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The bane of transmission line leakage can also be a surprise benefit.

When does an open-wire transmission line begin to lose its ideal characteristics and start to behave as an antenna? Is there such a thing as a balanced, but radiating transmission line in free space? If so, can we call this a line antenna? This article correlates twin-lead transmission line input impedance with antenna radiation resistance.

A discussion of transmission lines and radiation Electromagnetic radiation from an open-wire length of twin-lead transmission line in free space can exist even when its standing wave ratio (SWR) is 1.00; that is, when the line is terminated in its characteristic (surge) impedance, Z subscript o. This article will quantify when a particular open-wire transmission line becomes an antenna. Conventional wisdom asserts that when equal magnitude but opposite sign currents exist on the opposite sides of the twin-lead, radiation does not occur. But this rule is valid only when the two wires are electrically close together. In fact a loop antenna is a degenerate form of twin-lead, and it is interesting to discover what physical line dimensions encourage radiation.

The surge impedance of balanced twin-lead in free space with an air dielectric is:

Z subscript o = 120 ln (2S/d) W (1)

where: S = spacing between wire centers d = diameter of wire ln = natural log

superscript *note that the dimension units of S and d do not matter as long as they are the same.

In theory, when 1000 W are delivered to the input of a lossless and perfect transmission line, then 1000 W should appear at the termination impedance, Z subscript t, when Z subscript t = Z subscript o. However, if the transmission line behaves as an antenna, then the input resistance to the line also includes a radiation component, R subscript r such that R subscript in = R subscript line + Rr. An ideal transmission line without any leaks would have no radiation resistance. An ideal antenna, on the other hand would radiate all power delivered to it from a generator, such that Rline would be zero. A leaky transmission line would exhibit both transmission line and antenna characteristics. Normally, a physical termination impedance, Z subscript t is not added to an antenna except in special cases where bandwidth needs to be augmented, etc. Keep in mind that this article is not about a conventional transmission line feeding a conventional antenna, but rather about a hybrid transmission-line/antenna mixture.

Modeling an example using twin-lead line As an example, a 10 meter length of twin-lead is used as a pair of 5 mm diameter wires spaced one meter. Equation 1 suggests that Z subscript o should be about 720 W in this case. Method of moments (MoM) analysis using three current segments per meter predicts a 10 MHz input impedance to the twin-lead of 731 - j9.4 W when the line is terminated in a pure resistance of 720 W. If we use a 730 W termination resistor instead, then the 10 MHz input impedance is exactly 730 W, leading one to believe that Z subscript o is actually closer to 730 W in this case. But because of the accuracy limits of the MoM model, the Z subscript o can only be said to be about 720 W.

However, keep in mind that what is being delt with is not quite a perfect transmission line. At 10 MHz the electrical length of the line is 120, (1/3 of a wavelength), and the width of the line is 12 (about 3% of a wavelength). This width is actually not a terribly small value electrically, so it should be suspected that there is some radiation from this open-wire line at frequencies above 10 MHz. According to Table 1, for an input power of 1000 W, the radiated power, P subscript r, and the radiation resistance, R subscript r, increase with increasing frequency even when the line is properly terminated in Z subscript t = Z subscript o. One percent radiated power occurs when the electrical width of the example twin-lead is about 20, which is about 6% of a wavelength at 17 MHz. Recall that the one meter spacing between conductors is rather large for typical open-wire line. At low RF power levels, this large spacing would not be required. But the point of this exercise is to show that such a line can degenerate into an antenna when the frequency is high enough. And it was thought that transmission line surge impedance was not a function of frequency? Think again. The assumptions upon which many of our RF formulas are designed to keep life simple, but are based beyond those assumptions, then calculation accuracy begins to degrade.

Assuming an ideal, lossless transmission line, a 1000 W is delivered to the input of that line, but only 990 W is received in the load resistance, then the missing 10 W must appear as radiation. Table 2 demonstrates what to expect at 10 MHz for high SWR on the ten meter length of 720 W twin-lead. The radiation resistance can be determined from:

R subscript r = R subscript in P subscript r/P (2)

where:

R subscript in = input resistance to transmission line.

P subscript r = radiated power.

P subscript r = P - P subscript t = input power - load power.

P = input power to transmission line.

The average value for R subscript r is 3.3 W. Remember that accuracy is constrained by the MoM model, but the results are close enough to deduce some valid generalizations.

Table 3, where the electrical wire spacing is less than three degrees, conveys that the radiation from the line is not so much a function of mismatch or SWR, but more a function of wire spacing. However, this conclusion may not be true of unbalanced lines, and it is sometimes difficult to obtain perfectly balanced transmission lines in the real world anyway.

The transmission line as an antenna Consider the case where Z subscript t = 0 W. Referring to Figure 1, the antenna looks like two parallel horizontal dipoles, one of which is driven directly by a generator. The second dipole is driven by the ten meter length of transmission line connected to the first dipole. Thus, the phase of the current in the second dipole, hence the radiation pattern, are affected by this transmission line. There is no physical termination impedance placed in the second dipole, although it does have an operating impedance that is reflected back down the transmission line to the first dipole, which influences the value of Z subscript in.

The 10, 15 and 20 MHz radiation patterns in the free-space horizontal plane are described in Figures 2 through 4. Clearly the transmission line can behave as an antenna, especially when it is terminated in some impedance other than Z subscript 0. The operating particulars are listed in table 4.

Note that all these pattern shapes and gains are very different. As expected, the greatest power leak from the open-wire line occurs at the highest frequency, where the greatest electrical spacing between conductors exists. It is also important to note that the electrical size of the dipoles at each end of the transmission line is greatest at the highest frequency.

The current distribution along the transmission line is described in Figure 5. This is a standing wave, where the node repeats every half wavelength, which is 7.5 meters at 20 MHz. The transmission line currents are equal magnitude, but opposite in phase, and do not contribute too much to the radiated field intensity. It is the dipole currents that primarily affect the radiation pattern. The dipole currents exist at each end of the transmission line, which is 1.5 wavelengths long at 20 MHz. Thus the free-space coupling or mutual impedance between these two dipoles is small, and the current in dipole 2 is mostly contributed by the direct transmission line connection. Referring to Table 4 and Figure 5, one can see that most of the current exists in dipole 2.

The current in dipole 2 can be manipulated with a terminating reactance in order to reduce the radiation effects, if desired. In any case the physical wiring at the source and termination ends of the transmission line is a necessary part of the complete model. When the spacing between transmission line conductors becomes large electrically, the same is true of the wiring at the input and output ends of the transmission line, which can be considered as radiating dipoles.

Summary The article has shown that electromagnetic radiation exists from a balanced open-wire line when the electrical spacing between wires is large (dipole size of source and termination wires is large), regardless of the value of terminating impedance.

It has also shown that a transmission line can behave as an antenna, in which case its input resistance has separate line and radiation components.

Further, termination impedance does not significantly affect radiation from a balanced open-wire line unless the spacing between conductors (dipole size) is electrically large. And source and termination dipoles are a necessary part of the radiating transmission line model.

In today's age of computers it is advantageous to use one of the several available modeling programs to aid in the development of such anaylses. In this case EZNAC superscript 1 was used as the application tool. The appendix provides a technique and the equations used to develop a lossy transmission line model.

References 1) EZNEC version 3.0 is available from Roy Lewallen, www.eznec.com.

2) For further information about RF Network Designer, go to www.qsl.net/km5kg

Appendix - a lossy transmission line model A transmission line conveys an electromagnetic signal from its input to its output. In other words, a transmission line can transport RF power from a generator to a load. The power is developed in the resistive portion of the load impedance, Z subscript t. Some power is lost to heating of the transmission line. A transmission line can be as a simple two-port using either a lumped-parameter tee network or a pi network. Lumped-parameter include resistors, inductors and capacitors. A transmission line is defined by its length, surge impedance (Z subscript 0 or characteristic impedance), dielectric constant and loss. All other parameters can be calculated from this information.

The insulating material between the conductors in a transmission line has a dielectric constant greater than that of air, which determines the velocity of propagation of the electromagnetic signal through the line, and also determines the capacitance of the line. Sometimes the insulation is a foam, or a series of stand-offs, or some other mixture of various dielectrics. In such special cases, the effective dielectric constant may have to be estimated or measured.

Since the velocity of propagation of the electromagnetic energy along a transmission line is less than that of air, a wavelength in a transmission line is shorter than its free-space value. Normally assign 360 electrical degrees are assigned to a wavelength. Thus a physical quarter-wave section of transmission line with air insulation (velocity factor = 1.0) would be 90 long. However the same physical length of line with solid polyethylene insulation would be about 136 long, since its velocity factor is about 66%. The phase shift across this line is denoted as negative 136 to convey the fact that a time lag or delay is associated with the propagation of the energy from the input to the output. Note that the phase shift across a transmission line is dependent on Z subscript t, and the value of -136 in our example assumes that Z subscript t = Z subscript 0. Also keep in mind that the voltage phase shift and the current phase shift are the same only when Z subscript t = Z subscript o.

Losses in a transmission line consist of dielectric heating (the dissipation factor of the insulation determines this), and conductor heating (I superscript 2R) caused by resistance to the RF current. Sometimes when a transmission line is mis-terminated additional loss can appear as electromagnetic radiation from currents on the outside of a co-axial line, but this effect is not accommodated in the following equations.

The maximum rated power-handling capability of a transmission line is based on a fixed temperature increase, and assumes the line is terminated in its characteristic impedance (Z subscript t = Z subscript 0). In general, the larger the transmission line, the lower the losses, and the higher the frequency, the higher the losses.

A p network model of a lossy transmission line:

p network shunt arms: Z subscript a = Z subscript b = Z subscript 0/tanh (y/2)

p network series arm: Z subscript c = Z subscript 0sinh (y)

A Tee network model of a lossy transmission line:

Tee network series arms: Z subscript 1 = Z subscript 2 = Z subscript o tanh (y/2)

Tee network shunt arm: Z subscript 3 = Z subscript 0/sinh (y)

Input impedance to line: Z subscript in = [Z subscript t + Z subscript 0 tanh (y)]/[1 + (Z subscript t/Z subscript 0) tanh (y )] W

where:

the dielectric constant, e = 1/vel subscript 2

velocity factor relative to the speed of light, vel < 1.0

Distributed circuit model components of a transmission line:

C = 1016 sqrt (e)/Z subscript 0, pF/foot L = 0.001016 Z subscript 0 sqrt (e), uH/foot Z subscript 0 = characteristic impedance of line

note that Z subscript 0 has a reactance term when loss is substantial, but the line is still terminated in R subscript 0 for a perfect match.

and:

y = a + jb:

a = line loss (Nepers) b = electrical length of line

hyperbolic tangent:

tanh (y) = [sinh (2a) + jsin (2b)]/[cosh (2a) + cos (2b)] (1 Neper = 8.686 dB)