Power Measurement Theory

This page covers the mathematics behind calculating real power, apparent power, power factor, RMS voltage and RMS current from instantaneous Voltage and Current measurements of single phase AC electricity. Discrete time equations are detailed since the calculations are carried out in the Arduino in the digital domain. There are also code snippets of the equations included below:

Real Power

Real power (also known as active power) is defined as the power used by a device to produce useful work.

Mathematically it is the definite integral of voltage, \(u(t)\), times current, \(i(t)\), as follows:

\[P = \frac{1}{T} \int u(n)\times i(n) \, \mathrm{d}t\]

Equation 1. Real Power Definition.

\(U\) - Root-Mean-Square (RMS) voltage.

\(I\) - Root-Mean-Square (RMS) current.

\(cos(\varphi)\) - Power factor.

The discrete time equivalent is:

\[P = \frac{1}{T} \sum_{n=0}^{N-1} u(n)\times i(n)\]

Equation 2. Real Power Definition in Discrete Time.

\(u(n)\) - sampled instance of \(u(t)\)

\(i(n)\) - sampled instance of \(i(t)\)

\(N\) - number of samples.

Real power is calculated simply as the average of N voltage-current products. It can be shown that this method is valid for both sinusoidal and distorted waveforms.

Code example of equation 2:

for (n=0; n < numberOfSamples; n++)
{
    //instV and instI calculation from raw ADC input goes here.
    instP = instV * instI;
    sumP += instP;
}
realPower = sumP / numberOfSamples;

RMS Voltage and Current Measurement

An RMS value is defined as the square root of the mean value of the squares of the instantaneous values of a periodically varying quantity, averaged over one complete cycle. The discrete time equation for calculating voltage RMS is as follows:

\[U_{rms} = \sqrt{\frac{\sum_{n=0}^{N-1} u^2(n)}{N}}\]

Equation 3. Voltage RMS Calculation in Discrete Time Domain.

Current RMS is calculated using the same equation, only substituting voltage samples, \(u(n)\), for current samples, \(i(n)\).

Code example of equation 3:

for (n=0; n < numberOfSamples; n++)
{
    //instV calculation fsqV = instV * instV;
    sumV += sqV;
}
Vrms = sqrt(sumV / numberOfSamples);

Substitute V for I for current.

Apparent Power and Power Factor

\[P_{apparent} = V_{rms} \times I_{rms}\]\[cos(\varphi) = \frac{P_{real}}{P_{apparent}}\]

Bringing it all together

The following snippet carries out all the measurements above:

for (n=0; n < numberOfSamples; n++)
{
    //instV and instI calculation from raw ADC input goes here.
    sqV = instV * instV;
    sumV += sqV;
    sqI = instI * instI;
    sumI += sqI;
    instP = instV * instI;
    sumP +=instP;
}
Vrms = sqrt(sumV / numberOfSamples);
Irms = sqrt(sumI / numberOfSamples);
realPower = sumP / numberOfSamples;
apparentPower = Vrms * Irms;
powerFactor = realPower / apparentPower;

Thats it, thats the core of single phase AC power measurment calculation.

This page is based on Atmel’s AVR465 Application Note pp. 12-15 which can be found here.

Page referenced from OpenEnergyMonitor.