Electrical Machines
Electrical machines are everywhere - generating and using electrical power. They’re used more and more in transportation as well as industry.
Generation is done either with electromagnets or permanent magnets, and some hydro stations use induction machines.
In a generator, the mechanical power in is converted into electrical power out. Some energy is lost in the generation of power, and some energy is lost in the conversion of the generated power into a useful form. The same happens in reverse for a motor: Electrical power in is converted into mechanical power, with some losses in the power converter and the motor itself.
1 Electric machines
All machines consist of a stator and a rotor.
The stator is stationary, the rotor rotates. Either one can contain conductors or permanent magnets (or occasionally neither!).
2 Electromagnetic terms
There are four basic principles governing the production of a magnetic field in a device.
- A current-carrying wire produces a magnetic field in the area around it, according to the right hand rule.
- A time-changing magnetic field induces a voltage in a coil of wire if it passes through that coil. (basis of transformer action)
- A current-carying wire in the presence of a magnetic field has a force induced on it. (basis of motor action)
- A moving wire in the presence of a magnetic field has a voltage induced in it. (basis of generator action)
2.1 Flux
Symbol \phi
Measured in Webers (Wb)
Always flows in closed loops (the divergence \nabla \cdot \bf B at any point must be 0, like a fluid.)
The direction of flux is determined by the right hand rule. Put your thumb in the direction of current in a DC wire, and the fingers of your thumb match the direction of the flux.
In a coil, put your forefingers in the direction of the current and your thumb points north (where the flux exits the coil).
2.2 Magnetic field intensity
The magnetic field intensity is a vector H. It’s produced by a current according to Ampere’s law:
\oint \bf H \cdot \it d~\bf l =\it I_{net}
In a ferromagnetic core, most of the flux will pass through the core itself, so you can assume there is no leakage and the path of integration becomes the mean path of the core. The current passing through the path of integration I_{net} is equal to Ni: the number of turns in the coil cutting the path of integration times the current in that coil. Then
H l_c = Ni
where H is the magnitude of the magnetic field vector H.
2.3 Flux Density
This is a measure of how much flux flows through a unit cross sectional area. The symbol of flux density is \bf B.
Flux density is measured in Teslas (much like my net worth). \rm{T} = \frac{\rm{Wb}}{\rm{m}^2}
The relationship between magnetic field intensity H and resulting magnetic flux density B is given by
\bf B = \mu \bf H
The relationship between flux and flux density is
\phi = \iint_A \bf{B}\cdot \it d\bf{A}
If the flux density vector is perpendicular to a plane of area A, as usually happens in the internal cross-sections of motors, then the expression simplifies to
\phi = BA.
2.4 B-H curve
In a B-H curve, you plot flux density against magnetic field strength.
\bf B = \mu_0 \mu_r \bf H
where \mu_0 is the permeability of vacuum, and \mu_r is the relative permeability of the material.
The permeability of vacuum is about the same as the permeability of air:
\mu_0 = 4\pi \times 10^{-7} ~\rm {H/m}
2.5 Flux linkage
If that flux \phi is produced by a coil with a particular current I_f, and if the material of the machine is within the linear current-flux portion of its B-H curve, then we can say that \phi = k\ I_f.
2.6 Inductance
Inductance is the flux linkage per unit of current producing the flux.
When you have two coils linked by magnetic flux, you can define the Self Inductance as the inductance of the driving coil, and the mutualinductance is the inductance of the driven coil.
3 Faraday’s Law
Electromagnetic force is induced by a conductor in a variable magnetic field.
The driving terms are:
- Flux density B
- Length of conductor l
- Relative velocity of conductor and magnet v
The voltage induced is the derivative of the flux induced o.
4 Lorentz force
On the left hand:
First finger - Field
SeCond finger - Motion
Thumb - Direction of current
You can also remember this much more simply by the right hand rule for cross products - curl your fingers in the direction of the second vector with relation to the first, and your thumb points in the correct normal direction. The Lorentz force operating on a conductor in a magnetic field is the is the cross product of the length of the wire in the direction of current times the magnetic field.
To determine the power in an electrical machine, note that the force creates a shear stress acting on the air in the air gap, of area \pi \ D \ L.
P = 2 \ (B K_E) \left( \pi \frac{D^2}{4} L \right)\ \omega (W)
You can use this relation to determine the size of the rotor needed to achieve a certain RPM, given the magnetic loading, electrical loading, and shear stress.
Bigger power, lower working speed -> larger machine.
5 Machine operation
You can compare the speed and the torque of a machine to determine its type of operation. If the speed and the torque are in the same direction, the machine is in traction / motoring mode. If the speed and torque are in opposite directions, the machine is in generating / braking mode.
You can see this by imagining an electrical car. When you push on the accelerator, you use power to create torque in the direction of power. If you push on the brake, the direction of torque swaps.
Plot torque against speed. Electric motors have higher torque at low speeds, but their torque drops off at higher speeds (switching to constant power to protect the electric motor).