Friday, July 13, 2012

Capacitance, Inductance and Time


The title is "Capacitance, Inductance and Time " but "Time" here is used as Frequency. The unit of Frequency is Hz as well as 1/s (1/t in symbol).


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

 X= - \frac{1}{\omega C} = - \frac{1}{2\pi f C}

or no <minus sign>

 and

X_L = \omega L = 2\pi f L\quad


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


1) Capacitance and Frequency

X= - \frac{1}{\omega C} = - \frac{1}{2\pi f C}
where X is Capative Reactance (Xc)

This relationship can be derived from the following basic Capacitor equation related with i and v.
i(t)= \frac{\mathrm{d}q(t)}{\mathrm{d}t} = C\frac{\mathrm{d}v(t)}{\mathrm{d}t}.
Rearranged

i (t) x dt 
---------- =  C
dv(t)

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

t/R = C

R = t/C

as Frequency (f) is 1/t or t = 1/f

R = 1/Cf

where R can be regarded as Xc and f is used as Frequency in general.  ω = f

We need some conceptional jump to get into another world of frequency domain.


2) Inductance and Frequency

Likewise,

X_L = \omega L = 2\pi f L\quad
where XL is Indcutive Reactance.

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

v(t) = L \frac{di(t)}{dt}

Rearranged

v (t) x dt 
---------- =  L
di(t)

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

tR =L

R = L/t

as Frequency (f) is 1/t or t = 1/f

R = Lf

where R can be regarded as XL.


Here again we need some conceptional jump to get into another world of frequency domain.


ACT


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