Tuggle Antenna Tuner

David Wagner 2008/04/30 15:09

This page accompanies the online Tuggle Tuner Calculator.

Another antenna tuner calculator is also available online.

Resonance Formulæ

Tuggle Tuner Schematic

First, combine some components for convenience and determine the variable capacitor's values as proportional to how much the vanes are meshed.

  1. L_{ant,load} = L_ant +L_{load};
  2. C_{ant,load} = {1/{1/C_ant +1/C_load}};
  3. C_{gnd} = C_{gnd,min} + closed * (C_{gnd,max} - C_{gnd,min}) + C_{gnd,par};
  4. C_tnk = C_{tnk,min} + closed * (C_{tnk,max} - C_{tnk,min}) + C_{tnk,par}.

The following formulas (after Patron) give a good approximation, assuming the tank inductance is much greater than the total antenna and loading coil inductance.

  1. C = C_tnk + 1/{{1/C_{ant,load}}+{1/C_{gnd}}}.
  2. L_tnk <= L <= L_tnk + L_{ant,load}; Take L approx L_tnk for a good approximation using typical values.
  3. f_kHz approx 1/{1000*2*pi*sqrt{L_{mu H}*10^{-6} * C_pF *10^{-12}} }.

A more accurate solution involves solving a quadratic. Start by rearranging equation (2) from Analysis of the Tuggle Front End, Part 1, by Ramon Vargas Patron.

  • L_{ant,load} omega +({L_tnk omega}/{1-L_tnk C_tnk omega^2}) -(1/C_{ant,load} + 1/C_gnd){1/omega}=0.

Multiply both sides by omega(1-L_tnk C_tnk omega^2)=omega-L_tnk C_tnk omega^3.

  • L_{ant,load} omega^2 -L_tnk C_tnk L_{ant,load} omega^4 +L_tnk omega^2 +L_tnk C_tnk(1/C_{ant,load} + 1/C_gnd)omega^2 -(1/C_{ant,load} + 1/C_gnd)=0.

Combine like terms in a quadratic equation of ω²: a omega^4 + b omega^2 +c =0.

  • a=-L_tnk C_tnk L_{ant,load};
  • b=L_{ant,load} +L_tnk + L_tnk C_tnk(1/C_{ant,load} + 1/C_gnd);
  • c=-1/C_{ant,load} + 1/C_gnd.

Solve the quadratic for ω² and convert to Hz (f=ω/2π).

  • f={1/{2 pi}} sqrt{ {-b pm sqrt{b^2 - 4ac}}/{2a} }.

For values typical of crystal radios, both roots of this equation are positive, with one in the broadcast range and one at shortwave frequencies.

Reactance Formulæ

Each section has its own series reactance.

  1. X_{ant,load} = 2 pi f(L_ant + L_load) - 1/{2 pi f}(1/C_ant + 1/C_load )
  2. X_tnk = 1/{1/{2 pi f L_tnk} - {2 pi f} C_tnk }
  3. X_gnd = -1/{2 pi f C_gnd}

At resonance, these three should all add up to zero.

  • X_{ant,load} + X_tnk + X_gnd = 0

Also, note the reactance with respect to ground at the top and bottom of the tank both approach infinity.

  1. X_{tnk,top} = 1/{1/X_{ant,load} + 1/{X_tnk + X_gnd} } right infty
  2. X_{tnk,bot} = 1/{1/X_{gnd} + 1/{X_tnk + X_{ant,load}} } right infty

Inverted Tuggle

Another similar configuration is possible, and this may be a better arrangement for making a simple single-tuned set using typical components. Connect the antenna to the rotor of the series portion of the variable capacitor and the rotor of the tank capacitor to ground. Attach the detector to the stator.

In this configuration, the detector is attached at a point where, at resonance, the inductive reactance of the tank is balanced by the capacitive reactance of the antenna and what used to be the ground section of the variable capacitor. Also, the detector's DC return path simply goes to ground.

Estimating Unloaded Q

You can think of the Tuggle antenna tuner as a combined serial and parallel tuner. When tuned to lower frequencies it is more series-like, and when tuned to higher frequencies it is more parallel-like. The unloaded Q may be estimated by combining the parallel (circulating) and serial resonant Q values.

  • 1/Q_u = {Z_s/{Z_s + Z_circ}} {1/{Q_p}} + {{Z_circ/{Z_s + Z_circ}}} {1/Q_s}.

Assume, because of the relatively high serial antenna-ground resistance, the parallel Q_p is much higher than the serial Q_s. FIXME This is not always true at the high end of the band.

  • 1/Q_u approx {{Z_circ/{Z_s + Z_circ}}} {1/Q_s}|Q_p >> Q_sQ_u approx {Z_s + Z_circ}/Z_circ Q_s.

The current flow divides proportionally between the series impedance and the (assumed lossless) tank capacitor impedance. Calculate the serial impedance branch as usual and assume all other losses are negligible.

  • Z_s approx sqrt{{R_ant}^2 +(X_ant +  X_load + X_L_tnk + X_gnd)^2}.

Approximate the parallel circulating current path impedance and keep it positive.

  • Z_circ approx |X_C_tnk + X_L_tnk| + 10^{-9}.

Now calculate the series Q_s then plug everything you need into one of the overall unloaded Q_u formulas above.

  • L_0 = L_ant + L_load + L_tnk.
  • f_0 = 1/{2 pi sqrt{L_0 C_0}}C_0 = 1/{(2 pi {f_0})^2 L_0},
  • Q_s = 1/R_s sqrt{L_0 / C_0} = {2 pi f_0 L_0}/{R_ant}.

FIXME The following is wrong. The upper limit of the loaded Q may be estimated by calculating the serial loaded impedance (Z_s,loaded) to include the parallel load (R_load) and using this loaded serial impedance in one of the overall unloaded Q_u formulas above.

  • 1/{Z_{s,loaded}} = 1/Z_s + 1/R_load

References


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