Calculating the component values for an antenna tuner

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In this article we are going to match a parallel LC circuit to an antenna.
Matching means, connecting LC circuit and antenna in such a way that there is maximum power transfer from antenna to LC circuit.
This article makes use of complex impedances, a explanation about this you can find here (external link).

There are several ways of matching the antenna to the LC circuit, I discuss here only the matching via a variable capacitor between antenna and LC circuit.


With the next calculator, the component values of this antenna tuner can be calculated.
Enter: frequency, value of the coil in the LC circuit (L5) and the unloaded Q of the LC circuit L5, C6
Also enter the complex impedance of the antenna, or the series component values of the antenna.
(when you enter both complex impedance, and series components, the complex impedance is used, and not the series components).
If you don't know the antenna impedance, use the standard values for the series components: 25 Ω, 20 μH en 200 pF.

Click on "calculate".

Frequency: kHz  
Coil value (L5): μH  
Unloaded Q of LC circuit:  
Complex impedance of antenna: Ω
Series components of antenna: Ω (resistor)
μH (coil)
pF (capacitor)
     
Complex impedance of antenna Ω  
Parallel resistance of LC circuit Ω  
Value of C6 pF  
Value of C4 pF alternative: μH
Frequency shift when
disconnecting antenna		 kHz  

Extra information as used in the calculations below:

=X8 (Ω) =C6+CP (pF)
=XP (Ω) =XC4  (Ω)
=CP (pF) =Frequency without antenna (kHz)

Calculating the component values.

Here is explained what the calculator is calculating.
To calculate the component values, we need the following data:

The frequency.
The induction value of the coil (L5).
The unloaded Q of the LC circuit
The complex impedance of the antenna

The complex impedance of the antenna is build up with a certain resistor R1 which is in series with a coil (L2) and a capacitor (C3).
The values of R1, L2, and C3 are depending on many factors, like frequency, antenna length, height of antenna above ground, etc.
The impedance of the antenna is: Zantenna = R1+Z2+Z3 = R1+J(XL2 - XC3).
With a short antenna (shorter then 1/4 wavelength) XC3 will have a higher value then XL2, the result is a capacitive complex impedance for the antenna, so with a minus sign in front of the J.

 


The antenna tuner
The antenna tuner split into separate impedances.
R1, Z2 and Z3 represents the antenna.
R7 is not a real resistor, but represents the losses in parallel circuit L5, C6


Step 1
Choose a frequency, calculate the complex impedance of the coil: Z5=+J(2.pi.f.L5)
Calculate the parallel resistor (R7) of the circuit L5 C6 with the formula: R7=2.pi.f.L5.Q
Resistor R7 represents the losses occurring in L5 and C6.

Step 2
The antenna can be considered as 3 separate components, R1, Z2 and Z3
We take the sum of Z2, Z3 and variable capacitor Z4, we consider this sum as one capacitor with impedance value Z8.
So Z8=Z2+Z3+Z4= -JX8
Z8 must provide the match between resistor R1 and R7



Step 3
Z8 in series with R1 can be converted to a parallel circuit RP, XP with the formulas:

RP=(R1² + X8²)/R1        XP=(R1² + X8²)/X8    however, we can't use these formulas yet, because X8 not known yet.

For impedance match, parallel resistor RP must be equal to parallel resistor R7.
So:  RP=R7

From this follows:

(R1² + X8²)/R1 = R7

R1² + X8² = R7.R1

X8² = R7.R1 -R1²

X8= √ (R7.R1 -R1²)

Step 4
Now we have X8 we can calculate also XP with the formula:
Xp=(R1² + X8²)/X8

XP is a capacitor which is parallel to C6.
XP represents a capacitor value of: Cp=1/(2.pi.f.XP), and this is in parallel with C6

Step 5
For resonance of the circuit must apply:
f=1/(2.pi.√(L5.Ctotal)

Or:

Ctotal = C6+CP = (1/(2.pi.f))²/L5   if we subtract the value of CP, the value of  C6 is left.

Step 6
Finally we want to know the value of  C4.
We already had X8.

With the formula XC4=X8+XL2-XC3 we can calculate the impedance of C4.
And then with C4= 1/(2.pi.f.XC4) calculate the value of  C4.
Now we know all values of the matched circuit.

Sometimes, XC4 will have a negative value, in that case it is not possible to get a match by means of a variable capacitor on the place of C4.
As an alternative, we then can replace C4 by a coil with value:  L4= -XC4/(2.pi.f)
 

Frequency shift when disconnecting the antenna
When we disconnect the antenna from the antenna tuner, the resonance frequency of the circuit will increase to value f2.
This is because Cp is no longer part of the tuned circuit.
f2=1/(2.pi.√(L5.C6)
The frequency shift of the circuit is equal to: f2-f
 

The loaded Q of the circuit
If we connect the antenna (via C4) to the LC circuit, the Q of the circuit will degrease.
The Q we have then is called the Q of the loaded circuit (or the loaded Q).
If the LC circuit is well matched to the antenna, the loaded Q will be halve the value of the unloaded Q.

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