What is Synchronizing Torque Coefficient?

The synchronising torque coefficient is defined as the torque at which the synchronous speed gives the synchronising power. If represents the synchronising torque coefficient, then

$$\mathrm{𝜏_{𝑠𝑦𝑛} =\frac{1}{πœ”_{𝑠}}π‘š\frac{𝑑𝑃}{𝑑𝛿}\:Nm/electrical\:radian …(1)}$$


$$\mathrm{𝜏_{𝑠𝑦𝑛} =\left (\frac{1}{πœ”_{𝑠}}π‘š\frac{𝑑𝑃}{𝑑𝛿} \right)\cdot \frac{πœ‹π‘}{180°}\:Nm/mechanical \:degree …(2)}$$


  • m is the number of phase of the machine,

  • πœ”π‘  = 2πœ‹π‘›π‘  is the angular synchronous speed,

  • 𝑛𝑠 is the synchronous speed in r.p.s.

  • 𝑝 is the total number of pair of poles of the machine.

The synchronising torque coefficient may also be given by,

$$\mathrm{𝜏_{𝑠𝑦𝑛} =\frac{π‘‘πœ}{𝑑𝛿}=\frac{1}{2πœ‹π‘›_{𝑠}}\frac{𝑑𝑃}{𝑑𝛿}… (3)}$$

$$\mathrm{∡\:\frac{𝑑𝑃}{𝑑𝛿}=𝑃_{𝑠𝑦𝑛} =\frac{𝑉𝐸_{𝑓}}{𝑍_{𝑠}}sin(πœƒ_{𝑧} − 𝛿)}$$

$$\mathrm{∴\:𝜏_{𝑠𝑦𝑛} =\frac{𝑉𝐸_{𝑓}}{2πœ‹π‘›_{𝑠 }\cdot 𝑍_{𝑠}}sin(πœƒ_{𝑧} − 𝛿) … (4)}$$

In many synchronous machines, Xs >>R. Therefore, for a cylindrical rotor machine, neglecting the saturation and stator resistances, the eqn. (4) becomes,

$$\mathrm{𝜏_{𝑠𝑦𝑛} =\frac{𝑉𝐸_{𝑓}}{2πœ‹π‘›_{𝑠} \cdot 𝑋_{𝑠}}cos \:𝛿 … (5)}$$


$$\mathrm{𝑃_{𝑠𝑦𝑛} =\frac{𝑉𝐸_{𝑓}}{𝑍_{𝑠}}sin(πœƒ_{𝑧 }− 𝛿)\:\:\:and\:\:\:πœ”_{𝑠} = 2πœ‹π‘›_{𝑠}}$$

Hence, Eqn. (4) may also be written as,

$$\mathrm{𝜏_{𝑠𝑦𝑛} =\frac{𝑃_{𝑠𝑦𝑛}}{2πœ‹π‘›_{𝑠}}=\frac{𝑃_{𝑠𝑦𝑛}}{πœ”_{𝑠}}… (6)}$$

Numerical Example

A 4-pole, 50 Hz, 3-phase, turbo alternator is excited to generate the busbar voltage of 11 kV on no-load. The machine is star-connected and the synchronising power of the machine is 332.48 kW per mechanical degree. Calculate the synchronising torque of the machine.


The synchronous speed of the machine in r.p.s. is,

$$\mathrm{𝑛_{𝑠} =\frac{2𝑓}{𝑃}=\frac{2 × 50}{4}= 25 r. p. s.}$$

Synchronising torque,

$$\mathrm{𝜏_{𝑠𝑦𝑛} =\frac{𝑃_{𝑠𝑦𝑛}}{2πœ‹π‘›_{𝑠}}=\frac{332.48 × 1000}{2πœ‹ × 25}= 2117.71\:Nm}$$

Updated on: 13-Oct-2021


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