Question

I'm learning about AC right now, and I think I've got an OK grasp of complex numbers (not in their polar form, though), but I don't understand this step in a derivation at all:

Vi=V0ejωt
I=V0ejωtZ
I=V0ejωt|Z|∠θ
I=V0|Z|∠(ωt−θ)

I don't understand how my professor went from line 3 to line 4 in the above derivation.

Answer 1

The step from line 3 to 4 involves converting the expression of current from exponential to polar form using Ohm's Law for AC circuits and negative angle subtraction due to the division of complex numbers in phasor form.

Step-by-step explanation:

When transitioning from line 3 to line 4 in an AC circuit analysis involving complex numbers, your professor performed a transformation that simplified the expression of current in terms of phasor representation from its exponential form to its trigonometric (polar) form. Specifically, the relationship between current (I) and voltage (Vi) across an impedance (Z) is given by Ohm's Law for AC circuits: I = V0ejωt/Z. When impedance is expressed as a complex number |Z|∠θ, the division operation in the complex plane is equivalent to subtracting the angles. Thus, I = V0/|Z|∠(ωt-θ) indicates that the current phasor has a magnitude of V0/|Z| and is phase-shifted by (ωt-θ) with respect to the voltage phasor.