Moment of Inertia vs. Inertia Constant

[hansherman]

The following equations are found in the following reference (Page 119):

http://www.eeh.ee.ethz.ch/fileadmin/user_upload/eeh/studies/courses/power_system_dynamics_and_control/Documents/DynamicsPartI_lecture_notes_2012.pdf

By definition, the inertia constant for a synchronous machine is defined as

$$H = (1/2 J \omega_0^2) / S$$

where

$$a) H= \text{constant of inertia } (s)$$
$$b) S = \text{rated power of synchronous machine } (MW)$$
$$c) \omega_0 = \text{nominal angular frequency } (rad/s)$$
$$d) J = \text{moment of inertia for rotor } (kg m^2)$$

I.e.

$$J = 2HS/\omega_0^2$$

can be used to find the moment of inertia. Based on the units of a), b) and c) the unit of J is

$$s MW/(rad^2/s^2)$$

However, i cannot see that this is the same as kg/m^2, as the result is supposed to yield from d). Can anyone help me?

 

[D H]

Hint #1: Radians are unit less, so you can drop that term.
Hint #2: The watt (and hence megawatts) is a derived unit. What are its primitive units?

 

[dauto]

Hint #3: Instead of using MW (megawatts) you should just use W (watts). M is just a numerical factor of 1000000 and therefore is unitless. The Watt is the standard unit of power for the metric system.

 

[nasu]

There is no reference to this on page 119.

However, the quantity defined in your post have units of seconds (energy/power).
The confusion may be due to the fact that (at least) two different quantities may be called the same name: "inertia constant".

See for example here:
http://books.google.ca/books?id=Su3...onepage&q=inertia constant of machine&f=false

You are talking here about the second quantity, the H defined on page 540 of that book and not the first one (I*ω) which is also called inertia constant, on the same page.

 

[hansherman]

I do not understand why Radians are unit less. Can anyone explain this? Thanks for the answers.

 

[TheAustrian]

Because you divide [Length] by [Length]

θ = s /r

s is arc length (of a circle)
r is radius

EDIT:

This page has a nice graphical (animated) explanation about the radian:
http://ru.wikipedia.org/wiki/Радиан