A common scheme for narrow band match of an end fed high Z antenna gives a Simsmith model for the matching arrangement that follows.
The tapped coil could also be considered an autotransformer.
Above is a screenshot of the Simsmith model. The elements D1 and Coil will be explained.
Coupled coils – an equivalent circuit
The above figure shows two coupled coils with inductances L1 and L2, and mutual inductance M, and the equivalent T circuit.
Let’s remind ourselves of the meaning of the term autotransformer:
An autotransformer has only one winding with connections to three different points on the winding making two active portions. One portion is shared by both primary and secondary circuits, and the other portion of the winding is exclusive to the primary or secondary circuit (depending on whether it is a step-down or step-up transformer).
The first figure uses an autotransformer.
The above is not an autotransformer configuration as none of the winding portions are shared by both primary and secondary circuits.
Autotransformer equivalent circuit
Recognise that an autotransformer can be viewed as two partial winds magnetically coupled and sharing one terminal.
Now to apply the T equivalent circuit to the matching autotransformer, we must ‘rotate’ and flip the schematic so that the common terminal connects to the coax centre conductor, and L1 is from that point to ground, L2 is from the coax centre conductor to the top of the matching autotransformer.
The Simsmith model calculates the equivalent circuit values in the D1 element, it uses Wheeler’s Continuous Formula to calculate the inductance Lt of the whole winding, and L1 and L2 of the partial windings using the specified total turns n, pitch n and tapping %. From that, mutual inductance is calculated. The resistance of the whole winding is calculated (using Q etc) and the resistance is apportioned to the partial winds.
The equivalent circuit is implemented / evaluated in the Coil element.
Above is the Coil element schematic.
Note that because Simsmith does things backwards, the right hand side of the schematic (P2) is the source, the coax in the first figure.
It contains some redundancy in pursuit of clarity. The values L1_, L2_ and M_ are supplied from element D1.