mathematics proof that we can use ZL= jωL and Zc = 1/(jωC) for inductor and capacitor

[hanhan]

Can you tell me the mathematics proof that we can use ZL= jωL and Zc = 1/(jωC) for inductor and capacitor?

 

[MrAl]

Hi,

Here is one way...

Start with a simple RC voltage divider. In the time domain the capacitor voltage with a unit step excitation is:
vC(t)=1-e^(-t/RC)

Take the Laplace Transform of that:
L(vC(t))=vC(s)=1/(s^2*R*C+s) {with step input}

The unforced response in the frequency domain is therefore:
vC(s)=s/(s^2*R*C+s)=1/(s*R*C+1)

Now lets do the same thing but we'll start with what we think is the equivalent impedance for the capacitor which is zC=1/(s*C):

vC(s)=zC/(R+zC)

which equals:
vC(s)=(1/(s*C))/(R+1/(s*C))

Multiply top and bottom by s*C we get:
vC(s)=1/(s*R*C+1)

Comparing the two results we see that they are the same. So we got the same result calculating directly from the time response as we did calculating from the impedance.

Another way is...

Start with the voltage across the capacitor with unit step input:
vC=(1-e^(-t/RC))

take the Laplace Transform:
vC(s)=1/(s^2*C*R+s)

and the current through the capacitor is:
iC=e^(-t/RC)/R

and the Laplace Transform of that is:
iC(s)=C/(s*R*C+1)

Now divide vC(s) by iC(s) we get:
vC(s)/iC(s)=1/(s*C)

which is the impedance of the capacitor.