# Chapter 6:Instrument Transformers¶

## Example:6.1,Page No:367¶

In [19]:
import math

#variable declaration
Np     =  1;                      #number of primary turns
Ns     =  240;                    #number of secondary turns
Is     =  5;                      #secondary current in A
Re     =  1.2;                    #external burden in Ω
mmf    =  96;                     #magnetomotive force in AT
Ac     =  1200*10**-6;            #cross section area mm**2
f      = 50;                      #frequency in Hz

#calculation
Kt   =  Ns/float(Np);                           #turns ratio
Es   =  Is*Re;                                  #voltage induced in secondary winding in V
Im   =  mmf/float(Np);                          #secondary current in A
Ip   = math.sqrt(((Kt*Is)**2)+((Im)**2));       #primary current in A
Kact = Ip/float(Is);                            #actual transformation ratio
x    = Im/float(Kt*Is);                         #tangential component
theta = math.atan(x);                           #phase angle
phimax = Es/float(4.44*f*Ns);
Bmax   = phimax/float(Ac);

#result
print'actual tranformation ratio = %3.2f'%Kact;
print'phase angle = %3.2f'%((theta*180)/float(math.pi)),'° ';
print'maximum flux density in core = %3.4f'%Bmax,'Wb/m**2';

actual tranformation ratio = 240.77
phase angle = 4.57 °
maximum flux density in core = 0.0938 Wb/m**2


## Example:6.2,Page No:368¶

In [4]:
import math

#variable declaration
I0      = 1;                  #exciting current in  A
Knom    = 200;               #current transformer ratio
Re      = 1.1;               #non inductive resistance in Ω
p       = 0.45;              #power factor
delta   = 0;
Is      = 5;               #rated secondary winding current in A

#calculations
alpha  = 90-(((math.acos(p))*180)/float(math.pi));
Kt     = Knom                                                          #since no turn compenasation
y      = math.sin(((delta+alpha)*math.pi)/float(180));
Kact   = Kt+((I0/float(Is))*(y));                                      #actual transformation ratio
r      = ((Knom-Kact)/float(Kact))*100;                                #ratio error
k      =math.cos(((delta+alpha)*math.pi)/float(180));
theta  = (180/math.pi)*((I0*k)/float(Kt*Is));                          #phase angle degreess

#calculation
print'ratio error at full load = %3.4f'%r,'%';
print'phase angle  = %f'%(theta*100),'degrees(equal to (3 minutes 4 seconds))';


ratio error at full load = -0.0450 %
phase angle  = 5.116677 degrees(equal to (3 minutes 4 seconds))


## Example:6.3,Page No:369¶

In [21]:
import math

#variuable declaration
Knom   = 200;                #nominal ratio
Np     = 1;                  #number of primary turns
R      = 1.4;                #secondary impendance in Ω
L      = 1.4;                #iron loss in W
I      = 5;                  #current in A
d      = 0;                  #load angle when burden is pure resistive
mmf    = 80;                 #magnetomotive force in A
f      = 50;

#calculations
Kt       = Knom;                                                    #turns ratio
Ns       = Kt*Np;                                                   #number of secondary turns
Es       = I*R;                                                     #secondary induced voltage in V
phimax  = Es/float(4.44*f*Ns);                                      #flux in core Wb
Ep       = Es/float(Kt);                                            #primary induced voltage in V
Iw       = L/float(Ep);                                             #loss component of exciting current in A
Im       = mmf/float(Np);                                           #magnetising current
Kact     = Kt+(((Im*math.sin(d))+(Iw*math.cos(d)))/float(Is));      #actual ratio
r        =  (Knom-Kact)/float(Kact);                                #ratio error in %
r1       = r*100;

#result
print'flux in the core = %3.4e'%phimax,'wb';
print'ratio error = %3.3f'%r1,'%';

flux in the core = 1.5766e-04 wb
ratio error = -3.846 %


## Example:6.4,Page No:370¶

In [22]:
import math

#variable declaration
Np     = 1;                       #number of primary turns
Ns     =  250;                    #number of secondary turns
Rp     =  1.4;                    #resistance of secondary circuit in Ω
Xs     =  1.1;                    #reactance of secondary circuit in Ω
Is     =  5;                      #current in secondary winding in A
mmf    =  80;                     #magnetomotive force in A
L      = 1.1;                     #iron loss in W

#calculations
Kt      = Ns/float(Np);                                              #turns ratio
Knom    = Kt;
Rs      = math.sqrt((Rp**2)+(Xs**2));                                #secondary  circuit impedance
cosd    = Rp/float(Rs);
sind    = Xs/float(Rs);
Es      = Is*Rs;                                                     #secondary induced voltage in V
Ep      =  Es/float(Ns);                                             #primary induced voltage in V
Iw      = L/float(Ep);                                               #loss of component reffering to primary winding in A
Im      = mmf/float(Np);                                             #magnetising current in A
Kact    = Kt+(((Im*sind)+(Iw*cosd))/float(Is));                      #actual transformation ratio
r       =  ((Knom-Kact)/float(Kact))*100;                            #ratio error in %
theta   = (180/math.pi)*(((Im*cosd)-(Iw*sind))/float(Kt*Is));       #phase angle degreess

#result
print'ratio error = %3.2f'%r,'%';
print'phase angle =%3.2f'%theta,'°';

ratio error = -5.57 %
phase angle =2.01 °


## Example:6.5,Page No:371¶

In [23]:
import math

#variable declaration
Np   = 1;                           #number of primary windings
Ns   = 300;                         #umber of secondary windings
Re   = 1;                           #ammeter ressistance in Ω
Xe   = 0.55;                        #reactance in Ω
Rs   = 0.3;                         #resistance if secondary winding in Ω
Xs   = 0.25;                        #reactance of secondary winding in Ω
mmf  = 90;                          # mmf for magnetisation
mmfc = 45;                          #mmf for core loss
Is   = 5;                           #current in A

#calculations
R   = Rs+Re;                        #total secondarycircuit resistance in Ω
X   = Xs+Xe;                        #total secondarycircuit reactance in Ω
delta  = math.atan(X/float(R));     #secondary circuit phase angle
c      = math.cos(delta);
s      = math.sin(delta);
Kt     = Ns/float(Np);              #turn ratio
Im     = mmf/float(Np);             #magnetising current in A
Iw     = mmfc/float(Np);            #loss component in A
Kact   = Kt+(((Im*math.sin(delta))+(Iw*math.cos(delta)))/float(Is));  #actual ratio
Ip     = Kact*Is;                   #primary current A
Knom   = Kt;
y      = (((Im*math.sin(delta))+(Iw*math.cos(delta)))/float(Is));
Kt1    = (Knom)-(y);
Ns1    = Kt1*Np;                    #secondary winding turns
r      = Ns-Ns1;                    #reduction in secondary winding turns

#result
print'actual ratio = %3.2f'%Kact;
print'primary current =  %3.2f'%Ip,'A';
print'reduction in secondary winding turns = %3.0f'%r;

actual ratio = 317.10
primary current =  1585.49 A
reduction in secondary winding turns =  17


## Example:6.6,Page No:372¶

In [24]:
import math

#variable declaration
Np    = 1;                            #number of primary windings
Ns    = 100;                         #number of secondary windings
Knom  = 100;                         #nominal ratio
Re    = 1.45;                         #external burden non inductive in Ω
Rs    = 0.25;                         #winding resistance in Ω
I0    = 1.8;                          #current in A
l     = 38.4;                         #lagging angle with secondary voltage reversed in °
Is    =  1;                           #current in secondary winding in A
delta  = 0;

#calculations
Kt      = Ns/float(Np);                        #turn ratio
Rt      = Re+Rs;                               #totaal secondary circuit resistance in Ω
alpha   = 90-l;
x       = math.cos(((delta+alpha)*math.pi)/float(180));
Kact    = Kt+((I0/float(Is))*x);                        #actual ratio
y       = math.cos(((delta+alpha)*math.pi)/float(180));
theta   = (180/float(math.pi))*((I0*y/float(Kt*Is)));     #phase angle in °

#result
print'actual ratio %3.2f'%Kact,'°';
print'phase angle %3.3f'%theta,'°';

actual ratio 101.12 °
phase angle 0.641 °


## Example:6.7,Page No:373¶

In [25]:
import math

#variable declaration
Np       = 1;                  #number of primary windings
Ns       = 200;                #number of secondary winding
Kt       = 200;                #actual ratio
Im       = 8;                  #magnetising current  in A
Iw       = 5;                  #loss component in A
cosphi   = 0.8;                # leading by
Knom     = 200;                #transformer is rated
cosphi1  = 0.8;                #lagging by
Is       = 5;                  #current in A

#calculations
sinphi   = math.sqrt((1**2)-(cosphi**2));
Kact     = Kt+(((Im*sinphi)+(Iw*cosphi))/float(Is));               #actual ratio
er       = ((Knom-Kact)/float(Kact))*100;                           #error ratio
theta    = (180/float(math.pi))*(((Im*cosphi)-(Iw*sinphi))/float(Kt*Is));  #phase angle
sinphi1  = -math.sqrt((1**2)-(cosphi1**2));
Kact1    = Kt+(((Im*sinphi1)+(Iw*cosphi1))/float(Is));               #actual ratio
er1      = ((Knom-Kact1)/float(Kact1))*100;                          #ratio error
theta1   = (180/float(math.pi))*(((Im*cosphi1)-(Iw*sinphi1))/float(Kt*Is));    #phase angle

#result
print'ratio error %3.2f'%er,'%';
print'phase angle %3.4f'%theta;
print'ratio error %3.2f'%er1,'%';
print'phase angle %3.4f'%theta1,'°';

ratio error -0.87 %
phase angle 0.1948
ratio error 0.08 %
phase angle 0.5386 °


# Example:6.8,Page No:373¶

In [5]:
import math

#variable declaration
Np       = 1;                  #number of primary windings
Ns       = 99;                #number of secondary winding
Rs       = 0.4;               #secondary winding resistance in Ω
Xs       = 0.35;               #secondary winding reactance in Ω
Knom     = 100;                #ratio
mmf      = 6;                   #magnetising mmf in AT
lmmf     = 8;                   #loss mmf in AT
V        = 20;                  #voltage in VA

#calculations
Kt       = Ns/float(Np);                #actual ratio
Im       = mmf/float(Np);               #magnetising current in A
Iw       = lmmf/float(Np);              #loss component in A
Re       = V/float(Is**2);               #external reistance burden in Ω
R        = Rs+Re;                       #resistance of total seccondary circuit in Ω
#reactance is zero
Xe       = 0;
X        = Xs+Xe;                       #reactance of total secondary circcuit burden in Ω
delta    = ((math.atan(X/float(R))*180)/float(math.pi));       #phase angle
c        = math.cos((delta*math.pi)/float(180));
s        = math.sin((delta*math.pi)/float(180));
Kact     = Kt+(((Im*s)+(Iw*c))/float(Is));   #actual ratio
er       = ((Knom-Kact)/float(Kact))*100;                           #error ratio
theta    = (180/float(math.pi))*(((Im*c)-(Iw*s))/float(Kt*Is));    #phase angle

#result
print'ratio error %3.2f'%er,'%';
print'phase angle %3.4f'%theta;

ratio error -0.86 %
phase angle 0.4074


# Example:6.9,Page No:374¶

In [27]:
import math

#variable declaration
Knom   = 20;               #nominal ratio of 100/5A
V      = 20;               #rated load in VA
Il     = 0.18;             #iron loss in W
Im     =  1.4;             #magnetising current in A
x      = 4;                #ratio of reactance to resistance
Ip     = 100;              #primary currnt widing in A
Is     = 5;                #current in secondary winding in A

#calculations
Kt       = Knom;                                                 #assuming the value of Kt
Ep       = V/float(Ip);                                          #voltage across primary winding in V
Iw       = Il/float(Ep);                                         #loss current of exciting current in A
delta    = ((math.atan(1/float(x))*180)/float(math.pi));         #phase angle
c        = math.cos((delta*math.pi)/float(180));
s        = math.sin((delta*math.pi)/float(180));
Kact     = Kt+(((Im*s)+(Iw*c))/float(Is));                          #actual ratio
er       = ((Knom-Kact)/float(Kact))*100;                           #error ratio
theta    = (180/float(math.pi))*(((Im*c)-(Iw*s))/float(Kt*Is));     #phase angle

#result
print'ratio error %3.3f'%er,'%';
print'phase angle %3.4f'%theta,'°';

ratio error -1.198 %
phase angle 0.6531 °


# Example:6.10,Page No:382¶

In [6]:
import math

#variable declaration
Kt   = 10;              #ratio of 1000/100volts potentia meter
Rp    = 86.4;          #primary resistance in Ω
Rs    = 0.78;          #secondary resistance in Ω
Xp    = 62.5;          #primary reactance in Ω
Xs    = 102;           #total equivalent reactance in Ω
I0    = 0.03;          #no-load current in A
cosphi   = 0.42;          #power factor
cosgamma  = 1;          #since power factor = 1
Vs        = 100;          #voltage in V

#calculations

sinphi   = math.sqrt(1-(cosphi**2));
Im       = I0*sinphi;                          #magnetising current in A
Iw       = I0*cosphi;                          #loss current in A

#theta    = ((((Is/Kt)*((X*cosgamma)-(Rp*singamma)))+(Iw*Xp)-(Im*Rp))/float(Kt*Vs));
#since Is  =0

theta   = (((Iw*Xp)-(Im*Rp))/float(Kt*Vs));
singamma = math.sqrt(1-(cosgamma**2));

#burden in VA,theta1 = 0,thus ((((Is/Kt)*((X*cosgamma)-(Rp*singamma)))+(Iw*Xp)-(Im*Rp))/float(Kt*Vs))=0
#(((Is/Kt)*((X*cosgamma)-(Rp*singamma)))+(Iw*Xp)-(Im*Rp)) =0
#Is/Kt   = ((Im*Rp)-(Iw*Xp)))/float(((X*cosgamma)-(Rp*singamma)))
#assume x = ((X*cosgamma)-(Rp*singamma)),y = (Iw*Xp)-(Im*Rp)
#Is     = Kt*(y/x)

x        = ((Xs*cosgamma)-(Rp*singamma));
y        = (Im*Rp)-(Iw*Xp);
Is       =   Kt*(y/float(x));       #current in A
l        = Vs*Is;                   # burden load in VA

#result
print'phase angle error  at no load %3.5f'%theta,'°';
print'Note:printing mistake in textbook,theta value  is printed wrong';
print'burden load in VA %3.2f'%l,'V A'

phase angle error  at no load -0.00156 °
Note:printing mistake in textbook,theta value  is printed wrong
burden load in VA 15.34 V A


# Example:6.11,Page No:383¶

In [29]:
import math

#variable declartion
Kt      = 60.476;                 #turns ratio  3810/63 tranformer
Vs      = 63;                     #secondary voltage in V
Rs      = 2;                      #series resistance in Ω
Xs      = 1;                      #reactance in Ω
R       = 100;                    #resistance in Ω
X       = 200;                    #reactance in Ω

#calculations

delta    = ((math.atan(X/float(R))*180)/float(math.pi));         #phase angle
Z        = math.sqrt((R**2)+(X**2));                           #agnitude of impedance

#Kact     = Kt+(((Rs*c)+(Xs*s))/float(Vs/float(Is)));
#Vs/float(Is) = Z

c        = math.cos((delta*math.pi)/float(180));
s        = math.sin((delta*math.pi)/float(180));
x        =(Rs*c)+(Xs*s);
y        = ((x*Kt)/float(Z));
Kact     = Kt+y;                                                #actual ratio
Knom     = Kt;                                                   #nominal ration
er       = ((Knom-Kact)/float(Kact))*100;                        #error ratio
theta    = (180/float(math.pi))*(((Xs*c)-(Rs*s))/float(Z));      #phase angle

#result
print'ratio error %3.4f'%er,'%';
print'phase angle %3.4f'%theta,'°';

ratio error -0.7937 %
phase angle -0.3438 °