Dimensions of complex frequency 's'

[rsashwinkumar]

Hi

I have some fundamental doubts on the units of complex frequency 's' and dimensions of time constant τ.

Consider a first order RC network, where the low pass transfer function V2(s)/V1(s) = 1/(1+sτ).
Clearly the transfer function must be dimensionless, but the dimension of τ is 'seconds' and if 's' has a dimensions of radians/second, then the transfer function seems to have dimensions of radians.

So is the units of 's' 1/seconds?
(While taking laplace transform we consider exp(-s*t). The argument of exponential function must be dimensionless and so again this seems to say that 's' must have dimensions of 1/seconds)

Also even though τ=RC has dimensions of seconds, the corner frequency 1/(RC) has dimensions of radians/seconds. So what is the catch here?

Please help me out...

 

[LvW]

Due to the mathematical derivation of the complex frequency variable "s" the corresponding unit is [1/s].
Hence, also the unit of the imaginary part of (s=sigma+j*w) is [1/s].
However, with the aim not to mix the angular frequency w and the "classical" frequency f it was agreed to use [rad/s] if we speak about w=2*Pi*f.
But the addendum "rad" is to be considered as a dimensionless unit - and , thus, just disappears if we divide w by 2*Pi.
Its only sense is to be able to discriminate better between f and w.

(Remember the addendum "dB" which also must not be treated as a dimension, because - for example - the difference dBm-dBm=dB)

 

[Chips & Chips]

Radians are a natural measure of angle and is defined as circumference / radius. Hence the units for Radians are distance / distance which reduces to a pure, dimensionless number.

So when you say that "the corner frequency 1/(RC) has dimensions of radians/seconds" you are really saying that the units of frequency really is 1/s. And yes, your transfer function is dimensionless.

If the math is done properly, the dimensions always work out properly. I passed many an exam by remembering that fact. LvW complicates the situation far beyond any reason. It really is simple.

 

[LvW]

LvW complicates the situation far beyond any reason. It really is simple.

@chips: Was it too complicated for you?
It is a known fact that 2*Pi is a dimensionless number.
It was my only intention to explain WHY we use rad/s in conjunction with the complex frequency - although it wouldn´t be mathematically false to use 1/s only.
The only reason is to know what someone is talking about when saying: "The frequency is 750 Hertz". As you know - even in case of "omega" it is common to use the term "frequency" instead of "angular frequency".
(Typical example: Pole frequency wp=|sp|).