April 1953 QST
Table of Contents
These
articles are scanned and OCRed from old editions of the ARRL's QST magazine. Here is a list
of the QST articles I have already posted. All copyrights are hereby acknowledged.

Here is a fairly major treatise on folded and loaded antennas that
appeared in a 1953 issue of QST magazine, with "Suggestions
for Mobile and RestrictedSpace Radiators." It is not for the faint
of heart or anyone with mathphobia. Integral calculus is part of
the presentation, although an understanding of calculus is not required
to get the gist of the article. Equations for calculating the antenna
configuration radiation resistances are given for the 3λ/4wave
folded dipole, the λ/8wave folded monopole, the bottom, center
and toploaded λ/8wave monopole, the bottomloaded λ/16wave monopole,
and the λ/4wave monopole folded twice, to name a few.
Folded and Loaded Antennas
Suggestions for Mobile and RestrictedSpace Radiators
By William B. Wrigley,* W4UCW
Fig. 1  Current and voltage distribution
on half, one, and one and onehalf wavelength antennas, fed
at the center.
Fig. 2  Variation of reactance and resistance
at input terminals of centerfed antennas as the total length
is varied.
Fig. 3  The current fed fullwave antenna
(A) and its development into a folded dipole.
Using a simplified method of calculation, the author develops
values for the radiation resistance of various folded and loaded
forms of short antennas. Several interesting possibilities for small
radiating systems are discussed.
While we are all quite familiar with the halfwave folded dipole,
its radiation pattern, input or radiation impedance, and application
to amateur installations, it seems that there are many more folded
configurations which are not well known and which may prove quite
surprising in their usefulness. Most of us are also reasonably familiar
with the basic methods of loading mobile antennas, but we may be
surprised at what a few simple calculations can tell us about the
effects of various methods of loading.
First let us consider the basic halfwave thin dipole^{1}
with a theoretical balanced centerfeed impedance of about 72 ohms.
Fig. 1A shows such an antenna with its current distribution (dashed
line) and charge distribution (solid line). While these distributions
are not exactly sinusoidal as shown, the assumption that they are
so introduces negligible error in impedance and fieldpattern calculations,
and at the same time reduces these calculations from formidable
complexity to fairly simple operations. Now Fig. 1B shows what happens
if we attempt to operate this antenna at the second harmonic. We
now have a condition of antiresonance. The input resistance is much
higher and the reactance variation with frequency is much greater
than in the original resonant case at the fundamental frequency.
Fig. 10 shows the current distribution at the third harmonic, where
we once again have a reasonably broad resonant condition. Fig. 2
shows qualitatively this same information as resistance and reactance
plotted against antenna length in wavelengths.
We might conclude from all this that, at least in the symmetrical
case, an antenna will be reasonably broadband only at frequencies
where the length is an odd number of halfwavelengths or such that
the feed point is at a current maximum.
Why can we not simply move the feed point to a current maximum
in the secondharmonic case of Fig. 1B? We can, in fact, but then
things change somewhat since the nowcontinuous center cannot support
a discontinuity in charge. So we get distributions as shown in Fig.
3A, somewhat unbalanced, as would be expected. Since the charge
polarity is now the same at both ends, however, we can fold the
left current loop over the driven loop as shown in Fig. 3B and obtain
the familiar folded dipole of Fig. 3C, which is, of course, quite
symmetrical. The impedance of this folded arrangement can be found
by considering it as merely an impedance transformer between the
feeder line and free space. The farfield intensity normal to the
axis of the antenna is proportional to the total current added up
along both wires and, since we now have exactly double the current
producing the far field, as compared to that in either wire alone
(in particular the one being fed), we must have double the farfield
pattern strength. However, the transmission line furnishes power
that is exactly the same as in a simple dipole, hence the input
or radiation resistance must be directly proportional to the square
of the total farfield intensity as compared to that of the fed
wire only (W = E^{2} / R). In this case 2^{2} =
4 and 4 x 72 is 288 ohms, which is the approximate theoretical radiation
resistance of a thin folded halfwave dipole. It is well known that
the reactancefrequency variation of the antenna is, in this particular
case, partially cancelled out by the opposite variation of the two
transmission line stubs in series seen from the feed point such
that the folded dipole has, in fact, broader bandwidth than the
single thin dipole.
Other Folded and Loaded Systems
Since this folding operation has proved so attractive, let us
now investigate the possibility of folding the configuration of
Fig. 1A. Because of the mobile antenna application we shall consider
half the antenna of Fig. 1A against a ground plane and fed with
a coaxial cable as shown in Fig. 4A. We can fold the antenna as
in Fig. 4B and obtain the eighthwave folded monopole of Fig. 4C?
Since the opposite ends of the original dipole were at opposite
charge polarity (Fig. 1A), we must leave these ends unconnected
upon folding; or, in the ground plane case, the foldedover section
must not be allowed to contact the ground plane. For radiation purposes,
the current in the folded section is opposite in direction to that
of the unfolded half and one must be subtracted from the other,
resulting in the radiation current distribution shown in Fig. 4D.
Now in the folded dipole case we found the impedance by adding
(mathematically integrating) the current distribution along the
wire to obtain a figure proportional to farfield strength.
Actually, these figures of proportionality are only valid comparisons
of two antennas if the farfield patterns or current distributions
are identical. However, in all the cases we shall consider here,
there will be only one combined radiating current loop and hence
only one farfield pattern lobe. These lobes will not be exactly
the same shape, but to assume them so is a reasonable approximation
as evidenced by the fact that the farfield radiation pattern of
a halfwave dipole with sinusoidal current distribution is only
slightly more directive (78 degrees between halfpower points) than
that of a minutely short dipole with uniform current distribution
(90 degrees between halfpower points).
As shown by the calculations in the Appendix, the approximate
radiation impedance of the folded eighthwave monopole of Fig. 4D
is 6.2 ohms. A similar analysis of a bottomloaded eighthwave monopole,
Fig. 4E, shows that its radiation resistance also is 6.2 ohms, which
is the same as for the folded case! This identity holds for a quarterwave
monopole which is folded into any even number of elements as compared
to a bottomloaded single element of the same actual height.
Fig. 4F shows the current distribution of a toploaded eighthwave
monopole.^{3} The approximate radiation resistance, as shown
in the Appendix, is 18 ohms. For the centerloaded eighthwave monopole
of Fig. 4G the approximate method of calculation still applies and
leads to a theoretical radiation resistance of 11.5 ohms.
Folded Mobiles
Fig. 4 Coaxialfed quarter wave antenna
and ground plane (A); effect of folding (B, C and D); bottomloaded
eighthwave (E), toploaded eighthwave (F), and centerloaded
eighth wave (G).
Fig. 4D could be interpreted as a 20meter mobile antenna made
up of two adjacent eightfoot whips. One significant advantage of
this arrangement is that there is no loadingcoil loss to contend
with. A further advantage is in the realization that a shorted stub
of appropriate length (λ/4 at 20 meters) connected to the
mounting point of the folded or second whip will be an open circuit
at 20 meters and a closed circuit at 10 meters. At 10 meters the
system becomes a quarterwave folded monopole (half a folded dipole)
with an input impedance of a little over 100 ohms, while at 20 meters,
with no mechanical change, it becomes an eighthwave folded monopole
with an impedance of about 5 ohms. (Five ohms is probably closer
than the theoretical 6.2 ohms since mobile quarterwave whips look
more like 30 than 36 ohms. They are not "thin.") The rather severe
difference in impedance between the fundamental and secondharmonic
case can be taken care of by feeding the pair of whips with another
quarterwavelength of cable at 20 meters. Being a halfwave at 10
meters, this would give a load impedance at the transmitter of somewhat
over 100 ohms at 10 meters and Z_{o}^{2} / 5 ohms
at 20 meters, where Z_{o} is the characteristic impedance
of the cable used. This double whip 1020 system will be slightly
more selective, however, than either of the plain folded monopoles,
since the reactance deviation with frequency of the shorted stub
is opposite in sign from that required to counteract the reactance
deviation of the antenna.
If a 40meter quarterwave monopole were folded down to eightfoot
height there would be four sections. The resulting impedance, which
again is the same as that of a bottomloaded sixteenthwave monopole,
is 1.2 ohms (see Appendix). An 80meter arrangement would take eight
whips and would have an impedance of approximately 0.3 ohm. However,
these latter two extensions of the folding process do not immediately
appear very attractive. To complete our discussion of folded and
loaded monopoles we should include the radiation resistances of
40 and 80meter loaded antennas. Table I shows all of these figures
calculated by the farfield factor method and based on a nominal
quarterwave monopole impedance of 30 ohms, more realistic than
the theoretical "thin" monopole value of 36 ohms.
It is interesting to investigate the impedances of short monopoles
with optimum current distributions. This requires loading at both
top and bottom so as to center the current loop on the antenna as
shown in Fig. 5. The values of radiation resistance for 40 and 80
meters are also included in Table I.
A top and bottomloaded (current loop centered) 10meter quarterwave
monopole has the very attractive impedance of 120 ohms, calculated
by this method. The ground current losses in it would be considerably
less than the losses in the unloaded case for the same radiated
power.
Comparisons
Table I  Approximate radiation resistance
of various loaded and folded monopole antennas based on a quarterwave
value of 30 ohms
Fig. 5  Top and bottomloaded eighthwave
coaxfed antenna with ground plane.
Fig. 6  Quarterwave antenna with ground
plane (A); same folded to height of oneeighth wave (B); threewire
folded quarterwave monopole and ground plane (C); same folded
to oneeighthwave height, with dimensions for 1.85 Mc. (D).
The approximate radiation resistance for the last case is 9
X 5.7 = 51 ohms.
Fig. 7  Centerfed antenna one and onehalf
wavelength long (A) folded into a threequarterwave folded
dipole (B and C).
Fig. 8  An antenna system for 7, 14 and
21 Mc., using a shorted stub to act as an automatic switch for
various bands.
Fig. 9  A possible 40meter beam arrangement
using quarterwave folded elements.
We can now draw some very definite conclusions regarding the
merits of various loading schemes. Since the principal loss in a
vertical radiator (outside of the loadingcoil loss) is due to ground
currents, the efficiency rapidly decreases with decreasing radiation
resistance. For constant radiated power, the current must be greater
for smaller values of radiation resistance. Greater current means
greater loss and consequent reduction in efficiency; therefore,
the power loss in a centerloaded eighthwave monopole is more than
that of a toploaded equivalent antenna and the loss in the bottomloaded
case is more than that of the centerloaded case. A combination
of both top and bottom loading, however, gives a radiation impedance
which in some cases reduces the loss to an exceptionally low value
compared to that of the bottomonly loaded, folded, or unloaded
case. Sufficient top loading is usually impractical, however, particularly
in the case of very short monopoles.
The main conclusion we can draw from all these calculations is
that short antennas (monopoles less than onetenth wavelength) have
uncomfortably low radiation resistances and practically nothing
can be done to improve their efficiencies to a reasonable value,
except possibly by using a multiwire system to raise the impedance
as described earlier. On the other hand, the efficiencies of longer
(quarter or eighthwave) monopoles may be increased considerably
by proper loading, or folding. A practical example is the 160meter
folded eighthwavelength threewire monopole shown in Fig. 6B. Adding
a third wire to the folded quarterwave monopole, Fig. 6C, raises
the resistance to about 300 ohms, and when this antenna is folded
over as shown in Fig. 6D, the radiation resistance becomes about
50 ohms, a good match for coaxial cable. Suggested dimensions for
1850 kc. are given in the sketch.
3λ/4 Folded Dipole
Now, since we have pretty well folded and loaded Fig. 1A, let
us investigate the results of folding Fig. 1C. This process and
the resulting current distribution is shown in Fig. 7 where the
center line represents a ground plane for the vertical analogue
of the system. Calculation leads to an impedance for this threequarter
wave folded dipole of about 420 ohms. J. D. Kraus ^{5, 6}
(W8JK) has measured one of these to be about 450 ohms. The new 21Mc.
band makes this arrangement quite useful as can be seen in the following
scheme:
Suppose we start with the 20meter folded dipole of Fig. 8A and
opencircuit the top dipole opposite the feed point. We now have
a quarterwave folded dipole at 40 meters, the vertical analogue
of which we have already discussed. The impedance of this dipole
at 40 meters should be about 12 ohms. A λ/4 length of TwinLead
at 40 meters would then transform this to Z_{o}^{2}/12
at the transmitter. A shorted λ/4 stub connected to the open
ends of the dipole as shown in Fig. 8B would provide an open circuit
at 40 meters, a short circuit (λ/2) at 20 meters, and open
again (3λ/4) at 15 meters. At 20 meters we have our original
halfwave folded dipole and at 15 meters we have our 3λ/4
folded dipole. In this last case the now 3λ/4 feed line transforms
the impedance to Z_{o}^{2}/420 at the transmitter.
Of course, the shorted stub may be folded up in some convenient
way so as not to consume all the space indicated in Fig.8B.
The radiation patterns of the antenna are practically identical
at all three frequencies.
Other Possibilities
No doubt there are many more folded arrangements which may prove
attractive, such as the possibility of a 40meter closespaced beam
made up of quarterwave folded dipoles. Now, due to the coupling
of the parasitic elements, the impedance of the driven element would
be considerably lower than the 10 to 12 ohms of a folded quarterwave
dipole in free space. This may be raised to a more reasonable value,
however, by feeding at the current node (voltage feed) rather than
at the current loop.^{2} The driven element would then look
just like a 20meter folded halfwave dipole, but the current distribution
would be as shown in Fig. 9A and the entire array would look something
like Fig. 9B. A Tmatch or a "Q"section at the feed point may provide
an even better drivingpoint impedance. The approximate overall
lengths of the elements, before folding, may be determined from
the curves for 20meter elements given in The Radio Amateur's Handbook
or in the ARRL Antenna Book, simply by dividing the frequency scale
in half and doubling the length scale.
With some compromise in element spacing this array may even be
operated as a combination 402015 beam by the use of small lumped
networks in the parasitic elements in place of the rotationally
cumbersome stubs of Fig. 8B. In case you would like to try this
latter arrangement, Fig. 10A shows the impedance vs. frequency characteristic
required and Fig. 10B shows a suitable network to accomplish this.
The parameter K allows for variation in the overall impedance level,
but should be chosen so as to make C_{1} a conveniently
small capacitor and include the capacity between the two ends of
the element. Since harmonic antennas are not, in general, exact
multiples of length, all the network elements may require some adjustment
after final assembly to approach resonance on all three bands.
Fig. 10  Impedance variations in the frequency
range of interest (A) and a lumpedcircuit equivalent of the stub
shown in Fig. 8. Component values can be calculated from the following
formulas, where f is in megacycles:
C_{1} = K μμfd.
C_{2} = 2.4K μμfd.
L_{1} 4220/Kf^{2} μh.
L2 = 1.67 L_{1} μh.
The factor K may be chosen to make the inductances and capacitances
come out to convenient or constructionallyfeasible values.
Appendix A
λ/8 Folded Monopole Antenna Radiation Resistance
(Fig. 4C)
To find the radiation impedance of the eighthwave folded monopole
we must find the farfield figure of proportionality normal to the
axis of the antenna by subtracting the integrated current of the
foldedover half from the integrated current of the unfolded half.
The total length is
λ/4 or
π/4
radians so
= 0.707  0  1.00 +0.707 = 0.414.
This is the farfield proportionality constant whose square must
be compared with that of a known antenna to give the impedance.
The known standard is, of course, the thin quarterwave monopole
whose figure is
and whose theoretical input impedance is about 36 ohms.
Therefore, the approximate radiation impedance of the thin folded
eighthwave monopole is
BottomLoaded λ/8 Monopole Antenna Radiation Resistance
(Fig. 4E)
The current distribution on a bottomloaded eighthwave monopole,
Fig. 4E, is identical with the top half of the quarterwave monopole
which we folded over in the previous case, since the loading coil
merely replaces the missing half. We can calculate the radiation
resistance of the remaining half as follows:
1.00  0.707 = 0.293.
This farfield figure, however, concerns the impedance referred
to a current maximum point and since we are feeding at the
π/4
or 45degree point we must divide by the square of the cosine of
45 degrees (constant power impedance is inversely proportional to
the square of the current). So finally we get
TopLoaded λ/8 Monopole Antenna Radiation Resistance
(Fig. 4F)
The approximate radiation resistance can be calculated from the
cosine integral from 0 to π/4 which gives a farfield factor of
0.707.
CenterLoaded λ/8 Monopole Antenna Radiation Resistance
(Fig. 4G)
For the centerloaded case the calculation is a little more complicated.
but our approximate method still applies. The farfield factor includes
two additive components, the first of which comes from the bottom
section of the antenna and is merely the cosine integral from 0
to π/8.
The loading coil effectively replaces the missing center half of
the antenna so that the current distribution along the top section
is essentially the cosine curve from 3π/8
to π/2.
Since the current is continuous through the coil, however, this
second integral must be multiplied by the ratio of the cosines of
π/8
and 3π/8.
The approximate theoretical radiation resistance of a centerloaded
eighthwave monopole is then
= 36 (0.566)^{2} = 11.5 ohms.
BottomLoaded λ/16 Monopole or λ/4 Monopole
Folded Twice Antenna Radiation Resistance
The cosine integral is broken down into four equal parts between
0 and
π/2
or 90 degrees. Two alternate parts are added and the other two are
subtracted. The resulting impedance, which is the same as that of
a bottomloaded sixteenthwavelength monopole, is
λ/32 Monopole Antenna Radiation Resistance
(Eight integrals)
Top and BottomLoaded λ/8 Monopole Antenna Radiation
Resistance (Fig. 5)
In this case the farfield factor is twice the cosine integral
from 0 to
π/8.
In calculating the impedance, however, we must divide by the square
of the cosine of
π/8
or 22.5 degrees, since we are feeding at a point 22.5 degrees from
the current loop. This is similar to the case considered in equation
(5). Therefore,
= 30 (0.830)^{2} = 21 ohms.
3λ/4 Folded Dipole (Fig. 7C)
The farfield factor for this case is found by calculating the
difference between the cosine integral from 0 to 3π/4
and the integral from 3π/4
to 3π/2.
This figure is 2.414 and from this the impedance of the folded 3/4wave
dipole comes out to be about 420 ohms.
* Asst. Prof. Research, Georgia Institute of Technology.
1 Editor's Note: As some readers may not he familiar with the
terms used here, the following may be helpful:
A "thin" antenna is one having a very large ratio of length to conductor
diameter, approaching infinitelysmall diameter; practically, a
wire antenna at low frequencies is "thin" but a 10meter beam element
is fairly "thick." The thickness affects the resonant length, radiation
resistance, and sharpness of tuning.
"Antiresonance" is the same as parallel resonance; in the antenna
case, it is the condition that exists when a resonant antenna is
viewed at a voltage loop.
"Charge" and "charge distribution" are equivalent to "voltage"
and "voltage distribution."
The" far field" is the radiation field at a large distance from
the antenna  so far that the waves may be considered to be plane
waves, and, of course, far beyond the region where the induction
field is of any consequence.
A "monopole" is onehalf of a dipole; e.g., a grounded antenna
or one in which a ground plane is substituted for actual ground.
2 Lindenblad, "Television Transmitting Antenna for Empire State
Building." RCA Review, 3, p. 400, April, 1939.
3 Terman, Radio Enoineers' Handbook, McGrawHill Book Co., Inc.,
New York, 1943, par. 11, sec. 11.
4 Jordan, Electromagnetic Waves and Radiating Systems, Prentice
Hall, Inc., New York, 1950, pp. 510517.
5 Kraus, Antennas, McGrawHill Book Co., Inc., New York, 1950;
particularly Chapter 5 and par. 13 of Chapter 14.
6 Kraus, "Multiwire Dipole Antennas," Electronics, 13, pp. 2627,
Jan., 1940.
Posted April 19, 2016
