Question

Derive the emf equation of a DC Machine.

Answer 1

The EMF equation of a DC machine (either a DC motor or DC generator) is derived from Faraday’s Law of Electromagnetic Induction. The equation gives the induced electromotive force (emf) in the armature of a DC machine. Here's a step-by-step derivation of the EMF equation for a DC generator:

Step 1: Key Parameters

1. Z = Total number of armature conductors
   

1. N = Speed of the armature in revolutions per minute (RPM)
   

1. A = Number of parallel paths in the armature (depends on the winding)
   

1. Φ = Flux per pole (in Weber)
   

1. P = Number of poles
   

1. E = Induced EMF per phase (in Volts)
   

Step 2: Total Flux Linkage

The total flux (Φ) passing through the armature conductors can be considered as the flux produced by the magnetic field of the poles. Since each pole produces the same flux and there are P poles, the total flux passing through the armature is Φ.

Step 3: Number of Conductors Cutting the Magnetic Flux

As the armature rotates, each conductor moves through the magnetic flux. The number of conductors cutting through the magnetic flux depends on the number of turns and the speed of rotation.

1. The armature conducts current in Z conductors.
   

1. Each conductor cuts the flux generated by the pole as the armature rotates. The number of conductors cutting the magnetic flux depends on the speed of the armature.
   

Step 4: Faraday’s Law of Induction

According to Faraday's law, the induced EMF in a conductor moving through a magnetic field is given by:

\[
E_{\text{induced}} = \frac{d\Phi}{dt}
\]

where:

1. dΦ is the change in magnetic flux,
   

1. dt is the time taken for this change.
   

Step 5: EMF in One Conductor

For one conductor moving with the armature, the induced EMF can be calculated by the equation:

\[
E_{\text{conductor}} = P \cdot Φ \cdot N \cdot \frac{1}{60}
\]

This accounts for the number of poles (P), flux per pole (Φ), and speed in revolutions per minute (N), and converts time from seconds to minutes by dividing by 60.

Step 6: Total Induced EMF

Since there are Z conductors and A parallel paths, the total EMF generated can be calculated by multiplying the EMF generated per conductor by the total number of conductors per parallel path.

\[
E = \frac{P \cdot Φ \cdot N \cdot Z}{60 \cdot A}
\]

This is the EMF equation for a DC machine, where:

1. P = Number of poles,
   

1. Φ = Flux per pole (in Weber),
   

1. N = Speed of the armature in RPM,
   

1. Z = Total number of armature conductors,
   

1. A = Number of parallel paths.
   

Conclusion:

The induced EMF in a DC machine depends on the number of poles, the flux per pole, the speed of the armature, the total number of conductors, and the number of parallel paths in the armature winding.