Phasors — Question 2

Question 2

A regular phasor polygon cancels even when no pair is opposite

Three waves of equal amplitude $A$ have phases $0$, $2\pi/3$ and $4\pi/3$ at a common reference. Their frequencies and propagation wave numbers are equal.

1. Find their sum using complex amplitudes.

2. Generalize to $N\ge2$ equal amplitudes with phases $2\pi j/N$, $j=0,\ldots,N-1$. Explain why pairwise cancellation is not necessary.

Question 2 - Solution

A regular phasor polygon cancels even when no pair is opposite

Method and mathematical working

Use the finite geometric-series identity for equally spaced phase angles. The closed polygon provides the corresponding vector interpretation.$$\begin{aligned}\sum_{j=0}^{N-1}e^{2\pi i j/N}&=\frac{1-e^{2\pi i}}{1-e^{2\pi i/N}}=0\qquad(N>1).\\[4pt]1+e^{2\pi i/3}+e^{4\pi i/3}&=1+(-1/2+i\sqrt3/2)+(-1/2-i\sqrt3/2)=0.\end{aligned}$$

(a) Calculation and conclusion

$Z=A[1+e^{2\pi i/3}+e^{4\pi i/3}]=0$. The real parts $1,-1/2,-1/2$ and imaginary parts $0,\sqrt3/2,-\sqrt3/2$ cancel.

(b) Interpretation and limits

Let $q=e^{2\pi i/N}\ne1$. Then $\sum_{j=0}^{N-1}q^j=(1-q^N)/(1-q)=0$. The phasors form a closed regular polygon. For odd $N$ no vector needs a diametrically opposite partner; vector closure is the general cancellation criterion.

Physical check

The denominator is nonzero for $N>1$, and the numerator vanishes exactly. Equal amplitudes and uniform phase spacing are both essential to this cancellation.