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Mean Value of AC Signal- Formula, Practice Problems, FAQs

Mean Value of AC Signal- Formula, Practice Problems, FAQs

You must have come across the term mean value in mathematics. Let’s say there are 20 students in the class and you are asked to calculate the mean height of the students in the class. We know the mathematical formula to calculate the mean value. So that value gives us the general idea about the height of students in that class. Since the value of AC signal varies with respect to the time. So we use the mean value to represent the signal.

Table of Contents

  • Mean value
  • Mean value of sinusoidal function in complete cycle
  • Mean value of sinusoidal function in half cycle
  • 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,

iavg=t1t2i dtt1t2dt=t1t2f(t) dtt1t2dt=1Δtt1t2idt

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

<f(x)>=x1x2f(x)dxx1x2dx=Area under f(x) curve from x1 to x2Length of interval from x1 to x2

Mean value of sinusoidal function in complete cycle

iavg=1Δtt1t2idt=1Δt0Tiosin(ωt)dt

iavg=-1ωTio[cos(ωt)]0T

iavg=-ioωT[cos(ωT)-cos(0)]

We know, T=2ΠωωT=2Π

Therefore, iavg=-ioωT[cos2Π-cos0]

iavg=-ioωT[1-1]

iavg=0……for one period

So the average value of sinusoidal function is zero over one complete cycle. Even by observing we can conclude that the mean value of a perfectly sinusoidal signal is zero in the complete cycle, because the amount signal is positive is the same as negative so net will definitely be zero. So we calculate the average value for half a cycle.

Mean value of sinusoidal function in half cycle

iavg=1Δtt1t2idt=1Δt0T2iosin(ωt)dt

iavg=-1ωTio[cos(ωt)]0T2

iavg=-ioωT[cos(ωT2)-cos(0)]

We know, T=2ΠωωT2=Π

Therefore, iavg=-ioωT[cosΠ-cos0]

iavg=-ioωT[-1-1]

iavg=2Πio……for one period

Practice Problems

Q. Find the average value of the current, the graph is shown below with respect to time.

A. According to the formula,

iavg=1Δtt1t2idt

iavg=1T0Ti(t)dt

We can write it as ,

iavg=1Time Period×Area of graph

In given graph time period is 2 s.

Area=12×base×height

Area=12×2×10

Area=10

iavg=12×10

iavg=5 A

Q. Find the average value of the current, the graph is shown below with respect to time.

A. According to the formula,

iavg=1Δtt1t2idt

iavg=1T0Ti(t)dt

We can write it as ,

iavg=1Time Period×Area of graph

Area of the rectangular part is 10T

Area of the triangular part is 12×T×10=5T

So total area is 15T

iavg=1T×15T

iavg=15 A

Q. The electric current in a circuit is given by i(t)=io(tT) for some time. Calculate the mean value or average value of current for the periods t=0 to t=T.

A. iavg=1Δtt1t2idt

iavg=1T0Ti(t)dt

iavg=1T0Tio(tT)dt

iavg=ioT20Ttdt

iavg=ioT2[t22]0T

iavg=ioT2T22

iavg=io2

Q. Find the average value of the square waveform shown below till t=2T.

A. According to the formula,

vavg=1Δtt1t2vdt

vavg=1T0Tv(t)dt

We can write it as ,

vavg=1Time Period×Area of graph

Area of the graph in one time period,

Area=VoT2-VoT2

Area=0

So vavg=0

FAQs

Q. What is the significance of average value?
A.
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. What is the significance of RMS value?
A.
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. Can the form factor be less than unity?
A.
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 the square waveform.

Q. Why is the average value of sinusoidal signal calculated in half cycle?
A.
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.

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