Three voltages at the same angular frequency act in series:

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Foundations of mathematical language · Foundations of mathematical language

Mathematical Methods for Physics, Ch Complex numbers, the complex exponential and phasors

Problem

Three voltages at the same angular frequency act in series: V1=4.0cos⁡ωtV_1=4.0\cos\omega t, V2=3.0cos⁡(ωt+π/2)V_2=3.0\cos(\omega t+\pi/2) and V3=6.0cos⁡(ωt−π/3)V_3=6.0\cos(\omega t-\pi/3), all in volts. Use this chapter's convention, Acos⁡(ωt+ϕ)=Re[A~e−iωt]A\cos(\omega t+\phi)=\mathrm{Re}\big[\tilde Ae^{-i\omega t}\big] with A~=Ae−iϕ\tilde A=Ae^{-i\phi}. ((a)) Write down the three complex amplitudes and add them. ((b)) Write the total voltage as a single Acos⁡(ωt+ϕ)A\cos(\omega t+\phi). ((c)) Check the result at ωt=0\omega t=0 and at ωt=π/2\omega t=\pi/2.

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