Asnetcomp Tutorials: Angular frequency ω (which is equal to 2πf)

A part of the explanation of "AC current" of wiki (as of 29-April-2013) is as follows:

"
Mathematics of AC voltages

Alternating currents are accompanied (or caused) by alternating voltages. An AC voltage v can be described mathematically as a function of time by the following equation:

$v(t)=V_\mathrm{peak}\cdot\sin(\omega t)$ ,

where

$\displaystyle V_{\rm peak}$ is the peak voltage (unit: volt),
is the angular frequency (unit: radians per second)
- The angular frequency is related to the physical frequency, $\displaystyle f$ (unit = hertz), which represents the number of cycles per second, by the equation $\displaystyle\omega = 2\pi f$ .
$\displaystyle t$ is the time (unit: second).

"
The title of wiki is "AC current" but Mathematics is about "AC voltage", which is not a big issue and you can change the voltage to the current (I) - i(t) = Ipeak x sin (ωt).

The relation between ω (angular frequency) and f ("physical" or "ordinary" frequency) is stated above. Then, what is the difference between ω and f ?

A part of the explanation of "Angular frequency" of wiki (as of 29-April-2013) is as follows:

"
One revolution is equal to 2π radians, hence^[1]^[2]

$\omega = {{2 \pi} \over T} = {2 \pi f} ,$

where

ω is the angular frequency or angular speed (measured in radians per second),

T is the period (measured in seconds),

f is the ordinary frequency (measured in hertz) (sometimes symbolised with ν),

Units

In SI units, angular frequency is normally presented in radians per second, even when it does not express a rotational value. From the perspective of dimensional analysis, the unit Hertz (Hz) is also correct, but in practice it is only used for ordinary frequency f, and almost never for ω. This convention helps avoid confusion.^[3]

^"

Well the same thing except it says ＜From the perspective of dimensional analysis, the unit Herz (Hz) is also correct＞.The relation between ω (angular frequency) and f ("ordinary" frequency) is stated above. Then, again, what is the difference between ω and f ?

Number
Frequency (f) is simply ---------------------------------------------- . and concerns
Unit time (usually second in Electronics)

about the number of cycles and does not concern about the content of one cycle. It simply shows "how many times the same thing happens per unit time, say per second" or " how frequently the same thing happens per unit time (per second)." So it applies cycles other than sinusoidal cycles.

Meanwhile Capative Reactance and Inductive Reactance are expressed as follow:

Capacitive reactance

$X_C = \frac {1} {\omega C} = \frac {1} {2\pi f C}$

Inductive reactance

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

As far as ＜the unit Herz (Hz) is also correct＞without mathematical rigor or even being incorrect for the sake of simplicity or conceptual understanding, why not

            1
Xc = -------       and     XL = fL
           fC

In case of very low frequency like one (1) cycle per 10 seconds or 1/10 cycle per second, 2π does matter - - 2π (2 x 3.14 = 6.28) /0.1 = 62.8. But in case of high frequency like 1MHz (1,000,000 cycles per second), 2π does not make much difference comparatively (in ohm) - 2π (2 x 3.14 = 6.28) /1,000,000 = 0.00000628. 2π factor contributing to the reactance is large at very low frequencies while almost negligible at high frequencies. What does matter is just frequency ("how frequently he same thing happens per unit time). Meanwhile common capacitors range from 10pF to some thousand µF (the unit of C in the above formulas is F) Farad and common inductors ranges 10µH to some hundred mH (the unit of L in the above formulas is Henry).

sptt

Asnetcomp Tutorials

Monday, April 29, 2013

Angular frequency ω (which is equal to 2πf)

Units

No comments:

Post a Comment