7.3:
Effective Value of a Periodic Waveform
Consider an AC circuit with a periodic voltage source connected to a resistor.
The average power delivered to the resistor equals the integral of the instantaneous power over a period divided by the period.
When this resistor is connected to a DC source, the average power it absorbs is the product of the current squared and its resistance.
Considering that the average power delivered by a DC source to the resistor is equivalent to that provided by the periodic current, the direct current is equal to the effective value of the periodic current.
So, the effective value of current is the square root of the average of the squared instantaneous current values.
Similarly, the expression for the effective value of voltage can be obtained.
For any periodic signal, the effective value, also known as the root-mean-square value, is equal to the square root of the mean of the square of the periodic signal.
For a sinusoidal current and voltage, the RMS value equals the peak value divided by the square root of two.
7.3:
Effective Value of a Periodic Waveform
The concept of effective value, the root mean square (RMS) value, is crucial in understanding electrical circuits and power delivery. This idea emerges from the necessity to measure the effectiveness of a voltage or current source in supplying power to a resistive load.
The effective value of a periodic current represents the direct current (DC) that conveys the same average power to a resistor as the periodic current itself. This concept is crucial when assessing AC circuits. To determine the effective value of current, one must find the RMS current, which is the square root of the mean of the squared instantaneous current values over a period. This RMS value corresponds to the DC that delivers the same average power as the periodic current when applied to a resistor.
Similarly, the effective value of voltage is calculated in the same manner as the current. It represents the RMS voltage, essential for assessing power consumption in electrical systems.
The RMS value is a fundamental concept, not limited to sinusoidal signals. For any periodic function, the RMS value is calculated by finding the square of the function, determining the mean of the squares, and then taking the square root of that mean. In practical applications, voltage and current are often expressed in terms of their RMS values rather than peak values. This is because the average value of a sinusoidal signal is zero, making the RMS value a more useful metric for power analysis. As well as this, analog voltmeters and ammeters are designed to directly read the RMS values of voltage and current, making them indispensable tools in power measurements.