Excercises Electric Machinery Fundamentals

Chapter 1 Problem 1-16



In [1]:

    
%pylab notebook









    



Populating the interactive namespace from numpy and matplotlib

Description

The core shown in Figure P1-2:

has the flux $\phi$ shown in Figure P1-12:



In [2]:

    
N = 500
dphi = array([0.010, -0.020, 0.010, 0.010]) # [Wb]
dt   = array([2e-3 ,   3e-3,  2e-3,  1e-3]) # [s]

Sketch the voltage present at the terminals of the coil.

SOLUTION

By Lenz’ Law, an increasing flux in the direction shown on the core will produce a voltage that tends to oppose the increase. This voltage will be the same polarity as the direction shown on the core, so it will be positive.

The induced voltage in the core is given by the equation:

$$e_\text{ind} = N \frac{d\phi}{dt}$$

so the voltage in the windings will be:



In [3]:

    
e_ind = N * dphi/dt
e_ind









    Out[3]:





array([ 2500.        , -3333.33333333,  2500.        ,  5000.        ])

Lets pretty-print the result:



In [4]:

    
t = 0  # time [s], whose initial value is set here
# print the table head
print('''
|--------------+----------|
|     Time     |  e_ind   |
|--------------+----------|''')
# We use a simple for-loop to print a row per result:
for i in range(4):
    print('| {:.0f} < t < {:.0f} ms | {:>5.2f} kV |'.format(
              *( t*1000,       # start of time inteval [ms]
                (t+dt[i])*1000, # end time of the inteval [ms]
                e_ind[i]/1000 # value of e_ind [V]
                )))
    t = t + dt[i]                
print('|=========================|')









    



|--------------+----------|
|     Time     |  e_ind   |
|--------------+----------|
| 0 < t < 2 ms |  2.50 kV |
| 2 < t < 5 ms | -3.33 kV |
| 5 < t < 7 ms |  2.50 kV |
| 7 < t < 8 ms |  5.00 kV |
|=========================|

The resulting voltage is plotted below:



In [5]:

    
T = [0, dt[0], dt[0], dt[0]+dt[1], dt[0]+dt[1],
     dt[0]+dt[1]+dt[2], dt[0]+dt[1]+dt[2], dt[0]+dt[1]+dt[2]+dt[3]]
e = [e_ind[0], e_ind[0], e_ind[1], e_ind[1], 
     e_ind[2], e_ind[2], e_ind[3], e_ind[3]]

plot(array(T)*1000, array(e)/1000)
title('Plot of Voltage vs Time')
xlabel('Time (ms)')
ylabel('Induced Voltage (kV)')
axis([0,8,-4,6]) # set the axis range
grid()