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- Synchronous motors: An introduction
- Chapter I: Synchronous motors General Principles
- Efficiency Synchronous Motors and Experimental Properties
- Stalling of a Synchronous Motor
- Over-excited synchronous motor
- Necessity of synchronism and stability of synchronous operation
- Explanation of Single-Phase Synchronous Motors
- Equations of Synchronous Motors; Analytical Theory
- Symmetrical Polyphase Motors
- Equation of the Synchronous Motor by the Method of Complex Variables
- Excitation of Synchronous Motors
- Shunt-excitation
- Chapter II: Operation of synchronous motor
- Installation of Synchronous Motors
- Current controller
- Starting of Single-Phase Machine
- Starting of Machine with Laminated Field Poles
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Equations of Synchronous Motors; Analytical Theory
The E.M.F. which is acting in the circuit is equal to the algebraical sum of the opposing E.M.F.'s.
From this the current, i, may be deduced, by the well-known differential equation,
In this equation let i=X sin ωt+Y cos(ωt).
If this value be substituted in the equation, the values of X and Y can be determined by making the coefficients of the sine-terms and of the cosine-terms successively equal to zero. We can then obtain, by differentiation, substitution, etc., the following value for i:
This may also be written,
In this equation β=the phase-angle of the resultant E.M.F. and γ=the supplemental phase-difference of the current measured from this E.M.F. In the simple particular case where E1 = E2 this expression reduces to
and will be out of phase by the angle γ with respect to the resultant E.M.F., which is itself out of phase with respect to the mean of e1 and e2. This result is easily interpreted in Fig. 11, by drawing the resultant curve e1 + e2 obtained by taking the difference of the ordinates of the first two curves. It will be seen that the curve has actually a phase-difference equal to with regard to π/2 the mean of e1 and e2, and that it increases with the phase-difference of e2 with respect to e1. In consequence of the lag, γ, of the current, measured with respect to this re.sultant E.M.F. (when γ is near π/2 in value), it will be seen that the current is approximately in phase with this mean value; it would be completely so if there were no resistance-losses.
The power-outputs of the two machines will be obtained by multiplying the instantaneous current i by the E.M.F.'s. e1 and e2. For example, in the case where E1 =E2 , we have
These equations show that the power is not constant, in either case, but pulsating, i.e., it presents variations of frequency= 2 T, as represented in Fig. 11.
These variations constitute sine-functions having pulsations twice as rapid as those of the current, which have for their axes the horizontal lines (P1, P2) corresponding to the mean powers given by the first terms within the brackets in the following equations:
The very small difference between P1 and P2 represents the loss by resistance (Joule effect). The axis of the curve P1 is therefore a little more above the axis of zero power than the axis of symmetry of the curve P2 is below it.
The torque could be obtained, in each case, by dividing the power by the angular velocity. These expressions show that the current increases with Θ until Θ equals π, but the torques, which equal zero so long as the lag Θ=zero, increase with Θ only until the value Θ = γ; and they then decrease.
Stability will, therefore, exist only with Θ < γ having for its axis the exact opposition of E.M.F.'s.
The solution, in the case where E1 is different from E2, will be obtained in an analogous manner, by forming the products e1i - e2i; and it would still give pulsating values for p1 and p2 ; but, since these caculations are uselessly complicated, we will pass them by and turn to more simple methods.