# Efficiency and gain of Small Transmitting Loops (STL)

Small Transmitting Loops (STL) are loops of less than about 0.1λ in diameter or about 0.3λ in circumference. Below these limits, the current around the loop is almost uniform and this permits a simplified analysis.

STL are commonly known by Hams as “magnetic loops”, but that term is rarely used in recognised antenna text books.

The efficiency and free space gain of a circular STL can be easily estimated by calculation from simple measurements.

## Theory

The half power bandwidth BW1 of a lossless STL at some frequency f is f/Q1. Q1 is the ratio of inductive reactance Xl of the loop to radiation resistance Rr and these can readily be calculated using simple formulas.

BW1=f/(Xl/Rr)   (1)

The half power bandwidth BW2 of a practical STL at some frequency f is f/Q2. Q2 is the ratio of inductive reactance of the loop to total resistance (radiation resistance Rr plus loss resistance Rloss). The half power bandwidth of a practical STL can be easily measured by matching it at some frequency and measuring the VSWR=2.62 bandwidth (which corresponds to half power bandwidth or Zload=Zo±jZo).

BW2=f/(Xl/(Rr+Rloss))    (2)

It follows from (1) and (2) that Rr/(Rr+Rloss)=BW1/BW2   (3).

The efficiency η of an STL is Rr/(Rr+Rloss), so from (1) and (3) we can say

η=(f/(Xl/Rr))/BW2   (4)

The directivity of a small loop irrespective of size is 1.5 or 1.76dB.

The gain of a small loop is Directivity*Efficiency which can be stated in deciBels as gain=1.76+10*log((f/(Xl/Rr))/BW2)dB.

The calculator Calculate small transmitting loop gain from bandwidth measurement uses this method to predict efficiency and gain of a circular STL from the loop radius, conductor radius and measured half power bandwidth.

Note that the ARRL handbooks frequently use Q of an STL to mean the Q when loaded with a receiver. That quantity involves another variable (rx Zin) with uncertainty. ARRL observations about Q will usually be lower and bandwidth higher for that reason, but these are properties of the receive system and not the antenna system alone.