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First Order System's Time Constant 
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#1
Jul1911, 03:14 AM

P: 6

Hello,
I have a question on a the units of a first order system's time constant. If i have a first order system the basic transfer function will be: K/(tau*s+1) where K is the Gain, and tau is the system's time constant. tau's units, according to what i've learned, are [sec]. but aren't the s plane's units in [rad/sec] (s=jw+sigma)? That means that tau should be given in [sec/rad] to match the "1"'s units in the transfer function. I know that rad can be considered "unitless" but when dealing with actual numbers it matters if the system's time constant is 1 [sec] or 1[sec/rad]= 2*pi [sec]. My question is specifically about the units of tau in the transfer function, not when it is used in the decay rate of e (e^(t/tau)), there it has to be sec. I'll appreciate a clarification. Thanks 


#2
Jul1911, 07:41 AM

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P: 26,160

hello yanaibarr! welcome to pf!
tau is always in seconds … the difference between radians and (eg) degrees is absorbed into the k 


#3
Jul2011, 01:50 AM

P: 576




#4
Jul2311, 12:56 PM

P: 6

First Order System's Time Constant
[QUOTE=tinytim;3410695]
tau is always in seconds … Thanks for he replay. One more question about it, if tau's units should be seconds, then the splane units should be Hz [1/s]. According to what I've learned, the splane's units are [rad/sec] (s=jw+sigma). Can i take the splane's units as Hz? I tried working with an actual differential equation, and according to it the splane's units will always be [1/sec], because the s represents the derivative. If it's so, when do i use the [rad/sec] units and when [Hz] in the splane? Thanks, Yanai barr 


#5
Jul2311, 03:41 PM

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sorry, i don't know, i haven't come across the splane



#6
Jul2411, 12:59 AM

P: 273

[QUOTE=yanaibarr;3417840]



#7
Jul2411, 03:18 AM

P: 6




#8
Jul2411, 03:27 AM

P: 6

[QUOTE=viscousflow;3418676]
but Hz [1/s] and omega's units [rad/s] are not the same, u should divide\multiply it by 2*pi. This is exactly my question, the units don't match (according to the theory I've learned). In theoretical problems it doesn't matter, but when i use actual numbers i need to decide how to use the data, and how to convert the units accordingly. Yanai Barr 


#9
Oct2311, 01:09 PM

P: 76

I've stumbled at the same problem. All learning materials seem to expose the concept but none gives example with exact units.
So, if I want a frequency break at 1 Hz, should I write 1/(s+1) or 1/(s+2Pi)? Second seems more plausible. However, when Laplacetransfromed, it gives e^{2pi t} meaning that time constant is T = 1/2pi. Yet, I'm customed that periods are measured in seconds rather than seconds per radian. I mean that 2pi is not usually a part of period. But, wikipedia article on time constant does not clarify what are the units. There is the same unanswered question in this forum 


#10
Oct2411, 03:54 AM

P: 76

In other words, is it right that time constant 2Pi corresponds to frequency of 1 Hz?
How to Obtain DiscreteTime Impulse Response from Transfer Function guide saying that a pole of 1, H(s)=1/(s+1), corresponds to time constant of 1 sec, seems to agree with this. 


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