Induced EMF Per Phase in Synchronous Machines: A Detailed Mathematical Approach

 Induced EMF Per Phase in Synchronous Machines: A Detailed Mathematical Approach

Introduction

Electromotive force (EMF) is a fundamental concept in electrical machines, particularly in synchronous machines. The induced EMF per phase plays a crucial role in determining machine performance. This article provides a detailed mathematical derivation of the induced EMF equation for a synchronous machine.

Fundamental Principles

In a synchronous machine, the rotor carries the field winding, which produces a rotating magnetic field when excited by a DC source. The stator contains the armature windings where the EMF is induced due to the relative motion between the stator and rotor magnetic fields. The magnitude of the induced EMF depends on the machine’s design and operating parameters.

Derivation of Induced EMF per Phase

The induced EMF in a synchronous machine follows Faraday’s Law of Electromagnetic Induction, which states that:

$$E = -N \frac{d\Phi}{dt}$$

where:

• $E$ = Induced EMF
• $N$ = Number of turns per phase
• $\Phi$ = Magnetic flux per pole
• $t$ = Time

Step 1: Expression for Flux

The fundamental magnetic flux per pole varies sinusoidally and is given by:

$$\Phi = \Phi_m \cos(\omega t)$$

where:

• $\Phi_m$ = Maximum flux per pole
• $\omega$ = Angular velocity of the rotating field
• $t$ = Time

Step 2: Rate of Change of Flux

Differentiating the flux equation with respect to time:

$$\frac{d\Phi}{dt} = - \Phi_m \omega \sin(\omega t)$$

The negative sign indicates that the induced EMF opposes the change in flux, as per Lenz’s Law.

Step 3: Induced EMF Per Turn

By substituting this derivative into Faraday’s Law:

$$E_{turn} = N \omega \Phi_m \sin(\omega t)$$

Step 4: Induced EMF Per Phase

For a three-phase machine with a winding having $T_p$ turns per phase and distributed over the stator,

$$E_{ph} = 4.44 f \Phi_m T_p K_w$$

where:

• $f$ = Frequency of the induced EMF (in Hz)
• $K_w$ = Winding factor, which accounts for distribution and pitch effects
• $4.44$ arises from the RMS conversion of a sinusoidal waveform.

Conclusion

The induced EMF per phase in a synchronous machine depends on the maximum flux per pole, frequency, and winding parameters. This derivation provides insights into the fundamental working of synchronous machines and helps engineers optimize their design for efficiency and performance.

Stay tuned for further discussions on machine performance, harmonics, and real-world applications of synchronous machines!