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Circuit Theory/Phasors/proof7
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Circuit Theory
|
Phasors
g
(
t
)
=
G
m
c
o
s
(
ω
t
−
ϕ
)
{\displaystyle g(t)=G_{m}cos(\omega t-\phi )}
g
(
t
)
=
G
m
Re
(
e
j
(
ω
t
−
ϕ
)
)
{\displaystyle g(t)=G_{m}\operatorname {Re} (e^{j(\omega t-\phi )})}
g
(
t
)
=
G
m
Re
(
e
−
j
∗
ϕ
e
j
ω
t
)
{\displaystyle g(t)=G_{m}\operatorname {Re} (e^{-j*\phi }e^{j\omega t})}
g
(
t
)
=
Re
(
G
m
e
−
j
∗
ϕ
e
j
ω
t
)
{\displaystyle g(t)=\operatorname {Re} (G_{m}e^{-j*\phi }e^{j\omega t})}
g
(
t
)
=
Re
(
G
e
j
ω
t
)
{\displaystyle g(t)=\operatorname {Re} (\mathbb {G} e^{j\omega t})}
G
=
G
m
e
−
j
∗
ϕ
=
G
m
(
c
o
s
(
−
ϕ
)
+
j
∗
s
i
n
(
−
ϕ
)
)
=
G
m
c
o
s
(
ϕ
)
−
j
G
m
s
i
n
(
ϕ
)
{\displaystyle \mathbb {G} =G_{m}e^{-j*\phi }=G_{m}(cos(-\phi )+j*sin(-\phi ))=G_{m}cos(\phi )-jG_{m}sin(\phi )}