   Chapter 7.2, Problem 66E

Chapter
Section
Textbook Problem

# Household electricity is supplied in the form of alternating current that varies from 155 V to − 155  V with a frequency of 60 cycles per second (Hz). The voltage is thus given by the equation E ( t ) = 155 sin ( 120 π t ) where t is the time in seconds. Voltmeters read the RMS (root-mean-square) voltage, which is the square root of the average value of [ E ( t ) ] 2 over one cycle.(a) Calculate the RMS voltage of household current.(b) Many electric stoves require an RMS voltage of 220 V. Find the corresponding amplitude A needed for the voltage E ( t ) = A sin ( 120 π t ) .

To determine

(a)

To find: the RMS voltage of household current

Explanation

RMS voltage is the root mean squared value of voltage. So, first find the average value of square of function over one cycle and then take the square root of the result.

Formula used:

The average value of a function over the interval [a,b] is given by the following integral:

avg=1baabf(x)dx. Then, the RMS value of f(x) over one complete cycle T will be:

RMS=sqrt(1T0Tf2(x)dx)

Given:

The voltage, E(t)=155sin(120πt)

Calculation:

The voltage signal is E(t)=155sin(120πt), let the time period be T. Then, since there are sixty cycles in one second time period for one cycle is

T=160

Substitute the given voltage function into the formula for RMS:

RMS=sqrt(1T0TE2(t)dt)=sqrt(116001601552sin2(120πt)dt)=15560sqrt(0160sin2(120πt)dt)

Use the identity sin2x=1cos2x2 to reduce the power of sine:

RMS=15560sqrt(0160(1cos(2(120πt)))2dt)=155602sqrt(0160(1cos(240πt))dt)=155602sqrt([tsin240πt240π]0160)=155602sqrt(160sin(240π60)480π)

Simplify further to get:

RMS=155602sqrt(160sin(4π)140π)=155602160=1552

Conclusion: The RMS value of the voltage signal E(t)=155sin(120πt) is found to be 1552

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