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Dimensions of complex frequency 's'

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rsashwinkumar

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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...
 

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)
 
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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 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|).
 

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