### Physical background

## ELECTRICAL OSCILLATIONS

In Figure 1 is depicted series circuit built from the capacitor of the capacitance *C*, inductor of the inductance *L* and resistor with resistance *R*. We may consider the resistor as a load, but very often it represents both the losses in capacitor or inductor.

*Figure 1. The schematical representation of the RLC series circuit*

How are oscillations generated in such circuit ? Suppose the capacitor is charged before the switch K is closed by the voltage* U** _{0}*. The energy accumulated in the circuit is the electric field energy of the capacitor. If we in

*t*= 0 s close the switch, the time dependent

*I*

*(t)*current starts to flow in the circuit and voltages occur on individual elements of the circuit – across the resistor there is a voltage and across the inductor the electromotive force.

We can write for the instantaneous charge on the capacitor

(1) |

and corresponding voltage

(2) |

Acros the inductor is created the electromotive voltage according to Faraday’s law, and the corresponding voltage is

(3) |

Across the resistor is the voltage

(4) |

The sum of all these voltages should be, according to the II^{nd} Kirchhoff’s law, equal to zero

(5) |

or

(6) |

The first derivative with respect to time and after slight rearrangement we have the differential equation of electrical oscillations

(7) |

(8) |

as natural angular frequency of the for undamped oscillator ( with R =0), called **Thomson formula **.

Resonant angular frequency with the maximum current is

(9) |

In majority of real cases we can neglect the term *b*

(10) |

with respect to b < and the resonant frequency then may be put (8).

Connecting the series RLC circuit to the source of harmonic electromotive voltage with the arbitrary angular frequency *ω, *we have the differential equation of the forced oscillations

(11) |

The steady-state solution (opposite to transient solution) of this differential equation gives for the current the harmonic oscillations

(12) |

Where the amplitude of the current *I*_{m} and its initial phase *φ* is

(13) |

(14) |

Both the amplitude of the current *I*_{m} and initial phase *φ* are functions of the driving angular frequency *ω*. According to the angular frequency *ω* of the driving electromotive force the current may advance or lag behind the voltage. For a certain angular frequency *ω _{res}* the initial phase of the current equals to zero

*φ=0*and its amplitude is maximal, let us denote it

*I*

_{m res}. And this is the situation we speak about the

**of the series RLC circuit with respect to its current.**

*resonance**Figure 2. Amplitude and phase transfer characteristics of the series RLC circuit*

**TRANSFERRED POWER**

The main application of the discrete RLC resonant circuits is energy transfer (or power – as the transferred energy in unit time) from the source of energy to the load. This energy is given by the general formula

(15) |

Where *I(t)* is the instantaneous value of the current in the circuit, *RI*^{2} is the instantaneous power, the integral represents the average transferred power *P*_{a} transferred from the generator ( or antenna) to the load. This may be expressed by the equation` [4]`

(16) |

In Figure 3 is the dependence of the average transferred power *P*_{a} on the driving angular frequency *ω*.

Obr. 3: Stredný prenesený výkon zo zdroja na záťaž v obvode

**Literatúra **

[1]** **BESANÇON, R.M.: *The encyclopedia of physisc. *3th ed. New York : Van Nostrand Reinhold, 1990, 1378 p. ISBN 0-442-00522-9.

[2] HALLIDAY, D., RESNICK, R., KRANE, K.S.: *Physics* : extended version. Volume two – 4th ed. New York : John Wiley & Sons, Inc., 1992. 1216 p. ISBN 0-471-54804-9.

[3] TIRPÁK A.: *Elektromagnetizmus. *Bratislava : Polygrafia SAV, 1999. 710 s.

[4] LINDSAY, R.B.: *Physical Mechanics*, 3rd ed., New Jesrsey – Princeton : Van Nostrand, 1962. 436 s.