# Why complex functions are used to represent some passive components in an electric circuits?

while representing capacitors (1/jwc) and inductors (jwl). Here generally, j- complex number. Then what is the use for representing with j of these components? what is the link to use it?

• Are capacitors and inductances passive? resistors yes... Aug 25, 2017 at 10:25
• @SolarMike Yes, they are passive components. Aug 25, 2017 at 10:50
• The complex number in their impedance is related to the phase difference between current and voltage. Have you read a textbook that derives these impedances? See also this question. Aug 25, 2017 at 10:52

The use of complex functions in electrical engineering is precisely for this reason: it is a way to express a complicated idea concisely. It is entirely possible to derive a theory of electric components that does not use any complex functions. In that case, everything is a combination of sines and cosines. e.g. $V=A\cos(\omega t) + B\sin(\omega t)$ The problem with this notation is that it will get really cumbersome to describe how capacitors and inductors work. E.g. you need to say things like
"if the input current $i$ is a sine wave at frequency $\omega$, then the output voltage $V$ is a cosine wave at frequency $\omega$ with amplitude $V=i \omega L$".
While this is a correct description, it is long. Because inductors and capacitors occur all the time in nearly every circuit, we don't want to have to write that entire sentence every single time we have an inductor. If we did, an EE textbook would be ten thousand pages long. The complex notation allows us to be more compact. If all agree on the notation that we can represent combinations of sines and cosines by $e^{i\theta}=\cos\theta + i \sin\theta$, and then you say that an inductor has complex impedance $j \omega L$, and then "$V=i(j\omega L)$", means that exact same thing as my sentence above, but it's so much shorter.