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# Discussion Groups | Comp.DSP | Low Pass Filter: f = 1/ 2Pi RC Why 2Pi ?

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# Low Pass Filter: f = 1/ 2Pi RC Why 2Pi ? - ADI8421 - 2012-10-06 10:26:00

```1000 ohm resistor and 0.01 uF capacitor = Standard 2 element Low Pass
filter.

RC time constant = 10us

Low pass fc = 1/2PiRC = 15915.494 Hz = -3db corner frequency = Vout/Vin 0.707106...

I'm trying to comprehend the use of 2Pi in the above equation.
We're multiplying 2Pi by 10us period.
This gives a result of 62.831us period which when inverted ie. 1/62.831us
gives us 15915.494 Hz. Is my understanding correct here ?

My second train of thought was that 2Pi was a conversion of our 10us period
to angular velocity in radians as used in w(omega)= 2Pif = radians per
second repesentation of f.
But this second train of thought surely can't be correct, because our
original time constant of 10us hasn't changed, merely a new representation
of it in radians. But then, can you even describe a time period rather than
Please tell me that the second train of thought is crap and I'll be real
happy.

So, if the second train of thought is crap then this begs a further
question:

Why is it that 2Pi just happens to be the precise multiplier required, of
our 10us time period, or any time period in fact, to allow us to arrive at
our f-3db result ?
This can't just be coincidence. If we wanted to arrive at an f-2db result,
the multiplier would be some other unrecognisable figure. Why does 2Pi hold
this important -3db position ?

If I can get a grasp on this, then a lot of other maths related stuff will
fall into place.

Thanks for a marvelous forum. So many 'brainy' dudes!

Cheers Jim

```
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# Re: Low Pass Filter: f = 1/ 2Pi RC Why 2Pi ? - robert bristow-johnson - 2012-10-06 11:28:00

```On 10/6/12 10:26 AM, ADI8421 wrote:
> 1000 ohm resistor and 0.01 uF capacitor = Standard 2 element Low Pass
> filter.
>
> RC time constant = 10us
>
> Low pass fc = 1/2PiRC = 15915.494 Hz = -3db corner frequency = Vout/Vin > 0.707106...
>
> I'm trying to comprehend the use of 2Pi in the above equation.
> We're multiplying 2Pi by 10us period.
> This gives a result of 62.831us period which when inverted ie. 1/62.831us
> gives us 15915.494 Hz. Is my understanding correct here ?
>
> My second train of thought was that 2Pi was a conversion of our 10us period
> to angular velocity in radians as used in w(omega)= 2Pif = radians per
> second repesentation of f.

that's pretty much it.

> But this second train of thought surely can't be correct, because our
> original time constant of 10us hasn't changed, merely a new representation

tau = RC = natural time constant.  the impulse response

h(t) = h(0) e^(-t/tau)

decays to a relative value of e^(-1) by this time.

omega_c = 1/tau = *angular* cutoff frequency,

rads per unit time (whatever unit time that tau is measured in.

f_c = omega_c/(2 pi)

> But then, can you even describe a time period rather than
> Please tell me that the second train of thought is crap and I'll be real
> happy.

your second train of thought was closer to reality.

> So, if the second train of thought is crap then this begs a further
> question:
>
> Why is it that 2Pi just happens to be the precise multiplier required, of
> our 10us time period, or any time period in fact, to allow us to arrive at
> our f-3db result ?
> This can't just be coincidence. If we wanted to arrive at an f-2db result,
> the multiplier would be some other unrecognisable figure. Why does 2Pi hold
> this important -3db position ?
>

it doesn't, for angular frequency.  if your language is in angular
frequency, there is no 2*pi until you speak of the periods of sinusoids.

>

--

r b-j                  r...@audioimagination.com

"Imagination is more important than knowledge."

```
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# Re: Low Pass Filter: f = 1/ 2Pi RC Why 2Pi ? - mnentwig - 2012-10-06 11:31:00

```maybe there is a better answer, but I'd say the "2 pi" are tied to the
reactance of a capacitor X = 1 / (omega C) with omega = 2 pi f.
To get it to the same unit as the resistance R, "volts per ampere".
```
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# Re: Low Pass Filter: f = 1/ 2Pi RC Why 2Pi ? - Tim Wescott - 2012-10-06 11:48:00

```On Sat, 06 Oct 2012 09:26:16 -0500, ADI8421 wrote:

> 1000 ohm resistor and 0.01 uF capacitor = Standard 2 element Low Pass
> filter.
>
> RC time constant = 10us
>
> Low pass fc = 1/2PiRC = 15915.494 Hz = -3db corner frequency = Vout/Vin
> = 0.707106...
>
> I'm trying to comprehend the use of 2Pi in the above equation.
> We're multiplying 2Pi by 10us period.
> This gives a result of 62.831us period which when inverted ie.
> 1/62.831us gives us 15915.494 Hz. Is my understanding correct here ?
>

The short answer is that we use the 2 pi because that's what makes the
math work.

I think your biggest confusion is taking the time constant as a "period",
as if it's the cycle time of a periodic signal.  It's not.  What it _is_
is just a number that appears in the step response of the RC filter:

v(t) = V0 * (1 - e^(-t/tau)),

where tau is the time constant.

Where the 2 pi comes in is when you find the Laplace transform of the
system, and analyze its frequency response.  The transfer function of the
system is

H(s) = 1 / (RC * s + 1)

If you want to take its frequency response, then you use

H(omega) = 1 / (RC * j * omega + 1)

where omega -- because that's the way the math works -- is the radian
frequency of the sine wave you're shoving in, and j (I hope you already
know, or I'm losing you: sorry) is the mathematician's 'i', = sqrt(-1).

It turns out that omega = 1/RC is a very characteristic frequency for the
filter, because at that frequency H = 1/(j + 1).  And 1/(j + 1) has a
magnitude of sqrt(1/2), or -3dB down, or 1/2 power.

All that's left is to observe -- because that's the way the math works --

--
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com
```
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# Re: Low Pass Filter: f = 1/ 2Pi RC Why 2Pi ? - Jerry Avins - 2012-10-06 12:42:00

```On 10/6/2012 10:26 AM, ADI8421 wrote:
> 1000 ohm resistor and 0.01 uF capacitor = Standard 2 element Low Pass
> filter.

OK. Recognize the capacitors with tighter tolerance than +/- 5% are
specialty items, and they 1% resistors are sold for a premium.

> RC time constant = 10us

Nominally, yes. In practice, rarely.

> Low pass fc = 1/2PiRC = 15915.494 Hz = -3db corner frequency = Vout/Vin > 0.707106...

Given the precision of the components, it is more accurate to call that
approximately 16 KHz. In order avoid propagating errors, you might call
it 15.9 KHz. No greater specificity is warranted.

> I'm trying to comprehend the use of 2Pi in the above equation.
> We're multiplying 2Pi by 10us period.

10 us ti not the period. It is the time constant. Time constant is the
time required for a DC level to decay to 1/e of its original.

> This gives a result of 62.831us period which when inverted ie. 1/62.831us
> gives us 15915.494 Hz. Is my understanding correct here ?

No. See above for why not.

> My second train of thought was that 2Pi was a conversion of our 10us period
> to angular velocity in radians as used in w(omega)= 2Pif = radians per
> second repesentation of f.

Spot on.

> But this second train of thought surely can't be correct, because our
> original time constant of 10us hasn't changed, merely a new representation
> of it in radians. But then, can you even describe a time period rather than
> Please tell me that the second train of thought is crap and I'll be real
> happy.

:-(

> So, if the second train of thought is crap then this begs a further
> question:

You should also learn what it means to "beg the question". It means
embedding in the premises the result that you want to establish in a
logical argument. It's either a deliberate dirty trick or a failure of
logical analysis. You probably want proof of that, so here's the best I
can offer: http://en.wikipedia.org/wiki/Begging_the_question and
http://begthequestion.info/. As always, feel free to beg an answer.

> Why is it that 2Pi just happens to be the precise multiplier required, of
> our 10us time period, or any time period in fact, to allow us to arrive at
> our f-3db result ?

Because, as I wrote above, 10 us is a time constant, not a period. The
reciprocal of a time constant is a period in radians per unit time, but
most of us prefer degrees or whole turns.

> This can't just be coincidence. If we wanted to arrive at an f-2db result,
> the multiplier would be some other unrecognisable figure. Why does 2Pi hold
> this important -3db position ?
>
> If I can get a grasp on this, then a lot of other maths related stuff will
> fall into place.

I hope you're right. Also, try to keep a handle on what a number really
means. Would you claim that you got 31.6042 MPG from your last tankful,
just because that number came up on your calculator?

> Thanks for a marvelous forum. So many 'brainy' dudes!

Come back and keep it lively.

> Cheers Jim

Ta.

Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
```
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# Re: Low Pass Filter: f = 1/ 2Pi RC Why 2Pi ? - ADI8421 - 2012-10-06 19:25:00

```Tim Wescott Wrote:
>All that's left is to observe -- because that's the way the math works --

Thankyou Tim. I still don't quite grasp it, but I will eventually, given
the above statement to ponder.

Thankyou to all who offered guidance, even the ugly guy with the small
penis and deficient social skills. Many thanks.

Jim

```
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# Re: Low Pass Filter: f = 1/ 2Pi RC Why 2Pi ? - Tim Wescott - 2012-10-06 20:36:00

```On Sat, 06 Oct 2012 12:42:51 -0400, Jerry Avins wrote:

> On 10/6/2012 10:26 AM, ADI8421 wrote:

> Because, as I wrote above, 10 us is a time constant, not a period. The
> reciprocal of a time constant is a period in radians per unit time, but
> most of us prefer degrees or whole turns.

<sniff>

Surely you meant to say Napiers per second, since we're talking about an
argument to e, here.

(If you can get picky about the meaning of "to beg the question", I can
get picky about obscure units... :)

--
My liberal friends think I'm a conservative kook.
My conservative friends think I'm a liberal kook.
Why am I not happy that they have found common ground?

Tim Wescott, Communications, Control, Circuits & Software
http://www.wescottdesign.com
```
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# Re: Low Pass Filter: f = 1/ 2Pi RC Why 2Pi ? - Jerry Avins - 2012-10-06 22:28:00

```On 10/6/2012 8:36 PM, Tim Wescott wrote:
> On Sat, 06 Oct 2012 12:42:51 -0400, Jerry Avins wrote:
>
>> On 10/6/2012 10:26 AM, ADI8421 wrote:
>
>> Because, as I wrote above, 10 us is a time constant, not a period. The
>> reciprocal of a time constant is a period in radians per unit time, but
>> most of us prefer degrees or whole turns.
>
> <sniff>
>
> Surely you meant to say Napiers per second, since we're talking about
> an argument to e, here.
>
> (If you can get picky about the meaning of "to beg the question", I
> can get picky about obscure units...

I know what a neper is (I'm sure you don't mean that), but not a napier.
As for begging the question rather than raising it, it's the same order
of difference as that between a proposal and a proposition.

Jerry
--
Engineering is the art of making what you want from things you can get.
Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯
Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯
```
______________________________

# Re: Low Pass Filter: f = 1/ 2Pi RC Why 2Pi ? - robert bristow-johnson - 2012-10-06 23:22:00

```On 10/6/12 10:28 PM, Jerry Avins wrote:
> On 10/6/2012 8:36 PM, Tim Wescott wrote:
>  > On Sat, 06 Oct 2012 12:42:51 -0400, Jerry Avins wrote:
>  >
>  >> On 10/6/2012 10:26 AM, ADI8421 wrote:
>  >
>  >> Because, as I wrote above, 10 us is a time constant, not a period. The
>  >> reciprocal of a time constant is a period in radians per unit time, but
>  >> most of us prefer degrees or whole turns.
>  >
>  > <sniff>
>  >
>  > Surely you meant to say Napiers per second, since we're talking about
>  > an argument to e, here.
>  >
>  > (If you can get picky about the meaning of "to beg the question", I
>  > can get picky about obscure units...
>
> I know what a neper is (I'm sure you don't mean that), but not a napier.
> Can you enlighten me please?

he means neper.  a neper and a radian are both the same thing.  from a
physical POV (not a convention), they are both the dimensionless number
1.  from a pure calculus POV, 100 kHz is about 628/ms.  that is what
"omega" is inside the cos() function, if "t" is in ms.  same for nepers.

while "turns" is a natural measure, *humans* then create these units
called percentages, degrees.  but those units are also really just
dimensionless numbers.  a % is the real number: 1/100.  and a degree is
pi/180.  they are not physical units, but conventional units and
conventional units are nothing other than scale factors.  physical units
have *dimension* as a property.

--

r b-j                  r...@audioimagination.com

"Imagination is more important than knowledge."

```
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# Re: Low Pass Filter: f = 1/ 2Pi RC Why 2Pi ? - ADI8421 - 2012-10-07 05:01:00

```>On 10/6/12 10:28 PM, Jerry Avins wrote:
>>The reciprocal of a time constant is a period in radians per unit time,

The reciprocal of a cycle time period is frequency in Hertz.

So ... with a cycle time period of 1ms the reciprocal is 1000 Hz.
But if our 1ms figure is a time constant then the reciprocal result of 1000
is not Hertz but Radians per unit time ?

Cheers Jim

```
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