Root-Mean-Square Current & Voltage (Cambridge International A-Level Physics)

Root-mean-square current & voltage

• Root-mean-square (r.m.s) values of current, or voltage, are a useful way of comparing a.c current, or voltage, to its equivalent direct current, or voltage
• The r.m.s values represent the d.c current, or voltage, values that will produce the same heating effect, or power dissipation, as the alternating current, or voltage
• The r.m.s value of an alternating current is defined as: The value of a constant current that produces the same power in a resistor as the alternating current
• The r.m.s current I_r.m.s is defined by the equation:

$I_{r.m.s}=\frac{I_{0}}{\sqrt{2}}$

• The r.m.s value of an alternating voltage is defined as: The value of a constant voltage that produces the same power in a resistor as the alternating voltage
• The r.m.s voltage V_r.m.s is defined by the equation:

$V_{r.m.s}=\frac{V_{0}}{\sqrt{2}}$

• Where:

  • I₀ = peak current (A)
  • V₀ = peak voltage (V)

• So, r.m.s current is equal to 0.707 × I₀, which is about 70% of the peak current I₀

• The r.m.s value is therefore defined as: The steady direct current, or voltage that delivers the same average power in a resistor as the alternating current, or voltage

• A resistive load is an electrical component with resistance e.g. a lamp

Peak voltage and RMS voltage

V_r.m.sand peak voltage. The r.m.s voltage is about 70% of the peak voltage

Worked Example An alternating current is I is represented by the equation

I=410 sin (100 straight π t)

where I is measured in amps and t is in seconds.

For this alternating current, determine

(a) the r.m.s current

(b) the frequency.

Answer:

(a)

Step 1: Write out the equation for r.m.s current

$I_{r.m.s}=\frac{I_{0}}{\sqrt{2}}$

Step 2: Determine the peak voltage I₀

• The alternating current equation is in the form: I=I_ 0 sin (ω t)
• Comparing this to I=bold 410 sin (100 straight π t)means the peak current is I₀= 410 A

Step 3: Substitute into the I _r.m.s equation

$I_{r.m.s}=\frac{410}{\sqrt{2}}=290A$

(b)

Step 1: Write out the equation for angular frequency

$ω=2πf$

Step 2: Determine the angular frequency

• The alternating current equation is in the form: I=I_ 0 sin (ω t)
• Comparing this to I=410 sin (bold 100 bold π t)means the angular frequency is ω = 100π rad s⁻¹

Step 3: Rearrange and substitute into the equation to calculate frequency

$f=\frac{ω}{2π}=\frac{100π}{2π}=50\mathrm{Hz}$

Exam Hint These equations are not given on your data sheet, so make sure you learn them!