The Power-Angle Curve and Stability of the Synchronous Machine
What happens past a load angle of 90°?
The power is P = (VE/Xs)sinδ, following the sine of the load angle δ. More load means a larger δ and more power. Use the slider to raise the load angle and watch the operating point on the curve and the curve’s slope (the synchronizing power that restores the angle). What happens once the load angle passes 90°?
Power is the sine of the load angle
Solving the per-phase equivalent E = V + jIXs for power, the active power the machine exchanges is P = (V E / Xs) sin δ, where δ is the phase between the internal EMF E and the terminal voltage V, the load angle. In a generator E leads V and in a motor it lags, but the magnitude of power follows the sine of the load angle either way. More load means a larger angle and a larger power exchanged.
The synchronizing power keeps stability
If a small disturbance widens the load angle a little, the power rises a little and acts to pull the angle back down. This restoring force is the synchronizing power, the slope of the curve dP/dδ = (V E / Xs) cos δ. At a small angle the slope is large and restores strongly; near 90° it shrinks and weakens. At 90° the slope is zero and beyond it negative, so a widening angle lowers the power and widens further in a vicious circle. So normal operation sits at a load angle well below 90°, and the hunting of the angle under a sudden shock is damped by a damper winding on the rotor.
The V-curve shows power-factor control at a glance
Holding the load (power) constant and plotting the armature current magnitude against the field current as the excitation varies traces a V against the field current. The valley is the unity-power-factor point of least armature current; the left side is under-excited and lagging, the right over-excited and leading. Raising the load lifts the whole V-curve and shifts the valley to the right. This single picture captures the previous unit’s power-factor control quantitatively. The power-angle curve shows stability and the V-curve shows power factor — the two maps of the synchronous machine.
Back to the first screen
Past a load angle of 90° synchronism is lost, because the power P = (VE/Xs)sinδ peaks at 90° and then falls as the angle grows. The synchronizing power dP/dδ ∝ cosδ that restored the angle becomes zero at 90° and negative beyond, so any small disturbance makes the angle diverge. So 90° is the steady-state stability limit, and real operation sits well inside it. Meanwhile, holding the power fixed and changing only the excitation makes the armature current trace a V as the power factor moves from lagging to leading. The power-angle curve shows how far it can hold, the V-curve shows how to choose the power factor. The two curves gather the synchronous machine’s torque and power factor, and their limits, in one place.
That completes the synchronous machine. Spinning a magnet makes electricity whose frequency is locked to speed (D1), the armature reaction of load current is bundled into the synchronous reactance (D2), run as a motor it controls power factor by excitation (D3), and the power-angle and V-curves show stability and power factor at a glance (D4). DC, induction and synchronous — all three rotating machines are behind us. The last group E is the modern machine shaped by permanent magnets and power electronics. It begins with an overview of BLDC, stepper and servo.