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Different examples and videos have shown that: $$P_{complex} = V \cdot I^*$$
However, my textbook shows: $$S = \frac{V \cdot I^*}{2}$$
My professor has even using \$S = V \cdot I^*\$ himself during an example problem, yet other problems were solved using the other formula. Which formula is correct or, if both are correct, what circumstances do you use either formula?
Both answers are correct, it's simply a matter of what \$V\$ and \$I\$ are supposed to represent. If they are the RMS value of voltages and current, then complex power is \$|S|=V\times I\$. If they instead represent a voltage amplitude (i.e. \$v(t)=V\cos(...)\$ and \$i(t)=I\cos(...)\$), then you must first take the RMS value by dividing amplitude by square root of 2. This leaves you with \$|S|=\frac{V}{\sqrt{2}}\times\frac{I}{\sqrt{2}}=\frac{V\times I}{2}\$.
Well according to Fundamentals of Electric Circuits 5th ed Alexander/Sadiku, they define complex power as S=Vrms x Irms. However when they use peak values of V and I then the complex power is S=(V x I*)/2 using the fact that Vrms=V/sqrt(2) and Irms=I/sqrt(2). Perhaps your professor is using rms values and the book uses peak values.
S = Vrms * Irms
OR
S= Vp * Ip / 2
Where Vrms and Irms stands for root mean square values of voltage and current
And Vp and Ip stands for peak values of voltage and current.
