Mean value, RMS value, Sinusoidal AC signals, Practice Problems, FAQs

Sinusoidal AC signals keep changing their values. So when dealing with alternating voltages (or currents) we are faced with the problem of how to represent a voltage or signal magnitude. One simple method is to use the waveform's peak values. The common way to write the AC current or voltage is the root mean square value. For any appliance voltage and current ratings are written by default as RMS values. In this article we study how to calculate the mean and RMS value for given signals. 

Table of Contents

• Mean Value
• RMS Value
• Form Factor
• Practice Problems
• FAQs

Mean Value

In an AC circuit the values of current and voltage are a function of time. So that means they keep changing with respect to time. If someone asks the value of a signal he/she needs to specify the particular time. So the one parameter that is used to define the AC signal is Mean Value or Average Value. 

Average value of AC voltage is defined based on the charge transfer. It is the equivalent AC voltage at which the charge transfer in the AC circuit is equal to charge transfer in the DC circuit. 

By using the concept of integration from mathematics, we can calculate the mean value of current which is the function of time as given by the formula. Below is the graphical representation of alternating current i=f(t) as the function of time from time instant t₁ to t₂

So the average value of current over a certain interval of time is given by,

Between any two arbitrary interval x₁ and x₂, the average value of any function of 𝑥 is defined as:

Average value of sinusoidal function for one complete cycle.

So the average value of sinusoidal function is zero over one complete cycle. So we calculate the average value for half a cycle. 

Average value of sinusoidal function for half period

RMS Value

While finding the average value of sinusoidal AC over a complete cycle, we found that it was zero. So we can not differentiate two sinusoidal signals of different amplitude using average values because the average value will be zero in both cases. For that purpose, we need to find the RMS value of AC.

RMS value is defined according to heat created by the AC in resistor. It is the equivalent value of AC voltage at which heat dissipation in an AC circuit is equal to heat dissipation in a DC circuit.

The procedure of finding the RMS value of any function is just doing the mathematical operation in the reverse order of the name i.e., square of function ⇒ find its mean ⇒ find square root.

The RMS value of any function f(x) from x1 to x2 is given by,

RMS value of sinusoidal function for half and complete period is the same.So let’s calculate it.

RMS value of sinusoidal function

After squaring it will become like this:

The second term will be zero, as integration of cosine function over the complete period is zero.

Practice Problems

Q 1. The electric current in a circuit is given by i(t)=io(tT) for some time. Calculate the RMS current for the periods t=0 to t=T

Answer:

Q 2. In the previous question, find the average value of the current?

Answer: 

Q 3. The electric current in a circuit is given by i(t)=io(tT). Find the form factor of the waveform?

Answer: We have already calculated the RMS and Average value of the given waveform in previous questions.

Q 4. A 20 Ω resistance is connected to a source of 110 V, 50 Hz. Find RMS current and maximum current?

Answer:

FAQs

Q 1. What is the significance of average value?
Answer: Average value of AC voltage is defined based on the charge transfer. It is the equivalent AC voltage at which the charge transfer in the AC circuit is equal to charge transfer in the DC circuit. 

Q 2. What is the significance of RMS value?
Answer: RMS value is defined based on the heating effect of the waveform. It is the equivalent value of AC voltage at which heat dissipation in an AC circuit is equal to heat dissipation in a DC circuit.

Q 3. Can the form factor be less than unity?
Answer: No. As Form Factor=RMS ValueAverage Value. RMS value is always greater than or equal to one. So the form factor is always greater than or equal to one. Form factor is unity for square waveform.

Q 4. Why is the average value of sinusoidal signal calculated in half cycle?
Answer: The average value of a whole sinusoidal waveform over one complete cycle is zero as the two halves cancel each other out, so the average value is taken over half a cycle.