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My question is essentially about the polar representation of complex numbers used in steady state analysis of AC circuits.

As far as I know a complex number \$a+jb\$ is represented as \$M \angle \theta\$, where M is the magnitude of the complex number and \$\theta\$ is the angle that it makes with the positive x-axis.

But in circuit analysis we replace \$A\cos(\omega t+\theta)\$ by \$A\angle\theta\$, which does not agree with how complex numbers are represented. According to me, \$A\cos(\omega t+\theta)\$ should rather be written as \$Re(A\angle\theta)\$, because it is the real part of the complex number \$A e^{j(\omega t+\theta)}\$ (I know that we drop the \$\omega t\$ part as it remains the same for all the currents and voltages in a circuit consisting of linear components).

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  • \$\begingroup\$ Are you assuming that the circuit has no reactive components? \$\endgroup\$ Commented Sep 14, 2016 at 15:17
  • \$\begingroup\$ If you just look at the real part a of a+jb then you lose information on reactive currents etc. which are actually real (in the sense that they exist, just out of phase) and have real-world consequences (hence power factor correction, for example). \$\endgroup\$ Commented Sep 14, 2016 at 15:19
  • \$\begingroup\$ @IgnacioVazquez-Abrams No. \$\endgroup\$ Commented Sep 14, 2016 at 15:20
  • \$\begingroup\$ Please ask a specific question \$\endgroup\$ Commented Sep 14, 2016 at 15:20
  • \$\begingroup\$ You must be, because the only time you can ignore the imaginary component of the phasor is if the circuit is in resonance. \$\endgroup\$ Commented Sep 14, 2016 at 15:23

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The main point of the question is why it is valid to replace the term \$A\cos(\omega t+\theta)\$ by \$A\angle\theta\$.

The expression \$A \angle \theta\$ is a notation for \$ A e^{j\theta}\$, which is called the phasor representation.

Actually there is no identity that would directly result in this representation instead a phasor transform is defined as follows $$ A e^{j\theta} = {\cal P} \left\{A cos(\omega t + \theta)\right\} $$ along with inverse transform $$ {\cal P}^{-1}\{A e^{j\theta} \} = Re\left\{A e^{j\phi} e^{j\omega t}\right\} $$ Often a formal definition is omitted, since phasors are quite common and well-known, but good textbooks usually include them.

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    \$\begingroup\$ ...and that's called Steinmetz transform :) \$\endgroup\$ Commented Sep 14, 2016 at 19:12
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We use complex numbers to simplify the math in ac circuits (mostly for parallel and series/parallel circuits). Series ac circuits are typically done with Vector addition (Pythagoras and trigonometry).

If you substitute different values of t into A cos(ωt + θ), you will get a cosine wave, that has a magnitude of A and starts θ before the y-axis. But we cannot use this in calculations, so we turn it into a vector with magnitude A at a phase angle θ.

The reason complex numbers can be used to represent ac circuits is because resistance R acts along x-axis (real component) and capacitive reactance \$X_C \$ and inductive reactance \$X_L \$ act along y-axis.

This just happens to correspond to the complex number plane, so we can use complex numbers to simplify math in ac circuits. Z = R + j (\$X_L \$ - \$X_C \$)

You could also Pythagoras and trigonometry to come to the same answer.

$$ Z = \sqrt {R^2 + (X_L - X_C)^2}\ \ \ \ \theta = tan^{-1} \frac {(X_L - X_C)} {R}$$

The complexity goes up when you have to deal with parallel circuits.

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