In working with complex exponentials we need some simple rules and short cuts for doing the algebra.
Remember that a complex number z is a vector in the 2-dimensional plane. It has a length (also called "magnitude") |z| and an angle ∠(z) measured counterclockwise from the real axis. You need to know the following facts:
Re(a+jb)=a; Im(a+jb)=b
|a+jb|=a2+b2 ; ∠(a+jb)=atan2(b,a)
(atan2(y,x) is the same operation as arctan(yx), except our solution is from −π to π to retain phase information. For more information see here).
|z1z2|=|z1||z2| ; |z1z2|=|z1||z2|
∠(z1z2)=∠(z1)+∠(z2); ∠(z1z2)=∠(z1)−∠(z2)
|ejθ|=1; ∠(ejθ)=θ
Now let's practice on the problem before us. We have a complex particular solution:
vPS(t)=Vi11+jωRCejωt
Let's get its magnitude. The complex number before us is a product of factors, so its magnitude is the product of the magnitude of the factors. Vi is a real number, so its magnitude is its absolute value. The second factor is a quotient, so its magnitude is the quotient of the magnitudes of the numerator and the denominator. The third factor is a unit vector; so its magnitude is 1.
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