Units for a vector magnitude in the s-plane

[Joseph M. Zias]

In the S plane we have a real component, usually called sigma, and the imaginary component, jw, in radians/sec. The real component is sometimes called nepers per second, with nepers being dimensionless. However, if we draw a vector in the s-plane, say s - s1, in polar form, what are the units of the magnitude of that vector.
I have read one suggestion that both axis are frequencies with unit sec^-1. If we forget about the axis names we could end up with a magnitude of sec^-1. Then when going back to the component form assign so much to the nepers and so much to the radians. That seems a bit odd so I ask opinions. What is the unit of the magnitude of a vector in the S-plane?

 

[anuttarasammyak]

However, if we draw a vector in the s-plane, say s - s1, in polar form, what are the units of the magnitude of that vector.

$$e^{st}=e^{\sigma t}\ e^{j \omega t}$$
So all $s,\sigma,\omega$ have dimension T^-1, unit of sec^-1 in SI.
The difference of two vectors are so also, i. e,
So all $\triangle s, \triangle \sigma,\triangle \omega$ have dimension T^-1, unit of sec^-1 in SI.

We may write it in polar coordinate
$$\sigma + j\omega = \sqrt{\sigma^2+\omega^2}\ e^{j \ \tan^{-1} \frac{\omega}{\sigma}}$$
You see radial part has dimension of T^-1 as well as $\sigma, \omega$ have it , and angle is dimensionless.

 

[Joseph M. Zias]

Well, I see the logic as I noted earlier. However, the dimensions of the y axis are radians/sec. How are we ignoring that in the quantity "square root (a^2+w^2).

 

[Baluncore]

A complex frequency plot requires the x and y axes have the same unit.
radians/sec.

 

[anuttarasammyak]

neper and radian are dimensionless. unit neper/second, rad/second both have dimension T^-1 as well.
$$\sigma + j\omega = \sqrt{\sigma^2+\omega^2}\ e^{j \ \tan^{-1} \frac{\omega}{\sigma}}$$
The radial part has unit neper/sec AND rad/sec of physical dimension T^-1. The angular part has no dimension.$$=\sqrt{\sigma^2+\omega^2}\cos( \tan^{-1} \frac{\omega}{\sigma})+j \sqrt{\sigma^2+\omega^2}\sin( \tan^{-1} \frac{\omega}{\sigma})$$
The first term which is real has unit neper/sec of physical dimension T^-1.
The sencond term which is imaginary has unit rad/sec of physical dimension T^-1.
We see neper/sec or rad/sec depends on it is real term or imaginary term on shoulder of e. I would like to disregard unit neper and radian and regard them just dimensionless numbers.

 

[Joseph M. Zias]

Good point that radians are essentially dimensionless. In this case we have a magnitude of sec^-1 and then divide it up appropriately; as you note the real part gets nepers per second and the imaginary part gets radians per second. However, we could have also used frequency for the vertical axis, thus cycles per second. Cycles would not be dimensionless. By-the-way, I posted a similar problem in general physics using Velocity vs time.

 

[anuttarasammyak]

cycle is radian / 2$\pi$ so it is also dimensionless.

 

[Joseph M. Zias]

Well, very good comments and I think takes care of the S plane. Comments on the physics site are interesting.