Synchronous motor/generators, which power electrical grids world wide, are typically run in two of four possible quadrants of operation. For motor operation, this is called the fourth quadrant, by convention, and for generator operation, the first quadrant. The operating characteristics of these machines in the other two quadrants are rarely discussed in engineering classes, in part because of the inherent instability of these modes. By instability we mean that if a machine slips a pole and enters into one of these quadrants by mistake, they tend to become desynchronized with the grid voltage, draw large transient currents, trip breakers, and experience large torque pulsations – sometimes even “bouncing” off the next available pole. However, by solidly connecting the shaft of an unstable synchronous generator with the shaft of a stable synchronous motor operating in quadrant-four, we can meta-stabilize the synchronous generator, with interesting results with regards to over-unity power generation.

In this paper, a simple synchronous machine, with given input currents sent to the single rotor current loop (dc) and three stator current loops (ac), has its torque, power and voltage characteristics considered from the standpoint of standard electromagnetic theory. The stator consists of three circular current loops. Phase A is of radius b, and is situated in the y-z plane, centered around the x-axis. Phase B and C are identically sized current loops, only rotated 120 and 240 degrees counter-clockwise, respectively, about the z-axis. The rotor is a similar current loop, only of radius (b-dR), where dR is the vanishing calculus infinitesimal. The rotor rotates on the z-axis, with the angle in radians of the top edge measured from its home position in the y-z plane being defined by β = ωt + α, where ω is the angular velocity of the rotor in radians per second, and α is an offset angle which defines the region in which the machine operates.

Because we are using current sources to drive the stator coils, the dividing lines and angles of the “quadrants” are somewhat skewed from what engineering classes talk about when stator coils are connected to a constant ac voltage bus (or grid). Nevertheless, for the sake of clarity, we will not belabor or attempt to keep track of differences, but will instead rely on the expertise of engineers to consider the thrust of our presentation which is primarily based on phasor analysis.

We chose [(1+√3)] dc amps for the rotor and 1.0 rms amps for each of the three sinusoidal stator phases (A, B, and C), such that the associated rotating magnetic dipole moment of the stator is equal the magnitude of that of the rotor. The current in the stator phases and in the rotor are defined below.

Current in Stator Phases:

Ia=√2 cos(ωt)

Ib=√2 cos(ωt+[(120)/360]2π)

Ic=√2 cos(ωt+[(240)/360]2π)

Ir=[(1+√3)] dc

Figure 1 shows the equivalent circuit diagram for Phase A of the synchronous machine, with generator convention used to define the direction of current flow.

Equivalent Circuit Diagram for Synchronous Generator

The currents in the stator coils taken together comprise a rotating magnetic dipole moment and linked magnetic field which rotates with synchronous angular velocity and whose home position, from which phasor angles are measured, is oriented along the x-axis. Let the sinusoidal flux linking Loop A as a result of these currents be seen as a phasor of magnitude Φo rms and phase angle of 180 degrees. Φo, and the voltage associated with it for our machine, Vo, are defined as having magnitudes of 1.0 per unit.

The magnetic dipole moment of the rotor current also rotates at a synchronous angular velocity. With respect to the rotor flux linking stator Loop A, and as a result of our choice of current magnitudes, we again have a phasor of magnitude of Φo rms and, with consideration of the offset rotor angle which defines our quadrant of operation, a phase angle of α degrees.

The net voltage appearing across the current source supplying current Loop A is found from the superposition of the rotor and stator flux phasors, and is given by the general expression V=-jωΦ. The superposition of the two flux phasors is given by Φa= Φsa + Φra = Φo at 180 degrees + Φo at α degrees. The voltage phasor for the A Loop voltage, Va has expressions for magnitude and phase shown below, and, keeping in mind that α is a variable and not a fixed angle, we see a diagram of the rotor and stator flux phasors along with the net voltage phasor (Va) for a representative value of α in Figure 2.

Phasor Diagram for Synchronous Generator

Eq. 1 Va=√[((-1)+(cosα))^2 +((sinα))^2]

and the phase angle, A, for Va given by:

Eq. 2 A = atan[((1)-(cosα))/(sinα)]

The expressions for single-phase per unit power, using the generator convention for power, is given from the current and voltage phasors of Loop A taken together and multiplied by cos(A), with Io and Vo set nominally at 1.0 per unit as appears below:

Eq. 3 Pa single phase p.u. =P 3-phase p.u.= (Va p.u.)(Ia p.u.)(cos(A))

Eq. 4 P3-phase p.u. = [√(((-1)+(cosα))^2 + ((sinα))^2)][1.0][cos(atan[((1)-(cosα))/(sinα)])]

Expressions for machine torque are derived from energy considerations, with the torque (T) equal to the derivative with respect to rotor angle (α) of the sum total of magnetic energy in the magnetic field produced by the superposition of the rotor and three stator currents. The magnetic dipole moments producing the fields in such a simple machine can be both decomposed into orthogonal components and/or summed like phasors. Figure 3 shows the net superposition of rotor and stator dipole moments for a representative choice of rotor angle (α).

Flux Phasor Diagram for Synchronous Generator

One we have expressions for magnetic energy in terms of rotor angle (α), we can take a derivative to give (net shaft torque)(angular velocity) which is the input power to our generator provided by the rotating shaft. See Eq. 5 through Eq. 6.

Eq 5. The magnitude of Φa, Φa = √((1-cos(α))²+(sin(α))²)

Eq. 6 The magnetic energy associated with Φa is given by: E=½c(Φa)²

Eq. 7 Substituting Eq. 5 into Eq. 6, we have: E=½c((1-cos(α))²+(sin(α))²)

Eq. 8 Expanding Eq. 7, we have: E= ½c(1-2cos(α)+[cos²(α)+sin²(α)])

Eq. 9 Taking advantage of the Euler identity, and simplifying: E= ½c(1-2cos(α)+[1])= ½c(2-2cos(α))

Eq. 10 And simplifying further: E=c(1-cos(α))

To get torque from Eq. 10, we differentiate E with respect to (α), as shown in Eq. 11.

Eq. 11 T = dE/dα = c(sin(α))

We scale the torque curve, such that the slope at zero rotor angle (α) is equal to the slope of the power curve (Pout). This makes account for the well documented appearance of conservative operation of the generator near rotor angle equal zero.

Figure 4 shows an overlapping of the Powerin and Powerout curves as a function of rotor offset angle (α), with B set to a value such that the upward going slope of the Powerin curve is equal to the upward going slope of the Powerout curve for rotor offset angle equal to zero. This effectively sets Powerin = Powerout for the Conservative first quadrant of operation. Upon simple inspection we can see that these curves are not identical in the unstable region, as Conservation of Energy would predict. The Powerin curve is colored green, and the Powerout curve appears in blue.

Pin and Pout curves as a function of Rotor Offset Angle ( α)

Although we have used current sources to drive our synchronous generator in the demonstration of theoretical over-unity, normally a synchronous generator is connected to a constant voltage source at a reference phase angle of zero degrees, from which all other current and voltage angles are measured. Now that we have done the ground work of demonstrating the theoretical possibility of over-unity generation with current sources, we leave it to engineers to verify our work and hammer out the necessary permutations of meta-stabilized, over-unity operation and equipment/grid protection.

Thanks to Darren and Mark for some great discussions.

David M Boie

email: lilboie1970 @gmail.com