How to find the summation of phasors in exponential form

Question

Is there any expression in form of $A e^{j\phi}$ for the following summation: $A_1 e^{j\phi_1} + A_2 e^{j\phi_2} + ... + A_n e^{j\phi_n}$?

Answer 1

Not a very nice one in general, I'm afraid.

We can always express a complex number $z$ in the form $a + ib$, where $a$ and $b$ are real. Then $z = re^{i\theta}$, where $r = \sqrt{a^2 + b^2}$ and $$\theta = \begin{cases} 0 & \text{if }a = b = 0 \\ \pi/2 & \text{if }a = 0 \text{ and }b > 0 \\ -\pi/2 & \text{if }a = 0 \text{ and }b < 0 \\ \arctan(b/a) & \text{if }a > 0 \\ \arctan(b/a) + \pi & \text{if }a < 0 \\ \end{cases}$$

Now $$\begin{aligned} \sum_{k=1}^{n}A_k e^{i\phi_k} &= \sum_{k=1}^{n}A_k (\cos(\phi_k) + i\sin(\phi_k))\\ &= \sum_{k=1}^{n}A_k \cos(\phi_k) + i \sum_{k=1}^{n}A_k \sin(\phi_k) \\ \end{aligned}$$ so we can set $$a = \sum_{k=1}^{n}A_k \cos(\phi_k)$$ and $$b = \sum_{k=1}^{n}A_k \sin(\phi_k)$$ and plug into the above formulas for $r$ and $\theta$.