Phasors — Question 1

Question 1

Three wave phasors require a quadrant-aware resultant

Three same-frequency contributions share phase variable $\Phi$: $3\cos\Phi$, $4\cos(\Phi+\pi/2)$ and $5\cos(\Phi+\pi)$, all in millimeters. Represent each as the real part of $Z e^{i\Phi}$.

1. Find the resultant complex amplitude and write the summed displacement as $R\cos(\Phi+\phi)$.

2. Check the answer at $\Phi=0$ and explain why a one-argument arctangent can give the wrong phase.

Question 1 - Solution

Three wave phasors require a quadrant-aware resultant

Method and mathematical working

Add Cartesian components before computing magnitude and angle. Use a two-argument inverse tangent so that the quadrant is retained.$$\begin{aligned}Z&=3+4i-5=-2+4i\quad(\mathrm{mm}),\\|Z|&=\sqrt{(-2)^2+4^2}=\sqrt{20}\,\mathrm{mm},\\\arg Z&=\operatorname{atan2}(4,-2).\\[4pt]|Z|&=4.4721\,\mathrm{mm},\quad\arg Z=2.03444\,\mathrm{rad},\qquad y(0)=\operatorname{Re}Z=-2\,\mathrm{mm}.\end{aligned}$$

(a) Calculation and conclusion

The complex amplitudes add to $Z=3+4i-5=-2+4i\,\mathrm{mm}$. Thus $R=\sqrt{20}=4.472\,\mathrm{mm}$ and $\phi=\operatorname{atan2}(4,-2)=2.0344\,\mathrm{rad}$, in quadrant II.

(b) Interpretation and limits

At $\Phi=0$, the original sum is $3+0-5=-2\,\mathrm{mm}$, matching $R\cos\phi$. Using $\arctan(4/(-2))$ alone selects quadrant IV and would reverse the sign. Both real and imaginary signs are needed.

Physical check

The real component is negative and the imaginary component positive, placing the phasor in quadrant II. Reconstructing $|Z|\cos\phi$ and $|Z|\sin\phi$ must return $-2$ and $4$.