When a dipole is bent back on itself in order to reduce the length it occupies it also reduces its radiation resistance from around 50 Ohms to approximately 12  15 Ohms. This can be recouped by changing its construction to that of a folded dipole before it is bent, which increases the input impedance by 4 times (Ref.15). The practical way to do this is to use a two core cable but the surrounding insulation introduces a complication by increasing the capacitance between the wires. This is easiest to appreciate when the folded dipole is redrawn to show that as well as appearing as a radiating antenna it also shows up as two transmission lines connected in series across the input. If there was no insulation present the velocity factor of the transmission lines would be above 0.95 so that they would act like two 1/4 wave stubs shorted at their remote ends. Hence they would present a high impedance at the input in parallel with the antenna radiating resistance and everything would be OK. However the insulation significantly reduces the velocity factor so that the transmission line stubs are now electrically longer than 1/4 wavelength when shorted at the remote ends and present a much reduced impedance in parallel with the radiation resistance, and so they can no longer be ignored.
Fortunately the radiation performance of a folded dipole is very tolerant of the positions that the two parallel wires are shorted together. This allows us to place the shorts at a distance of 1/4 wavelength of the antenna x the velocity factor of the cable from the feed point on each side. This restores the two transmission line stubs to be electrically equivalent to a 1/4 wavelength in the cable so that they again present a high impedance in parallel with the radiation resistance at the input to the antenna, and hence can again be ignored.
The velocity factor of the cable is an important parameter in the design of doubly folded dipoles, there are three basic ways of determining it :
The measurement can be taken on a few metres of the cable by leaving the far end open circuit and using an antenna analyser, a noise bridge or a grid dip oscillator to determine the lowest frequency that gives a resonant minimum impedance between the pair of wires at the other end. The transmission line formed by the sample length of cable is then 1/4 wavelength long at that frequency.
Velocity factor then = (freq x len) / 75
where freq is the frequency in MHz
and len is the length in metres
Take a few metres sample of the cable and measure the capacitance between the pair of wires, many modern multimeters have this facility. Use this value to derive the capacitance per metre (pF/m). Now examine a clean cut crosssection of the cable and carefully measure the diameter of the conductors and determine their radii, also measure the distance between their centres. Go onto the web site given in Ref.16 and enter the cable data in order to determine the inductance per metre (nH/m) for the transmission line represented by the cable (Note: Use 1.0 for µr  the relative permeability of the medium).
Velocity factor then = 105 / √(L x C)
where L is the inductance in nH/m
and C is the capacitance in pF/m
The velocity factors were measured for a variety of different types of cable. It can seen from the table of the results that the type of construction used for the cable gives a strong indication of its likely value. This is because the permittivity of pvc is approximately 4x greater than air. Hence the more pvc there is surrounding the wires, the greater the capacitance for a given arrangement, and the lower the velocity factor.

A steady DC current will flow uniformly across the whole cross section of a wire. However an AC current will only occupy an increasingly shallow band at the surface at RF frequencies. The current density (Amps/m^{2}) falls off exponentially from the surface of the conductor towards the centre. At a critical depth called the depth of penetration the current density has reduced to 0.368 of the surface density. For example the depth is only 0.02mm at 10MHz for a round copper conductor. This produces a significant increase in the resistance at the RF frequency (Ref.17). The critical depth δ is calculated in the following way:
δ = √ (ρ / (π x f x μ) metres
where ρ = Permeability = 4 x π x 10^{7} (Henries/metre)
π = pi
f = frequency (Hz)
μ = resistivity Ohms/metre = 1.69 x 10^{8} Ohms/metre for copper
The approximate ratio of the DC resistance to the RF resistance can then be obtained by using the ratio of the area of the shallow band over the area of the whole conductor:
R_{dc} / R_{rf} = (2 x π x δ x r) / ( π x r^{2}) = (2 x δ) / r
where r = radius of the solid conductor, or the outer radius of a bundle of strands (metres)
The current derating factor is therefore = √ (R_{dc} / R_{rf} )
There is an additional effect when two current carrying conductors are in close proximity as in two core cables. If the RF current is flowing in opposite directions in the two conductors more current tends to flow on the surfaces facing each other than on the opposite sides. The reverse happens if the current flows in the same direction in the pair of conductors. This can reduce the derating factor by approximately 5% for the cables listed below (Ref.18).
Tests were carried out by the Radio Club on some typical BS6500 flexible PVC insulated 2core cables to check the formulae. The heat runs were performed at 29.5 MHz with the cables raised 100cm clear of the bench to better simulate attic conditions. The continuous RF current ratings were based on a 30°C rise in a 30°C ambient. The dc current ratings were based on the conductor cross sections for the 30°C temperature rise prescribed for flexible cables in BS6500.

It can be seen from the above table that the skin depth formula yields a reasonable estimate for the RF derating factors for the cables. The following table therefore gives the calculated derating factors for the current for other typical conductors covering 7, 10 & 30 MHz:

Note: The current in the antenna is maximum at the feed point and can be calculated by:
Current = √ (Watts/feed_point_resistance)
For example: The feed point resistance of the double folded loop at resonace is 50 Ω
Therefore for 100 Watts the input current = √ (100/50) Arms
Hence the current in each conductor of the double folder antenna = 1.41 Arms.
The following table 1 has been based on staying within the European continuous exposure field strength limits for the operator and his family at a distance of 1m below the ceiling under the nested loops, which places it well above the seated position in the room. The NEC simulation data files supplied excite the antenna with an RF voltage having a magnitude (peak) of 1 Volt. The NEC program was run to determine the maximum peak values for the E and H fields at a distance of 1m below the ceiling. The power delivered to the antenna at resonance is given by the Vpeak^{2}/(2R) where R is the input resistance of the antenna at the feed point. The field strengths are proportional to the applied voltage. The applied voltage can therefore be scaled to the maximum permitted in the following manner:
Vpeak_max = Vapplied x [RMS_field_limit x √2/Peak_field_simulated]
The maximum power is then given by:
Max_continuous_power = Vpeak_max^{2}/2R
Note: The NEC antenna data files represent the loops as single wires instead of the folded dipole arrangement for ease of simulation. The feeder impedance should therefore be reduced to 12.5 Ω in the simulations so that the correct SWR is displayed.

Note: To down load the nec file, right click on the file name and choose 'save link as ...'.
Table 2 applies if there are neighbouring properties of the same height within 1.5m of the sides of the nested loops. It takes account of the lower European limits on continuous exposure levels for the general public. The power reductions from the previous table are shown in bold.

Note: To down load the nec file, right click on the file name and choose 'save link as ...'.
Table 3 is to be used if there are neighbouring properties that have living spaces that are at the same height as the nested loops or are taller. The forward and reverse distances are measured from the roof ridge, whilst the sideways clearances are measured from the midpoint of the nested loops. This ensures that the power levels given in Table 1 do not expose the general public to excessive levels of E & H fields (European Limits).

Normal steady state theory applies to circuits where the propagation time through them is much quicker than the period of the conducted signal, as in the output stage of the transmitter. A common rule of thumb is that cables or wires should be treated as transmission lines if their lengths are greater than 1/10 of the wavelength. Antenna feeders usually have significant propagation times and therefore have to be treated as transmission lines where energy is reflected back and forth between any impedance mismatch at their ends setting up standing waves.
Table 4 shows the bandwidths of the nested loops corresponding to an SWR of ≤ 2 and ≤ 4 when fed with a coaxial cable having a characteristic impedance of 50 Ohms. When there is a mismatch at the antenna it causes some of the energy to be reflected back towards the transmitter. An ATU is then used to cancel any reactance and transform the resistance seen at the connection with the antenna feeder to 50 Ω presented to the transmitter to meet its loading requirements.
Variations in the voltage at the collector (or drain) of the output transistor in the transmitter has very little influence on its current, as illustrated by the characteristics for a VN66AF MOSFET transistor shown here. It therefore acts like a current source with its accompanying high dynamic resistance. Matching circuits in the transmitter output convert an applied 50 ohms load into the optimum load at the transistor. If this load resistance is less than the optimum the transmitter cannot deliver the nominal full power within its maximum current rating. If the load resistance is greater than the optimum then the output current (and hence power) has to be reduced to prevent the transistor being destroyed by over voltage.
The relatively high dynamic resistance of the output transistor results in a large mismatch with the characteristic impedance of the coaxial cable joining the transmitter to the ATU. This mismatch is passed on by the ATU to its connection with the antenna feeder. Hence virtually all of the reflected energy reaching the ATU is reflected back towards the antenna, minus any power lost in the ATU (Ref.19). Energy is also lost in the feeder during each transit. If the feeder is made out of 10m of RG56 with a loss of 0.045dB/m and the tuned ATU has a typical insertion loss of 0.5dB and is situated close to the transmitter then:
Note: If the ATU is situated at the antenna end of the feeder then 80% of the transmitter power is delivered to the antenna largely independent of the antenna missmatch.
Table 4: Bandwidths for SWR ≤ 2 and ≤ 4

The lower the frequency the more compromises that have to be accepted when the antennas are confined to small attics. Worthwhile performances can still be obtained but the construction becomes more complex and the bandwidth, gain, efficiency and safe operating powers are all reduced. There is not enough space in the example attics chosen for a bent dipole without adding loading inductors to lower the resonance. This reduces the input impedance to around 6.0 Ohms depending on the height above ground. It therefore requires approximately 3x as much current in the antenna compared to a simple dipole for the same radiated power. A trifilar arrangement using a 3core flexible pvc cable is therefore employed which provides a 9:1 impedance transformation in order to achieve a better match with the standard 50 Ω coaxial feeder cable (Ref.15). This arrangement also allows the increased current to be shared equally between the the 3cores so that there is no loss in efficiency. The dipole is shown installed on the opposite side of the roof ridge to the nested loop antennas and requires a separate coaxial feed otherwise it would interfere with the performance of the 29MHz antenna.
The velocity factor of the cable has to be taken into account in the construction of the trifilar bent dipole. A round 3core 300/500 Volt flexible pvc cable with 24/0.2 (0.7mm^{2}) cores has sufficient rating for the performances in the following tables. This has a velocity factor of approximately 0.52. The connections between the fed core and one of the other cores are made at the positions 1/4 wavelength (in the cable) from the centre. This corresponds to a distance of 3.9m either side of the feed point for 10.125MHz. The two unfed cores are connected together at their remote ends. The arrangement is show in the following diagram.

Each loading coil inductance is formed by winding the round 3core 24/0.2 pvc 300/500 volt flexible cable into a flat spiral to form a rosette. This shape makes it easier to adjust the inductance and to fix it onto a roof rafter afterwards. The leads are brought out radially to minimise additional coupling. The formers are made out of 3mm thick 3core plywood as shown in the following diagram. The ring of 3mm diameter holes are there to allow string to be threaded through to tie down the end of the coil. The coils are then fixed to the rafters using the centre holes, 1cm spacers, and any type of wood screw.
The flat coil inductance can be determined by using the the formula derived by Harold A. Wheeler. This has been programmed into the calculator available on the web site given in Ref.20. Weaving the cable between the fingers of the coil former was found to increase both the winding length and the inductance by 5% more than the calculated values for the idealised flat shape. The following table gives the details of the resulting loading coils.
Table 6: 10.125 MHz Dipole Loading Coil Details

A certain amount of trial and adjust is required when setting up the dipole because of the restricted bandwidth. The two halves of the dipole need to be kept balanced. Remember to short the fed line and the other line together at the distance corresponding to 3.9m from the feed point as it will lie within each coil. The lengths of the arms can be adjusted by bending the wire back along themselves. Shortening each end by 10cm will raise the resonant frequency by approximately 25kHz. This is equivalent to removing a quarter of a turn from each loading coil.
The following Table 7 gives the performance details and the maximum continuous powers that comply with the safe operating limits for the living spaces below. It assumes that the neighbouring properties of the same height are at least 2.5m away from either side of the dipole, or at least 3.3m from the roof ridge. Neighbouring properties that are taller need to be at least 3.7m away from either side, or 4.5m away from the ridge.
The longer wavelength at 10MHz means that the power limits and the antenna gains are more greatly influenced by the height of the dipole. There is no practical difference between the forward and backward propagation directions, but most of the radiated power tends to be directed upwards rather than horizontal. The gains are given for 60° when the maximums occur at that angle or above relative to the horizon. The bandwidths are 57kHz for the 3.6m ridge design increasing to 100kHz for the 5.4m ridge. This is based on the the SWR being ≤4.
Even though the continuous power limits are restricted they are still comparable with the power ratings of most standard amateur transceivers when the duty factors for the various operating modes are taken into account see Application of the limits.
Note: The NEC antenna data files represent the loops as single wires instead of the trifilar dipole arrangement for ease of simulation. The feeder impedance should therefore be reduced to 5.5 Ω in the simulations so that the correct SWR is displayed.
Table 7: 10.125 MHz  Continuous Power Limits  No Close Neighbours

Note: To down load the nec file, right click on the file name and choose 'save link as ...'.
However the continuous power limits are further restricted if the neighbouring properties on either side are attached and at the same height, as shown in the following Table 8 i.e. if the houses are semidetached or part of a terraced row. This is in order to stay within the exposure limits for uncontrolled areas.
Table 8: 10.125 MHz  Attached Neighbours at the same Height

Ref.15  ARRL Antenna Book (1991) section 233, figure 28.
Ref.16  http://emclab.mst.edu/resources/tools/inductancecalculator/
Ref.17  http://www.rfcafe.com/references/electrical/skindepth.htm
Ref.18  Glenn Smith; The Proximity Effect in Systems of Parallel Conductors... http://www.dtic.mil/cgibin/GetTRDoc?AD=AD0736984
Ref.19  RF Design Basics by J.Fielding (ZS5JF), published by the RSGB.
Ref.20  http://www.circuits.dk/calculator_flat_spiral_coil_inductor.htm