GATE Electrical Engineering (EE) 2020 Solved Paper

Question : 30 of 65

Marks: +1, -0

{X}_{R}

and

{X}_{A}

are, respectively, the rms and average values of

{x}( {t})= {x}( {t}- {T})

, and similarly,

{y}_{R}

and

y_{A}

are, respectively, the rms and average values of

y(t)=k x(t) . k, T

are independent of

t

. Which of the following is true?

Solution:

Given that:

{X}_{R}= {rms}

value of

{x}( {t})

X_{A}=

average value of

x(t)

also, as

x(t)=x(t-T), i . e ., x(t)

is a periodic waveform with period

T

therefore, the rms value of

x_{R}

is given as

{x}_{ {R}}=√{\;{1}/{ {T}} ∫↙{ {T}} {x}^{2}( {t}) {dt}}

And the average value of

X_{A}

is given as

{x}_{ {A}}=\;{∫↙{ {T}} {x}( {t}) {dt}}/{ {T}}

Now, therefore

y(t)

is also periodic signal with period "T",
Now, rms value is

y_{R}=√{\;{1}/{T} ∫↙{T} y^{2}(t) d t}=√{\;{1}/{T} ∫↙{T} k^{2} x^{2}(t) d t}

y_{R}=√{k^{2} \;{1}/{T} ∫↙{T} x^{2}(t) d t}=k √{\;{1}/{T} ∫↙{T} x^{2}(t) d t}=k x_{R}

Now, average value is

{y}_{ {A}}=\;{∫↙{ {T}} {y}( {t}) {dt}}/{ {T}}=\;{∫ {ky}( {t}) {dt}}/{ {T}}= {k} \;{∫ {y}( {t}) {dt}}/{ {T}}= {kx}_{ {A}}

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