Excercises Electric Machinery Fundamentals

Chapter 2 Problem 2-5



In [1]:

    
%pylab inline
%precision 4
from scipy import constants as c   # we like to use some constants









    



Populating the interactive namespace from numpy and matplotlib

Description

When travelers from the USA and Canada visit Europe, they encounter a different power distribution system. Wall voltages in North America are 120 V rms at 60 Hz, while typical wall voltages in Europe are 230 V at 50 Hz, which means:



In [2]:

    
#60Hz side (North America)
Vrms60 = 120 # [V]
freq60 = 60  # [Hz]

#50Hz side (Europe)
Vrms50 = 240 # [V]
freq50 = 50  # [Hz]

Many travelers carry small step-up / step-down transformers so that they can use their appliances in the countries that they are visiting. A typical transformer might be rated at 1-kVA and 115/230 V. It has 500 turns of wire on the 115-V side and 1000 turns of wire on the 230-V side, through which it's known that:



In [3]:

    
S = 1000     # Apparent power (VA)
NP60 = 500   # Primary turns at 115V side
NP50 = 1000  # Primary turns at 230V side

The magnetization curve for this transformer is shown in Figure P2-2, and can be found in p22_mag.dat at this book's Web site.



In [4]:

    
#Load the magnetization curve data
import pandas as pd
# The data file is stored in the repository
fileUrl = 'data/p22_mag.dat'
data = pd.read_csv(fileUrl,     # the address where to download the datafile from
                   sep=' ',     # our data source uses a blank space as separation
                   comment='%', # ignore lines starting with a "%" 
                   skipinitialspace = True,  # ignore intital spaces
                   header=None, # we don't have a header line defined...
                   names=['mmf_data', 'flux_data'] # ...instead we define the names here
                   )

(a)

Suppose that this transformer is connected to a 120-V, 60 Hz power source with no load connected to the 240-V side.

Sketch the magnetization current that would flow in the transformer.
What is the rms amplitude of the magnetization current?
What percentage of full-load current is the magnetization current?

(b)

Now suppose that this transformer is connected to a 240-V, 50 Hz power source with no load connected to the 120-V side.

Sketch the magnetization current that would flow in the transformer.
What is the rms amplitude of the magnetization current?
What percentage of full-load current is the magnetization current?

(c)

In which case is the magnetization current a higher percentage of full-load current? Why?

SOLUTION

(a)

When this transformer is connected to a 120-V 60 Hz source, the flux in the core will be given by the equation $$\phi(t) = - \frac{V_M}{\omega N_P}\cos(\omega t)$$

Calculate the angular velocity $\omega$:



In [5]:

    
w60 = 2 * pi * freq60
print('w = {:.4f} rad/s'.format(w60))









    



w = 376.9911 rad/s

Calculate the maximum voltage $V_M$:



In [6]:

    
VM60 = Vrms60 * sqrt(2)
print('VM = {:.4f} V'.format(VM60) )









    



VM = 169.7056 V

Calculate flux versus time $\phi(t)$ (saved as a vector):



In [7]:

    
time = linspace(0, 1./30, 100)              # 0 to 1/30 sec
flux60 = -VM60 / (w60 * NP60) * cos(w60 * time)

The magnetization current required for a given flux $\phi(t)$ can be found from Figure P2-2 or from the equivalent table in file p22_mag.dat by using the interpolation function:



In [8]:

    
mmf60 = interp(flux60, data['flux_data'], data['mmf_data'])

Calculate the magnetization current $i_m$:



In [9]:

    
im60 = mmf60 / NP60

Calculate the rms value of the current $i_\text{rms}$:



In [10]:

    
irms60 = sqrt(sum(im60**2) / im60.size)
print('The rms current at 120 V and 60 Hz is {:.4f} A'.format(irms60))









    



The rms current at 120 V and 60 Hz is 0.3186 A

Calculate the full-load current:



In [11]:

    
i_fl60 = S / Vrms60

Calculate the percentage of full-load current:



In [12]:

    
percnt60 = irms60 / i_fl60 * 100
print('The magnetization current is {:.3f}% of full-load current.'.format(percnt60))









    



The magnetization current is 3.823% of full-load current.

Sketch the magnetization current $i_m$ that would flow in the transformer:



In [13]:

    
rc('text', usetex=True)   # enable LaTeX commands for plot
title(r'\bf Magnetization current at 60 Hz')
xlabel(r'\bf Time (s)')
ylabel(r'$\mathbf{I_m}$ \textbf{(A)}')
axis([0,0.04,-0.5,0.5])   #set the axis range
plot(time,im60)
legend(('$60 Hz,\,I_{{RMS}} = {:.3f}\,A$'.format(irms60),), loc=4);
grid()

(b)

When this transformer is connected to a 240-V 50 Hz source, the flux in the core will be given by the equation $$\phi(t) = - \frac{V_M}{\omega N_S}\cos(\omega t)$$