Negative Resistance Antenna Elements
Grant Bingeman, KM5KG
 

A single antenna may have many radiating elements, and the designer may choose to drive more than one of those elements. When this is the case, it is possible for a "negative" driven element to exist. A careful designer needs to know how to treat negative elements if he expects to obtain the desired antenna performance.

An individual element in a multi-element, multi-source antenna can have a negative input resistance, in which case that input should actually be treated as an output. A thorough understanding of negative antenna elements is necessary for successful design and adjustment of phased arrays. The concept of negative resistance is somewhat arcane, but if you consider a negative resistor simply as a source instead of a load, you will be starting on the right foot.

When an antenna has more than one driven element, it is possible to have more RF power in one element than is actually delivered to the antenna, in which case the remaining elements have a negative power equivalent to the difference between the total antenna input power and the power in the positive-resistance elements. In other words, it is possible for operating impedances in a phased array to have a negative resistance. This means that the negative element is receiving power from the other elements, rather than radiating power. I guess we could call this negative resistance element a collector instead of a radiator.

There are two ways to treat a negative radiator: you can tune out the reactance of the negative element (resonate it) and terminate with a resistor whose value is equal to the absolute value of the operating resistance (Figure 7), or you can match the operating impedance of the negative element to a transmission line and bring the power back to the power divider (Figure 4).

This article will show you how to create a negative resistance in a 10 meter ham band antenna consisting of two vertical monopoles erected over an abbreviated ground system. You can build this array and measure its impedance and currents to verify the conclusions of this article. In doing so you will learn how to match the impedance of a negative resistance element, and how to obtain a low VSWR on the transmission line connected to that element. Negative operating resistances can occur in almost any type of antenna where more than one of its elements are driven.1 Negative resistances don't appear too often in amateur radio antennas, because ham antennas typically consist of only one driven element while the remaining elements are parasitic and passively tuned.


 

The simple example antenna chosen for this article is a pair of ten foot monopoles spaced ten feet apart. This is considered a two-element antenna. Each monopole is mounted over a two-wire, five-foot-radius counterpoise raised a foot off ground per Figure 1. All materials are half-inch type M copper pipe, which has an outside diameter of 0.6 inches. The operating frequency is 28 MHz, which has a wavelength of about 35 feet in free space. Thus the monopoles are about 103 degrees long, and the counterpoise wires are about 51 degrees long physically. These wires may look a bit longer electrically, but that subject is beyond the scope of this article.

The example antenna was not chosen for its radiation pattern, but rather to highlight the effects of a negative element and the methods used to deal with negative power. In general antennas with negative elements may tend to have a relatively narrower bandwidth, because they have higher circulating currents than antennas that can produce the same pattern without a negative element. Typically the closer the elements are spaced in an antenna, the greater likelihood of having a negative operating resistance when more than one element is driven. If only one element is driven, you can never have a negative input resistance. If your antenna analysis program tells you that you have a negative input resistance when only one element in your antenna is driven, then there is either a mistake in your antenna model or an error in the program.

The method of moments analysis software I used was EZNEC 2.0, the length of the current segments was one foot, the ground conductivity was 5 mS/m, and the dielectric constant of the earth was 13. The self impedance of these copper monopoles is 56.0 - j36.1 ohms, and the mutual impedance between them is 13.0 - j27.2 ohms (30.1 / -64.5). We could tune out the capacitive self reactance (the -36.1 ohms) by increasing the counterpoise pipe length to about six feet. However, because the standard length of half inch copper water pipe in my locale was ten feet, this was not convenient. Besides I like to keep my antenna input reactance a bit negative, because series resonating with a coil is easier than using a capacitor.

In the context of this antenna, the terms pipe, wire and tubing are synonymous. The schematic diagrams in this article are labeled with voltage nodes (circled) and current branches (underlined). To keep things simple, all discrete network components are assumed to be lossless, unless noted otherwise.

The two-port model of the coupled antenna elements looks like the tee network in Figure 2. Note that sometimes the resistances in this model will be negative. Many canned network analysis programs will not accept negative resistor values. I wrote my own program 20 years ago that does not have this problem.2 We can use the Figure 3 antenna impedance model in a network analysis program to determine what goes on when we vary the transmission line length, or adjust any of the capacitors and inductors in the feeder system. Remember that what we seek is a low VSWR on the transmission line and the desired relative currents in the antenna elements. When your antenna contains a negative element, intuition can lead to some very wrong designs, so a good network analysis program can save you a lot of grief. It is a brute-force means to check your feeder network design. Refer to my earlier treatment of phased arrays in the Summer 1998 issue of Communications Quarterly for a detailed description of multi-port antenna modeling when more than two elements are involved.
 

where:

Z11 = self impedance of element 1

Z22 = self impedance of element 2

Z12 = mutual impedance between both elements

Z1 = operating input impedance to element 1

Z2 = operating input impedance to element 2
 

In this case Z11 = Z22, since the antenna array is bilaterally symmetrical
 

Z1 = Z11 + Z12 ( I2 / I1)     Equation 3

Z2 = Z22 + Z12 ( I1 / I2)     Equation 4
 

| I2 |        | Z12 | cos (U1 - U2 + U12)
-----  =  ------------------------------     Equation 5
| I1 |                    -R22
 

where:

U1 = element 1 current phase

U2 = element 2 current phase

U12 = phase of mutual impedance between elements 1 and 2

In general, the closer the radiating elements are spaced electrically, the more likely you are to have a negative operating resistance when more than one element is driven. This is because the magnitude of the mutual impedance is greater at closer spacings. A larger mutual impedance is more likely to overwhelm the self impedance per Equations 3 and 4, when the resistance of the second term of the equations is negative. This depends on the phase of the element currents and the phase of the mutual impedance, of course. And it should be clear upon inspecting the equations that if the current magnitude ratio is large, this will also make the second term larger. Thus it is a simple matter to find what current ratio will create a negative operating resistance if you know the self and mutual impedances. In our example antenna when the Element Two current leads the Element One current by 120 degrees, we have a negative operating resistance for Element Two when the ratio of the element current magnitudes ( I2 / I1 ) is greater than 0.54 (Equation 5). The general operating impedance equation for multiple elements is described by Equation 6.
 

Zi = (Ij/Ii) Zij, for j = 1 to n     Equation 6
 

where:

Zi = operating impedance of element i

Ii = complex current in element i

Ij = complex current in element j

Zij = mutual impedance between elements i and j

n = number of elements in the antenna
 

The correct formula for determining suitable phase shifts for the transmission lines and networks when element 2 has a negative resistance is Equation 1, and the formula to use when no negative resistances are involved is Equation 2. Note that they are quite different from each other.
 

negative R: theta1 - phi1 + phi2 + theta2 = 180     Equation 1

positive R: theta1 - phi1 + phi2 - theta2 = 0     Equation 2
 

where:

theta1 = the total phase shift of all networks and lines from the common point to element 1

theta2 = the total phase shift of all networks and lines from the common point to element 2

phi1 = the desired phase of the current in element 1

phi2 = the desired phase of the current in element 2
 

The specific phase shift numbers for Figure 4 then become 0 - 0 + 120 - 300 = -180 degrees. Of course + 180 degrees is the same as -180 degrees in the context of this article. In steady state systems you can always add or subtract 360 degrees from a phase shift value, since the waveform repeats every wavelength.

The feeder system consists of a power divider, a transmission line, and an impedance-matching/phasing network (Figure 4). L1 simply resonates the first monopole, and C1 resonates the input to the system. The currents in the two monopoles are unequal in magnitude and have a relative phase of 120 degrees. Table 1 shows what happens to the input impedance to each monopole as the magnitude ratio of the radiated currents (I9/I3) is varied, and the relative phase is maintained at 120 degrees. The power in each monopole is based on a fixed input power of 1000 watts at the overall input to the feeder networks.
 

 

Table 1 Monopole Pair Feed-point Parameters for Various Current Ratios

I9 / I3

Mag Phase    Z1 (Ohms)    P1 (Watts)    Z2 (Ohms)    P2 (Watts)

0.3 120    61.1 - j28.6     1070     -44.7 - j28.9     -70
0.4 120    62.9 - j26.1     1052     -19.5 - j30.7     -52
0.5 120    64.5 - j23.6     1017      -4.4 - j31.8     -17
0.6 120    66.2 - j21.0      970       5.7 - j32.5      30
0.7 120    67.9 - j18.3      915      12.9 - j33.0      85
0.8 120    69.6 - j16.0      856      18.3 - j33.4     144
0.9 120     7.1 - j13.5      797      22.5 - j33.7     203
1.0 120    73.0 - j11.0      739      25.8 - j34.0     261
 

Note that Element Two becomes parasitic (its operating resistance becomes zero) when its current is about 54 percent of Element One's current. In other words we could tune Element Two as a parasitic reflector if we simply placed a 32 ohm inductor between its base and its counterpoise (Figure 5). But let's push a bit beyond the parasitic case, and choose a current ratio of 30 percent, where we have 1070 watts in Element One and -70 watts in Element Two. The vertical pattern is mildly directional (Figure 6A).


 

What happens to that pattern if we drive Element One only and terminate Element Two in 45 + j29 ohms per Figure 7? We get the same pattern shape, but it is slightly smaller in size (Figure 6B). The maximum gain is reduced by 0.3 dB because of the losses in the resistor. We have 1000 watts in Element One, and 66 watts lost in our 45 ohm resistor. The current in the base of Element One is 4.0 amperes at zero degrees, while the current in the base of Element Two is 1.2 amperes at 120 degrees (Appendix Table 3). The element currents are slightly lower than those of the ideal situation shown in Figure 4.

Table 3 proves that passively terminating a negative antenna element in the negative of its operating impedance is a viable solution if some loss in radiated power can be tolerated. This method is certainly the simplest way to deal with a negative element in a phased array, and deserves serious consideration because it is much easier to adjust for the desired pattern. In fact you are well advised to begin tune-up of an array having negative elements with a passive termination on each negative element, even if you plan to graduate to the more complex configuration typified by Figure 4.
 

 

We can eliminate this 0.3 dB loss associated with Figure 7 by replacing Element Two's terminating resistor with a tee network to match the operating impedance to the 50 ohms of a transmission line brought back to a power divider per Figure 4. Note that Element One's reactance is tuned out with inductor L1. Element Two's reactance is tuned out by L3, plus a little more inductance is added to L1 as part of the tee network to match the 45 ohms up to 50 ohms at a specific phase shift. Then this 50 ohms is presented at the other end of the transmission line to the power divider, L4, which is a tapped coil. The circled numbers in the various network figures represent the voltage nodes as listed in the network analysis output tables in the Appendix. Node zero is ground. The underlined numbers represent the individual component currents, starting with the current source driving the common point of the networks, which is branch number one. In all the network analysis print-outs listed in the appendix, branch number 1 is the RF current generator or transmitter, which is connected to node 1.


 

We need to pay close attention to the phase shifts across the transmission line and the various networks in order to bring the negative power back at the correct phase at the top of the power divider. Note that the total phase shift from the top of the power divider to Element One is zero degrees in Figure 4. At first glance, you might then assume that you need +120 or -240 degrees of phase shift in the feeder branch to Element Two. If we change the Figure 4 tee network phase shift from -95 to -35 degrees, however, the results are rather poor (Figure 8, Appendix Table 7). The relative current in Element Two is 0.6 at 99 degrees, which is a far cry from the desired 0.3 at 120 degrees. The operating impedances are also radically different from the expected values, of course, since they are a function of the element currents per Equation 6. And the actual phase shifts across the RG-8 and the networks also differ from the expected values, because nothing within the feeder system is matched anymore. So what we thought was a -35 degree tee network, for example, is actually something else.


 

Referring to Equation 1, the correct design phase shift from the top of the power divider to Element Two is about -300 degrees, which is the same as +60 degrees. This 60 degree phase shift consists of about -34 degrees across the power divider, -171 degrees across the transmission line (velocity factor = 0.66), and -95 degrees across the tee network. We want a relative phase between element currents of 120 degrees, which is the same as -240 degrees. According to my lumped-parameter network analysis program, the network of Figure 4 attached to the antenna impedance model of Figure 2 produces the desired currents in the two radiating elements with low VSWR on the transmission line (Appendix Table 2).

As you can see in Table 2, the input impedance to the feeder network of Figure 4 is quite good, about 50 - j1 ohms. The RG-8 impedance match is also good, about -47 + j5 ohms at either end. Don't let the negative resistance bother you; it simply means that power is flowing from right to left instead of the conventional left to right on the drawing.

Table 2 verifies Equation 1, even though intuition suggests that the relative phase shift between element currents produced by the networks of Figure 4 should be 60 degrees instead of 120 degrees. So what is really happening here? Let's take another look at the phase shift selection equations, and compare the negative resistance case (Equation 1) to the positive case (Equation 2). For the case of a negative resistance we change the sign of the negative element's feeder phase shift (2) to account for the fact that the power is flowing "backwards." The 180 term is added because we want the returned power to subtract from the forward power to produce a net input power of 1000 Watts. This is similar to the way you read a reflectometer. In physics, energy is conserved, and you don't get something for nothing. So we are not putting in 1000 watts and getting out 1070 watts. If you add up the power in the Figure 3 antenna model resistors, you will see in all cases that it comes to 1000 watts (see Appendix).

As another test of our phase shift selection method, let's add equal amounts of phase shift to each element's feeder branch. Intuition tells us that if we "balance the equation" by doing the same to each feeder branch, the relative current phase in the radiating elements will remain the same. For example, if we add two feet of 60 ohm transmission line in Branch One of Figure 4 between L4 and L1, and two feet of RG-8 to the existing 50 ohm line in Branch Two, we introduce an additional -30 degrees of phase shift to each element, assuming a velocity factor of 0.66 (Figure 9). Do we still get the desired element currents and a good impedance match on the RG-8?

The answer is a resounding "NO!" per Table 4 (see Appendix). The relative phase of the current in Element Two is okay, but the magnitude is now 0.55 instead of the desired 0.3. In fact, Element Two no longer has a negative resistance. The operating impedance has changed very significantly, and the VSWR on the RG-8 is quite high (in this case the co-ax won't melt, since the power in Element Two is now only ten Watts). But the point is that everything has changed dramatically from the desired operating conditions. I also want you to take note of the fact that the input impedance to the feeder network is only a bit off. If you were relying on a change in this input impedance to tell you that something was wrong with your antenna, you would never know, would you?

Remember that we are dealing with a negative power in Element Two, and the simple fact is that if we add two feet of transmission line to Element One, then we must subtract two feet from Element Two. This derives from the fact that the power is traveling in opposite directions on the two transmission lines. In other words, if we were to add a -60 degree tee network at the base of Element One, then our existing -95 degree tee network at the base of Element Two would have to be changed to -95 + 60 = -35 degrees. Similarly if we add -30 degrees of transmission line to Element One's feeder then we must either add +30 degrees of phase shift to Element Two's tee networks (-95 + 30 = -65), or reduce the length of Element Two's transmission line by 30 degrees.


 

If instead we subtract two feet of RG-8 from the original 11.5 foot length (Figure 9 with 9.5 feet of RG-8), our antenna behaves the way we want it to, the VSWR on the transmission lines becomes reasonable, the monopole operating impedances and currents are close to the design values, etc (Appendix Table 5). Again, we could have accomplished the same effect by leaving the 11.5 foot length of RG-8 intact, and compensated for the additional -30 degrees of phase shift in feeder one by adding +30 degrees to the feeder two tee network, making it -65 degrees (Figure 10, Appendix Table 6).
 

Now one might ask why go to all the bother of bringing back the negative element's power to the antenna input, since it requires several additional components, and only improves the radiated field intensity by 0.3 dB? In fact, since we now have transmission line losses and extra component losses to worry about, we may actually end up with less radiated power compared to the simple coil and resistor termination of the negative element in Figure 7. If we assume all coils and capacitors have a Q of 200, and the RG-8 loss is 1.1 dB / 100 feet at 28 MHz, then Table 8 tells us that we only lose 11 watts in the feeder network, which is .05 dB. Thus the complete network of Figure 4 has a 0.25 dB gain compared to the simpler network of Figure 7.


 

Also keep in mind that there may be times when you have a lot more power in the negative element than the seven percent in our example dual monopole array. Thus the loss in a simple resistor termination may be high enough to force you to bring the negative power back to the power divider if you expect any gain from your antenna. If you don't know how to bring that power back to the antenna input properly, you will never get the desired radiation pattern shape and size, and you may even damage your network components. Phased array adjustment can be tricky, and it is easy to get lost in the woods. But if you have an understanding of Equations 1 and 2, at least you will be starting out in the right direction.

I have used Equation 1 successfully in a 200 kW, four tower array with two negative towers, and have every confidence that Equation 1 will work for you as well. But because there are a number of antenna designers who are still using Equation 2 instead of Equation 1 to accommodate a negative element, I have compiled the rather lengthy appendix that you see below. It is still possible to obtain the desired currents in an antenna having negative elements if one is willing to settle for significant impedance mismatches within the feeder system. However, if you want to do it right, use Equation 1, unless you really like flying by the seat of your pants!
 
This article first appeared in the Winter 1999 issue of Communications Quarterly
 

References

1) Grant Bingeman, "Phased Array Adjustment," Summer 1998 Communications Quarterly.

2) a copy of the network analysis program used in this article which can model transformers, lossy transmission lines, negative resistances, lossy inductors and capacitors, and transfer functions, is available for $20 from:
 

LYSIS2
Grant Bingeman
1908 Paris Ave
Plano, TX 75025.
 

Biography

Grant Bingeman is a registered Professional Engineer in Texas and a Principal Engineer with Continental Electronics in Dallas, Texas. His Amateur Extra call sign is KM5KG. He has been in the broadcast industry for about 25 years.
 

Appendix:         Computer Print-outs of Network Analysis Program LYSIS2
 

LEGEND:

NODE ZERO IS GROUND

I = CURRENT SOURCE
R = RESISTOR
C = CAPACITOR
L = INDUCTOR
Z = IMPEDANCE
X = TRANSMISSION LINE
M = MUTUAL INDUCTANCE
K = COUPLING COEFFICIENT
T = TRANSFORMER
G = TRANSCONDUCTANCE
 

Table 2 Computer Analysis of Figure 4 Feeder Network Including Coupled Antenna Model

INPUT PARAMETERS

FREQUENCY: 28000.0 KHZ
RESISTANCE: 50.4 OHMS
REACTANCE: -0.9 OHMS
CURRENT: 4.5 AMPS RMS
VOLTAGE: 224.5 VOLTS RMS
VSWR: 1.02
POWER: 1000.0 WATTS
 

NODE VOLTAGE-TO-GROUND

      POLAR          CARTESIAN

1   224.5  -1.0     224.4   -4.0    NETWORK COMPONENTS
2   255.7  28.7     224.4  122.6
3   282.9   4.1     282.2   20.0
4   57.6   54.9      33.1   47.1
5   57.8 -126.8     -34.6  -46.3
6   80.3  -76.6      18.6  -78.1
7   68.9    1.0      68.9    1.2

8   116.1 -13.3     112.9  -26.7    ANTENNA MODEL
9   113.0 -18.5     107.2  -35.8
10  142.8 -28.5     125.4  -68.2
11  101.9 -44.1      73.1  -70.9
 

BRANCH  NODES  BRANCH-VOLTAGE  BRANCH-CURRENT  FROM-IMPEDANCE   TO-IMPEDANCE
        FR TO   MAG    PHASE    MAG    PHASE   R OHMS  X OHMS  R OHMS  X OHMS

2 C      1  2  126.6   -90.0    4.46     0.0    50.4    -0.9    50.4    27.5
3 L      2  3  117.8   119.4    4.18    29.4    61.1    -0.8    61.1   -28.9
4 L      2  4  205.7    21.5    2.21   -68.5   -14.3   115.0   -14.3    21.8
5 L      4  0   57.6    54.9    1.72   -35.1     0.0    33.4     0.0     0.0
6 X      4  5    0.0     0.0    1.22  -119.5   -47.0     4.7   -47.3     4.8
7 L      5  6   62.0   149.1    1.22    59.1   -47.3     4.8   -47.3   -46.2
8 C      6  0   80.3   -76.6    1.70    13.4     0.0   -47.4     0.0     0.0
9 L      6  7   93.9  -122.4    1.21   147.6   -47.4    46.2   -47.4   -31.2
 

10 R     7  8   52.2   147.6    1.21   147.6   -47.4   -31.2   -90.4   -31.2
11 C     8  9   10.8    57.6    1.21   147.6   -90.4   -31.2   -90.4   -22.3
12 R     3 10  179.9    29.4    4.18    29.4    61.1   -28.9    18.1   -28.9
13 C    10  9   37.2   -60.6    4.18    29.4    18.1   -28.9    18.1   -20.0
14 R     9 11   48.9    45.9    3.76    45.9    13.0   -27.1     0.0   -27.1
15 C    11  0  101.9   -44.1    3.76    45.9     0.0   -27.1     0.0     0.0
 

        NODES   BRANCH       NODE         NODE     BRANCH
BRANCH  FR TO  REACTANCE  FROM-POWER    TO-POWER   POWER

 2  C    1  2      0.0      1000.0      1000.0      0.0     NETWORK COMPONENTS
 3  L    2  3     28.1      1069.8      1069.8      0.0
 4  L    2  4     93.2       -69.8       -69.8      0.0
 5  L    4  0     33.4         0.0         0.0      0.0
 7  L    5  6     51.0       -69.8       -69.8      0.0
 8  C    6  0      0.0         0.0         0.0      0.0
 9  L    6  7     77.4       -69.8       -69.8      0.0
 
10  R    7  8     43.0       -69.8      -133.1     63.3     ANTENNA MODEL
11  C    8  9      0.0      -133.1      -133.1      0.0
12  R    3 10     43.0      1069.8       317.3    752.6     total radiated
13  C   10  9      0.0       317.3       317.3      0.0     power = 1000 Watts
14  R    9 11     13.0       184.1         0.0    184.1
15  C   11  0      0.0         0.0         0.0      0.0
 

TRANSFER FUNCTION   MAG   PHASE     DB

1      I 9 / I 3   0.290  118.3   -10.8
 

Table 3 Computer Analysis of Figure 7 Network Including Coupled Antenna Model with Passive Termination of Negative Element

INPUT PARAMETERS

FREQUENCY: 28000.0 KHZ
RESISTANCE: 61.0 OHMS
REACTANCE: -0.4 OHMS
CURRENT: 4.0 AMPS RMS
VOLTAGE: 247.0 VOLTS RMS
VSWR: 1.22
POWER: 1000.0 WATTS
 

NODE        VOLTAGE-TO-GROUND
         POLAR        CARTESIAN

1     247.0  -0.4   247.0   -1.8     NETWORK COMPONENTS
2     272.8 -25.1   247.0 -115.8
3      54.0 -59.9    27.1  -46.7
4      63.7 -27.9    56.3  -29.8

5     110.9 -42.2    82.1  -74.5     ANTENNA MODEL
6     108.1 -47.6    72.9  -79.8
7     136.8 -57.8    72.9 -115.8
8      97.4 -73.2    28.1  -93.3
 

BRANCH  NODES  BRANCH-VOLTAGE  BRANCH-CURRENT  FROM-IMPEDANCE   TO-IMPEDANCE
        FR TO   MAG    PHASE    MAG    PHASE   R OHMS  X OHMS  R OHMS  X OHMS

 2  L    1  2  114.0    90.0   4.05     0.0      61.0    -0.4    61.0   -28.6
 3  L    3  4   33.8  -149.9   1.20   120.1     -45.0     0.0   -45.0   -28.1
 4  R    3  0   54.0   -59.9   1.20   -59.9      45.0     0.0     0.0     0.0
 
 5  R    4  5   51.6   120.1   1.20   120.1     -45.0   -28.1   -88.0   -28.1
 6  C    5  6   10.7    30.1   1.20   120.1     -88.0   -28.1   -88.0   -19.3
 7  R    2  7  174.1     0.0   4.05     0.0      61.0   -28.6    18.0   -28.6
 8  C    7  6   36.0   -90.0   4.05     0.0      18.0   -28.6    18.0   -19.7
 9  R    6  8   46.8    16.8   3.60    16.8      13.0   -27.1     0.0   -27.1
10  C    8  0   97.4   -73.2   3.60    16.8       0.0   -27.1     0.0     0.0
 

        NODES   BRANCH       NODE         NODE     BRANCH
BRANCH  FR TO  REACTANCE  FROM-POWER    TO-POWER   POWER

 2  L    1  2    28.1      1000.0       1000.0      0.0
 3  L    3  4    28.1       -64.8        -64.8      0.0
 4  R    3  0    45.0        64.8          0.0     64.8     65 watts lost

 5  R    4  5    43.0       -64.8       -126.7     61.9     total of
 6  C    5  6     0.0      -126.7       -126.7      0.0     935 watts
 7  R    2  7    43.0      1000.0        295.2    704.8     radiated
 8  C    7  6     0.0       295.2        295.2      0.0
 9  R    6  8    13.0       168.5          0.0    168.5
10  C    8  0     0.0         0.0          0.0      0.0
 

TRANSFER FUNCTION   MAG    PHASE    DB

1    I 3 / I 2     0.296   120.1  -10.6
 

Table 4 Computer Analysis of Figure 9 Network Including Coupled Antenna Model with 13.5 feet of RG-8
 
INPUT PARAMETERS

FREQUENCY: 28000.0 KHZ
RESISTANCE: 56.7 OHMS
REACTANCE: -4.0 OHMS
CURRENT: 4.2 AMPS RMS
VOLTAGE: 238.8 VOLTS RMS
VSWR: 1.16
POWER: 1000.0 WATTS
 

NODE        VOLTAGE-TO-GROUND
         POLAR        CARTESIAN

 1     238.8   -4.1   238.2   -16.9     NETWORK COMPONENTS
 2     259.3   23.3   238.2   102.4
 3     269.2  -28.3   237.2  -127.5
 4      91.1   21.6    84.7    33.5
 5     101.9 -159.5   -95.5   -35.7
 6      97.3 -161.7   -92.4   -30.6
 7      68.9   23.2    63.3    27.1
 8     255.8   -4.3   255.1   -19.3
 
 9     112.3  -31.9    95.4   -59.4     ANTENNA MODEL
10     101.8  -40.4    77.5   -66.0
11     123.3  -54.3    71.9  -100.1
12      91.8  -66.1    37.2   -83.9
 

BRANCH  NODES  BRANCH-VOLTAGE  BRANCH-CURRENT  FROM-IMPEDANCE   TO-IMPEDANCE
        FR TO   MAG    PHASE    MAG    PHASE   R OHMS  X OHMS  R OHMS  X OHMS

 2  C    1  2  119.3   -90.0    4.20    0.0     56.7    -4.0    56.7    24.4
 3  X    2  8    0.0     0.0    3.83   25.5     67.6    -2.6    65.4     5.8
 4  L    8  3  109.7    80.6    3.90   -9.4     65.4     5.8    65.4   -22.3
 5  L    2  4  168.3    24.2    1.80  -65.8      2.3   143.6     2.3    50.4
 6  L    4  0   91.1    21.6    2.72  -68.4      0.0    33.4     0.0     0.0
 7  X    4  5    0.0     0.0    0.92  106.6      8.6   -98.1   533.8   686.1
 8  L    5  6    6.0   -121.6   0.12  148.4    533.8   686.1   533.8   635.0

 9  C    6  0   97.3   -161.7   2.05  -71.7      0.0   -47.4     0.0     0.0
10  L    6  7  166.0   -159.7   2.15  110.3      1.6    45.3     1.6   -32.1
11  R    7  9   92.2    110.3   2.15  110.3      1.6   -32.1   -41.4   -32.1
12  C    9 10   19.1     20.3   2.15  110.3    -41.4   -32.1   -41.4   -23.2
13  R    3 11  167.5     -9.4   3.90   -9.4     65.4   -22.3    22.4   -22.3
14  C   11 10   34.6    -99.4   3.90   -9.4     22.4   -22.3    22.4   -13.5
15  R   10 12   44.1     23.9   3.39   23.9     13.0   -27.1     0.0   -27.1
16  C   12  0   91.8    -66.1   3.39   23.9      0.0   -27.1     0.0     0.0
 
 
        NODES   BRANCH       NODE         NODE     BRANCH
BRANCH  FR TO  REACTANCE  FROM-POWER    TO-POWER   POWER

 2  C    1  2     0.0       1000.0       1000.0     0.0     NETWORK COMPONENTS
 4  L    8  3    28.1        992.7        992.7     0.0
 5  L    2  4    93.2          7.3          7.3     0.0
 6  L    4  0    33.4          0.0          0.0     0.0
 8  L    5  6    51.0          7.3          7.3     0.0
 9  C    6  0     0.0          0.0          0.0     0.0
10  L    6  7    77.4          7.3          7.3     0.0

11  R    7  9    43.0          7.3       -190.5   197.8     ANTENNA MODEL
12  C    9 10     0.0       -190.5       -190.5     0.0
13  R    3 11    43.0        992.7        339.9   652.7
14  C   11 10     0.0        339.9        339.9     0.0
15  R   10 12    13.0        149.4          0.0   149.4
16  C   12  0     0.0          0.0          0.0     0.0
 

TRANSFER FUNCTION  MAG   PHASE   DB

1    I10 / I 4    0.551  119.7  -5.2
 

Table 5 Computer Analysis of Figure 9 Network Including Coupled Antenna Model with 9.5 feet of RG-8

INPUT PARAMETERS

FREQUENCY: 28000.0 KHZ
RESISTANCE: 50.4 OHMS
REACTANCE: -1.8 OHMS
CURRENT: 4.5 AMPS RMS
VOLTAGE: 224.7 VOLTS RMS
VSWR: 1.04
POWER: 1000.0 WATTS
 

NODE        VOLTAGE-TO-GROUND
         POLAR         CARTESIAN

 1     224.7   -2.0    224.6   -8.0     NETWORK COMPONENTS
 2     253.9   27.8    224.6  118.5
 3     282.5  -27.3    251.1 -129.4
 4      56.7   54.1    33.2    45.9
 5      60.9 -158.2   -56.5   -22.6
 6      79.1 -111.7   -29.3   -73.5
 
 7      67.2  -27.0    59.9   -30.6     ANTENNA MODEL
 8     256.2   -2.7   255.9   -12.1
 9     115.9  -43.7    83.7   -80.1
10     112.4  -49.2    73.5   -85.0
11     141.7  -59.5    72.0  -122.0
12     101.3  -74.8    26.5   -97.8
 

BRANCH  NODES  BRANCH-VOLTAGE  BRANCH-CURRENT  FROM-IMPEDANCE   TO-IMPEDANCE
        FR TO   MAG    PHASE    MAG    PHASE   R OHMS  X OHMS  R OHMS  X OHMS

 2  C    1  2  126.5   -90.0   4.45     0.0     50.4    -1.8    50.4    26.6
 3  X    2  8    0.0     0.0   4.21    29.2     60.3    -1.4    61.4    -0.4
 4  L    8  3  117.4    87.6   4.17    -2.4     61.4    -0.4    61.4   -28.5
 5  L    2  4  204.7    20.8   2.20   -69.2    -14.2   114.8   -14.2    21.6
 6  L    4  0   56.7    54.1   1.70   -35.9      0.0    33.4     0.0     0.0
 7  X    4  5    0.0     0.0   1.21  -119.4    -46.4     5.3   -53.5     5.9
 8  L    5  6   57.7   118.2   1.13    28.2    -53.5     5.9   -53.5   -45.1
 9  C    6  0   79.1  -111.7   1.67   -21.7      0.0   -47.4     0.0     0.0
10  L    6  7   99.0  -154.3   1.28   115.7    -41.9    45.6   -41.9   -31.8
 
11  R    7  9   55.0   115.7   1.28   115.7    -41.9   -31.8   -84.9   -31.8
12  C    9 10   11.4    25.7   1.28   115.7    -84.9   -31.8   -84.9   -22.9
13  R    3 11  179.3    -2.4   4.17    -2.4     61.4   -28.5    18.4   -28.5
14  C   11 10   37.0   -92.4   4.17    -2.4     18.4   -28.5    18.4   -19.6
15  R   10 12   48.6    15.2   3.74    15.2     13.0   -27.1     0.0   -27.1
16  C   12  0  101.3   -74.8   3.74    15.2      0.0   -27.1     0.0     0.0
 
 
        NODES   BRANCH       NODE         NODE     BRANCH
BRANCH  FR TO  REACTANCE  FROM-POWER    TO-POWER   POWER

 2  C    1  2     0.0       1000.0      1000.0      0.0     NETWORK COMPONENTS
 4  L    8  3    28.1       1068.4      1068.4      0.0
 5  L    2  4    93.2        -68.4       -68.4      0.0
 6  L    4  0    33.4          0.0         0.0      0.0
 8  L    5  6    51.0        -68.4       -68.4      0.0
 9  C    6  0     0.0          0.0         0.0      0.0
10  L    6  7    77.4        -68.4       -68.4      0.0
 
11  R    7  9    43.0        -68.4      -138.7     70.3     ANTENNA MODEL
12  C    9 10     0.0        -138.7     -138.7      0.0
13  R    3 11    43.0        1068.4      320.7    747.7
14  C   11 10     0.0         320.7      320.7      0.0
15  R   10 12    13.0         182.1        0.0    182.1
16  C   12  0     0.0           0.0        0.0      0.0
 

TRANSFER FUNCTION   MAG   PHASE    DB

1    I10 / I 4    0.307   118.1   -10.3
 

Table 6 Computer Analysis of Figure 10 Network Including Coupled Antenna Model 2 feet 60 ohm line, 11.5 feet RG-8, tee network phase shift -65

INPUT PARAMETERS

FREQUENCY: 28000.0 KHZ
RESISTANCE: 50.6 OHMS
REACTANCE: -1.9 OHMS
CURRENT: 4.4 AMPS RMS
VOLTAGE: 225.1 VOLTS RMS
VSWR: 1.04
POWER: 1000.0 WATTS
 

NODE         VOLTAGE-TO-GROUND
          POLAR         CARTESIAN

 1     225.1   -2.2    224.9   -8.6     NETWORK COMPONENTS
 2     253.9   27.6    224.9  117.8
 3     282.3  -27.5    250.3 -130.5
 4      58.7   53.0     35.3   46.9
 5      58.8 -128.5    -36.6  -46.0
 6      65.6  -98.5     -9.7  -64.9
 7      66.4  -28.0     58.6  -31.2
 8     256.0   -2.9    255.6  -13.1
 
 9     115.3  -44.1     82.8  -80.3     ANTENNA MODEL
10     112.0  -49.6     72.6  -85.3
11     141.4  -59.9     71.0 -122.3
12     101.0  -75.2     25.8  -97.6
 

BRANCH  NODES  BRANCH-VOLTAGE  BRANCH-CURRENT  FROM-IMPEDANCE   TO-IMPEDANCE
        FR TO   MAG    PHASE    MAG    PHASE   R OHMS  X OHMS  R OHMS  X OHMS

 2  C    1  2  126.4   -90.0    4.45    0.0     50.6    -1.9    50.6    26.5
 3  X    2  8    0.0     0.0    4.21   28.9     60.3    -1.3    61.3    -0.3
 4  L    8  3  117.5    87.4    4.17   -2.6     61.3    -0.3    61.3   -28.5
 5  L    2  4  202.4    20.5    2.17  -69.5    -14.6   116.0   -14.6    22.8
 6  L    4  0   58.7    53.0    1.76  -37.0      0.0    33.4     0.0     0.0
 7  X    4  5    0.0     0.0    1.17 -123.4    -50.1     3.1   -50.3     3.1
 8  L    5  6   32.9   145.0    1.17   55.0    -50.3     3.1   -50.3   -25.0
 9  C    6  0   65.6   -98.5    1.25   -8.5      0.0   -52.6     0.0     0.0
10  L    6  7   76.1  -153.7    1.27  116.3    -42.3    29.4   -42.3   -30.4

11  R    7  9   54.7   116.3    1.27  116.3    -42.3   -30.4   -85.3   -30.4
12  C    9 10   11.3    26.3    1.27  116.3    -85.3   -30.4   -85.3   -21.6
13  R    3 11  179.5    -2.6    4.17   -2.6     61.3   -28.5    18.3   -28.5
14  C   11 10   37.1   -92.6    4.17   -2.6     18.3   -28.5    18.3   -19.6
15  R   10 12   48.5    14.8    3.73   14.8     13.0   -27.1     0.0   -27.1
16  C   12  0  101.0   -75.2    3.73   14.8      0.0   -27.1     0.0     0.0
 

        NODES   BRANCH       NODE         NODE     BRANCH
BRANCH  FR TO  REACTANCE  FROM-POWER    TO-POWER   POWER

 2  C    1  2     0.0       1000.0      1000.0      0.0     NETWORK COMPONENTS
 4  L    8  3    28.1       1068.6      1068.6      0.0
 5  L    2  4    93.2        -68.6       -68.6      0.0
 6  L    4  0    33.4          0.0         0.0      0.0
 8  L    5  6    28.1        -68.6       -68.6      0.0
 9  C    6  0     0.0          0.0         0.0      0.0
10  L    6  7    59.8        -68.6       -68.6      0.0
 
11  R    7  9    43.0        -68.6      -138.2     69.7     ANTENNA MODEL
12  C    9 10     0.0       -138.2      -138.2      0.0
13  R    3 11    43.0       1068.6       319.2    749.4
14  C   11 10     0.0        319.2       319.2      0.0
15  R   10 12    13.0        180.9         0.0    180.9
16  C   12  0     0.0          0.0         0.0      0.0
 

TRANSFER FUNCTION    MAG    PHASE    DB

1    I10 / I 4     0.305    118.9  -10.3
 

Table 7 Computer Analysis of Figure 8 Network Including Coupled Antenna Model Incorrectly using Positive Design Equation for Negative Case
 
INPUT PARAMETERS

FREQUENCY: 28000.0 KHZ
RESISTANCE: 45.0 OHMS
REACTANCE: 4.0 OHMS
CURRENT: 4.7 AMPS RMS
VOLTAGE: 212.9 VOLTS RMS
VSWR: 1.14
POWER: 1000.0 WATTS
 

NODE         VOLTAGE-TO-GROUND
         POLAR         CARTESIAN

 1     212.9    5.1    212.1   18.8     NETWORK COMPONENTS
 2     261.4   35.8    212.1  152.8
 3     278.7   13.7    270.7   66.2
 4      18.0   -0.8     18.0   -0.3
 5      20.2 -174.3    -20.1   -2.0
 6      15.9  104.3     -3.9   15.4
 7     110.8   50.1     71.1   85.0

 8     135.1    7.2    134.1   17.0      ANTENNA MODEL
 9     120.1    1.9    120.0    4.0
10     140.5   -9.5    138.5  -23.3
11     108.3  -23.7     99.1  -43.6
 

BRANCH  NODES  BRANCH-VOLTAGE  BRANCH-CURRENT  FROM-IMPEDANCE   TO-IMPEDANCE
        FR TO   MAG    PHASE    MAG    PHASE   R OHMS  X OHMS  R OHMS  X OHMS

 2  C    1  2  134.0   -90.0    4.71     0.0    45.0     4.0    45.0    32.4
 3  L    2  3  104.5   124.1    3.71    34.1    70.4     2.0    70.4   -26.1
 4  L    2  4  247.2    38.3    2.65   -51.7     4.3    98.5     4.3     5.3
 5  L    4  0   18.0    -0.8    0.54   -90.8     0.0    33.4     0.0     0.0
 6  X    4  5    0.0     0.0    2.26   -43.1     5.9     5.4     5.9     6.7
 7  L    5  6   23.8  -132.9    2.25   137.1     5.9     6.7     5.9    -3.8
 8  C    6  0   15.9   104.3    0.19  -165.7     0.0   -83.6     0.0     0.0
 9  L    6  7  102.3  -137.2    2.15   132.8     6.5    -3.5     6.5   -51.0
 
10  R    7  8   92.6   132.8    2.15   132.8     6.5   -51.0   -36.5   -51.0
11  C    8  9   19.1    42.8    2.15   132.8   -36.5   -51.0   -36.5   -42.1
12  R    3 10  159.6    34.1    3.71    34.1    70.4   -26.1    27.4   -26.1
13  C   10  9   33.0   -55.9    3.71    34.1    27.4   -26.1    27.4   -17.2
14  R    9 11   52.0    66.3    4.00    66.3    13.0   -27.1     0.0   -27.1
15  C   11  0  108.3   -23.7    4.00    66.3     0.0   -27.1     0.0     0.0
 

        NODES   BRANCH       NODE         NODE     BRANCH
BRANCH  FR TO  REACTANCE  FROM-POWER    TO-POWER   POWER

 2  C    1  2     0.0       1000.0      1000.0      0.0     NETWORK COMPONENTS
 3  L    2  3    28.1        969.9       969.9      0.0
 4  L    2  4    93.2         30.1        30.1      0.0
 5  L    4  0    33.4          0.0         0.0      0.0
 7  L    5  6    10.6         30.1        30.1      0.0
 8  C    6  0     0.0          0.0         0.0      0.0
 9  L    6  7    47.5         30.1        30.1      0.0

10  R    7  8    43.0         30.1      -169.4    199.5     ANTENNA MODEL
11  C    8  9     0.0       -169.4      -169.4      0.0
12  R    3 10    43.0        969.9       377.3    592.6
13  C   10  9     0.0        377.3       377.3      0.0
14  R    9 11    13.0        207.9         0.0    207.9
15  C   11  0     0.0          0.0         0.0     0.0
 

TRANSFER FUNCTION    MAG    PHASE     DB

1     I 9 / I 3    0.580     98.7   -4.7
 

Table 8 Computer Analysis of Figure 4 Network with Lossy Components
 
INPUT PARAMETERS

FREQUENCY: 28000.0 KHZ
RESISTANCE: 50.5 OHMS
REACTANCE: -0.9 OHMS
CURRENT: 4.5 AMPS RMS
VOLTAGE: 224.7 VOLTS RMS
VSWR: 1.02
POWER: 1000.0 WATTS
 

NODE         VOLTAGE-TO-GROUND
          POLAR         CARTESIAN

 1     224.7   -1.0    224.7   -3.9     NETWORK COMPONENTS
 2     255.4   28.7    224.0  122.6
 3     281.7    4.2    280.9   20.5
 4      56.6   53.7     33.5   45.6
 5      57.7 -127.7    -35.3  -45.6
 6      77.1  -77.3     16.9  -75.2
 7      69.8    2.5     69.7    3.1

 8     116.3  -13.0    113.4  -26.1     ANTENNA MODEL
 9     112.9  -18.1    107.4  -35.1
10     142.3  -28.2    125.5  -67.2
11     101.8  -43.8     73.5  -70.4
 

BRANCH  NODES  BRANCH-VOLTAGE  BRANCH-CURRENT  FROM-IMPEDANCE   TO-IMPEDANCE
        FR TO   MAG    PHASE    MAG    PHASE   R OHMS  X OHMS  R OHMS  X OHMS

 2  C    1  2  126.5   -89.7    4.45     0.0    50.5    -0.9     50.3    27.5
 3  L    2  3  116.8   119.1    4.15    29.4    61.5    -0.8     61.4   -28.9
 4  L    2  4  205.5    22.0    2.20   -67.7   -12.9   115.1    -13.4    21.9
 5  L    4  0   56.6    53.7    1.69   -36.0     0.2    33.4      0.0     0.0
 6  X    4  5    0.0     0.0    1.17  -117.1   -47.6     7.7    -48.5     7.2
 7  L    5  6   60.0   150.5    1.18    60.7   -48.5     7.2    -48.7   -43.8
 8  C    6  0   77.1   -77.3    1.63    12.4     0.2   -47.4      0.0     0.0
 9  L    6  7   94.4  -124.0    1.22   146.3   -45.8    43.6    -46.2   -33.8
 
10  R    7  8   52.5   146.3    1.22   146.3   -46.2   -33.8    -89.2   -33.8
11  C    8  9   10.8    56.3    1.22   146.3   -89.2   -33.8    -89.2   -24.9
12  R    3 10  178.5    29.4    4.15    29.4    61.4   -28.9     18.4   -28.9
13  C   10  9   36.9   -60.6    4.15    29.4    18.4   -28.9     18.4   -20.1
14  R    9 11   48.9    46.2    3.76    46.2    13.0   -27.1      0.0   -27.1
15  C   11  0  101.8   -43.8    3.76    46.2     0.0   -27.1      0.0     0.0
 

        NODES   BRANCH       NODE         NODE     BRANCH
BRANCH  FR TO  REACTANCE  FROM-POWER    TO-POWER   POWER

 2  C    1  2     0.0      1000.0        997.2      2.8     NETWORK COMPONENTS
 3  L    2  3    28.1      1060.0       1057.6      2.4
 4  L    2  4    93.2       -62.8        -65.1      2.3
 5  L    4  0    33.4         0.5          0.0      0.5
 7  L    5  6    51.0       -67.1        -67.5      0.4
 8  C    6  0     0.0         0.6          0.0      0.6
 9  L    6  7    77.4       -68.1        -68.7      0.6

10  R    7  8    43.0       -68.7       -132.7     64.0     ANTENNA MODEL
11  C    8  9     0.0      -132.7       -132.7      0.0
12  R    3 10    43.0      1057.6        316.6    741.0
13  C   10  9     0.0       316.6        316.6      0.0
14  R    9 11    13.0       183.9          0.0    183.9
15  C   11  0     0.0         0.0          0.0      0.0
 

TRANSFER FUNCTION   MAG   PHASE    DB

1    I 9 / I 3    0.294   116.8  -10.6