Non-Foster and connected planar arrays



[1] A planar array of dipoles, with contiguous collinear dipole ends connected by a fixed inductor or capacitor, is evaluated. A Moment Method simulation of a 20 × 20 array provides Scan Element Pattern (gain per element), Scan Impedance, and current distributions. Cases considered, over a 10:1 bandwidth (dipole length .05λ to .5λ), include zero impedance coupling, capacitor coupling, inductor coupling, and Non-Foster negative inductor coupling. The last is outstanding, probably because the physics of the Non-Foster coupling makes sense.

1. Introduction

[2] Finite arrays of dipoles, arranged in a regular lattice, with or without a ground plane, have been analyzed and used with great success for decades. However, their wideband performance has been limited by the variations in Scan Impedance with frequency, primarily the reactive part [Hansen, 1999]. Interconnection of collinear linear or planar arrays was considered at least as early as 1970, by Carl Baum, and it was recognized that the current continuity provided adequate low frequency performance [Baum, 1997]. Such arrays have applications for wideband electronic scanning [McGrath and Baum, 1999; Friederich et al., 2001; Kesler et al., 2000]. The dipoles can also be connected by capacitors [Munk, 2003, section 6.4; Taylor et al., 2003], or by Non-Foster elements [Hansen, 2003a]. It is the purpose of this paper to evaluate the wideband capabilities of each type of connection.

[3] A large planar array of dipoles, with and without a ground plane, is simulated by the Moment Method. This simulation provides current distributions on the dipoles, Scan Impedance, and Scan Element Pattern [Hansen, 1998]. Computer simulation is important as it is difficult to measure Scan Impedance [Hansen, 2002]. All elements must be excited with the appropriate amplitude and phase, but a direct measurement requires a technique such as the load pull method recently suggested [Van Wagoner and Hansen, 2003].

2. Computer Simulation

[4] The array is square, with N elements per row and N columns; the dipoles are equi-spaced. Dipoles are moderately fat. Piecewise sinusoidal expansion and test functions are used in a Galerkin Moment Method formulation. Because all dipoles and images in the array are straight and parallel, the Carter mutual impedances are used. A double precision version of a compact code [Hansen, 1972] is used; requisite Sine and Cosine Integrals are computed using a double precision routine involving Chebyshev polynomials [Luke, 1975, Table 4.4]. The array lattice is square, and the dipole length equals the lattice spacing. Each dipole has a feed point at its center. Contiguous collinear dipole ends are connected by a reactance or short. Ground plane effects are simulated by out of phase images as usual. All dipoles are excited with unity amplitude; drive phases correspond to scanning in any desired plane. The matrix equation consisting of this drive voltage vector and the array impedance matrix is solved for the current vector. From this, the Scan Impedance is immediately available. Scan Impedance behavior is critical, as any impedance mismatch to the transmitter or receiver results in a loss of gain. For the finite arrays the Scan Impedance is obtained directly from the currents. Scan Element Pattern is obtained by calculating radiated power and the array pattern.

[5] As mentioned, measurement of Scan Impedance is difficult but in a computer model all quantities are available. The load resistance is applied at the center (feed point) of each dipole and is added to each feed self impedance term. Total power is calculated by taking the real part of the summation of VI* over the current at the center of each piecewise sinusoidal expansion function. Load power is found from the summation of II* over the dipole feed points, multiplied by the load impedance Zo (real). Finally radiated power is equal to total power minus load power. Scan Element Pattern (gain per element) is calculated from the intensity at the main beam peak, divided by radiated power, times the impedance mismatch loss (1 − ∣Γ∣2), divided by the number of elements. All calculations are in double precision. Results of the code are validated against codes for conventional dipole arrays by inserting large values of resistance (105 ohms) in place of the coupling impedances, thus simulating a conventional planar array. The simulation code allows results to be calculated for a number of angles with a specified feed spacing, or for a number of feed spacings in wavelengths at a specific scan angle. Parameters calculated are the magnitude and phase of a current distribution along the wire, the impedance at the center element and the gain per element (Scan Element Pattern).

[6] Six piecewise sinusoidal expansion and test functions are used per dipole; expansion functions extend across one dipole to the next contiguous dipole, allowing current to flow between dipoles. Thus a row of 10 collinear dipoles utilizes 59 unknowns. The 20 × 20 array has 119 unknowns per row, for a total of 2,380 unknowns. Calculations are made for a 10:1 bandwidth, for dipole lengths from .05λ to .5λ. Thus the array length over the bandwidth, is 1 to 10 wavelengths, for the 20 × 20 array. Figure 1 sketches a portion of the dipole, and expansion functions, in one row of the planar array.

Figure 1.

Portion of one row of planar array.

[7] Adding the load resistance to the feed terms in the impedance matrix makes no difference in dipole arrays. However, for connected arrays, it does markedly change the shape, magnitude and phase of the currents, and the Scan Impedance and Scan Element Pattern. This is probably due to the resonant edge currents on connected arrays as seen in the current amplitude plots that follow, and the ability of currents to flow from one feed to another.

[8] The center frequency was chosen such that the dipoles were λ/4 long, allowing an octave up to λ/2 length, and one or more octaves to .125λ and shorter. Grating lobes may appear for element spacings just above .5λ, so this is a useful upper limit. Dipole length/diameter is 125; for 6 segments per dipole this allows segment length/diameter to be a comfortable 10 .4. Several values of a load characteristic impedance were used. A frequency range of 10:1 is used throughout.

3. Connected Arrays

[9] The arrays will be evaluated in terms of Scan Element Pattern (gain per element), Scan Impedance, and dipole currents. Figure 2 shows Scan Element Pattern (SEP) at broadside for the 20 × 20 array, for dipole lengths from .05λ to .5λ. SEP includes impedance mismatch. Figure 2 uses Zo = 400 ohms. This provides a better match over the upper part of the band than 200 ohms; both are good over the lower part of the band. The small oscillations are due to edge effects. Also shown is the area gain 2πA/N2λ2 for an N × N array. When the Scan Impedance mismatch loss (in dB) is subtracted from the area gain per element (in dB), the result is generally within one dB of the SEP value. This indicates the correctness of the SEP calculation.

Figure 2.

The 20 × 20 connected array at broadside, Zo = 400 ohms.

[10] For isolated short dipoles the Scan Reactance (at broadside) is 240/π = 76 ohms; for connected dipoles, where the current is uniform, the Fourier transform of the isolated dipole cosine distribution (2/π COSC ka/2) in equation (7.41) of Hansen [1998] is replaced by the FT of a constant, which is SINC ka/2. For short dipoles this gives 240/π × (π/2)2 = 60π. Note a typo in equation (7.41): the next last factor should be Sin (2kzo h).

[11] Similar calculations were made for the 20 × 20 array with ground screen. In Figure 3 the Scan Element Pattern shows gain 4 to 5 dB below area gain, which is now 4πA/N2λ2 for an N × N array. Note the rapid falloff of gain at the bottom of the band.

Figure 3.

The 20 × 20 connected array/screen at broadside, Zo = 400 ohms.

[12] Scan Impedance (SI) behavior is critical, as any impedance mismatch to the transmitter or receiver results in a loss of gain. Figure 4 is a Smith chart plot of the 20 × 20 array impedance for the match resistance of 400 ohms over a 10:1 bandwidth. The inner red circle is VSWR = 2 ( .51 dB loss). The other two red circles are VSWR = 3 (1.25 dB loss) and VSWR = 5.828 (3 dB loss). The upper half of the band is poorly matched, even to 400 ohms. With a ground screen, spaced at a distance equal to half the dipole length, the impedance at the lower frequencies is much worse; see Figure 5. A modest increase in fatness of the dipoles has only a small effect upon SEP over the lower part of the 10:1 band. Doubling the radius gains a little more than a dB of SEP at .5λ, and less than half a dB at midband. Use of fat bowtie dipoles would likely offer significant improvement in impedance match.

Figure 4.

The 20 × 20 coupled array, Zo = 400 ohms, broadside.

Figure 5.

The 20 × 20 coupled array/screen, Zo = 400 ohms, broadside.

[13] The results are based on the center element of a row next to the array centerline. Both SEP and SI will vary from element to element. To test the sensitivity of element choice, SEP and SI are calculated for an adjacent element. The small oscillations in Scan Element Pattern shifted, but the results are very close to those of Figure 2. Similarly the Smith Chart plot for the adjacent element is very close to that of Figure 4. It may be concluded that the figures are representative of large arrays.

[14] Currents along the center row of the array have been plotted in Figure 6 for Zo = 400 ohms. They are typical. In this and in all following current plots, six values are computed along each dipole; the points are connected by straight lines. This plot shows current for .5λ dipole spacing. Current peaks occur at every other dipole, with smaller peaks in between. When Zo is lowered to 200 ohms, the intermediate peaks become even smaller. Similar peaks spaced 1λ apart occur when a finite length wire is illuminated by a plane wave [Hansen, 2003b]. When the 20 × 20 array is shortened by λ/4 in each dimension, current peaks are still at every other dipole, but the values in between are small. Edge effects are more pronounced. With shortening of array length by λ/2, current peaks are still at every other dipole, but values in between are several dB down. For dipole length of .25λ, Figure 7 shows current peaks at each element, but the amplitudes show a modulation near the edges typical of the edge effects [Hansen, 1996]; the array is now only 5λ long (and wide). More important, the current nulls are shallow. This trend of current continuation continues with dipole lengths of .1λ and .05λ. Figure 8, for an array size of 2λ × 2λ, shows a nearly smooth current from feed to feed. Finally Figure 9, for dipole spacing of .05λ, and array size of 1λ × 1λ, has an almost continuous current amplitude (and constant phase) over the entire array. This current continuity at low frequencies greatly reduces Scan Reactance, and allows a Scan Element Pattern close to area gain per element. Thus when the array length is a wavelength or less, the connected array approximates the hypothetical Wheeler current sheet [Wheeler, 1965].

Figure 6.

The 20 × 20 array of connected dipoles, L = .5λ.

Figure 7.

The 20 × 20 array of connected dipoles, L = .25λ.

Figure 8.

The 20 × 20 array of connected dipoles, L = .1λ.

Figure 9.

The 20 × 20 array of connected dipoles, L = .05λ.

4. Capacitor Coupled Arrays

[15] Another type of array uses capacitors, either lumped or distributed, to connect the contiguous collinear dipole ends [Munk, 2003]. Since the value of the capacitance is fixed, it is chosen to make the dipole impedance real at one frequency. Above and below that frequency the reactance varies as expected for a fixed capacitance. The match frequency used is for a dipole length of λ/4. Many values of coupling reactance have been tried for the 20 × 20 array, from 1000 to 0 ohms. A value of 428 ohms yields a real impedance of 154 ohms. Figure 10 shows Scan Element Pattern for Zo of 200 ohms. This value appears to provide a good compromise between performance at the two ends of the 10:1 frequency band. Again SEP includes impedance mismatch. The gain per element (SEP) falls off severely below roughly .2λ, and above roughly .35λ, primarily due to a poor match of the impedance. Since SEP also includes the frequency squared factor (G = 2πA/λ2), it drops rapidly at low frequencies. The dashed line represents area gain per element, 2πA/N2λ2. The SEP is close to area gain/N2 at midband, but shows large departures at upper and at lower frequencies. Results for 10 × 10 and 14 × 14 arrays are similar. When the match point is moved to a dipole length of .15λ, SEP at broadside is improved from .1λ to .2λ, but it is degraded from .2λ to .4λ. A higher match, at dipole length .35λ, makes a significant improvement from .35λ to .5λ, but major degradation below .3λ. Even with the improvements, the SEP is 2 dB below area gain at .5λ. Changing Z0 does not help this. Over a narrower band, the match frequency and the Z0 could be chosen to optimize the performance.

Figure 10.

The 20 × 20 cap coupled array at broadside, Lmatch = .25λ.

[16] When a ground screen is added, with spacing h equal to half the dipole length, the matching reactance (at L = λ/4) is −819 ohms, and the resistance there is 145 ohms. The Scan Element Pattern has a rapid fall off for small dipole spacings, as expected. See Figure 11. Now the dashed line represents area gain per element: 4πA/N2 λ2.

Figure 11.

The 20 × 20 cap coupled array at broadside, Lmatch = .25λ, h/2.

[17] Smith chart impedance is shown in Figure 12; only a small frequency range, from .2λ to .3λ, is within the VSWR equation image 2 circle. With a ground screen the impedance performance at upper and at lower frequencies is poor; see Figure 13; the impedance mismatch below .2λ is very large; the VSWR equation image 2 region is less than for Figure 12.

Figure 12.

The 20 × 20 cap coupled aray at broadside, Lmatch = .25λ.

Figure 13.

The 20 × 20 cap coupled array/screen at broadside, Lmatch = .25λ.

[18] Current across the center row of the 20 × 20 array is shown in Figure 14 for dipole length of .5λ, and in Figure 15 for dipole length of .05λ. This last figure shows that the capacitive coupling does not produce a current sheet. The current distribution is close to that expected for isolated short dipoles. The physics is wrong: to produce current continuity at low frequencies the reactance should be small; near the frequency of half-wave the array works well without any connection, so the reactance should be high. Inductive coupling, which appears to provide the right behavior, was tried, from +1000 to 0 ohms. But it produced many sharp resonances in impedance. Next it will be shown that a negative inductive coupling provides excellent results.

Figure 14.

The 20 × 20 array of cap coupled dipoles, L = .05λ.

Figure 15.

The 20 × 20 array of cap coupled dipoles, L = .5λ.

5. Non-Foster Coupled Arrays

[19] As discussed in the previous section, a capacitive reactance can match the Scan Reactance at one frequency, but the frequency behavior of a capacitor is wrong. A possible solution lies in the use of negative inductances to couple the collinear dipole ends. This type of Non-Foster behavior can be produced by a Negative Impedance Convertor circuit. Such circuits were used circa 1930 for telephone long line compensation. Modern transistors have made negative inductance circuits feasible even at microwave frequencies. See Hansen [2003a] for an introductory bibliography.

[20] In the 20 × 20 array the contiguous dipoles are connected (in the computer model) by negative reactances; the reactance increases with frequency. Again the Scan Reactance (at broadside) is matched for dipole length λ/4; the matching reactance is −428 ohms as it was in section 4. Figure 16 shows Scan Element Pattern over the .05λ to .5λ band of 10:1. It is remarkable that the performance is very close to area gain per element over most of the band, with only a 2 dB drop at the upper end. These results used a characteristic impedance of 200 ohms, which was also used in section 4; for reactive coupling this provides better overall performance than 400 ohms. The Non-Foster negative inductance cancels the Scan Reactance except near λ/2 where the Scan Reactance no longer varies as 1/f. When a ground plane is added, the performance below about .2λ drops as expected. Near .5λ the performance is improved, as seen in Figure 17.

Figure 16.

The 20 × 20 array, Non-Foster coupling, Lmatch = .25λ.

Figure 17.

The 20 × 20 array/screen, Non-Foster coupling, Lmatch = .25 λ, h = L/2.

[21] Figure 18 shows the Smith chart locus, and the VSWR is below 2 over an 8:1 bandwidth, starting at .05λ up. When the ground screen is added the impedance is poor at low frequencies, as seen in Figure 19. Now the VSWR is below 2 over a 2 .5:1 bandwidth, starting at .5λ down.

Figure 18.

The 20 × 20 Non-Foster coupled array, broadside.

Figure 19.

The 20 × 20 Non-Foster coupled array/screen, broadside, h = L/2.

[22] Plots of the current distribution along the center row are illuminating. In Figure 20 for dipole length .05λ the current is nearly constant over the entire 1λ long array. Thus the Non-Foster coupling does indeed produce the Wheeler current sheet, at low frequencies. Figure 21, for .1λ dipoles, shows a modest dip, but the current continuity is maintained. Finally at .5λ dipole length, the results are close to isolated dipole currents. As in the capacitor coupled case, the larger coupling reactance at .5λ produces a modest dip at the feed points. See Figure 22.

Figure 20.

The 20 × 20 array of Non-Foster coupled dipoles, L = .05λ.

Figure 21.

The 20 × 20 array of Non-Foster coupled dipoles, L = .1λ.

Figure 22.

The 20 × 20 array of Non-Foster coupled dipoles, L = .5λ.

6. Conclusions

[23] Arrays of dipoles connected end-to-end by reactance are evaluated over a 10:1 band, from dipole lengths of .05λ to .5λ. Load characteristic impedance must be included in the Moment Method simulation, as Zo affects the current distribution. Zero or low impedance coupling at low frequencies produces current continuity along each row of collinear dipoles, thus approximately realizing the current sheet hypothesis of Wheeler. At midband and high band the zero impedance coupling produces large Scan Impedance (broadside) and large mismatch loss. Capacitor coupling does not produce current continuity at low band, thus large reactance mismatch occurs. Also, at high band significant mismatch occurs. Only in midband is the performance good. Inductive coupling produced many high Q resonances versus frequency. Non-Foster negative inductive coupling gave amazingly good Scan Element Pattern: very close match to area gain per element from low band to almost high band; only 2 dB below at .5λ.

[24] When a ground screen is added, Scan Element Pattern degrades for all couplings at low band. Capacitor coupling is the worst performer there; zero impedance coupling is also poor. Non-Foster coupling shows the least low-band drop-off.


[25] The suggestions of a reviewer are gratefully acknowledged.