[SOLVED] Why using angular frequency (w) instead of ordinary frequency (f)?

In the text books it's usually the angular frequency used for frequency response, can anyone tell me why? what is the advantages of using (w) instead of (f)?

If there is an advantage it is a small one. Because it is more intuitive to say: I am working at 500 Hz rather than 3141.59265 rad/s. Frequency gives you a intuitive way to know how many times your wave is repeating per second.

Hi,

W = 2 x Pi x f

The 2 x Pi come from a circle. 2 x Pi is one revolution. It is the same as 360°.

When we speak of sine full wave it is like one complete revolution.

****
Both values say different things.
Frequency is somehow an expression of time. Exactely 1/time. And the unit is 1/s.
Where w is an expression of angle/time. Unit is rad/s.

So you could also ask: why do we sometimes talk about "s" (= seconds = time) and another time we talk about "m/s" (= meter/second = speed).
But speed and time are different things, they must not be mixed.

Klaus

Okay so frequency (f) and angular frequency (w) are two different measures of the repetation of say a sine wave, am I correct?

but CataM make a good point isn't it more intuitive to speak in terms of frequency rather than angular frequency?

ammar_kurd said:

In the text books it's usually the angular frequency used for frequency response, can anyone tell me why? what is the advantages of using (w) instead of (f)?

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You said for frequency response, so I automatically thought about bode plot in which my post #2 is still valid.

Of course "f" and "ω" are different things (just by looking at their units you see that). As Kaluss said too, "ω" in physics is used as angular speed which is not the same as cycles per second (frequency).

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ammar_kurd said:

Okay so frequency (f) and angular frequency (w) are two different measures of the repetation of say a sine wave, am I correct?

Click to expand...

Frequency speaks about repetition of whatever wave or whatever thing.
Angular frequency tells you the same as when you hear a car is moving at 5 m/s with the only difference that now is a circular motion and knowing there are 2Π rad you can know how many times passes through a point in 1 second meaning you are calculating frequency. So, as said before, the one who gives us more intuitive information is frequency.

I think the main reason to use "ω" is that it avoids the 2pi factor in the frequency response equations, so they look less messy.

They are the same quantity expressed *slightly* differently. Frequency can be cycles (dimensionless) per second OR 2*pi*cycles radians (dimensionless) per second. Obviously you can clearly see that one cycle is 2*pi and one is called linear frequency (simple harmonic) and the other is called angular frequency (going round in circles). The equivalence is seen that a simple harmonic motion (sine or cosine wave) can be considered a the addition of two circular waves in opposite directions with same amplitude and period. In the same way, a circular motion is the sum of two simple harmonic motions in perpendicular direction (with same amplitude and frequency).

Why two different nomenclature for the same thing? Angular frequency is expressed in terms of the phase angle and one cycle is 2*pi and lambda/2 is just pi and so on...

When you will see computations involved, you will find that it is omega that automatically pops up. The angular frequency is mathematically more fundamental in nature.

KlausST said:

Hi,

W = 2 x Pi x f

The 2 x Pi come from a circle. 2 x Pi is one revolution. It is the same as 360°.

When we speak of sine full wave it is like one complete revolution.

****
Both values say different things.
Frequency is somehow an expression of time. Exactely 1/time. And the unit is 1/s.
Where w is an expression of angle/time. Unit is rad/s.

So you could also ask: why do we sometimes talk about "s" (= seconds = time) and another time we talk about "m/s" (= meter/second = speed).
But speed and time are different things, they must not be mixed.

Klaus

Click to expand...

Klaus - I cannot fully agree to you.
I think, "frequency" is NOT an "expression of time". It is rather a number of occurences per time unit (frequency: periods per second) . And that is a very important difference!.
And "angular frequency" is nothing else than "angles per time unit" given in rad/s.
Both are very close to each other and are parameters in the frequency domain.
(That means: Yoiu example with "time" and "speed" seems to be not applicable)
The invention of the angular frequency is primarily motivated by mathematical reasons (description of frequencies in the complex plane using exp(jwt).

c_mitra said:

The equivalence is seen that a simple harmonic motion (sine or cosine wave) can be considered a the addition of two circular waves in opposite directions with same amplitude and period. In the same way, a circular motion is the sum of two simple harmonic motions in perpendicular direction (with same amplitude and frequency).

Click to expand...

I did not see that, could you give me an mathematical example ?

Hi,

LvW said:

Klaus - I cannot fully agree to you.
I think, "frequency" is NOT an "expression of time". It is rather a number of occurences per time unit (frequency: periods per second) . And that is a very important difference!.

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..But i can fully agree with you. ;-)
I know that frequency is not exactely an expression of time. I tried to point this out with "somehow" and "Exactely 1/time. And the unit is 1/s."

But your "number of occurences" is a much better description.

****

Both are very close to each other and are parameters in the frequency domain.

Click to expand...

Maybe.
I like your expression "number of occurences per time unit".
..While one may be able to convert rad/s back to frequency (I wonder if this is always true)..
..the other way round is not true: you can not always convert frequency ro rad/s.

With frequency in 1/s there is one information missing. "What" is happening in periodic distance of time.
An example that comes into my mind: 0.0000116 1/s (Hz) means once per day. So rad/s may be applied to the rotation of the earth.
But it could also mean a chicken lays an egg per day. (where i find it difficult to express this in rad/s)
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The invention of the angular frequency is primarily motivated by mathematical reasons (description of frequencies in the complex plane using exp(jwt).

Click to expand...

This is definitely true.

Klaus

The maths involving angular notation, such as sin(ωt+ϕ ), computationally brings some benefits, such as reducing the amount of 2PI multiplying opperations to scale to the second (s) unity of measurement, but perhaps the most advantageous aspect is that many trigonometric functions can be straightfowardly synthesized by Taylor's series in that notation, such as sinϕ = ϕ³/3!+ϕ⁵/5!+...

KlausST said:

An example that comes into my mind: 0.0000116 1/s (Hz) means once per day. So rad/s may be applied to the rotation of the earth.
But it could also mean a chicken lays an egg per day. (where i find it difficult to express this in rad/s)

Click to expand...

OK - Klaus, I know what you mean.
However, if somebody uses a number (e.g. 0.00000116) together with the unit 1/s(Hz) everybod - I think - should automatically realize that we speak about a frequency (and not about an egg per day).
More than that, even if such an occurence is periodic - according to my understanding of the definition - the unit Hz ist to be used for sinusoidal waveforms only.
With other words: I prefer to use "repetitions per second" for all periodic events which are NOT sinusoidal.

CataM said:

I did not see that, could you give me an mathematical example ?

Click to expand...

Sure. A simple harmonic motion is x=a*sin(wt) // x is displacement; a is amplitude; w is ang freq and t is the time (obviously)

Another SHM in the perpendicular direction will be y=a*sin(wt) // this is in y axis but has the same freq and amplitude

If we combine these two: x^2+y^2=a^2 // this is a circle

(if they do not have the same phase, you will not get the same result; you can even get a diagonal line)

The reverse is also true: a circular motion in x=a*sin(wt); y=a*cos(wt) // this is an anticlock wise rotation
another circular motion in the opp sense will be x=a*sin(-wt); y=a*cos(-wt) // this motion is clock wise...

Add the two and you will get 2*y=2*a*cos(wt) // this is also a SHM

A plane polarised light can be considered a sum of left circularly polarised light PLUS a right circularly polarised light. Some satellite (TV) transmissions use circular polarisation...