Hi guys,

I'm currently confused over the expression ofimpedance on both capacitors and inductors in complex numbers.

From what i know, Capacitor impedance = Xc<90° and Inductor impedance = Xl<-90°

i know that the ±90° comes from the leading and lagging of the current.

So, am i right to say that Impedance in complex form is basically focused on Voltage magnitude with the current's angle?

If so, how is this related to the impedance, Ω?

Would someone please explain this in a clearer context for impedance in complex numbers ( rect and polar form)?

P.S I'm also confused on how the impedance is drawn on a phasor diagram, wouldn't it basically be a drawing of the current's magnitude with its lagging/leading angle?

Thanks!

 

A scan from a very old text book that has a decent take on how to think about complex numbers which represent inductive reactance.

 

So, am i right to say that Impedance in complex form is basically focused on Voltage magnitude with the current's angle?

If so, how is this related to the impedance, Ω?

I think you are right to say that.

To understand how this is related to impedance in Ohms, let's back up a bit.

I'm sure you understand Ohms's law. The basic idea is that an ideal resistor has a linear relationship between voltage and current. In fact the ratio of V/I is the resistance, or impedance if you like the more general term. Note that Ohms law works for DC and AC signals, and when I say AC signals, I don't just mean sinusoidal signals. It works for any signal at any point in time.

So, what is all this stuff about reactance? If you think about it, neither a coil nor a capacitor obeys Ohms law. The general circuit relations between voltage and current involve derivatives which are very different from Ohms law. The relations are as follows.

V=L dI/dt and I=C dV/dt

However, what if we want to somehow fit coils, capacitors and resistors into one general "Ohm's Law" type of relation for sinusoidal signals? We may notice that the magnitudes and phases of voltages and currents seem to take on a vector-like quality. If we study this more and more we might eventually notice that the use of complex numbers allows a general formulation to be made if we use a complex sinusoid rather that a real sinusoid. So, instead of thinking of driving a system with cos(wt) or sin(wt) or sin(wt+theta), we can drive the system with exp(jwt)=cos(wt)+j*sin(wt). Now work out the relations using this.

V=L dI/dt=L j w I and I=C dV/dt = C j w V

so we get

V= jwLI and V=-jI/(wC)

Now notice that j=exp(j pi/2) and -j=exp(-j pi/2)

From there think about the angle in the exp function as the phase and the V and the I as the magnitudes.

Basically, it's mathematical trickery that allows the complex numbers to keep track of magnitude and phase relations between voltage and current and you end up with an "Ohm's Law" type of relation that can be applied to sinusoidal signals. In order for the linear relation to hold between voltage and current, we have to think of the impedance as complex.

 

Thanks guys i kinda understand it now. Really appreciate your help!