# Reconciling W5QJR’s loop formulas

Ted Hart inspired interest in loops for transmitting applications with his article “Small, high efficiency loop antennas” (Hart 1986).

He included a table of recommended designs, the following is an extract of the table rows relating to an octagonal loop with perimeter=20′ (6.1m). The tube specified was 3/4 copper pipe which has an OD of 22mm. Also given are his design equations:

## NEC model

I have constructed an NEC-4.2 model of the physical structure of the 20′ perimeter octagon on 7.1MHz at 2m centre height above 0.007/17 ground, and included 0.050Ω ESR for the air spaced tuning cap shown in his article.

## Reconciliaton

Attempts at reconciliation of Hart’s table, Hart’s formulas and my NEC model have not been entirely successful.

Lets go through the formulas one by one.

### Equation 1

Hart’s formula gives Rr=0.078Ω, NEC yields Rr=0.074Ω, not much difference. (See Accuracy of estimation of radiation resistance of small transmitting loops  for further discussion.)

### Equation 2 (incorrectly labelled 5)

Hart’s formula gives total loss resistance=0.013Ω, NEC yields total loss resistance (conductor, capacitor, ground)=0.196Ω, a huge difference. (See Small transmitting loop – ground loss relationship to radiation resistance for discussion.)

### Equation 3

Hart’s formula gives efficiency=86.1%, his table gives -2.7dB (53.7%), NEC yields 27.5%. Harts table and formula are inconsistent, and both way above the NEC results.

### Equation 4

Hart’s formula gives L=5.52µH, NEC doesn’t directly calculate the inductance but using Calculate small transmitting loop gain from bandwidth measurement which uses a formula from Grover, we get 5.12µH. A small difference.

### Equation 5

Hart’s formula gives Xl=246Ω, using NEC gives 263Ω (partly because it captures the effect of self capacitance, eg 5.12µH || 10.3pF gives 255Ω). A small difference.

### Equation 6

Hart’s formula gives Ct=91pF, NEC needs 85pF to tune the loop. A small difference, again as NEC captures the self capacitance effect.

### Equation 7

Hart’s formula gives Q=1353, NEC doesn’t directly calculate Q, but Q calculated from its results is 848. A large difference, Hart’s had a different meaning for Q. (See Antennas and Q for further discussion.)

### Equation 8

Hart’s formula gives BW=5.25kHz, his table gives 7kHz. NEC doesn’t directly calculate BW, but BW calculated from its results is 8.37kHz. A large difference.

### Equation 8.5 (not labelled)

Hart’s formula gives Cd=16.4pF, NEC doesn’t directly calculate the equivalent capacitance, but using Calculate small transmitting loop gain from bandwidth measurement we get 10.3pF. A large difference.

### Equation 9

Hart’s formula gives Vc=5772V (presumably RMS), NEC doesn’t directly calculate the voltage, but Vc calculated from its results is 4407. A large difference.

## Summary

There is significant difference between Harts table of recommended antennas, most notably the efficiency figure is internally inconsistent, and unrealistically large… no wonder people are excited by the claims.

Many of these formulas are in (Straw 2007) and many other ARRL Antenna Books.

The formulas have also been used in many online loop calculators and design spreadsheets, time honored, but wrong nevertheless.