# First Order System's Time Constant

 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
 Sci Advisor HW Helper Thanks P: 26,148 hello yanaibarr! welcome to pf! tau is always in seconds … the difference between radians and (eg) degrees is absorbed into the k
P: 581
 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.

 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
 Sci Advisor HW Helper Thanks P: 26,148 sorry, i don't know, i haven't come across the s-plane
P: 273
[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.
P: 6
 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.
P: 6
[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
 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
 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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