Question

If \( \delta \) is load angle, the real power of 3-\(\phi\) synchronous motor is

• A. \( \text{directly proportional to } \delta \)
• B. \( \text{directly proportional to } \sin\delta \)
• C. \( \text{inversely proportional to } \sin\delta \)
• D. \( \text{inversely proportional to } \delta \)

Hint:

The power-angle curve of a cylindrical synchronous machine is purely sinusoidal ($P = P_{\max}\sin\delta$). Maximum power occurs at $\delta = 90^\circ$. For small values of $\delta$, $\sin\delta \approx \delta$, but the exact universal relationship is always direct proportionality to $\sin\delta$.

Answer 1

The Correct Option is B

Concept: The steady-state real power ($P$) developed per phase by a cylindrical-rotor synchronous machine (neglecting armature resistance $R_a$) is derived from its equivalent electrical circuit model and phasor diagram. The mathematical expression for the total three-phase active real power is given by: $$P = \frac{3 \cdot V \cdot E}{X_s} \cdot \sin\delta$$ Where:
• $V$ = Terminal phase voltage
• $E$ = Induced excitation EMF per phase
• $X_s$ = Synchronous reactance per phase
• $\delta$ = Load angle (or power angle/torque angle), which represents the physical magnetic displacement between the rotor and stator magnetic fields.

Step 1: Examine the variables in the power equation.
In standard operating conditions, the terminal voltage $V$ provided by the grid is constant, the synchronous reactance $X_s$ is a constant parameter of the winding geometry, and the excitation voltage $E$ is held constant for a fixed field current. Therefore, the term $\frac{3VE}{X_s}$ can be replaced by a constant value $P_{\max}$: $$P = P_{\max} \cdot \sin\delta$$

Step 2: Determine the direct proportionality relation.
From the simplified equation, it is clear that the real power $P$ varies dynamically with the sine of the load angle: $$P \propto \sin\delta$$ This shows that real power is directly proportional to $\sin\delta$, which corresponds to option (2).