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16.16 | The Series RLC Circuit |
In these revision notes for The Series RLC Circuit, we cover the following key points:
The maximum voltage in a series RLC circuit produced by the source is not the arithmetic sum of the three maximum individual voltages, i.e.
This is because voltages are not in phase with each other. However, when considering the instantaneous voltages ΔV(t) for each component, we have
Since current and voltage across the resistor in a series RLC circuit are in phase, the phasor arrow of the resistive voltage will be collinear with that of current.
On the other hand, the current in the capacitor leads the voltage by π/2 (a quarter of a cycle, or rotation). Therefore, the capacitive voltage phasor is displaced by π/2 radians anticlockwise to the current phasor because capacitive voltage is quarter a cycle behind the current.
Moreover, since the current is behind by π/2 to the voltage (it lags voltage by quarter of a cycle), the inductive voltage phasor is displaced by π/2 clockwise to the current phasor.
Using the expression (ΔVL(max) - ΔVC(max)) instead of their separate notation, we obtain for the net voltage (when expressed in a vector form):
The last equation is obtained by applying the Pythagorean Theorem. Using the Ohm's Law for each component, we obtain
where XL and XC are the inductive and capacitive reactances in the circuit respectively.
The maximum current flowing in a series RLC circuit is
The expression √R2 + (XL-Xc )2 is known as impedance Z of the RLC circuit for the given driving angular frequency ωd. It represents the total opposition a RLC circuit presents to current flow. The unit of impedance is Ohm, Ω. Hence, we have
Thus, from the Ohm's Law we can write
If we substitute the reactances XL and XC with their corresponding expressions found earlier, we obtain
The only voltage in phase with current is the resistive voltage. This means for resistive voltage the phase constant is zero (φ = 0). As for the other two voltages, the phase constant is -π/2 for ΔVC and + π/2 for ΔVL.
The general phase constant φ will be the angle formed by the resistive voltage and the total voltage phasors. Thus, we have
As special cases, in purely inductive circuits (XL ≠ 0 and XC = R = 0), we have the maximum value possible of phase constant (φ = π/2); in purely capacitive circuits (XC ≠ 0 and XL = R = 0), the phase constant is minimum (φ = π/2); while in purely resistive circuits (R ≠ 0 and XL = XC = 0) the phase constant is zero because φ = 0 and therefore, tan φ = 0.
Resonance is a state of a RLC circuit in which the current and net emf are in phase. This occurs when XL = XC.
The driving frequency during resonance is
Giving that ωd = 2π ∙ f, we obtain for the frequency f in a series RLC circuit:
The effective (rms) values of current and voltage in an alternating circuit are
and
respectively. This is because both these quantities oscillate in a sinusoidal fashion.
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