Asnetcomp Tutorials: 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 = $2π$ 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

Asnetcomp Tutorials

Friday, July 13, 2012

Capacitance, Inductance and Time

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