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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
$$\mathrm{π_{π π¦π} =\frac{1}{π_{π }}π\frac{ππ}{ππΏ}\:Nm/electrical\:radian …(1)}$$
Also,
$$\mathrm{π_{π π¦π} =\left (\frac{1}{π_{π }}π\frac{ππ}{ππΏ} \right)\cdot \frac{ππ}{180°}\:Nm/mechanical \:degree …(2)}$$
Where,
m is the number of phase of the machine,
ππ = 2πππ is the angular synchronous speed,
ππ is the synchronous speed in r.p.s.
π is the total number of pair of poles of the machine.
The synchronising torque coefficient may also be given by,
$$\mathrm{π_{π π¦π} =\frac{ππ}{ππΏ}=\frac{1}{2ππ_{π }}\frac{ππ}{ππΏ}… (3)}$$
$$\mathrm{β΅\:\frac{ππ}{ππΏ}=π_{π π¦π} =\frac{ππΈ_{π}}{π_{π }}sin(π_{π§} − πΏ)}$$
$$\mathrm{∴\:π_{π π¦π} =\frac{ππΈ_{π}}{2ππ_{π }\cdot π_{π }}sin(π_{π§} − πΏ) … (4)}$$
In many synchronous machines, Xs >>R. Therefore, for a cylindrical rotor machine, neglecting the saturation and stator resistances, the eqn. (4) becomes,
$$\mathrm{π_{π π¦π} =\frac{ππΈ_{π}}{2ππ_{π } \cdot π_{π }}cos \:πΏ … (5)}$$
Since,
$$\mathrm{π_{π π¦π} =\frac{ππΈ_{π}}{π_{π }}sin(π_{π§ }− πΏ)\:\:\:and\:\:\:π_{π } = 2ππ_{π }}$$
Hence, Eqn. (4) may also be written as,
$$\mathrm{π_{π π¦π} =\frac{π_{π π¦π}}{2ππ_{π }}=\frac{π_{π π¦π}}{π_{π }}… (6)}$$
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.
Solution
The synchronous speed of the machine in r.p.s. is,
$$\mathrm{π_{π } =\frac{2π}{π}=\frac{2 × 50}{4}= 25 r. p. s.}$$
Synchronising torque,
$$\mathrm{π_{π π¦π} =\frac{π_{π π¦π}}{2ππ_{π }}=\frac{332.48 × 1000}{2π × 25}= 2117.71\:Nm}$$
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