# First Order System's Time Constant

1. Jul 19, 2011

### yanaibarr

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. Jul 19, 2011

### tiny-tim

welcome to pf!

hello yanaibarr! welcome to pf!

tau is always in seconds …

the difference between radians and (eg) degrees is absorbed into the k

3. Jul 20, 2011

### MisterX

Re: welcome to pf!

No, one may use any unit for tau. For exponential decay, Ae^(-t/tau), the exponent (-t/tau) should be unit-less.

Last edited: Jul 20, 2011
4. Jul 23, 2011

### yanaibarr

Re: welcome to pf!

5. Jul 23, 2011

### tiny-tim

sorry, i don't know, i haven't come across the s-plane

6. Jul 24, 2011

### viscousflow

Re: welcome to pf!

7. Jul 24, 2011

### yanaibarr

The s-plane is what u get after using the Laplace Transform.

8. Jul 24, 2011

### yanaibarr

Re: welcome to pf!

9. Oct 23, 2011

### valjok

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.