Capacitance, Inductance and Frequency Domain (s-domain)

This post is a continuation of the last post Capacitance, Inductance and Time＞.

The relations between Capacitance and Frequency, Inductance and Frequency are commonly stated as below usually with no explanation.

 

or no <minus sign>

 and




Two (2) very simple calculations lead to the above two equations.


1) Capacitance and Frequency

where X is Capative Reactance (Xc)

This relationship can be derived from the following basic Capacitor equation related with i and v.

Now we regard


                            d
Now we regard   ---   as a symbol  s
                           dt 

 Rearranged

i (t) 
---------- =  C
v(t) s

as I/V (or i/v) = 1/R

1/R s = C

R = 1/C s


Now if  s  is 2πf

R = 1/ 2πf C


                         d
As we regard   ---   as a symbol  s,
                        dt


                  d
If we see    ---   = 2πf  (also = symbol  s ),
                 dt

Frequency (f) is 1/second.


Then what does 2π mean here ?  2π is one cycle in radian or in angular frequency or we could say angular frequency factor. So 2πf is angular frequency but this is also frequency (1/sec). The key concept of s is

                                       d
We repeat we regards    ---   as a symbol  s or 2πf
                                      dt

We must think of frequency as an independent unit, not number/second.  And we forget the change happened in one angular frequency cycle (which simply repeat the exactly same change per any one cycle), and simply think of how many cycles.


                                                                                          dv
Meanwhile, when the frequency is high the derivative of v,  -----, is high, so i becomes high although a
                                                                                          dt
very short period. The combination (multiplication) of a high current (though short period) and high frequency (many repetitions per unit time) becomes high and R at a capacitor becomes low.



2) Inductance and Frequency

Likewise,

where XL is Inductive Reactance.

This relationship can be derived from the following basic Inductor equation related with i and v.




Now we regard


                           d
Now we regard   ---   as a symbol  s
                           dt 

 Rearranged

v (t) 
---------- =  L
i(t) s


as V/I V (or v/i) = R

R s = L

R = L s


Now if  s  is 2πf

R = 2πf L


                         d
As we regard   ---   as a symbol  s,
                        dt


                  d
If we see    ---   = 2πf  (also = symbol  s ),
                 dt


Frequency (f) is 1/second. 2πf is frequency in radian.


                                       d
We repeat we regards    ---   as a symbol  s or 2πf
                                      dt

Again - We must think of frequency as an independent unit, not number/second.  And we forget the change happened in one angular frequency cycle (which simply repeats the exactly same change per any one cycle), and simply think of how many cycles.




ACT