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The Series RLC Circuit Revision Notes

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16.16The Series RLC Circuit


In these revision notes for The Series RLC Circuit, we cover the following key points:

  • Why the maximum voltage in a series RLC circuit is not the arithmetic sum of maximum voltages in each component?
  • How can we use the help of phasors to understand how a series RLC circuit works?
  • What is the phase relationship between current and voltage across the resistor in a series RLC circuit?
  • The same for the inductor and capacitor in series RLC circuits
  • What is the impedance of an electrical circuit? What is its unit?
  • What is phase constant? How can we calculate it?
  • What is resonance? What does the phase constant tells us about resonance in a RLC circuit?
  • What are the effective (rms) values of current and voltage in an alternating circuit? How to calculate them?

The Series RLC Circuit Revision Notes

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.

∆Vsource(max) ≠ ∆VR(max) + ∆VL(max) + ∆VC(max)

This is because voltages are not in phase with each other. However, when considering the instantaneous voltages ΔV(t) for each component, we have

∆V(t)source = ∆Vr (t) + ∆VL (t) + ∆Vc (t)

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):

εsource(max)2 = ∆VR(max)2 + (ΔVL(max) - ΔVC(max) )2

The last equation is obtained by applying the Pythagorean Theorem. Using the Ohm's Law for each component, we obtain

εsource(max)2 = (imax ∙ R)2 + (imax ∙ XL-imax ∙ Xc)2

where XL and XC are the inductive and capacitive reactances in the circuit respectively.

The maximum current flowing in a series RLC circuit is

imax = εsource(max)/R2 + (XL-Xc )2

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

Z = √R2 + (XL-Xc )2

Thus, from the Ohm's Law we can write

imax = εsource(max)/Z

If we substitute the reactances XL and XC with their corresponding expressions found earlier, we obtain

imax =εsource(max)/R2 + (ωd ∙ L-1/ωd ∙ C)2

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

tan⁡φ = XL-Xc/R
  1. If XL > XC, the circuit is more inductive than capacitive. The phase constant φ is positive, so the phasor εmax is ahead of the current phasor imax.
  2. If XL < XC, the circuit is more capacitive than inductive. The phase constant φ is negative, so the phasor εmax is behind the current phasor imax.
  3. If XL < XC, we say the circuit is in resonance, for which we will discuss in the next paragraph. The phase constant is zero, so the current and voltage rotate together.

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

ωd = √1/L ∙ C = 1/L ∙ C

Giving that ωd = 2π ∙ f, we obtain for the frequency f in a series RLC circuit:

f = 1/2π ∙ √L ∙ C

The effective (rms) values of current and voltage in an alternating circuit are

irms = imax/2

and

∆Vrms = ∆Vmax/2

respectively. This is because both these quantities oscillate in a sinusoidal fashion.

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