Thevenin intuitive approach

[MrAl]

Not sure what you are trying to say here.  Two numbers that form a vector have amplitude and angle.  If we draw a triangle we can show the relationship.  We can find triangular structures in real life.

Complex numbers arise from trying to find the roots of polynomials when some or all of the roots are not real numbers. The result is the invention of a new kind of number that can be expressed as an ordered pair of reals (x, y) and some definitions for how to do arithmetic with these numbers. An ordered pair of reals can also be used to represent points on a 2D plane and this presents a way of visualizing complex numbers.

It is not obvious that solving linear problems in AC circuit theory has anything to do with finding the roots of polynomials, however it was discovered that if such problems are expressed as complex numbers then the rules of complex arithmetic are the right way to solve these AC circuit problems. Again, it is not obvious why the rules of complex arithmetic are the right way to go here rather than other rules (e.g. dot products, cross products, or combinations thereof), but serendipitously or otherwise, they are.

Hi,

Are you saying that complex numbers work because complex numbers happen to work?
My opinion is just because someone started using them and didnt know the whole physical meaning doesnt mean they are somehow just happenstance.

It becomes more obvious when we look at the physical interpretation.  A sine wave is a side view of a helix and a helix is generated using the exponential form.  I have a nice graph showing this somewhere i'll have to dig it up.
Stated a little better, the sine wave is a projection of the helix onto a side plane, and the helix is generated partly by using the imaginary operator usually denoted by "j" in electronic work.
In still other words, if you take an ordinary spring and step on it to flatten it out you get a sine wave shape on the floor.

[Vtile]

Not sure what you are trying to say here.  Two numbers that form a vector have amplitude and angle.  If we draw a triangle we can show the relationship.  We can find triangular structures in real life.

Complex numbers arise from trying to find the roots of polynomials when some or all of the roots are not real numbers. The result is the invention of a new kind of number that can be expressed as an ordered pair of reals (x, y) and some definitions for how to do arithmetic with these numbers. An ordered pair of reals can also be used to represent points on a 2D plane and this presents a way of visualizing complex numbers.

It is not obvious that solving linear problems in AC circuit theory has anything to do with finding the roots of polynomials, however it was discovered that if such problems are expressed as complex numbers then the rules of complex arithmetic are the right way to solve these AC circuit problems. Again, it is not obvious why the rules of complex arithmetic are the right way to go here rather than other rules (e.g. dot products, cross products, or combinations thereof), but serendipitously or otherwise, they are.

Hi,

Are you saying that complex numbers work because complex numbers happen to work?
My opinion is just because someone started using them and didnt know the whole physical meaning doesnt mean they are somehow just happenstance.

It becomes more obvious when we look at the physical interpretation.  A sine wave is a side view of a helix and a helix is generated using the exponential form.  I have a nice graph showing this somewhere i'll have to dig it up.
Stated a little better, the sine wave is a projection of the helix onto a side plane, and the helix is generated partly by using the imaginary operator usually denoted by "j" in electronic work.
In still other words, if you take an ordinary spring and step on it to flatten it out you get a sine wave shape on the floor.

There is indeed sine, but there is also the cosine, which is just projection of the same helix (spring)(where the length on number of turns are equivalent of time and speed of rotation[frequency]) from another direction.  Of course both are sinusoidal by shape.

[nForce]

Ok, thank you. But I would like to see the hard way. I know what Fourier transform is, but I don't know what to insert instead of f(t).

There isn't a "hard way". Complex impedances are not a physical thing, and they are not derived from frequency analysis.

The use of complex numbers in AC circuit analysis is nothing more than a mathematical trick. You convert the problem to complex numbers, do the complex number arithmetic, and then extract the real answers (magnitude and phase angle) from the complex number result.

Many people who use this trick have not necessarily seen or have derived the proof of why it works, they just take it as a given and use the results.

Oh, I thought that complex impedance (j*w) came from frequency analysis. How is then derived? For example inductor: jwL?

[rstofer]

Here is a discussion:

http://scipp.ucsc.edu/~johnson/phys160/ComplexNumbers.pdf

At the bottom of page 2 the author states that using complex number notation results in easier math since we can uuse Euler's Formula to convert between notations.

The more important reading starts on page 4.