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1. Topic 3 Application: Complex Numbers (2 pages) An electrical engineer is designing a circuit with alternating current. There will be two voltage sources, paired together in a series. The two voltage sources are given by the sinusoidal expressions: v₁ = 28 sin(8nt +30°) and v₂ = 46 sin(6nt+ 60°) where t is time in seconds. What is a phasor? A phasor is a line used to represent a complex electrical quantity as a vector. It indicates both magnitude (peak amplitude) and direction (phase). Given the sinusoidal expression v₁ = 15 sin(8nt + 40°), the phasor would be identified as: 15240° We can convert the phasor to a complex number in rectangular form (a+jb); notice we use j instead of i as the imaginary unit as this is the convention for electrical circuitry. a = 15cos40° = 11.49. b= 15sin40° = 9.64 a + jb = 11.49 + j9.64 a. Identify the phasors for these expressions. (2 points) b. Convert the two phasors to complex numbers in rectangular form. Use the letter j as the imaginary unit, which is the convention for electrical circuitry. (2 points)
c. In a series circuit, the voltages are added to determine the total voltage. Add these two voltages, in rectangular form, to determine the total circuit voltage. (1 point) d. Use the rectangular coordinates to write a sinusoidal function, v = r sin(8πt + 0), for the total voltage of the circuit. (2 points) 2. Critical Thinking: Explain why the rectangular forms of complex numbers are unique, but the polar forms of complex numbers are not. (2 points)