Generating AC Voltages
Generator Principles
Voltage induced in a single phase
The rotor in the generator illustrated above produces a flux which passes radially from the rotor surface into the stator. The circumferential variation of this radial flux density around the rotor can be described using
where θ is the angle around the surface. If the rotor of the machine rotates at constant speed ωt then the flux density seen at a position θ0 on the stator will be given by
Using the generator law
the voltage induced in each side of the coil can be obtained. Considering the conductor at θ0=0, the velocity of the conductor with respect to the flux is down (the flux density is moving up relative to the conductor), the flux density is left to right, the cross product is parallel with the conductor. If we take the conductor to be oriented with positve direction out of the page then the induced voltage is given by
Relating linear velocity to angular velocity
the induced voltage becomes
For the return side of the conductor at θ0=π,
Therfore, the total voltage induced in the loop formed by the two conductors is
Now, if there are NT loops forming a coil,
Noting that the area enclosed by the coil is given by 2rl the voltage can also be written as
A multi-coil machine
In the simple example with only one coil, there is a significant waste of space in the machine. It is possible to add additional coils to the stator and make full use of the available space. If there are Ncoils coils, all capable of carrying equal current, I, the maximum available power is available in an Np phase machine with Np = Ncoils. In this csae, the maximum available power is given by given by
Unfortunately, to access this power, each phase (each coil) would need to be connected to a load by its own transmission line. This is unacceptably expensive for a transmission and distribution system. As an alternative, all the coils could be connected together in series to form a single phase voltage. If all the coils are connected in series the total available voltage is given by the phasor summation of the individal coil voltages:
and the available power is
In the case where Ncoils is large, the series connected power is 2/π times the availble maxiumum power (about 64%).
A Three-phase machine
Clearly some compromise between generator utilization and power delivery cost must be reached. This compromise is the three-phase power system. In a three-phase generator, there are still many coils in the machine, but the coils are now connected in three distinct groups, called phases. Noting that the sum of all the coil voltages must equal the sum of the phases, the rms phase voltage can be obtained:
Now, if each phase is connected to a load by its own transmission line, the total available power is given by
and if the number of coils is large:
General 3-phase voltages
In general in the course we will think of a three-phase winding as being made up of only three coils, separated by 120 degrees. However, it is important to realise that a real machine has many coils which are grouped in to three phases. As shown above, the fundamental voltage induced in a phase comprised of a number of distributed coils will be slightly lower than that produced by a single coil with the same total number of turns. Distributed windings can also reduce the magnitude of higher harmonics which may be induced in a coil. Detailed infomration about distributed windings is beyond the scope of the course, more information can be found in the appendices of the textbook.
In a simple ac machine constucted using only three coils, the three-phase induced voltages can be written as:
