Encontre os valores médio e eficaz da função g(t) = Gp cos(ωt). Lembre- se que ω = 2πf = 2π T

valor eficaz vc multiplica o valor de Gp por raiz de 3, e o valor medio e igual a 0

Considerando a função \(g(t) = G_P \cos(\omega t)\), seu período é igual a \(T={2 \pi \over \omega }\). Portanto, sendo \(G_P\) um valor constante, o valor médio \(V_m\) é:

\(\Longrightarrow V_m = {1 \over T} \int \limits_0^{T }g(t) \, \partial t\)

\(\Longrightarrow V_m = {1 \over T} \int \limits_0^{T }G_P \cos( \omega t ) \, \partial t\)

\(\Longrightarrow V_m = {G_P \over T} \int \limits_0^{T } \cos( \omega t ) \, \partial t\)

\(\Longrightarrow V_m = {G_P \over T} \Big [ {1 \over \omega } \sin( \omega t ) \Big ] \bigg |_0^{T}\)

\(\Longrightarrow V_m = {G_P \over \omega T} \Big [ \sin( \omega T ) - \sin( \omega \cdot 0 ) \Big ] \)

\(\Longrightarrow V_m = {G_P \over \omega T} \Big [ \sin( \omega \cdot {2 \pi \over \omega }) - \sin( 0 ) \Big ] \)

\(\Longrightarrow V_m = {G_P \over \omega T} \Big [ 0-0 \Big ] \)

\(\Longrightarrow \fbox {$ V_m = 0 $}\)

E o valor eficaz \(V_{ef}\) é:

\(\Longrightarrow V_{ef} = \sqrt { {1 \over T} \int \limits_0^{T }g^2(t) \, \partial t }\)

\(\Longrightarrow V_{ef} = \sqrt { {1 \over T} \int \limits_0^{T }\Big [ G_P \cos(\omega t) \Big ] ^2 \, \partial t }\)

\(\Longrightarrow V_{ef} = \sqrt { {G_P^2 \over T} \int \limits_0^{T } \cos^2(\omega t) \, \partial t }\)

\(\Longrightarrow V_{ef} = \sqrt { {G_P^2 \over T} \int \limits_0^{T } {1 \over 2} \Big [ 1 + \cos(2\omega t) \Big ]\, \partial t }\)

\(\Longrightarrow V_{ef} = \sqrt { {G_P^2 \over 2T} \int \limits_0^{T } \Big [ 1 + \cos(2\omega t) \Big ]\, \partial t }\)

\(\Longrightarrow V_{ef} = \sqrt { {G_P^2 \over 2T} \Big [ t + {1 \over 2 \pi }\sin(2\omega t) \Big ] \bigg |_0^T }\)

\(\Longrightarrow V_{ef} = \sqrt { {G_P^2 \over 2T} \Bigg \{ \Big [ T + {1 \over 2 \pi }\sin(2\omega T) \Big ] - \Big [ 0 + {1 \over 2 \pi }\sin(2\omega \cdot 0) \Big ] \Bigg \} }\)

Substituindo \(T={2 \pi \over \omega }\), o valor de \(V_{ef}\) é:

\(\Longrightarrow V_{ef} = \sqrt { {G_P^2 \over 2T} \Bigg \{ \Big [ T + {1 \over 2 \pi }\sin(2\omega \cdot {2 \pi \over \omega }) \Big ] - \Big [ 0 + {1 \over 2 \pi }\sin( 0) \Big ] \Bigg \} }\)

\(\Longrightarrow V_{ef} = \sqrt { {G_P^2 \over 2T} \Bigg \{ \Big [ T + {1 \over 2 \pi }\cdot 0\Big ] - \Big [ 0-0 \Big ] \Bigg \} }\)

\(\Longrightarrow V_{ef} = \sqrt { {G_P^2 \over 2T} \cdot T }\)

\(\Longrightarrow V_{ef} = \sqrt { {G_P^2 \over 2} }\)

\(\Longrightarrow \fbox {$ V_{ef} = {G_P \over \sqrt{2} } $}\)