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NBSIR 78-898

ANALYTICAL AND NUMERICAL TECHNIQUES FOR ANALYZING

AN ELECTRICALLY SHORT DIPOLE WITH A NONLINEAR LOAD

Motohisa Kanda

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

^lovember 1978

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I 100

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78-898

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APR 1 7 1979

NBSIR 78-898

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ANALYTICAL AND NUMERICAL TECHNIQUES FOR ANALYZING

AN ELECTRICALLY SHORT DIPOLE WITH A NONLINEAR LOAD

Motohisa Kanda

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

November 1978

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Director

Contents

Abstract 1

1. Introduction —— —~ 1

2. Theory • 2

2.1 Analytical Technique- 3

2.2 Numferical Technique 5

3. Results 7

4. Conclusion 8

Acknowledgments 8

References • 9

List of Figures 10

ill

Analytical and numerical techniques

FOR analyzing AN ELECTRICALLY SHORT DIPOLE WITH A NONLINEAR LOAD

Motohlsa Kanda

Electromagnetic Fields Division

National Bureau of Standards

Boulder, Colorado 80303

An electrically short dipole with a nonlinear dipole load is analyzed

theoretically using both the analytical and numerical techniques. The

analytical solution is given in terms of the Anger function of imaginary

order and imaginary argument, and is derived from the nonlinear differential equation for the Thevenin's equivalent circuit of a dipole with

a diode. The numerical technique used was to solve the nodal equation

using a time-stepping finite difference equation method. The nonlinear

resistance of the diode is treated using an iteration technique. A

comparison between the analytical and numerical solutions is given.

Key words: Anger function; electrically-short dipole; iteration method;

nonlinear differential equation; nonlinear load; time-stepping finite

difference equation technique.

1. Introduction

It is well accepted that microwave radiation can produce biological effects. Although

it is very difficult to d(_Cermine the biological hazards associated with electromagnetic (KM)

fields, the biological effects resulting from the EM fields can be adequately described from

knowledge of one or more parameters that characterize the EM field. One of the advocated

field parameters for quantifying hazardous EM fields is an electric energy density which can easily be calculated from the electric field strength [1]. For this reason, the National

Bureau of Standards has recently developed a broadband, isotropic electric energy density

meter (EDM). The EDM consists of three orthogonal, electrically short dipoles with diode

detectors between the arms of the dipoles. To synthesize sul i an EDM in terms of its frequency response and its dynamic range, one needs to analyze theoretically an electrically

short dipole with a nonlinear load such as a diode.

Traditionally, the characteristics of an antenna with a nonlinear load have been

analyzed in the frequency domain by considering the spectral components the solutions at harmonic frequencies [2]. For example, Sarker and Weiner [3] have used the Volterra series

analysis to obtain the scattering due to nonlinearly loaded antennas. The nonlinear

transfer function of a nonlinearly loaded antenna was determined at several harmonic frequencies. However, the calculation of the nonlinear transfer function is generally very

tedious, particularly when the circuit model of the nonlinear load is complicated and its

nonlinearity is strong. For this reason, recent analyses of the nonlinearly loaded antennas

have been considered using direct time-domain techniques. Schuman [4] has described the

application of the time-domain method of moments technique to determine the scattering

current on a thin wire with discrete nonlinear resistive loadin^v Lin and Tesche [5,6] have

used frequency-domain data to compute the time-dependent currents and voltages across the

nonlinear load by means of the Laplace transform. A unified numerical procedure was recently proposed by Landt [7]. The antenna characteristics were derived from a time-domain

electric field integral equation, whereas the nonlinear network analysis was performed in a simple time-domain nodal analysis.

In this paper, two techniques for analyzing an electrically short dipole with a nonlinear load are described. The first technique, described in Section 2.1, gives an analytical solution to the first-order, nonlinear time-domain differential equation in terms of

Anger functions. The second technique, described in Section 2.2, is a time-stepping, finite

difference solution technique for obtaining the numerical solution to the time-domain nonlinear differential equation. The nonlinear effect due to a diode is solved by a conventional iteration method. This numerical technique gives the physical insight for the
nonlinear load effects on antennas in terms of the time-domain waveform, and also permits

the consideration of certain problems which are too complicated to be treated by an analytical technique.

The nonlinear effects on an electrically short dipole are first investigated in

Section 3 in the time domain using a time-stepping, finite difference solution technique.

The frequency responses and the dynamic ranges of the dipole with a nonlinear diode are then compared using the two different techniques described above.

2. Theory

Using the frequency-domain concept of the effective length and the driving point

Impedance of an electrically short dipole without a nonlinear load, the th^venin's equivalent circuit for a diode with a nonlinear load is shown in Fig. 1. The element v^(t) is

the induced, open-circuit voltage at the dipole terminal and is given by

:: ,

• V -i(^> = ^inc^^) • K • <i>

where e. is the normal, incident electric field strength and h is the effective length

inc e of the dipole. The element is the equivalent driving point capacitance of the dipole.

A parallel combination of a linear capacitance and a nonlinear resistance represents

a simplified model of a diode.

For an electrically short dipole antenna (i.e., kh<l where k is free-space wave number),

the effective length h and the driving point capacitance C of the antenna are given

by [81 , ^ . ^

• : ^

^ ^e " 2(n-2+£n 4)

and ,

. ,.

4ttc h

^a "

4)

f^"'^^'

The symbols have the following meanings: h is half the physical length of a dipole antenna

in meters, is the free-space permittivity in farads/meter, and J2 is the antenna thickness

factor (i.e., f2 = 2 £.n(2h/a) where a is the antenna radius in meters).

When h = 0.02 m and a = 2.84 x 10 ^ m (i.e., the antenna thickness factor Q = 14.50),

the effective length h and the equivalent antenna input capacitance of the antenna C

^ -3 -12 ^ become, respectively, 9.72 x 10 m and 0.2 x 10 F.

When an electrically short dipole is terminated with a nonlinear load such as a beam

lead Schottky barrier diode, the effect of loading on the antenna can be analyzed using the

simple equivalent circuit shown in Fig. 1, which consists of a parallel combination of a nonlinear resistance R,adand a linear capacitance C.. Here, the nonlinear resistance R. of

d

the diode is characterized by its V-i characteristic, i.e.,

i(t) = I^(e«''°^^> - 1) (4>

2
The symbols have the following meanings: i(t) is the current (ampere) and V^(t) is the

voltage (volt) across the diode Junction; is the saturation current which is assumed to be 2 X 10 ^ A; and a => = 38 V

^ where q is the electronic charge (1.6 x 10 ^ coulomb), n is the diode ideal ity factor ('v 1.05), k is Boltzmanr.'s constant (1.38 x 10~^^ J/K) , and

T is the temperature (-v 290 K)

The Junction capacitance and the package capacitance are combined and are shown

as C, in Fig. 1. The exact value of C, varies from diode to diode and is very difficult to d cl determine. In this paper, is assumed to be constant and equal to 0.34 pF for the beam

lead Schottky barrier diode. In a more elaborate diode model, the Junction capacitance

is nonlinear and is a function of the built-in potential V, as

D

C (0)

^ (1-f-)^^

b

for a step Junction. The package capacitance is generally constant. The more general

treatment of an antenna with linear and nonlinear loads, such as a nonlinear resistance and

a nonlinear junction capacitance as well as a linear package inductance and a linear series

resistance of a diode, is being pursued.

In this paper, a sf.mple nonlinear resistance with a constant diode capacitance

for the beam lead Schottky barrier diode shown in Fig. 1 is used in the following sections

for analyzing the loading effect of an electrically short dipole with a nonlinear diode

using analytical and numerical techniques.

2.1. Analytical Technique

Using th^venln's equivalent nonlinear circuit shown in Fig. 1, the voltage equation and

the corresponding current equation are given below:

q.(t) q.(t)

v,(t) = ^= v^(t) (6)

and

d

dq (t) dq (t)

^f— + -^+i(t) - 0 , (7)

where q and q, are the charges on C and C. , respectively, and i is the current through

the nonlinear resistance R,

d' By substituting

av (t)

y(t) - e ° , (8)

one gets

+ ay^(t) + y(t)f(t) - 0 (9)

3
where

al

a d

f = - ^

(1 + c,/c ) d a

I dv. - - -^j (11)

a

Equation (9) given above is a first-order, nonlinear differential equation. The

detailed mathematical steps for solving the equation have been performed by P. F. Wacker.

When the induced voltage v^ft) is a periodic sinusoid, i.e., v^(t) = V^sin oit the detected

dc voltage averaged over a complete cycle is given by

, 2iT T J._ (jU) V - - - an „ ^ •'^

, (12)

e - 1

where Jj^(jU) is the Anger function of imaginary order (jT) and imaginary argument (jU); T is the normalized period as

cil s

03(C + C.) a d

(13)

and U is the normalized induced voltage as

aV C

a d

Using the series representation for the Anger function, one gets

where

and

S, - 1 + I (16)

"^'^ n (k^ + T^)

^^^"k=2

even

S .

I K . (17)

m-1 /, 2 , ^2. odd > k-l

odd

4
At high frequencies where

T = al

s

aj(C + C.) a a

<< 1 (18)

then one can show that, for small V.

1 + C,/C

d a

(19)

and, for large

V = -

1 + C,/C

Q a

(20)

Equation (19) indicates that, for the small induced steady-state voltage V^, the output dc voltage is a square-law function of the induced voltage On the other hand, eq (20) indicates that, for large induced voltage V., the output dc voltage is proportional to the induced voltage V^.

2.2 Numerical Technique

A time-stepping difference equation technique can be used for solving the nonlinear

network, shown in Fig. 1. The basic idea of a time-stepping, finite difference equation

technique is briefly discussed below. More detailed discussion on this subject is given

by Calahan [9]

The linear and nonlinear elements are converted into resistance-current source

equivalent networks in the nodal equation method. For instance, in a regular R, C, and

L circuit, we have

J = R i n n

(21)

n+1 (22)

and

n+1

n+1 (23)

where t is a sampling time interval. Once initial v^ or i^ is given, one can determine v^ and i.|, then and etc.; such a method is, therefore, called a time-stepping, finite

difference equation method.

In order to deal vjlth nonlinear equations which result from nonlinear elements such as

a diode, the general approach to a solution of such problems is by iteration. The basic

technique used is discussed below. Given a nonlinear system f(i,v), the solution of f(i,v) = 0 yields the solution for the system response.

5
First f(i,v) is expanded at an initial solution

that is.

f(i,v) = f (i^.v"") + J[f (i,v)]

where J is the Jacobian of f and has the form

(24)

'3f, 3f

,

3i, 3vn+m

J =

3f

r 3i,n-hn

3f, 3vn+m

n+m

(25)

Now f(i,v) = 0 determines

Ai^

J[f(i°,v°)] - , . o o

.

f(l ,v ) Av

(26)

The solution of

fAi°

updates the initial value of

f.n+1

1n+1

Av

(27)

6
The operation is repeated for n = 0, ••• until the change

Ai

is significantly small.

Results

In this section the time-domain waveforms of a sinusoidal wave at various nodes are shown first using the time-stepping difference equation technique discussed in Section 2.2.

Here, a single time-domain sinusoidal wave is divided into 16 discrete digitized points

(17 points including both ends) in the analysis. Then, the detected sinusoidal wave, averaged over many cycles which corresponds to dc output of the electrically short dipole with

the beam lead Schottky barrier diode, is given.

Figure 2 shows the time-domain waveform of the first sinusoidal wave at 100 MHz at various nodes. At node 1, an applied sinusoidal wave Vj^ is shown with unit amplitude. At

node 2, the detected sinusoidal wave, which is skewed or distorted due to nonlinearity of

the diode, is shown. It is obvious that, at node 2, the detected voltage starts developing

in the negative polarity, which eventually leads to dc negative output for the dipole with

the diode.

Figure 3 shows the time-domain waveform of the dipole with the diode due to the 10th

sinusoidal wave driving voltage. Again, a sinusoidal wave at node 1 is an applied driving

voltage with unit amplitude. The detected time-domain waveform at node 2 after diode detection indicates a much more pronounced negative charge accumulation. Finally, an almost dc

detected voltage starts appearing in the negative polarity after 10 cycle time average.

Figure 4 shows the time-domain waveforms of the dipole with the diode due to the 100th

sinusoidal wave induced voltage. The negative charge accumulation is much more pronounced,

and constant detected dc voltage appears after 100 cycle time average.

To arrive at the steady-state time average of these sinusoidal wave excitations using

the time-stepping, finite difference equation technique along with the iteration method,

400 sinusoidal waves, which correspond to 6400 discrete points (16 points per one cycle),

are applied successfully to compute 6400 discrete output voltages, which are then nximerIcally time-averaged.

Figure 5 shows the detected dc voltage from the dipole with the diode as a function

of Induced voltage V.(= e, x |h I, where e, is the normal incident electric field and * 1 inc ' e ' inc

h^ Is the dipole effective length). The detected dc voltages as a function of the

Induced voltage are calculated using both the analytical technique given in Section 2.1

and the numerical technique described in Section 2.2 Generally, the agreement between the

analytical and numerical solutions is very good. Below an induced voltage of about 0.1

volt, the detected dc voltage is equal to square of the induced voltage V^. On the

other hand, above an induced voltage V of about 0.1 volt, the detected dc voltage V is

• 1 o proportional to the induced voltage V^. Thus, the diode detection is square-law at a small

signal level, but becomes linear at a large signal level.

Figure 6 shows the transfei function of an electrically short dipole with the beam lead

Schottky barrier diode as a function of frequency. Here, the transfer function is defined

as a ratio of the detected dc voltage to the amplitude of the induced voltage

V.("= s<ri„*h ) expressed in dB when e Is equal to 1 V/ra. Thus, the transfer function so

7
defined is for a detected dc voltage of several mV, which corresponds to a square-law

signal level. As indicated in eq (2), the effective length of an electrically short dipole

is independent of frequency. The transfer functions of the dipole with the diode are caldulated both analytically and numerically. Except at frequencies below 10 KHz, the

analytical and numerical transfer functions agree to within 8 dB, and are constant with

frequency up to several gigahertz.

The discrepancy between the analytical and numerical results of the transfer function at the extremely low-frequency range could be due to insufficient time-averaging in the numerical solution; that is, although successive 400 sinusoidal waves are used in the numerical

computation, the convergence at the extremely low-frequency range may still be poor.

The sharp cut-off (20 40 dB per octave) below 10 kHz in the transfer function

predicted from both analytical and numerical results can be explained as follows. Figure 7 shov7s the detected time-domriin sinusoidal waveforms at node 2 at various frequencies. It is

clearly indicated that, for example the detected time-domain sinusoidal waveform at 1 kHz

(whose amplitude is magnified by a factor of 10) is very similar to the original sine wave

excitation, whereas the time-domain waveforms at I and 100 MHz are more skewed (or distorted)

compared to the original sinusoidal wave. The detected time-domain sinusoidal waveform is

less strongly skewed (or distorted) at lower frequencies than at higher frequencies. Since

the skewness of the detected waveform and the rate of charge accumulation decrease in the

lower frequency range below 10 kHz, so does the transfer function of the dipole with the

diode, as shown in Fig. 6.

4. Conclusion

This paper introduces two independent techniques to analyze the electrically short

dipole with a nonlinear load. The nonlinear load considered in this paper is a beam lead

Schottky barrier diode. The analytical solution, given in Section 2.1 in terms of the x\nger

function of imaginary order and imaginary argument, was derived from the nonlinear differential eqation for the thevenin's equivalent circuit of the dipole with the diode. The

numerical technique, explained in Section 2.2, is basically to solve nodal equations using

the timestepping finite difference equations technique. The nonlinear resistance of a diode was treated by an iteration method.

Both analytical and numerical solutions basically agree very well. The transition from

the square-law detection region to the linear detection region was observed as the induced

voltage was varied. The transfer function of the electrically short dipole with a diode was also investigated. The decrease in the transfer function at frequencies Helow 10 kHz was explained through the time-domain sinusoidal waveforms obtained from the time-stepping,

finite difference equation technique.

One of the advantages of using the analytical solution in terms of the Anger function

of imaginary order and imaginary magnitude is that the solution is given in the closed form,

and is very easy to evaluate. However, it is very difficult, or may be even impossible, to find the closed-form solution of a nonlinear differential equation for much more complicated

models of an antenna and a diode, e.g., including a nonlinear capacitance, a linear inductance, as well as nonlinear and linear resistances, and a linear capacitance. In such

cases, a time-stepping finite difference equation technique along with an iteration method

provides an accurate time-domain solution for more general nodal equations.

The analysis of a linear antenna with a nonlinear load, in which a diode model consists

of a parallel combination of a nonlinear resistance and a nonlinear junction capacitance

along with a linear series inductance, a linear series resistance, and a linear package

capacitance, has been carried out using a time-stepping, finite difference equation technique

along with an iteration method and will be presented in the future.

Acknowledgments

The author wishes to express his appreciation to P. F. Wacker, N. S. Nahman,

R, R, Bowman, S. M. Rlad, C. K. S. Miller, and F. X. Ries for many stimulating discussions

and suggestions in the preparation of this paper.

8
References

] Wacker, P. F. and Bowman, R. R. , Quantifying hazardous electromagnetic fields:

Scientific basis and practical considerations, IEEE Trans, on Microwave Theory and

Technique, MTT-19 , No. 2, pp. 178-187, February 1971.

] Wiener, N., Nonlinear Problems in Random Theory, Cambridge, Mass.: MIT Press, 1959.

] Sarkar, T. K. and Weiner, D. D., Scattering analysis of nonlinearly loaded antennas,

IEEE Trans on Antennas and Propagation, AP-24 , No. 2 pp. 125-131, March 1976.

] Schuman, H., Time-domain scattering from a nonlinearly loaded wire, IEEE Trans, on Antennas and Propagation, AP-22 , pp, 611-613, July 1974.

] Liu, T. K. and F. M. Tesche, Analysis of antennas and scatterers with nonlinear loads,

IEEE Trans, on Antennas and Propagation, AP-24 , No. 2, pp, 131-139, March 1976.

] Lied, T. K., Tesche, F. M. and Deadrick, F. J., Transient excitation of an antenna with

a nonlinear load: Numerical and experimental results, IEEE Trans, on Antennas and

Propagation, AP-25 , No. 4, pp. 539-541, July 1977.

] Landt, J. A., A unified time-domain electromagnetic and circuit analysis code,

1977 USNC/URSI Meeting, Standford, California, pp. 153, June 1977.

] Kanda, M., The characteristics of broadband, isotropic, electric field and magnetic

field probes, NBSIR 77-868, November 1977.

] Calahan, D. A., Computer-aided Network Design, revised edition. New York: McGraw-Hill,

Inc., 1972.
List of Figures

Figure 1. Thevenin's equivalent circuit of an electrically short dipdle with a diode.

Figure 2. Time-domain waveform of 1st detected 100 MHz sinusoid.

Figure 3. Time-domain waveform of 10th detected 100 MHz sinusoid.

Figure 4. Time-domain waveform of 100th detected 100 MHz sinusoid.

Figure 5. Detector voltage response at 100 MHz.

Figure 6. Transfer function of an electrically short dipole with a nonlinear load.

Figure 7. Time-domain waveform of detected sinuosids at various frequencies.

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10

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-2

10

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10

-4

10

-5

10

-6

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10

-7

10

-8

10

-9

lO"* 10'

J L

10

-2 in-l 10 ' 10 1 INDUCED VOLTAGE (V.)

Figure 5. Detector voltage response at 100 MHz,
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NBS.114A <R^v. 7-V3>

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBSIR 78-898

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Analytical and Numerical Techniques for Analyzing

an Electrically Short Dipole With a Nonlinear Load

5. Publication Date November 1978 »

6. Performing Organization Code

723.03

7. AUTHOR(S)

M. Kanda.

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

Boulder, Colorado 80303

10. Project/Task/Work Unit No. 7233281 \ 11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, Stste, ZIP)

Same Item 9.

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or /ess factual summary of most si^ificant information. If document includes a significant

bibliography or literature survey, mention it here.)

An electrically short dipole with a nonlinear dipole load

is analyzed theoretically using both the analytical and numerical

techniques. The analytical solution is given in terms of the

Anger function of imaginary order and Imaginary argument, and is

derived from the nonlinear differential equation for the Thevenin's

equivalent circuit of a dipole with a diode. The numerical technique used was to solve the nodal equation using a time-stepping

finite difference equation method. The nonlinear resistance of

the diode is treated using an iteration technique. A comparison

between the analytical and numerical solutions is given.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Anger function; electrically-short dipole; iteration method; nonlinear differential

equation; nonlinear load; time-stepping finite difference equation technique.

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