH- I charge measurement field tests [8]. 

In these field experiments, we assumed a constant 

calibration factor for both locations. The published 

aircraft potential data shows a pronounced oscillation at 

the main rotor blade beat frequency for the "top" sensor 

that is not present in the data for the "bottom" sensor, as 

we would have predicted using this type of model. 

0,8 - - tail (front, near top) --. tail boomJop, afl end) 

bottom (centerof fuselage) top (behind~ai? rotor hub) 

- - 
Computation of External ELF E-fields 

We defined our model so that the total aircraft charge is 

constant, with respect to time. We computed the charge 

centroid as a function of time, and we note from figure 3 

that the modulation of the centroid is approximately 

sinusoidal at the beat frequency of the main rotor. We 

also note that, for the surveillance scenario, typical 

aircraft-to-sensor surveillance distances (r = 1 km) are 

much larger than the source dimensions (d = 0.1 m for 

the centroid or d=lO m for the entire aircraft), and are 

much smaller than the wavelengths of the frequencies of 

interest (h = 10 Mm). Therefore, as a first approximation, it makes sense to treat the entire aircraft as an 

oscillating point source (or current element), and to 

apply the harmonic analysis suggested by Jackson [9] 

for the near (static) zone (where d << r c< I). The 

dominant E-fields due to the oscillating charge have 

dipole characteristics; for this reason, the current 

element is sometimes called a Hertziun dipole [lo]. In 

the near zone, the quasi-stationary E-fields are: 

and 

where 1 is the "length" of the Hertzian dipole, 8 is the 

angle between the dipole direction and the field direction, a is the angular frequency of the oscillation, and P 

is a phase term. Assuming a total charge Q of 5.1 pC 

(as in table 1) and a dipole length 1 of 0.15 m (from 

figure 3), we estimate the largest field component at a 

sensor 1 km in front of the helicopter (r, = 1000 m) to 

be Ex; we expect Ex(t) (= E,(t) here) to have a peak 

amplitude of 14 pV/m, and to be in-phase with Q,(t). 

It is interesting to compare the computed "dipole" fields 

(due to the current, or movement of charge in our 

simplified point-charge model) with the quasi-static 

fields (due to the changing position of the charge in our 

high-resolution model). Once the quasi-static charge 

distribution (i.e., the charge on each element) is known 

at each point in time, the quasi-static E-fields can be 

computed by summing the contributions from each 

charged element. Figure 7 shows an example of these 

computations for a sensor 1 km in front of the helicopter. 

-1 0 r I I I I I 

0 50 100 150 200 250 

Time (ms) 

From figure 3, we see that the movement of the charge 

centroid is not sinusoidal. Furthermore, the point-charge 

model cannot account for any higher-order chargedistribution statistics. Therefore, it may be desirable to 

develop models that can account for the resulting 

higher-order current statistics. The model discussed 

here is a step in that direction. 

Conclusions 

ARL developed and has used high-resolution quasistatic modeling techniques to investigate electrostatic 

fields for some time. Recent extensions to these 

techniques allow the examination of certain ELF fields 

for sensor applications, both on and off airborne platforms. These fields have radial components and 

variation with distance that depend on detailed 

properties of the source. Detailed, dynamic source 

models allow us to model and investigate these fields at 

a level not possible with previous techniques. 

Acknowledgements 

The development of the dynamic Hind-D model, as well 

as other dynamic models not discussed in this paper, 

would not have been possible without the exceptional 

talent and hard work of two undergraduate summer 

students, Greg Werner and Lisa Zorn. The animated 

movie on the Web showing the time-varying 3-D charge 

distribution is a direct result of their efforts, and the 

underlying models provided a framework for the analysis in this paper. 

References 

(1) R. Harrington, "Matrix Methods for Field Problems," 

Proc. IEEE, Vol. 55, No. 2 (Feb 1967), pp 136-149. 

(2) D. T. Paris and F. K. Hurd, Basic Electromugnetic 

Theory, McGraw-Hill, New York (1969), pp 181-184. 

(3) John D. Kraus, Electromugnetics, 4th Edition, 

McGraw-Hill, New York (1992), pp 81-84. 

(4) Zbid, pp 52-58. 

(5) P. McKenna, R. Dalke, R. Perala, and D. Steffen, 

"Evaluation of the Observability/Detectability of Electrostatically Charged Rotorcraft," presented at the 1991 

Int. Conf. on Lightning and Static Electricity, Cocoa 

Beach, Florida (April 16-19, 1991). [The paper is not 

included in the proceedings; more details are included 

in Electro Magnetic Applications, Inc. Report # EMA89-R-36, SBIR contract # N00019-88-C-0301 (March 

1989).] 

(6) http://w3.arl.mil/tto/ARLDTT/FocusTech.html. 

(7) K. Hewitt, J. Kositsky, R. Maffione, and J. Thayer, 

"On the Accuracy of an Aircraft-Borne Ambient 

Electric-Field Measuring System," presented at the 1989 

Int. Conf. on Lightning and Static Electricity, Bath, 

England, SFU International paper p89-009 (June 1989). 

(8) D. Hull and M. Wrenn, "Helicopter Charge 

Measurements," Harry Diamond Laboratories internal 

report # R-ST-SA-91-01 (20 Jun 91). 

(9) J. D. Jackson, Clussicul electrodynamic.^, 2nd Ed., 

John Wiley & Sons, Inc., New York (1975), pp 391-397. 

(10) Paris and Hurd, op. cit., pp 458-463. 

Figure 7. Modeled E-field of Hind-D. 
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B. DATE Report Downloaded From the Internet 1/5/99 

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