Excercises Electric Machinery Fundamentals

Chapter 1 Problem 1-12



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

The core shown in Figure P1-4 below:

and is made of a steel whose magnetization curve is shown in Figure P1-9 below:

Repeat Problem 1-7, but this time do not assume a constant value of $\mu_r$.

How much flux is produced in the core by the currents specified?
What is the relative permeability of this core under these conditions?
Was the assumption in Problem 1-7 that the relative permeability was equal to 1200 a good assumption for these conditions?
Is it a good assumption in general?



In [2]:

    
N1 = 600
N2 = 200
i1 = 0.5           # A
i2 = 1.0           # A

SOLUTION

The two coils on this core are wound so that their magnetomotive forces are additive, so the total magnetomotive force on this core is $$\mathcal{F}_\text{TOT} = N_1 i_1 + N_2 I_2$$



In [3]:

    
F_tot = N1 * i1 + N2 * i2 
print('F_tot = {:.1f} At'.format(F_tot))









    



F_tot = 500.0 At

Therefore, the magnetizing intensity $H$ is: $$ H = \frac{\mathcal{F_\text{TOT}}}{l_c}$$



In [4]:

    
lc = 4 * (0.075 + 0.5 + 0.075)   # [m] core length on all 4 sides.
H = F_tot / lc
print('H = {:.1f} At/m'.format(H))









    



H = 192.3 At/m

From the magnetization curve:



In [5]:

    
B = 0.17 # [T]

and the total flux in the core is: $$\phi_\text{TOT} = BA$$



In [6]:

    
A = 0.15**2  # [m²] 
phi_tot = B*A
print('ϕ = {:.3f} mWb'.format(phi_tot*1000))









    



ϕ = 3.825 mWb

The relative permeability of the core can be found from the reluctance as follows:

$$\mathcal{R}_\text{TOT} = \frac{\mathcal{F}_\text{TOT}}{\mathcal{\phi}_\text{TOT}} = \frac{l_c}{\mu_0 \mu_r A}$$

Solving for $\mu_r$ yields:

$$\mu_r = \frac{\mathcal{\phi}_\text{TOT}l_c}{\mathcal{F}_\text{TOT}\mu_0 A}$$



In [7]:

    
mu_r = phi_tot * lc / (F_tot * c.mu_0 * A)
print('μ = {:.1f}'.format(mu_r))

The assumption that $\mu_r = 1200$ is not very good here. Actually, it is not very good in general.