Question

A dc generator running at 30 rev/s generates an e.m.f. of 200 V. Determine the percentage increase in the flux per pole required to generate 250 V at 20 rev/s.

• A. 87.5%
• B. 85.5%
• C. 75.5%
• D. 70.5%

Hint:

In DC machines, emf varies directly with both speed and magnetic flux. Any reduction in speed must be compensated by an increase in flux.

Answer 1

The Correct Option is A

Step 1: Write the emf equation of a DC generator.
For a DC generator, the generated emf is directly proportional to flux per pole and speed.
\[ E \propto \Phi N \]
Step 2: Form the ratio of emfs under two operating conditions.
\[ \frac{E_2}{E_1} = \frac{\Phi_2 N_2}{\Phi_1 N_1} \]
Step 3: Substitute the given values.
\[ \frac{250}{200} = \frac{\Phi_2 \times 20}{\Phi_1 \times 30} \]
Step 4: Simplify the equation.
\[ 1.25 = \frac{2}{3} \times \frac{\Phi_2}{\Phi_1} \]
\[ \frac{\Phi_2}{\Phi_1} = 1.875 \]
Step 5: Calculate percentage increase in flux.
\[ \text{Percentage increase} = (1.875 - 1) \times 100 = 87.5% \]
Step 6: Conclusion.
The flux per pole must be increased by
\[ \boxed{87.5%} \]

Answer 2

Since generated emf is proportional to both flux per pole and speed for a DC generator, we can set the flux per pole in the first condition to a convenient reference value of \(1\) unit and solve for what the second condition's flux must be, then compare with each option.

With \( \Phi_1 = 1 \), \( N_1 = 30 \) rev/s, \( E_1 = 200 \) V, the constant of proportionality is \( k = \dfrac{E_1}{\Phi_1 N_1} = \dfrac{200}{1 \times 30} = 6.667 \).

In the second condition, \( E_2 = 250 \) V, \( N_2 = 20 \) rev/s, so:

\[ \Phi_2 = \frac{E_2}{k \, N_2} = \frac{250}{6.667 \times 20} = \frac{250}{133.33} \approx 1.875 \]

Since \( \Phi_1 = 1 \), this means the flux must become \(1.875\) times its original value, an increase of \(0.875\), i.e. \(87.5\%\).

1. 87.5%: Matches the increase of \(0.875\) units on a starting flux of \(1\) unit computed above.
2. 85.5%: Would require a final flux of \(1.855\), which would only generate \( 6.667 \times 1.855 \times 20 \approx 247.3 \) V, short of the required \(250\) V.
3. 75.5%: Would require a final flux of \(1.755\), generating only about \(6.667 \times 1.755 \times 20 \approx 234.0\) V, well below the target \(250\) V.
4. 70.5%: Would require a final flux of \(1.705\), generating about \(6.667 \times 1.705 \times 20 \approx 227.3\) V, the furthest from the required \(250\) V among the options.

Only a \(87.5\%\) increase in flux produces exactly the \(250\) V required at \(20\) rev/s.

Therefore, the correct answer is 87.5%.