§ 7-4 The complex forcing function
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)co s ( ?? ?tV m
)s in ( ?? ?tjV m
)s in ( ?? ?tjI m
)c os ( ?? ?tI mN
)co s ( ?? ?tV m
)s i n ( ?? ?tjV m
)c o s ( ?? ?tI m
)s i n ( ?? ?tjI m
?? )co s ( ?? tV m )s i n ( ?? ?tjV m ?? )co s ( ?? tI m )s i n ( ?? ?tjI m
A real,an imaginary,or a complex forcing function will
produce a real,an imaginary,or complex response,respectively.
Instead of applying a real forcing function to obtain the desired
real response,we apply a complex forcing function whose real
part is the given real forcing function; we obtain a complex
response whose real part is the desired real response.
)t(jm eV ?? ?
)t(jm eI ?? ?
tjte tj ??? s i nc o s ??
]R e [c o s tjet ?? ?
The complex source is,
tjmeV ?
The complex response is, )( ?? ?tj
m eI
sdt
diLRiKVL ???,tj
mtjmtjm eVeLIjeRI ????? ? ?? ?? )()(
mjmjm VeLIjeRI ?? ?? ?
)t a n(
222
1
R
Lj
mmj
m e
LR
V
LjR
VeIor ??
??
??
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???
mjm VLjReI ?? )( ??
R
L?? 1t a n ???
222 LR
VI m
m ???
)t a nc o s ()c o s ()( 1222 R Lt
LR
VtIti m
m
??
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?? ??
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???
tV ms ?? c o s?
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