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Physics Lesson 16.16.3 - The Phase Constant

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Welcome to our Physics lesson on The Phase Constant, this is the third lesson of our suite of physics lessons covering the topic of The Series RLC Circuit, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

The Phase Constant

We have explained earlier that 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.

However, we are more interested about the phase constant of the total voltage, than for individual voltages. In this regard, the general phase constant φ will be the angle formed by the resistive voltage and the total voltage phasors, as shown in the figure.

Physics Tutorials: This image provides visual information for the physics tutorial The Series RLC Circuit

Applying the trigonometry rules, we have:

tan⁡φ = opposite/adjacent
= ΔVL(max) - ΔVC(max)/ΔVR(max)
= imax ∙ XL-imax ∙ Xc/imax ∙ R
= XL-Xc/R

This formula obtained above is very important, as we don't have to know the amplitudes of current and potential difference in a RLC circuit to calculate the initial phase. It is enough knowing the values of resistance and the two reactances for this.

The following cases are present when considering the last equation of phase constant φ in a series RLC circuit:

  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.

Example 2

The equation of voltage in a series RLC circuit is

ε(t) = 40 ∙ sin⁡(100π ∙ t)

and the values of resistance, inductance and capacitance of the corresponding circuit elements (resistor, inductor and capacitor) are 10Ω, 40mH and 80μF respectively.

  1. What kind of circuit is it (more resistive, more inductive or in resonance)?
  2. What is the phase constant in radians and degrees?
  3. What is the maximum current in the circuit?
  4. What is the voltage in the circuit at t = 0.506 s after the switch turns on?
Physics Tutorials: This image provides visual information for the physics tutorial The Series RLC Circuit

Solution 2

First, let's write some useful clues. Thus, from the equation of voltage, we see that

εmax = 40 V
ωd = 2π ∙ f = 100π rad/s = > f = 50 Hz
R = 10 Ω
L = 40 mH = 4 × 10-2 H
C = 80 μF = 8 × 10-5 F

  1. To know what kind of circuit is it, we have to find the two reactances. Thus,
    XL = ωd ∙ L
    = (100π rad/s) ∙ (4 × 10-2 H)
    = 4π Ω
    = 12.56Ω
    and
    Xc = 1/ωd ∙ C
    = 1/(100π rad/s) ∙ (8 × 10-5 F)
    = 39.81 Ω
    Thus, since XC > XL and R ≠ 0, the circuit is more capacitive than inductive.
  2. The phase constant is calculated by
    tan⁡φ = XL-Xc/R
    = 12.56 Ω - 39.81 Ω/10 Ω
    = -2.725
    Therefore, the phase constant φ is
    φ = tan-1 (-2.725)
    = -1.219 rad
    = -69.80
    The negative sign means the phasor εmax is behind the current phasor imax by 69.80.
  3. The maximum current in the circuits is
    imax = εmax/Z
    = εmax/√R2 + (XL-Xc )2
    = 40 V/√(10 Ω)2 + (12.56 Ω-39.81 Ω)2
    = 40 V/29 Ω
    = 1.38 A
  4. The voltage in the circuit at t = 0.506 s is
    ΔV(t) = ε(t) = 40 ∙ sin (100π ∙ t)
    Thus,
    ΔV(0.506) = 40 ∙ sin (100π ∙ 0.506)
    = 40 ∙ sin (50.6π)
    = 40 ∙ sin (0.6π)
    = 40V ∙ 0.951
    = 38.04 V

You have reached the end of Physics lesson 16.16.3 The Phase Constant. There are 5 lessons in this physics tutorial covering The Series RLC Circuit, you can access all the lessons from this tutorial below.

More The Series RLC Circuit Lessons and Learning Resources

Magnetism Learning Material
Tutorial IDPhysics Tutorial TitleTutorialVideo
Tutorial		Revision
Notes		Revision
Questions
16.16The Series RLC Circuit
Lesson IDPhysics Lesson TitleLessonVideo
Lesson
16.16.1Recap on the Series RLC Circuit
16.16.2The Current Amplitude
16.16.3The Phase Constant
16.16.4Resonance in a Series RLC Circuit
16.16.5Effective Values of Alternating Current and Voltage

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