Theoretical part. Forced oscillations appear in the contour when alternating current from an extraneous source (in our case from a generator) is following through serially

Forced oscillations appear in the contour when alternating current from an extraneous source (in our case from a generator) is following through serially connected inductance – L, capacity – C and resistance – R.

Differential equation which describes the change of charge on a condenser with time looks like:

. (22.2)

where - attenuation coefficient; - own cyclic frequency of non-attenuating oscillations in the contour when .

The solution of equation (22.2) is

, (22.3)

where ω is cyclic frequency of free oscillations

, .

After some time the first element in equation (22.3) will become infinitely small due to exponent in a negative power. The contour will get into the permanent mode of the forced oscillations with voltage frequency . Condenser voltage:

. (22.4)

Amplitude of these oscillations depends on frequency of the enclosed difference of potential

. (22.5)

This function has an extremum (max) at frequency

, (22.6)

which is named resonance frequency. Consequently, the phenomenon of resonance consists in achieving of maximal amplitude of the forced oscillations at the change of frequency of external action. Resonance frequency does not coincide with the own frequency of oscillations ω_о as shown in (22.6).

From fig. 22.2 it is evident that voltage higher than the amplitude of the enclosed signal appears at two values of frequency.

Figure 22.2

From (22.5):

.

After simplifications obtain a correlation which enables to calculate resonance frequency by a value and for any voltage

. (22.7)