Question:** An electrical engineer studies a alternating current signal modeled by \( z(t) = 4e^{i\omega t} + 3e^{-i\omega t} \), where \( z(t) \) represents voltage in complex form. Compute the maximum real value of \( |z(t)| \).
["Optimize AC Signal Voltage: Maximum Real Magnitude of Complex Envelope ( z(t) = 4e^{i\omega t} + 3e^{-i\omega t} )", "When analyzing alternating current (AC) signals in complex form, an electrical engineer often encounters expressions like ( z(t) = 4e^{i\omega t} + 3e^{-i\omega t} ), where ( z(t) ) models the instantaneous voltage envelope. Understanding the peak magnitude of this signal—crucial for assessing circuit stress and power delivery—is key. This article computes the maximum real value of ( |z(t)| ), providing both mathematical rigor and practical insight.", "---", "### Understanding the Model", "The given expression:", "[\nz(t) = 4e^{i\omega t} + 3e^{-i\omega t}\n]", "represents a sum of two complex exponentials. Physically, this models an AC voltage composed of a clockwise rotating (forward phase ( e^{i\omega t} )) and an counterclockwise rotating (backward phase ( e^{-i\omega t} )) component, with magnitudes proportional to 4 V and 3 V, respectively. We seek the maximum real-valued magnitude:", "[\n\max_{t} |z(t)| = \max_{t} \left| 4e^{i\omega t} + 3e^{-i\omega t} \right|\n]", "---", "### Rewrite in Terms of Trigonometric Functions", "Using Euler’s formula:", "[\ne^{i\ heta} = \cos\ heta + i\sin\ heta\n]", "we compute:", "[\n\begin{aligned}\nz(t) &= 4(\cos\omega t + i\sin\omega t) + 3(\cos(-\omega t) + i\sin(-\omega t)) \\n&= 4(\cos\omega t + i\sin\omega t) + 3(\cos\omega t - i\sin\omega t) \quad \ ext{(since } \cos(-\ heta) = \cos\ heta, \sin(-\ heta) = -\sin\ heta) \\n&= (4 + 3)\cos\omega t + i(4 - 3)\sin\omega t \\n&= 7\cos\omega t + i\cdot1\cdot\sin\omega t\n\end{aligned}\n]", "Thus,", "[\nz(t) = 7\cos\omega t + i\sin\omega t\n]", "---", "### Compute the Magnitude ( |z(t)| )", "The magnitude squared is:", "[\n|z(t)|^2 = (7\cos\omega t)^2 + (\sin\omega t)^2 = 49\cos^2\omega t + \sin^2\omega t\n]", "Use identity ( \cos^2\ heta + \sin^2\ heta = 1 ):", "[\n|z(t)|^2 = 49\cos^2\omega t + (1 - \cos^2\omega t) = 48\cos^2\omega t + 1\n]", "---", "### Maximize the Expression", "Since ( \cos^2\omega t ) ranges from 0 to 1, the maximum occurs when ( \cos^2\omega t = 1 ):", "[\n|z(t)|^2_{\ ext{max}} = 48(1) + 1 = 49\n]", "Thus,", "[\n\max |z(t)| = \sqrt{49} = 7\n]", "---", "### Interpretation and Application", "The maximum real-valued magnitude of the AC voltage signal is 7 volts. This result tells the electrical engineer that even though the instantaneous value ( z(t) ) rotates in the complex plane, its peak real-line magnitude—representing the most intense instantaneous voltage—reaches a clean maximum of 7 V. This is vital for selecting surge ratings, designing filters, and ensuring component reliability under peak stress.", "---", "### Final Answer", "[\n\boxed{7}\n]", "This elegant result confirms the photon-like efficiency of complex phasor analysis: while the signal oscillates in phase space, its peak envelope magnitude remains analytically tractable and physically meaningful."]
