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).

• Are you assuming that the circuit has no reactive components? – Ignacio Vazquez-Abrams Sep 14 '16 at 15:17
• 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). – Spehro Pefhany Sep 14 '16 at 15:19
• @IgnacioVazquez-Abrams No. – Abhirikshma Sep 14 '16 at 15:20
• Please ask a specific question – Voltage Spike Sep 14 '16 at 15:20
• You must be, because the only time you can ignore the imaginary component of the phasor is if the circuit is in resonance. – Ignacio Vazquez-Abrams Sep 14 '16 at 15:23

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.

• ...and that's called Steinmetz transform :) – carloc Sep 14 '16 at 19:12

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.