Phasors — Question 4

Question 4

Changing the phase reference rotates every phasor together

A sum is represented by $y=\operatorname{Re}[Ze^{i\Phi}]$. A new phase coordinate is defined by $\Phi^{\prime}=\Phi+\chi$.

1. Find the new complex amplitude $Z^{\prime}$ that represents the identical physical field.

2. For $Z=3+4i\,\mathrm{mm}$ and $\chi=\pi/2$, compute $Z^{\prime}$. Which measurable quantities remain unchanged?

Question 4 - Solution

Changing the phase reference rotates every phasor together

Method and mathematical working

Demand that the product of phasor and oscillation factor represent the same real motion after the reference phase is changed. This fixes the compensating phasor rotation.$$\begin{aligned}Ze^{i\Phi}&=Z^{\prime}e^{i(\Phi+\chi)},\\Z^{\prime}&=Ze^{-i\chi},\qquad (3+4i)e^{-i\pi/2}=4-3i.\\[4pt]|3+4i|&=|4-3i|=5,\qquad\arg Z^{\prime}=\arg Z-\pi/2.\end{aligned}$$

(a) Calculation and conclusion

Since $e^{i\Phi}=e^{-i\chi}e^{i\Phi^{\prime}}$, $Z^{\prime}=Ze^{-i\chi}$. The phasor rotates opposite to the change in phase coordinate so the physical displacement remains identical.

(b) Interpretation and limits

$Z^{\prime}=(3+4i)(-i)=4-3i\,\mathrm{mm}$. Its magnitude remains $5\,\mathrm{mm}$. Relative phasor angles, resultant amplitude and carried power are unchanged by a common reference rotation; absolute phasor angle depends on convention.

Physical check

The modulus is still $5$. Only the coordinates relative to the chosen phase origin change; the measured amplitude and the actual time-dependent displacement do not.