V-curve and inverted V-curve

For a synchronous motor connected to a supply of constant voltage and frequency and driving a constant load, the V-curve is the plot of armature current I_a against field current I_f. It is called a V-curve because of its V-like shape. The inverted V-curve is the plot of power factor against field current under the same conditions. It has an inverted V (arch) shape.

Curves

 V-curves (Ia against If)

 Ia
  |  \     \     \       / / /
  |   \     \     \     / / /
  |    \     \     \   / / /
  |     \     \     \ / / /       full load
  |      \     \     o / /
  |       \     \     / /
  |        \     \   / /          half load
  |         \     \ / /
  |          \     o /
  |           \     /
  |            \   /              no load
  |             \ /
  |              o
  +--------------------------------------> If
    under-excited      over-excited
    (lagging pf)       (leading pf)
  o = minimum Ia (unity pf); the line through the o's is the
      unity-pf compounding curve, sloping to the right.

 Inverted V-curve (pf against If, one load)

 pf
  |       unity
  |         *
  |       *   *
  |     *       *
  |   *           *
  +---------------------> If
    lagging  |  leading

• The lowest point of every V-curve is where I_a is minimum and the power factor is unity. The line joining these minima (the compounding curve) slopes to the right as the load rises, since more excitation is needed to keep unity pf at a higher load.
• Left of the minimum: the motor is under-excited, I_a lags V (lagging pf), and the current rises as I_f falls.
• Right of the minimum: the motor is over-excited, I_a leads V (leading pf), and the current rises as I_f rises.
• A higher load gives a curve higher up (larger I_a for every I_f), because the in-phase component I_a cos φ is larger. The no-load curve has its minimum close to the I_f axis: the minimum current is only the small current needed to supply the no-load losses.

Reason

At constant load, the power input 3 V I_a cos φ is constant, so I_a cos φ is constant. Changing I_f only changes the reactive component I_a sin φ. At minimum I_a, sin φ = 0 and I_a = I_a cos φ, so the pf is unity. On either side, the extra reactive component makes I_a larger, which gives the V shape. The pf is lagging on the left, rises to unity at the V minimum, and then falls again (now leading) on the right, giving an inverted V. For a higher load the pf curve is flatter and its peak moves to the right.

Use

The curves show how the motor can be used for power-factor correction (on the over-excited side) and let the operator find the excitation needed for a required pf.