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Units and Significance of Synchronizing Power Coefficient
Units of Synchronizing Power Coefficient (????)
Generally, the synchronizing power coefficient is expressed in Watts per electrical radian, i.e.,
$$\mathrm{?_{syn} =\frac{? ?_{?}}{?_{?}}cos\:? \:\:Watts/electrical\:radian …(1)}$$
$$\mathrm{? \:?\:radians = 180\:degrees}$$
$$\mathrm{\Rightarrow\:1\:radian =\frac{180}{?}\:degrees}$$
$$\mathrm{? \:?_{syn}=\frac{??}{??}\:\:Watts/ \left(\frac{180}{?}\:degrees \right)}$$
$$\mathrm{\Rightarrow\:?_{syn}=\left( \frac{??}{??}\right)\left(\frac{?}{180}\right)\:\:Watt/electrical\:degree …(2)}$$
If p is the total number of pole pairs in the machine, then
$$\mathrm{?_{electrical} = ? \cdot ?_{mechanical}}$$
Therefore, the synchronizing power coefficient per mechanical radian is given by,
$$\mathrm{?_{syn} = ? \cdot\left( \frac{??}{??}\right)\:\:Watts/mech. radian …(3)}$$
And, the synchronizing power coefficient per mechanical degree is given by,
$$\mathrm{?_{syn} =\left( \frac{??}{??}\right)\left(\frac{?\:?}{180}\right)\:Watts/mech.degree …(4)}$$
Significance of Synchronizing Power Coefficient
The synchronizing power coefficient ($?_{syn}$) is the measure of the stiffness of the electromagnetic coupling between the stator and the rotor. A large value of the synchronizing coefficient ($?_{syn}$) shows that the electromagnetic coupling is rigid. The synchronising power coefficient is given by,
$$\mathrm{?_{syn} =\frac{3\:?\:?_{?}}{?_{?}}cos\:? … (5)}$$
Equation (5) indicates that $?_{syn}$ is inversely proportional to the synchronous reactance of the machine. The synchronous machines with large air gaps have relatively small reactances. Thus, the coupling in a synchronous machine with large air gap is more rigid than a machine with smaller air gap.
Also, the $?_{syn}$ is directly proportional to $?_{?}$, hence, an over-excited synchronous machine is more stiff than an under-excited synchronous machine.
From Eqn. (5), it can also be seen that the restoring action is maximum when the load angle $\delta$ = 0°,, i.e., at no-load while the restoring action is zero when the load angle $\delta$ = ±90°. At these values of load angle ($\delta$), the machine would be at the steady state limit of stability and in the condition of unstable equilibrium. Thus, it is impossible to run a synchronous machine at the steady-state limit of stability because its ability to resist small changes is zero.
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