Hartley oscillators (inductances)

[watari]

Hi everyone,

I ended up in this forum (that seems pretty interesting, btw) looking for information about Hartley oscillators. I'm new in this field that is electronic, and to tell the truth I don't know so much about it. So here is (one of)  my question.

   If I want an output frecuency for a Hartley oscillator, let's say 1MHz, I have a given equation that can provide the parameter L and C for the tank circuit. That is:

         f= 1/ (2*Pi*sqrtLC)    => LC = 1 / (2*Pi*f)^2

now I can get LC for that frecuency (1MHz), and according to my maths it is

         LC= 2.53*10^-14

so at a given C (let's say 4nF) I can get L, that is L= 6.325 uH ~ 6.33 uH.

But, because L= L1+L2, now I have to give some number to those two coils.  Using a N3904 transistor, I take its beta (at 40 mA current collector) as 200. The gain for a Hartley oscillator must be:

      beta > or = L2/L1

So to solve this problem I deciced to stablish a system of equations (or unequestions):

      
      L1+L2= 6.33
      
      L2/L1<or=200

So my results are    L1= 0.032 uH and   L2<or=6.3uH

But because  (beta=200),  200 >or= L2/L1 this means 200 is the top limit, therefore whichever proportion between L2 and L1 which result is less than 200 (always respecting that L1+L2=6.33uH) will be ok to make the Hartley oscillator oscillate at 1MHz. So for instance, if I chose:

   L1= 1.032uH and  L2= 5.3uH will be ok, as

      L1+L2= 1.032+5.3= 6.332 uH      and

      L2/L1= 5.3/1.032= 5.136  which is less than 200

These two conditions are satisfied, but if I chose any other L1 and L2 which satisfies this as well, it will be ok too. So finally, here is my question: Even though I can satisfy those two conditions with differents values of L1 and L2, How can this affect to the gain of the circuit? I mean, the closer this proportion L2/L1 is to 200, the better gain will I have and the better the circuit will run? Or how does it work instead? For example, if I get a gain (L2/L1) =1.000000000000001, will it be as good as having a gain like 199.9999999 to get the circuit working ok?

Thanks!!

[gyulasun]

Normally the gain of the circuit does not depend on the ratio of the two inductances.  Rather, the gain mainly depends on other components, namely collector impedance and on either dynamic emitter resistance (r_d=26mV/I_E) or the inserted emitter resistance if you use bipolar transistor (like you referred to, 2N3904).   
The collector impedance may include a normal resistor or a choke coil but you have to consider the load resistance value too, so gain becomes the ratio of the collector impedance divided by the emitter impedance. If this value is say 200 then you may wish to choose the ratio of L2 and L1 to be also around 200.   You may know that the gain normally should be slightly higher than 1 to maintain a positive feedback for the continuous oscillation but component losses and changes in temperature may also influence gain so it should be chosen higher than 1. 
(I assumed a decent Beta for the transistor of course, in the some hundred range, and feedback should be used to lower it.)

[watari]

First of all, thanks for answering, gyulasun and also i apologize for taking so long to respond you back.

So, if i did not understand wrong, the gain depends on the ratio collector impedance/emitter impedance (for a BJT which is what i have) and once i get this, i have to chose a ratio L2/L1~ 200.

Am i right?

I'll try to work on this and i'll report my results

Thanks!!

[watari]

I forgot!! L2/L1 ~200 if i get a ratio collecto impedance/ emtter impedance=200.

By the way, is this valid for every kind of transistor configuration or just for common emitter one?

[gyulasun]

Well the 200 for the gain is not 'obligatory', any smaller number will be good as long as it is higher than 1 and just compensate for the losses in the components. Depends also on the supply voltage whether it is changing and also on the expected ambient temperature change. 
Practical consideration for the gain also includes the inductance value for the coils because if the smaller coil becomes too low value due to a higher gain number, then it may not be practical to wind (you wish to use at least 5 or 10 turns rather than much smaller number of turns) so you can avoid this by thinking in advance.

This is valid for grounded emitter and grounded base configuration only, (provided there is no resistor feedback between collector and base).