Calculating DC Generator Field Coil Induced Voltage Magnitude

The magnitude of the average voltage induced across coils is calculated using Faraday's Law of Electromagnetic Induction. The specific formula used is:

$ \mathcal{E}_{avg} = -N \frac{\Delta\Phi}{\Delta t} $

Where:

• $ \mathcal{E}_{avg} $ represents the average induced electromotive force (voltage).
• $ N $ is the total number of turns in the coil(s) where the voltage is induced.
• $ \Delta\Phi $ is the total change in magnetic flux linkage.
• $ \Delta t $ is the time taken for this change in flux to occur.

From the problem statement, we have the following information:

• Number of poles in the DC generator, $ P = 6 $.
• Number of turns in each field coil = 500.
• The field coils are connected in series. This means the total number of turns involved is the sum of turns in all coils: $ N = P \times (\text{Turns per coil}) = 6 \times 500 = 3000 $ turns.
• The magnetic flux per pole when the field is excited is $ \Phi_{initial} = 0.03 $ Wb/pole.
• The residual magnetism per pole after the field circuit is opened is $ \Phi_{residual} = 0.003 $ Wb/pole.
• The time taken for the flux to change is $ \Delta t = 0.02 $ seconds.

First, we calculate the change in magnetic flux per pole:

$ \Delta\Phi_{pole} = \Phi_{initial} - \Phi_{residual} = 0.03 \text{ Wb/pole} - 0.003 \text{ Wb/pole} = 0.027 \text{ Wb/pole} $

To find the total voltage induced across all series-connected field coils, we consider the total change in flux linkage. This is calculated by summing the flux change from all poles:

$ \Delta\Phi_{total} = P \times \Delta\Phi_{pole} = 6 \times 0.027 \text{ Wb} = 0.162 \text{ Wb} $

Now, we can calculate the average induced voltage using the total number of turns and the total change in flux:

$ \mathcal{E}_{avg} = N \times \frac{\Delta\Phi_{total}}{\Delta t} $

$ \mathcal{E}_{avg} = (3000 \text{ turns}) \times \frac{0.162 \text{ Wb}}{0.02 \text{ s}} $

$ \mathcal{E}_{avg} = 3000 \times 8.1 \text{ V} $

$ \mathcal{E}_{avg} = 24300 \text{ V} $

So, the magnitude of the average induced voltage is 24300 V.

Determining DC Generator Induced Voltage Direction

The direction of the induced voltage is determined by Lenz's Law. Lenz's Law states that an induced voltage generates a current whose magnetic field opposes the change in magnetic flux that created it.

Here's how it applies:

• The field coils were initially excited, carrying a current that established a magnetic flux.
• When the field circuit is opened, the current decreases rapidly, causing the magnetic flux to collapse.
• This collapse represents a reduction (decrease) in the magnetic flux.
• To oppose this decrease, the induced voltage must create a magnetic field in the same direction as the original magnetic field.
• Therefore, the induced voltage drives a current in the same direction as the initial exciting current.

This means the direction of the induced voltage is the same as the initial direction of the exciting current.

Summary of Induced Voltage

Combining the magnitude and direction:

• Magnitude of average induced voltage = 24300 V.
• Direction of induced voltage = Same as the initial direction of the exciting current.

This matches the characteristics provided in the correct answer.