Root-Mean-Square Current & Voltage
- Direct current sources provide a constant voltage and current over time, making it easy to measure
- In situations involving alternating voltage and current, the average values of voltage and current will always be zero
- This can make it difficult to measure
The mean value for alternating current and voltage is always zero
- The use of root mean square values gets around this problem
- First remove all the negative signs by simply squaring the peak current, or voltage
- Find the average of the squared value
- And finally, take the square root
- Root-mean-square (rms) values of current, or voltage, are a useful way of comparing a.c current, or voltage, to its equivalent direct current (d.c), or voltage
- The rms values represent the direct current, or voltage, values that will produce the same heating effect, or power dissipation, as the alternating current, or voltage
- The rms value of an alternating current is defined as:
The square root of the mean of squares of all the values of the current in one cycle
- An alternate definition is:
The equivalent direct current that produces the same power
- The rms current Irms is defined by the equation:
- Where:
-
- I0 = peak current (A)
- The rms value of an alternating voltage is defined as:
The square root of the mean of squares of all the values of the voltage in one cycle
- An alternate definition is:
The equivalent dc voltage that produces the same power
- The rms voltage Vrms is defined by the equation:
- Where:
- V0 = peak voltage (V)
- Rms current is equal to 0.707 × I0, which is about 70% of the peak current I0
- This is also the case for rms voltage
- The rms 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 any electrical component with resistance eg. a lamp
Vrms and peak voltage. The rms voltage is about 70% of the peak voltage
Worked Example
An electric oven is connected to a 230 V root mean square (rms) mains supply using a cable of negligible resistance.
Calculate the peak-to-peak voltage of the mains supply.
Step 1: Write down the Vrms equation
Step 2: Rearrange for the peak voltage, V0
V0 = √2 × Vrms
Step 3: Substitute in the values
V0 = √2 × 230
Step 4: Calculate the peak-to-peak voltage
-
- The peak-to-peak voltage is the peak voltage (V0) × 2
- Peak-to-peak voltage = (√2 × 230) × 2 = 650.538 = 651 V (3 s.f)
Exam Tip
Average Power Calculations
- The average power of a supply is the product of the rms current and voltage:
Average power = Irms × Vrms
Worked Example
What is the maximum current supplied to a 2300 W kettle which is connected to an a.c. supply of peak voltage 325 V?
Step 1: Write down the Vrms equation
Step 2: Substitute in the values and calculate Vrms
= 230 V
Step 3: Write down the equation for average power
Average power = Irms × Vrms
Step 4: Rearrange the equation for Irms
Step 5: Substitute in the values and calculate Irms
= 10 A
Step 6: Write down the equation for I0
Step 7: Rearrange for I0 and substitute in the values
= 14.1 A