# Difference between revisions of "Inertia"

In power systems engineering, "inertia" is a concept that typically refers to rotational inertia or rotational kinetic energy. For synchronous systems that run at some nominal frequency (i.e. 50Hz or 60Hz), inertia is the energy that is stored in the rotating masses of equipment electro-mechanically coupled to the system, e.g. generator rotors, fly wheels, turbine shafts.

## Derivation

Cross-section of a cylindrical body rotating about the axis of its centre of mass

Below is a basic derivation of power system rotational inertia from first principles, starting from the basics of circle geometry and ending at the definition of moment of inertia (and it's relationship to kinetic energy).

The length of a circle arc is given by:

$L = \theta r$

where $L$ is the length of the arc (m)

$\theta$ is the angle of the arc (radians)
$r$ is the radius of the circle (m)

A cylindrical body rotating about the axis of its centre of mass therefore has a rotational velocity of:

$v = \frac{\theta r}{t}$

where $v$ is the rotational velocity (m/s)

$t$ is the time it takes for the mass to rotate L metres (s)

Alternatively, rotational velocity can be expressed as:

$v = \omega r$

where $\omega = \frac{\theta}{t} = \frac{2 \pi \times n}{60}$ is the angular velocity (rad/s)

$n$ is the speed in revolutions per minute (rpm)

The kinetic energy of a circular rotating mass can be derived from the classical Newtonian expression for the kinetic energy of rigid bodies:

$KE = \frac{1}{2} mv^{2} = \frac{1}{2} m(\omega r)^{2}$

where $KE$ is the rotational kinetic energy (Joules or kg.m2/s2 or MW.s, all of which are equivalent)

$m$ is the mass of the rotating body (kg)

Alternatively, rotational kinetic energy can be expressed as:

$KE = \frac{1}{2} J\omega^{2}$

where $J = mr^{2}$ is called the moment of inertia (kg.m2).

Note that in physics, the moment of inertia $J$ is denoted as $I$. In electrical engineering, the convention is that the letter "i" is always reserved for current, and is often replaced by the letter "j", e.g. the complex number operator i is j in electrical engineering).

TBA

## Generator Inertia

The moment of inertia for a generator is dependent on its mass and apparent radius, which in turn is largely driven by its prime mover type.

 Steam turbine Hydro turbine Combustion engine
 Aeroderivative gas turbine Industrial gas turbine