5. For the impedance matching, I will select a transformer so that 55MHz is matched. In contrast I will leave two lower resonances as they are.
This is because 11MHz and 29.5MHz usually tend to have higher impedance than 55MHz. In this case, even if the impedance is mismatched, the gain for these can be kept higher than 11.
I will post the detail for this mismatched case tomorrow.
Here the technique of the impedance matching for the triple resonant circuit are explained.
In our case, the transformer should be chosen so that the impedance of the resonance at 55MHz is matched.
We are going to use the transformer to step up the voltage applied onto the EOM.
To obtain the maximum step-up-gain, it is better to think about the behavior of the transformer.
When using the transformer there are two different cases practically. And each case requires different optimization technique. This is the key point.
By considering these two cases, we can finally select the most appropriate transformer and obtain the maximum gain.
( how to maximize the gain ?)
Let us consider about the transformer. The gain of the circuit by using the transformer is represented by
Where ZL is the impedance of the load (i.e. impedance of the circuit without the transformer ) and n is the turn ratio.
It is apparent that G is the function of two parameters, ZL and n. This leads to two different solutions for maximizing the gain practically.
- case.1 : The turn ratio n is fixed.
In this case, the tunable parameter is the impedance ZL. The gain as a function of ZL is shown in the left figure above.
In order to maximize the gain, Z must be as high as possible. The gain G get close to 2n when the impedance ZL goes to infinity.
There also is another important thing; If the impedance ZL is bigger than the matched impedance (i.e. ZL = 50 * n^2 ), the gain can get higher than n.
- case.2 : The impedance ZL is fixed.
In contrast to case1, once the impedance ZL is fixed, the tunable parameter is n. The gain as a function of n is shown in the right figure above.
In this case the impedance matched condition is the best solution, where ZL=50*n^2. ( indicated as red arrow in the figure )
The gain can not go higher than n somehow. This is clearly different from case1.
( Application to the triple resonant circuit )
Here we can define the goal as "all three resonances have gain of more than n, while n is set to be as high as possible"
According to consideration of case1, if each resonance has an impedance of greater than 50*n^2 (matched condition) it looks fine, but not enough in fact.
For example if we choose n=2, it corresponds to the matched impedance of 50*n^2 = 200 Ohm. Typically every three resonance has several kOhm which is clearly bigger than the matched impedance successfully.
However no matter how big impedance we try to make, the gains can not be greater than G=2n=4 for all the three resonance. This is ridiculous.
What we have to do is to choose n so that it matches the impedance of the resonance which has the smallest impedance.
In our case, usually the resonance at 55MHz tends to have the smallest impedance in those three. According to this if we choose n correctly, the other two is mismatched.
However they can still have the gain of more than n, because their impedance is bigger than the matching impedance. This can be easily understand by recalling the case1.
(expected optimum gain of designed circuit)
By using the equation (1), the expected gain of the triple resonant circuit including the losses is calculated. The parameters can be found in last entry.
The turn ratio is set as n=11, which matches the impedance of the resonance at 55MHz. Therefore 55MHz has the gain of 11.
The gain at 11MHz is bigger than n=11, this corresponds to the case1. Thus the impedance at 11MHz can go close to gain of 22, if we can make the impedance much big.