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Physics Lesson 16.16.5 - Effective Values of Alternating Current and Voltage

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Welcome to our Physics lesson on Effective Values of Alternating Current and Voltage, this is the fifth 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.

Effective Values of Alternating Current and Voltage

From all discussed so far about the current and voltage in an alternating circuit, we can point out two important features:

  1. Both the current and voltage oscillate in a sinusoidal fashion
  2. Current and voltage can be either positive or negative

If we are asked to find other related quantities in an AC circuit such as power or energy, we cannot use the maximum values of current and voltage as this gives a considerable error; all values would be higher than actually they are. We cannot consider other known methods to find the average values such as the arithmetic mean, or approximations of the sinusoids to obtain a series or rectangles for example. Therefore, the only available method remains the root mean square method, similar to that discussed in the kinetic theory of gases, in which we calculated the rms speed of an ideal gas molecule.

Giving that

i(t) = imax ∙ sin⁡(ωd ∙ t)

we obtain for the rms current

irms = √[imax ∙ sin⁡(ωd ∙ t) ]2ave
= √i2max ∙ sin2 (ωd ∙ t)ave

Since from trigonometry it is known that

sin2 (ωd ∙ t)ave = 1/2

we can write

irms = √i2max ∙ 1/2
irms = imax ∙ √1/2
= imax ∙ 1/r = √2
= imax/√2
≈ 0.707 ∙ imax

Thus,

iave = irms = imax/√2

We have used this value earlier in some exercises but now you know what does it mean.

Likewise, for the average voltage, we have

∆Vave = ∆Vrms = irms ∙ R
= imax/√2 ∙ R
= ∆Vmax/√2

Example 3

A series RLC circuit is shown in the figure.

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

Calculate:

  1. The resonance frequency of the circuit
  2. The root mean square of current when the circuit is in resonance
  3. The maximum voltages across the resistor, inductor and capacitor when the circuit is in resonance

Solution 3

We have

εmax = 250 V
R = 5 Ω
L = 5 mH = 5 × 10-3 H
C = 4 μF = 4 × 10-6 F

  1. The resonance frequency of the circuit is obtained when XL - XC = 0 or when XL = XC. Thus, we have
    XL = Xc
    ωd ∙ L = 1/ωd ∙ C
    ωd2 = 1/L ∙ C
    2π ∙ f = 1/√L ∙ C
    f = 1/2π ∙ √L ∙ C
    = 1/2 ∙ 3.14 ∙ √(5 × 10-3 ) ∙ (4 ∙ 10-6 )
    = 1/6.28 ∙ √2 ∙ 10-8
    = 1/6.28 ∙ 1.414 × 10-4
    = 1126 Hz
  2. First, let's calculate the impedance in the circuit. If the circuit is in resonance, we have
    Z = √R2 + (XL - Xc )2
    = √R2 + 02
    = √R2
    = R
    = 5 Ω
    The maximum current in the circuit therefore is
    imax = εmax/Z
    = 250 V/5 Ω
    = 50 A
    Hence, the rms current in the circuit is
    irms = imax/√2
    = 50 A/1.414
    = 35.4 A
  3. The maximum voltages in each component of the circuit are
    ΔVR(max) = imax ∙ R
    = (50 A) ∙ (5Ω)
    = 250 V
    ΔVL(max) = imax ∙ XL
    = imax ∙ 2πf ∙ L
    = (50 A) ∙ (2 ∙ 3.14 ∙ 1126 ∙ 0.005 Ω)
    = 1768 V
    ΔVC(max) = imax ∙ Xc
    = imax ∙ 1/2πf ∙ C
    = (50 A) ∙ (1/2 ∙ 3.14 ∙ 1126 ∙ (4 × 10-6 ) )
    = 1768 V

You have reached the end of Physics lesson 16.16.5 Effective Values of Alternating Current and Voltage. 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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