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Forced Oscillations in RLC Circuit
 
The settled oscillations in a circuit in the presence of a harmonic voltage Vcosωt from a generator are called forced oscillations. Forced oscillations are always of the same frequency ω as those of the generator.
If the external harmonic voltage is in the L - R - C circuit, the amplitude of the forced oscillations (for example, the amplitude of the voltage on the capacitor) greatly depends on the relationship between the frequency ω of the generator and the natural frequency ω0 = 1/(LC) 1/2 of the circuit. At ω = ω0, a resonance occurs. Resonance is defined as a dramatic increase in the ratio of amplitude of the forced oscillations to the amplitude of the external generator. The graphic depiction of this ratio as a function of the generator?s frequency is called the resonance curve. When the resistance of the circuit increases, and the loss of energy increases with it, the slope of the resonance curve becomes flatter.
A phase shift between the source voltage and the voltage on the capacitor will occur depending on the relationship between ω and ω0. At resonance, the shift is 90°.
The vector diagrams graphically represent and analyze the relationships between the amplitudes of voltages and currents. Two or more harmonic oscillations with the same generator's frequency ω are represented on the diagram by the vector diagrams. Their lengths are equal to the amplitudes of the oscillations, and the angles between them are equal to the phase shift.
Turn the vector diagram that represents the voltage of the coil in the counter-clockwise direction in respect to the vector diagram that represents the current in the coil by 90°. The voltage of the coil now leads the current by 90°. At the same time, the current leads the voltage by 90° in the capacitor. In the circuit, the vector sum of all the voltages of elements connected in series must equal the magnitude of the vector diagram that represents the external generator's voltage.
If the L-R-C series circuit is at resonance, the amplitudes of the voltages VC and VL of the capacitor and the coil, respectively, are the same. The phase shift between them is equal to 180°. This explains why the magnitude of the vector diagram of the external voltage coincides with the magnitude of the vector diagram of the resistor.
 
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