Radio and Electronics (DED Philippinen, 66 p.) 
8. PASSIVE COMPONENTS 
8.4. COMBINATION OF PASSIVE COMPONENTS 
8.4.3. TUNED CIRCUIT CONNECTED TO AN ACVOLTAGE 

SERIES TUNED CIRCUIT AT AC
If we connect an acvoltage source across an LC combination as shown in fig. 79 This can be looked at as a series connection of two impedances connected to an acvoltage source.
fig. 79
To simplify the problem we can ommit first the ohmic parts of the two components and concentrate on the reactances only. We know, that their reactances are depending on the frequency of the voltage they are connected to. Refering to a series connection of resistors and the rule, that the overall resistance of two resistors connected in series is the some of the two original resistances, we can easily derive that the overall impedance of that series connection will be the sum of the two original reactances.
fig. 80
If we do the addition of the two reactances by graphical means, as shown in fig. 80 we find, that the overall impedance will have high values at deep and at high frequencies, and it will have a minimum at a certain frequency. This certain frequency is called the RESONANT FREQUENCY or the TUNED FREQUENCY and it will be exactly that frequency at which the reactance of the inductor and the reactance of the capacitor will be equal. Summing up our findings we can also say: the current in this circuit will be maximum at the resonant frequency.
PARALLELTUNED CIRCUIT AT AC
If we want to derive, what happens in a parallel combination of an inductor and a capacitor, connected to an acvoltage (as shown in fig. 81), we can again use the graphical method.
fig. 81
But this time we have to add the ADMITTANCES of the reactances of the two components to find the overall admittance in that circuit.
The admittances equal the reciprocal of the reactance. Adding the values of the admittances we get as the overall admittance Ytot. But we should not forget, that this represents the overall admittance, and in order to be able to compare it with our findings in the chapter before we have to turn this graph into a graph representing the overall impedance which is again the reciprocal value of the admittance. The result is shown in fig. 81. And we find that here the impedance has a peak value exactly at the so called resonant frequency.
SUMMARIZING
SERIES TUNED CIRCUITS HAVE A MINIMUM IMPEDANCE AT RESONANT FREQUENCY!
PARALLEL TUNED CIRCUITS HAVE A MAXIMUM IMPEDANCE AT RESONANT FREQUENCY!
fig. 82
THE RESONANT FREQUENCY
Up to now we do not know, how to calculate the resonant frequency, for any combination of an inductor and a capacitor. As we stated above, the resonant frequency appears if the reactance of the inductor equals the reactance of the capacitor.
Therefore it is easy to derive the formula for the resonant circuit.
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