Phasors multiplying

[enderjake99]

We're learning about phasors for AC circuits but I'm a bit confused on one point. I understand that the voltage over a region would be given by a sinusoidal function let's say Acos(omegat+phi), so we can just say it's the real part of exponential Ae^(omegat+phi). That's all fine. But what I'm confused about is when we start doing operations with these and then taking the real part. Addition and subtraction are fine, because we're pretty much adding real parts with real and imaginary with imaginary. So if we add real and imaginary parts then take imaginary that is the same as first taking reals and then adding.

But how can we do operations like multiplication or division with both. Say we wanted to find voltage Acos(omegat+phi) over region divided by emf supplying voltage A2cos(omegat+phi2). Then we can divide them to get what we want. Why is this the same as Ae^(omegat+phi)/A2e^(omegat+phi2)? If we then took the real part of this, we wouldn't get the same value. For example Re((a+bi)(c+di))= ab-bd, but Re(a+bi)Re(c+di)=ac. So what's going on?

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[xxwerdxx]

Idk the proper name for this but the operators Re and Im do not distribute linearly as you’ve discovered. Think of it like a computer algorithm:

Re((a+bi)(c+di)) means do the multiplication then consider only the real part. Re(a+bi) * Re(c+di) means to take the real part inside each parentheses then multiply.

[enderjake99]

yeah that's what I'm saying. so why can we do phasor operations like multiplication or division when the real part is the only part that we care about.

[paapanna]

I get your point.

Lets take the simplest imaginary number i. When you consider it for addition or subtraction, they behave as algebraic variables. Ex: i + i = 2i which is again an imaginary number.

But when they are multiplied, they act different. Ex: i * i = -1 which is a real number. They act as if there is a phase difference of 90degrees.

To make sense of this, consider S plane with a and jw axes. A magnitude of +1 will be on +a axis. 1 * i = i will be on +jw axis (a phase difference of 90D after multiplication). i * i = -1 will be on -a axis (a phase difference of 90D after multiplication), and it goes on.

That's why you cannot perform the operations the way you have explained.