This article describes a linear circuit model for a small transmitting loop with auxiliary loop feed.
The main loop is taken to be 1m in diameter, 20mm diameter aluminium tube and to have an inductance of 2.5µH, total resistance (loss +radiation) of 0.04Ω. The feed loop is a fifth the diameter and taken to have inductance of 0.5µH and negligible resistance.
The inductive reactance at 7.5MHz is 117Ω, Q is 117/0.04=2945, and half power bandwidth BW=7500kHz/2945=2.55kHz.
The two loops are coupled, so they have mutual inductance, and it is the extent of the flux coupling and resulting mutual inductance M that performs the impedance transformation of 0.04Ω of the main loop to the desired 50Ω feed point impedance.
The above screen shot shows to the right the loop inductance and 180pF tuning capacitance, and port 2 contains the 0.04Ω main loop resistance.
A tee equivalent circuit for the coupled loops is used, and it was found that the system matches with M=-0.033µH, so -0.033µH is deducted from the 0.5µH and 2.5µH physical inductors in the tee model, and -0.033µH inserted in the tee branch.
Above is a plot of S11, and it can be seen that ReturnLoss (-S11) is very large at resonance at about 7.508MHz, the system offers a very good match (RL>40dB, VSWR<1.02). The cursor is set to RL=7dB, the lower half power point of the antenna.
Above is a Smith chart plot of the same scenario. R is hardly constant, but note that the red curve is a circular arc of about the same radius as the R=50 circle, and it is rotated about the chart centre. This hints a transformation like that of a length of 50Ω transmission line.
The value of S21 at resonance is 1λ -116°, suggesting that 180-116° further delay of 50Ω line will rotate the arc onto the R=50 circle. In fact, 7.1m or 64° of line did just that.
Above, the circuit is amended to include a length of 50Ω line to build out the impedance transformation. The length of line was chose to rotate the Smith chart curve above to lie over the R=50 circle.
Above, the desired rotation performed. Note that the curve lies almost perfectly over the R=50 circle, R is approximately constant.
Above is the S11 curve. The half power bandwidth of the antenna (between RL=7dB points) can be seen to be approximately 2.6kHz which reconciles with the prediction made earlier.