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

RMS v Peak grap, downloadable AS & A Level Physics revision notes

V_r.m.s and 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

$I = 410 \sin (100 π t)$

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 $I = 410 \sin (100 π t)$ means the peak current is I₀ = 410 A

Step 3: Substitute into the I_r.m.s equation

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

(b)

Step 1: Write out the equation for angular frequency

$ω = 2 π f$

Step 2: Determine the angular frequency

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

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

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

Examiner Tip

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

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

Revision Note