Understanding DC Generator Speed Calculation

This question asks us to determine the operating speed of an 8-pole, lap-wound DC generator given its specifications: the number of armature conductors, the flux per pole, and the generated electromotive force (e.m.f.). To solve this, we need to use the fundamental formula for the e.m.f. generated in a DC generator.

Formula for Generated E.M.F. in a DC Generator

The e.m.f. generated in the armature of a DC generator is given by the formula:

E = \frac{P \phi Z N}{60 A}

Where:

• \(E\) is the generated e.m.f. in volts (V).
• \(P\) is the number of poles.
• \(\phi\) is the flux per pole in webers (Wb).
• \(Z\) is the total number of armature conductors.
• \(N\) is the speed of the armature in revolutions per minute (r.p.m.).
• \(A\) is the number of parallel paths in the armature winding.

Parallel Paths in Lap Winding

For a lap-wound DC machine, the number of parallel paths (\(A\)) is always equal to the number of poles (\(P\)).

A = P

Substituting this into the e.m.f. formula, we get:

E = \frac{P \phi Z N}{60 P}

The \(P\) terms cancel out, simplifying the formula for a lap-wound machine:

E = \frac{\phi Z N}{60}

Solving for the Speed (N)

We are given the generated e.m.f. (\(E\)), the flux per pole (\(\phi\)), and the total number of conductors (\(Z\)). We need to find the speed (\(N\)). Let's rearrange the simplified formula to solve for \(N\):

60 E = \phi Z N

N = \frac{60 E}{\phi Z}

Applying the Given Values

The given values are:

• Number of poles, \(P = 8\) (Used to determine \(A\) but not directly in the final lap-wound formula).
• Armature winding type: Lap-wound (\(A = P = 8\)).
• Number of armature conductors, \(Z = 1000\).
• Flux per pole, \(\phi = 20 \text{ mWb}\). Note that 1 mWb = \(10^{-3}\) Wb, so \(\phi = 20 \times 10^{-3} \text{ Wb}\).
• Generated e.m.f., \(E = 200 \text{ V}\).

Now, substitute these values into the formula for \(N\):

N = \frac{60 \times 200}{(20 \times 10^{-3}) \times 1000}

Let's simplify the denominator first:

(20 \times 10^{-3}) \times 1000 = 20 \times 10^{-3} \times 10^3 = 20 \times 10^{(-3+3)} = 20 \times 10^0 = 20 \times 1 = 20

So, the formula for \(N\) becomes:

N = \frac{60 \times 200}{20}

Now, perform the final calculation:

N = \frac{12000}{20}

N = 600

The speed of the machine is 600 r.p.m.

Conclusion

Based on the calculations using the e.m.f. equation for a lap-wound DC generator, the speed of the machine is 600 r.p.m.

Revision Table: DC Generator Parameters

Additional Information: DC Generator Windings

DC machines use armature windings that are typically either lap-wound or wave-wound. The choice of winding affects the number of parallel paths and thus the characteristics of the machine.

• Lap Winding:
  • The conductors are connected in such a way that they overlap, resembling laps.
  • The number of parallel paths (\(A\)) is equal to the number of poles (\(P\)).
  • Lap winding is suitable for machines designed for high current and low voltage.
  • It requires more brushes than wave winding, typically one brush per pole.
• Wave Winding:
  • The conductors are connected in series, forming a continuous wave.
  • The number of parallel paths (\(A\)) is always 2, regardless of the number of poles.
  • Wave winding is suitable for machines designed for high voltage and low current.
  • It requires only two brushes, irrespective of the number of poles.

Understanding the winding type is crucial because it determines the value of \(A\), which is a key parameter in the generated e.m.f. formula.