The ARRL Antenna Book 21 (Straw 2007) gives equations and advice for design of small transmitting loops.
Many Hams have used these formulas and method for their designs, and some have even produced design tools (online calculators and spreadsheets) implementing these formulas.
This article reviews these formulas and advice.
In Table 3, (Straw 2007) gives the following equations.
Reviewing the equation above for Q, (Terman 1955) tells us that circuit Q is 2π(energy stored in the circuit)/(energy dissipated in circuit in one cycle) which if you consider the loop circuit to have a lossless capacitor, at the moment when the capacitor voltage is zero, all stored energy is in the inductor and energy stored is I^2*L/2 where I is the peak current, and energy lost in a cycle is I^2*R/2f, so for the circuit comprising the loop inductance, total resistance (R=Rr+Rl), and capacitance:
The formula for Q in (Straw 2007) is wrong, it underestimates Q by a factor of 2. Error in Q may roll up into error calculated bandwidth, voltage, current, and efficiency.
(Straw 2007) states:
Note that the RL term above also includes the effect of the tuning capacitor’s loss. Normally, the unloaded Q of a capacitor can be considered to be so high that any loss in the tuning capacitor can be neglected. For example, a very high-quality tuning capacitor with no mechanical wiping contacts, such as a vacuum-variable or a transmitting butterfly capacitor, might have an unloaded Q of about 5000.
Even if the capacitor had an intrinsic (or unloaded) Q of 5000 can it be ignored?
A 1m diameter loop of 20m copper tube at 7MHz (including the effect of radiation resistance) has a Ql of close to 3000 if properly implemented with low resistance joints. The Q of the loop when tuned with a capacitor with Qc=5000 will be 1/(1/Ql+1/Qc)=1875, just 63% of Ql alone so it can be a significant error to ignore capacitor loss.
(Jennings nd) states for their vacuum capacitors:
The “Q” factor, or ratio of stored energy to dissipated energy, is typically in the order of 1000 or 5000 or higher
so they are not as optimistic as (Straw 2007).
(Payne 2013) gives models and measurements for some of the types of air variable capacitors that appear in many loop projects, and Q is of the order of 1000 for typical capacitors at 3.5MHz.
Some implementations have used capacitors fabricated from double sided FR-4 PCB. FR-4 has a quite high dissipation factor (~0.18) and makes very low Q capacitors (~50).
Some have used open circuit coaxial transmission line stubs, and these may have Q as low as 200, possibly up to 1000 depending on line length, line type and frequency.
It is unsafe to ignore capacitor loss, even in high performance loops.
The power developed in some conductance G by some RMS voltage V is given by P=V^2*G W.
This can be rearranged to give V for some power P as V=(P/G)^0.5 Vrms (1).
In a high Q loop circuit, G is approximately 1/(Xl*Q) (2).
Substituting (2) into (1) we get V=(P*Xl*Q)^0.5 Vrms (3).
(3) is essentially the formula for V given by (Straw 2007), but:
- his value for Q is wrong by a factor of 2 (as discussed above);
- Q that ignores capacitor loss may significantly underestimate circuit Q; and
- the voltage is not qualified as RMS (peak voltage would be 1.414 time RMS).
Just as calculated voltage depends on Q, so does current and the error in Q discussed earlier rolls up into error in calculated current.
Again, ignoring capacitor loss may result in significant error in calculating current.
(Straw 2007) gives an equation for loop efficiency.
As discussed above, capacitor loss may not only be significant, but in many low end designs it may dominate loss and actual efficiency might well be a third or less of that calculated by ignoring capacitor loss.
As mentioned, the ARRL Antenna Book 21 (Straw 2007) has been used by some calculation / design tools, apparently without verification that the formulas are indeed correct. Be wary of the tools you use, validate them.
The formulas given in the ARRL Antenna book 21 bear a striking resemblance to those in (Hart 1986) which gives the Q formula with no development or explanation. It seems that these formulas have been accepted without verification for almost 30 years.
- Hart, Ted (W5QJR). 1986. Small, high efficiency loop antennas In QST June 1986.
- Jennings. nd. Vacuum capacitor characteristics. http://www.jenningstech.com/technotes/vchrctr.shtml (accessed 23/05/14).
- Payne, A. 2013. Measuring the loss in variable air capacitors. http://g3rbj.co.uk/wp-content/uploads/2013/10/Measurements_of_Loss_in_Variable_Capacitors_issue_2.pdf (accessed 23/05/14).
- Straw, Dean ed. 2007. The ARRL Antenna Book. 21st ed. Newington: ARRL. Ch21.
- Terman, Frederick. 1955. Electronic and Radio Engineering – 4th ed. New York: McGraw-Hill.