# About use of complex number in AC steady state analysis

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

## 2 Answers

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.