Phase in RLC

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 U0. 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 IInd 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)

where and the quantity

 (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 Im and its initial phase φ is

 (13)
 (14)

Both the amplitude of the current Im 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 ωres the initial phase of the current equals to zero φ=0 and its amplitude is maximal, let us denote it Im res. And this is the situation we speak about the resonance of the series RLC circuit with respect to its current.

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, RI2 is the instantaneous power, the integral represents the average transferred power Pa 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 Pa 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.