# Impedance Transforming T-Networks - Antenna Tuner Application

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

C1, pF ¦ C2, pF

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From transmitter. $ To antenna feeder.

Required network $ L, uH Input Z = R+/-jX

input resistance ----> $ ----->

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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.

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