Impedance Transforming T-Networks - Antenna Tuner Application

Author: R.J.Edwards G4FGQ © 3nd June 2001

              o--------¦¦-------o-------¦¦-------o
                     C1, pF     ¦     C2, pF
                                ¦
 From transmitter.              $                To antenna feeder.
 Required network               $  L, uH         Input Z = R+/-jX
 input resistance ---->         $                ----->
                                ¦
                                ¦
              o-----------------o----------------o

For given input resistance, load impedance and frequency the program computes the values of C1, C2 and L. For given coil Q, capacitor Q and RF power input, the power dissipated in each of the 3 components and overall efficiency is computed, together with the peak voltages across C1 and C2.

This program can be used to design antenna tuning and similar networks to cover given ranges of load resistance and reactance. It is usual for both capacitors to be continuously variable. The inductor L may be variable or band-switched.

Arbitrary Parameter "A"
The load is defined by two parameters, R and jX, so in principle only two variables are needed in the network to obtain a match. But there are three variable components, one L and two C's, so the extra degree of freedom can be used to vary the network's L/C ratio. The extra input parameter is named "A".

"A" is variable from 1 to 20. For a given Z match, when A = 1 capacitor values are largest and power loss is least. As "A" increases so does coil inductance, capacitor sizes decrease and, most important, component power losses increase. When the program user sets a limit to capacitor size the program automatically increases "A" and the inductance value such that a Z-match is obtainable with smaller capacitors. But a compromise versus power efficiency must be made.

As a matter of interest a simple L network always has lower loss than a T. A T-network is used in antenna applications because a much wider range of network terminating impedances can be matched without any need for circuit-switching.

Volts, Amps and Watts
A loss resistance = X/Q ohms is considered to be in series with each component. For given network input power it is a straightforward matter to calculate current through each component and the power dissipated in it.

How The Program Works
The classical formula for calculating L & C values for a T-match is used first. But this assumes components are lossless, i.e., they have infinite Q values. Once approximate L and C values are known, with the inputted coil and capacitor Q's, the loss resistances associated with the three components are estimated and are included in a more complete and accurate model of the network.

Loss resistances are then transformed to values they would have if they were external to the network and resulted in the same power loss. The values of external resistances are then combined with the specified terminating impedances.

L and C network values to match the modified terminations are then recalculated and displayed on the screen. Depending on Q values they will differ by small amounts from the classical values which are assumed to be lossless. Classical L and C values are obtainable by setting Q to 10^8. Hit key S(et Q=inf.)

Network Image Impedances
Input and output image impedances are computed as a check on reliability of the main program and for interest. Note the approximate conjugate match between the output image and load impedances. The input image is not exactly equal to the required pure input resistance because transformation of loss resistances to other values external to the network is not an exact procedure. Larger discrepancies in the output image are due to the fact that if there is a conjugate match at the input end of a lossy network there cannot simultaneously be a conjugate match at the other end. But observe what happens when Q is set to 10^8

NOTE: The Input Image Impedance of a 4-terminal network is that looking into the input terminals when the output is closed with *its* Image Impedance. And vice-versa for the Output Image. In practice, the internal resistance of the transmitter is undefined and large mis-matches may occur at network terminals.

Coil and Capacitor Q's at HF
A solenoid, 50mm diameter, 100mm long, not close wound, has Q roughly 250 at 2 MHz and increases proportional to overall dimensions and to sqrt(F). So at 30 MHz a coil 25mm by 50mm has a Q of roughly 500. Q will decrease when a coil is in a small screening enclosure or near to other conductors or materials.

Air-spaced capacitor Q is an order of magnitude higher. It tends to decrease as frequency increases. It reduces when in contact or close proximity to insulating materials. In the present application loss in switch contacts, bearings and wiring can be lumped with capacitor loss. Q is in the range 800 to 5000. If nothing is known enter a typical air capacitor value: Q = 1500.

Miscellaneous Jottings
If C2 is computed to have a negative value it can be replaced by an inductance which has the same reactance as C2 at the operating frequency. The required Z-match will then be obtained. But computed component losses will be incorrect.

Usually, when a negative C value occurs, by increasing parameter "A" which in in turn increases coil inductance, the capacitor will acquire a positive value and the Z-match can be obtained without changing the network configuration. But it may be found the higher value of "A" causes transmission efficiency to be excessively degraded. Preferably "A" should not exceed 2.

If the inductor should have a negative value it can be replaced by a capacitor of the same reactance. The foregoing comments apply.

Computed phase shift is output volts relative to input volts. It always leads.

Unreasonable Data
There are many ways in which unreasonable data can be entered in this program. It is not possible to guard against all of them. Occasionally the program may abort. To re-run the program just type against the dos prompt the program name T_TUNER. To return to Windows type "exit".

Run this Program from the Web or Download and Run it from Your Computer
This program is self-contained and ready to use. It does not require installation. Click this link T_Tuner then click Open to run from the web or Save to save the program to your hard drive. If you save it to your hard drive, double-click the file name from Windows Explorer (Right-click Start then left-click Explore to start Windows Explorer) and it will run.