MC-D4 · V-curve & stability

The Power-Angle Curve and Stability of the Synchronous Machine

The output of a synchronous machine follows the sine of the load angle. Raise the load angle, watch how the output and the force that pulls the load angle back change, and pin down for yourself where the limit of losing synchronism lies.

What happens past a load angle of 90°?

The output of a synchronous machine is P = (VE/Xs)sinδ, following the sine of the load angle δ, so as the load grows and δ grows, the output grows too. Raising the load angle with the slider moves the operating point along the curve together with the slope of the tangent drawn at that point (the synchronizing power that pulls the load angle back). Pick first what happens once the load angle passes 90°, then move the slider.

Load angle δδ = 35°

The slider changes the load angle continuously. Dashed line = 90° limit, tangent slope = synchronizing power; a green tangent pulls the angle back, a red one pushes it further away. Past the limit the operating point turns red too.

Power and synchronizing power

P/Pmax = 0.57 · Ps ∝ cosδ = 0.82

Past 90° the torque does not get any larger. sinδ peaks at 90° and falls after that, so the larger the load angle, the smaller the output becomes, and the machine cannot carry the load.

Past δ = 90°?

Misaligned

Power is the sine of the load angle

Solving the per-phase equivalent circuit E = V + jIXs for the output, the real power a synchronous machine exchanges per phase is P = (V E / Xs) sin δ, written with phase values, so the three-phase total is three times that. Here δ is the phase difference 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 in both the size of the output follows the sine of the load angle, so a larger load means a larger load angle and a correspondingly larger output exchanged.

ObserveP = (\rac{VE}{Xs}) sin δ

The output is proportional to the sine of the load angle.

Choosemax P: δ = ?

The output is largest at a load angle of 90°.

The synchronizing power keeps stability

When a small disturbance increases the load angle a little, the output also rises a little and acts to bring the load 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 load angle the slope is large and the pull back is strong, but closer to 90° the slope shrinks and the pull weakens; at 90° the slope becomes 0, and beyond it the slope is negative, so a growing load angle reduces the output and opens the angle further in a vicious circle. That is why normal operation runs at a load angle well below 90°, and hunting, the swinging of the load angle after a sudden shock, is damped out by the damper winding in the rotor. A large shock such as a fault makes the load angle swing widely once, so whether the machine survives it is studied separately as transient stability.

Fill inPs = dP/dδ ∝ ?

The synchronizing power is the slope of the curve, proportional to the cosine.

On your ownstable limit: Ps = ?

The stability limit is 90°, where the synchronizing power becomes 0.

The V-curve shows power-factor control at a glance

Holding the load (output) constant, changing the field excitation and plotting the size of the armature current gives a V shape, with armature current on the vertical axis against field current on the horizontal. The bottom of the V is the point of power factor 1, where the armature current is smallest; the left side is underexcited and lagging, the right side overexcited and leading. Raising the load lifts the whole V-curve and moves its bottom to the right, because a larger output needs more excitation to reach unity power factor. This one graph puts numbers on the power-factor control of the previous unit: how far the machine can hold is read from the power-angle curve, and how to choose the power factor from the V-curve, so the two curves serve as two maps for reading a synchronous machine.

Back to the first screen

Once the load angle passes 90°, synchronism is lost, because the output P = (VE/Xs)sinδ peaks at 90° and then falls as the load angle grows. The synchronizing power dP/dδ ∝ cosδ, which had been pulling the load angle back, becomes 0 at 90° and negative beyond it, so even a small disturbance opens the load angle; 90° is the steady-state stability limit, and real operation stays well inside it. If the same machine keeps its output fixed and only the excitation is changed, the armature current traces a V and the power factor moves from lagging to leading. How far the machine can hold is read from the power-angle curve, and how to choose the power factor from the V-curve, so the torque, the power factor and the limits of the synchronous machine are gathered in these two curves.

Power-angle curve: the output of a synchronous machine is P = (VE/Xs)sinδ, largest at δ=90°. The synchronizing power dP/dδ ∝ cosδ that pulls the load angle back becomes 0 at 90° and negative beyond it, so δ=90° is the steady-state stability limit (pull-out beyond it). The V-curve is the curve of armature current against field current at fixed output; its bottom is power factor 1 and minimum current, its left side underexcited (lagging), its right side overexcited (leading).

Closing out group D

That completes the synchronous machine. Spinning a magnet made electricity whose frequency is tied to speed (D1), the armature reaction of load current was bundled into a synchronous reactance (D2), running it as a motor let the excitation adjust the power factor (D3), and the power-angle curve and the V-curve summed up stability and power factor at a glance (D4). With that, all three rotating machines, DC, induction and synchronous, are behind us. The last group, E, covers today's machines shaped together by permanent magnets and power electronics, starting with an overview of BLDC, stepper and servo motors.