[SOLVED] basic question about time constant

[stanleyche]

Hi, I have a question about time constant. For a 1st order RC filter, time constant=RC, which obviously has the unit of [second]. The bandwidth of this filter is 1/(2*pi*RC), which has the unit of [Hz]=[cycle/second]. I just don't know how this unit comes about. Here is my confusion:

If RC has the unit of [second], then 2*pi must be 1/[cycle] to make 1/(2*pi*RC) have the unit of [cycle/second]. But it seems weird to me that 2*pi is in unit of 1/[cycle].

Thanks for any help!

 

[keith1200rs]

One cycle = 360 degrees = 2*Pi

Keith

 

[LvW]

If RC has the unit of [second], then 2*pi must be 1/[cycle] to make 1/(2*pi*RC) have the unit of [cycle/second]. But it seems weird to me that 2*pi is in unit of 1/[cycle].

2*Pi is an angle and is given in radian (rad).
1/RC is given in rad/sec.
1/(2*Pi*RC) is given in rad/(rad*sec)=1/sec=Hz.

 

[stanleyche]

If RC has the unit of [second], then 2*pi must be 1/[cycle] to make 1/(2*pi*RC) have the unit of [cycle/second]. But it seems weird to me that 2*pi is in unit of 1/[cycle].

2*Pi is an angle and is given in radian (rad).
1/RC is given in rad/sec.
1/(2*Pi*RC) is given in rad/(rad*sec)=1/sec=Hz.

Based on your 2nd line, you are then saying that the number "1" in the numerator has the unit of "rad" (because we all know that RC at the bottom is in "sec"). If so, why? Could you please follow up? Thanks.

 

[LvW]

Hi Stanleyche,

I see your point and I think you are confused because we discuss a mixture between formal definition and common agreement.
Let me explain:
*The frequency f of a sinusoidal wave is defined as "N cycles per second". Thus: f in 1/sec defined as Hz.
*Now it turns out, that in many formulas in the field of electronics the frequency always appears with a factor of 2*Pi.
*Thus, it makes sense to define a new symbol w for this expression: w=2*Pi*f (formally still in1/sec). This is only a symbol and a name for this new parameter is found next.
*It turns out, that this expression is identical to the "angular velocity" of a rotating phasor, which leads to the formal definition: w=d(phi)/dt. This velocity parameter (not yet: frequency) is given in rad/sec.
*Although all electronic formulas contain the first abbreviation (w in 1/sec) and in order to avoid a mix-up with the "normal" frequency f, it makes sense to call this new parameter "angular frequency" and to give it in rad/sec (instead of 1/sec).
*However, if you want to check units you have to use 1/sec.
*By the way, in textbooks/articles you can find for w also the unit "1/sec" if a mix-up between f and w is not probable.

Summary: Under formal aspects, w must be given in 1/sec; however for practical reasons (discrimination between f and w) the unit "rad/sec" is used very often.

LvW

 

[stanleyche]

Hi LvW, I digged deeper and I think I understand my issue better now, with your help, of course.

So the -3dB frequency for any 1st order RC filter is found by setting 1+j*w*RC=1+1j. This frequency is in rad/sec because wRC must have the unit of "rad" given that it is a phasor. Therefore, w=1/RC and "1" means "1 radian". Now, when we find its equivalent -3dB in Hz, we do f=1/(2*pi*RC).

Is my understanding right?

 

[LvW]

No, the expression wRC certainly has not the unit rad.
The reason is that radian must be considered as a "dimensionless unit" that sometimes is included in an dimensional expression just for clarity purposes (as explained above).
Example: In the mathematical expression vmax*sin(wt+phi), which describes a sinusoidal voltage, the part "wt" must be interpreted as an angle (because you can find the sin only if the argument is considered to be an angle).Thus, w must be seen as a number with the unit "rad/sec.
However, for instance, to find a capacitive current you have to differentiate this expression yielding
i=Cdv/dt=wC*vmax*(cos(wt+phi).
Now w appears as a factor and must be interpreted as an angular frequency in 1/sec. Otherwise the resulting current is not given in A (and the resulting capacitive resistance 1/wC not in ohms)

 

[stanleyche]

Hi LvW, I think I finally get you now.

When we derive the bandwidth of 1st-order RC filter, the "w" in the jwRC=j actually is w=2*pi*f where the 2*pi is JUST A FACTOR which is unitless. Therefore, w has the unit of 1/sec.

In order to extract the f in the w=2*pi*f, we need to divide 1/RC by 2*pi where the 2*pi is again unitless.

So formally:
2*Pi is unitless.
1/RC is given in 1/sec.
1/(2*Pi*RC) is given in 1/(1/sec)=Hz.

But in common-sense:
2*Pi is an angle and is given in radian (rad).
1/RC is given in rad/sec.
1/(2*Pi*RC) is given in rad/(rad*sec)=1/sec=Hz.

Am I right?

Thanks again.

 

[LvW]

Yes, I think we have come to an agreement now.
Thanks
lvW