I'm having trouble understanding something, so I would really appreciate it if you could find the time to really see what will complete the gap in my understanding and not just state facts
so, impedance and reactance. Something just doesnt fit completely in my mind as to why use complex numbers to represent these. I know about complex numbers, I can see how they represent phase-shifts, I know how to solve with them, I can see that it works and I can see that it makes seemingly difficult phase shift problems very easy. But what I'm missing is a validation a proof, or a source of a proof, as to why using them is adequate representation of reality and why it still goes easily with ohm's law..
I know complex numbers came about at the first place as a way to solve problems where till now your roots of a negative number would be invalid answers. Where does the root of a negative hide in our case of reactance ? Where does this substitution to i (or j) happens ?
I mean, do we have to use complex numbers to solve these problems ? or is it merely a choice to represent it in this manner to make our lives easier if so, what other choices do we have ? how can we prove that the complex calculations are equivalent and will hold true ?
If I came across the problem of representing a capacitor's impedance I would first go at the time domain. Impedance = voltage / current. Very quickly I would get to the expression : (sin(wt))/(cos(wt) * wc) .. right ? so this gives us (tan(wt) / wc). But now we have time varying impedance and undefined impedance when wt is half pi (where the limit is undefined aswell). So while looking at this I don't know if it's wrong, or just messy or simply unsolvable?
So let say I can define my reactance as the voltage-current ratio of the amplitude only this will give us a frequecy depended (1/wc). And then I need to express the 90 degrees phase shift somehow.
complex numbers work well here. But why ? this is what I don't understand completely why do they work? I've looked into Euler's formula to maybe find what bridges it together, but I didn't really maybe I missed something, or maybe its just difficult for me to grasp.
how can you help me ?
thanks a bunch.
so, impedance and reactance. Something just doesnt fit completely in my mind as to why use complex numbers to represent these. I know about complex numbers, I can see how they represent phase-shifts, I know how to solve with them, I can see that it works and I can see that it makes seemingly difficult phase shift problems very easy. But what I'm missing is a validation a proof, or a source of a proof, as to why using them is adequate representation of reality and why it still goes easily with ohm's law..
I know complex numbers came about at the first place as a way to solve problems where till now your roots of a negative number would be invalid answers. Where does the root of a negative hide in our case of reactance ? Where does this substitution to i (or j) happens ?
I mean, do we have to use complex numbers to solve these problems ? or is it merely a choice to represent it in this manner to make our lives easier if so, what other choices do we have ? how can we prove that the complex calculations are equivalent and will hold true ?
If I came across the problem of representing a capacitor's impedance I would first go at the time domain. Impedance = voltage / current. Very quickly I would get to the expression : (sin(wt))/(cos(wt) * wc) .. right ? so this gives us (tan(wt) / wc). But now we have time varying impedance and undefined impedance when wt is half pi (where the limit is undefined aswell). So while looking at this I don't know if it's wrong, or just messy or simply unsolvable?
So let say I can define my reactance as the voltage-current ratio of the amplitude only this will give us a frequecy depended (1/wc). And then I need to express the 90 degrees phase shift somehow.
complex numbers work well here. But why ? this is what I don't understand completely why do they work? I've looked into Euler's formula to maybe find what bridges it together, but I didn't really maybe I missed something, or maybe its just difficult for me to grasp.
how can you help me ?
thanks a bunch.