Flux through a rotating loop
The flux is defined as the field times the perpendicular area over which the field interacts. This interaction is at maximum when the area and the field are orientated at 90 degrees. At other angles the component of the area that is perpendicular, is what contributes the size of the flux.
Indicate the direction the current must be flowing in this position.
As the loop rotates in the field the we can examine the angle of rotation by considering the loop from one end.
The perpendicular area exposed to the field can be expressed as a function of sin θ
thus if the loop has dimension l × w then
Φ = B × l w sin θ
= BA sin θ
In the table below state the size of θ for the following scenarios
In our model for a generator you will start with a model for flux through a square loop rotating in a uniform field. You will set up the sheet to enable you to change the dimensions of the loop and the strength of the field.
The sheet will also allow you to change the angle increment
By setting the first value as 0 and then adding the next value as the sum on the increment.
To generate our flux curve, we will use the SIN function in Excel. Remember this must be expressed in radians so you will need to convert your angle value. On the sheet below this done with:
=$F$8*SIN(B13/180*PI())
You should now label the axis correctly and change the x – axis scale to reflect angles to 360.
Investigate what happens when you change the angle increment to 10 or 20. What do you notice about the dot separate? This separate is very important to our next step.
Faraday stated that the EMF produced is related to the change in flux.
Considering how the dots appear on your flux diagram, where is the greatest change in flux?
How does this relate to the rotation of the loop?
Make a copy of your first sheet and rename it EMF.
We now need to review Faraday’s and Lenz’ relationship
To set up our model correctly we need to find the change in flux for each increment in angle change. We also need to calculate the corresponding time change.
Consider:
If we use increments of 10 degrees. How many intervals will we examine of a full rotation?
The change in time for each interval can be determined simply enough. If it takes time T, for one full rotation, which represents 360 degrees then the timing for each increment will be (increment/360)*T.
Calculate the timing for a loop with a period of 0.15s and using increments of 10 degrees.
We must also consider that often the loop rotation is given as a frequency rather than a period. Remember
f= 1/T
What frequency does the above period represent?
We will design our model to take inputs of Number of loops N, and frequency, f (Hz).
To calculate the change in flux we will simply subtract flux values between intervals.
Plot your EMF vs degrees graph.
Comment on the shape of the EMF graph
How does it compare to the Flux graph?
How does the gradient of the flux graph relate to the EMF graph?
How does your answer to Q3 relate the Faraday’s and Lenz’ law?
Most models would use the timing on the x-axis rather than degrees, can you change this on your model?
Generate some data
Use your model to generate the following data to answer the following questions..
How does increasing the length of the loop change the EMF production?
How does changing both the length and the width of the loop?
What set of values would give you 240V RMS at 50 Hz?