Understanding the Phazor Equation: I_s(t)=sin(t)

[electron2]

for this signal
<br /> I_s(t)=sin(t)<br />
i have this equation
<br /> I_L(1-CL+jCR+jcz)=I_s<br /> [/tex]<br /> then the next line is<br /> &lt;br /&gt; I_L(1-CL+jCR+jcz)=j&lt;br /&gt;<br /> <br /> why they substitute j with I_s

 

[electron2]

<br /> sin(t)=cos(\frac{\pi}{2}-t)=cos(t-\frac{\pi}{2})<br />
the formula of the signal is
<br /> Is=Acos(\omega t+\phi)<br />
then we transform it to the phasor representation formula
<br /> Is=Ae^{j\phi}<br />
so we get
<br /> Is=1e^{-j\frac{\pi}{2}}<br />
and when we look at this expression as oilers formula we get
the Is=-j

so why its written Is=j
?

 

[CEL]

<br /> sin(t)=cos(\frac{\pi}{2}-t)=cos(t-\frac{\pi}{2})<br />
the formula of the signal is
<br /> Is=Acos(\omega t+\phi)<br />
then we transform it to the phasor representation formula
<br /> Is=Ae^{j\phi}<br />
so we get
<br /> Is=1e^{-j\frac{\pi}{2}}<br />
and when we look at this expression as oilers formula we get
the Is=-j

so why its written Is=j
?

e^{j\theta} = cos(\theta) + j sin(\theta)
for \theta = \frac{\pi}{2}
e^{j\frac{\pi}{2}} = cos(\frac{\pi}{2}) + j sin(\frac{\pi}{2}) = 0 + j.1 = j