# TUNED CIRCUITSby Harry Lythall - SM0VPO

## Introduction

I have just been in hospital for a few more weeks in January and February 2008. Although I could not do any home construction work, I was still mentally active, figuring coils and capacitors.

One of the larget problems in home construction is to "guestimate" the capacitance and inductance for a tuned circuit. Fear not, for this page will simplify that guesswork so you can estimate the correct value for coils and capacitances. I will give a rough guess-table from 10kHz to 10MHz. Please be aware that with any project you will always have to experiment a little. Your component layout and choice of materials will affect the results, but this page will get you in the right area.

## Method #1

This method I have used for years, and seems to work with ten fingers: no need to take off your socks.

Find out the wavelength, in metres, then express this as turns, and picofarads. For example, you want a tuned circuit for 3.5MHz. This is the Amateur 80-metre band. Use a capacitor of 80pf and wind 80 turns on a 5mm Diameter former, without ferrite. The frequency will be a little low, but you then remove a few turns to get the exact frequency you want. The tuned circuit impedance will be about 500 Ohms(ish)

## Method #2

To work out a tuned circuit using maths is always a bit of a bind, especially since you do not use it every day. But the usual method is to take the usual frequency formula and transpose it to find capacitance, and inductance.

Frequency = 1 / (2 x Pi x root(L x C))

Where Pi = 3.1415927, L = inductance in Henries, C = capacitance in Farads, and Frequency is expressed in Hertz. I used the old-fashioned "x" to mean multiply.

## Method #3

The reactance of a capacitor and inductor are equal at resonance, which means that can use the more simple inductive and capacitove reactance formulas:

Inductive reactance = 2 x Pi x F x L

Capacitive reactance = 1 / (2 x Pi x F x C)

where the reactances are in Ohms, Pi = 3.1415927, L = inductance in Henries, C = capacitacne in Farads, and F is the frequency in Hertz.

Easy, uh? Not quite. I always seem to end up a decimal point or two out, and I do lot of "figuring" and approximating. So is there some form of "crib-sheet" (lathund)? Yes, of course there is.

If you calculate the capacitor and inductor for one frequency, you can multiply them both together to re-create that "L x C" in the frequency formula. The LC-product will remain the same for any combination of inductance and capacitance for any given frequency. At 3.5MHz the LC-index is 2041. So if I divide 2041 by 100pf, then the result 20.41 is the inductance in micro-Henries.

It is also interesting that if you take the square-root of 2041 = 45.177 then 45.177pf and 45.177uH then the impedance of the tuned circuit is 1000 Ohms at resonance. With 90pf and 22uH you will have 500 Ohms. What could be easier? All you now need is that table if LC indexes I calculated while laying in that hospital bed:

 Chart of LC-index for LF & HF Bands kHz nF * mH kHz nF * uH MHz pF * uH 10 250.0 100 2500 1.0 25000 15 111.1 150 1111 1.5 11111 20 626.5 200 625 2.0 6250 25 40.0 250 400 2.5 4000 30 27.7 300 277 3.0 2777 35 20.4 350 204 3.5 2041 40 15.6 400 156 4.0 1562 45 12.3 450 123 4.5 1234 50 10.0 500 100 5.0 1000 60 6.9 600 69.4 6.0 694 70 5.1 700 51.0 7.0 510 80 3.9 800 39.0 8.0 390 90 3.1 900 30.8 9.0 308

Note that the left-hand column begins with an index number of 250,000,000 where inductance is in uH and capacitance is in pf. In this case it is convenient to divide by 1,000,000 (remove "000,000") and increase the capacitance values from pf to nf, and the inductance units from uH to mH. Remember too, that it is when the uH and pf (or mH and nf) values are identical that the impedance at resonance is equal to 1kOhms. The is the situation ONLY in the left and right-hand columns.

By arranging these columns you see a nice pattern to the numbers, from which you can extrapolate down to 1kHz, or up to 100MHz. The index valus is multiplied by 100 when the frequency is divided by 10, and the index number is divided by 100 when the frequency is multiplied by 10.

### Example calculation

I want to make a parallel tuned circuit for 100kHz, with 50-Ohm output tap, and an impedance of 10kOhms.

With reference to the table center column, I see that 100kHz has a nf/uh index of 2500. I want the pf/uh index (so that Z = 1kOhms), so I multiply this figure by 1000 = 2500 000 pf/uH. If I take the square-root of 2500000 I will get an equal value for both pf and uH = 1581pf and 1581uH. I now have a set of L and C values for 100kHz. The parallel resonance impedance is 1kOhms.

By dividing the pf by 10, and multiplying the uH by 10, we keep the same frequency, but increase the impedance by 10 and get 158pf + 15810uH (16mH). The impedance is now 10kOhms at 100kHz. The end impedance is 10K-Ohms and I want a 50-Ohm tapping. The impedance ratio is 10000:50, which = 200:1 ratio. Square root of 200 is 14 so the tapping must be 1/14th = 7% of the total number of turns: 7 turns in every 100 turns.

Very best regards from Harry Lythall