## First Order System's Time Constant

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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 Blog Entries: 27 Recognitions: Gold Member Homework Help Science Advisor hello yanaibarr! welcome to pf! tau is always in seconds … the difference between radians and (eg) degrees is absorbed into the k

 Quote by tiny-tim 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.

## First Order System's Time Constant

[QUOTE=tiny-tim;3410695]

tau is always in seconds …

Thanks for he replay.
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
 Blog Entries: 27 Recognitions: Gold Member Homework Help Science Advisor sorry, i don't know, i haven't come across the s-plane

[QUOTE=yanaibarr;3417840]
 Quote by tiny-tim 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] $$(s=j\omega+\sigma)$$. 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
$$\omega$$ has units of $$\frac{rad}{sec}$$ (s = jw+sigma) , Hz has units of $$\frac{1}{s}$$ so the connection you made between the derivative, 1/s and, Hz for the s domain is correct.

 Quote by tiny-tim 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.

[QUOTE=viscousflow;3418676]
 Quote by yanaibarr $$\omega$$ has units of $$\frac{rad}{sec}$$ (s = jw+sigma) , Hz has units of $$\frac{1}{s}$$ 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
 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
 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.

 Tags first order system, time constant, transfer function