First Order System's Time Constant


by yanaibarr
Tags: first order system, time constant, transfer function
yanaibarr
yanaibarr is offline
#1
Jul19-11, 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
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tiny-tim
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#2
Jul19-11, 07:41 AM
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hello yanaibarr! welcome to pf!

tau is always in seconds

the difference between radians and (eg) degrees is absorbed into the k
MisterX
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#3
Jul20-11, 01:50 AM
P: 537
Quote Quote by tiny-tim View Post
tau is always in seconds …
No, one may use any unit for tau. For exponential decay, Ae^(-t/tau), the exponent (-t/tau) should be unit-less.

yanaibarr
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#4
Jul23-11, 12:56 PM
P: 6

First Order System's Time Constant


[QUOTE=tiny-tim;3410695]

tau is always in seconds


Thanks for he replay.
One more question about it,
if tau's units should be seconds, then the s-plane units should be Hz [1/s].
According to what I've learned, the s-plane's units are [rad/sec] (s=jw+sigma).
Can i take the s-plane's units as Hz?

I tried working with an actual differential equation, and according to it the s-plane'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 s-plane?

Thanks,

Yanai barr
tiny-tim
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#5
Jul23-11, 03:41 PM
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sorry, i don't know, i haven't come across the s-plane
viscousflow
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#6
Jul24-11, 12:59 AM
P: 273
[QUOTE=yanaibarr;3417840]
Quote Quote by tiny-tim View Post

tau is always in seconds …


Thanks for he replay.
One more question about it,
if tau's units should be seconds, then the s-plane units should be Hz [1/s].
According to what I've learned, the s-plane's units are [rad/sec] [tex](s=j\omega+\sigma)[/tex].
Can i take the s-plane's units as Hz?

I tried working with an actual differential equation, and according to it the s-plane'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 s-plane?

Thanks,

Yanai barr
[tex]\omega [/tex] has units of [tex] \frac{rad}{sec}[/tex] (s = jw+sigma) , Hz has units of [tex] \frac{1}{s}[/tex] so the connection you made between the derivative, 1/s and, Hz for the s domain is correct.
yanaibarr
yanaibarr is offline
#7
Jul24-11, 03:18 AM
P: 6
Quote Quote by tiny-tim View Post
sorry, i don't know, i haven't come across the s-plane
The s-plane is what u get after using the Laplace Transform.
yanaibarr
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#8
Jul24-11, 03:27 AM
P: 6
[QUOTE=viscousflow;3418676]
Quote Quote by yanaibarr View Post

[tex]\omega [/tex] has units of [tex] \frac{rad}{sec}[/tex] (s = jw+sigma) , Hz has units of [tex] \frac{1}{s}[/tex] so the connection you made between the derivative, 1/s and, Hz for the s domain is correct.
Thank u for the reply ,
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
valjok
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#9
Oct23-11, 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 Laplace-transfromed, 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
valjok
valjok is offline
#10
Oct24-11, 03:54 AM
P: 76
In other words, is it right that time constant 2Pi corresponds to frequency of 1 Hz?
How to Obtain Discrete-Time 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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