# Chapter 4: Transformers¶

## Example 4.1, Page 210¶

In [1]:
#Variable declaration
Nab = 200   #flux between points a and b
Ncd = 500   #flux between points c and d

#Calculations&Results
#From 0 to 0.06s
do_by_dt_1 = 0.15         #rate of change of flux
e_ab_1 = -Nab*do_by_dt_1  #V
e_cd_1 = Ncd*do_by_dt_1   #V
print "Part a: Induced emf's are %d V and %d V"%(e_ab_1,e_cd_1)

#From 0.06 to 0.1s
do_by_dt_2 = -0.45         #rate of change of flux
e_ab_2 = -Nab*do_by_dt_2  #V
e_cd_2 = Ncd*do_by_dt_2   #V
print "Part b: Induced emf's are %d V and %d V"%(e_ab_2,e_cd_2)

Part a: Induced emf's are -30 V and 75 V
Part b: Induced emf's are 90 V and -225 V


## Example 4.2, Page 212¶

In [7]:
#Variable declaration
N_p=150.;#no. of turns in primary winding
N_s=750;#no. of turns in secondary winding
f=50;#frequency in Hz
I_2=4;#loa.d current (in Amperes)
V_1=240;#voltage on primary side (in Volts)
pf=0.8;#power factor

#Calculations&Results
a=N_p/N_s;
print '(a) a-ratio=%.1f'%a
I_1=I_2/a;
print '(b) current in primary (in Amperes)=%.f'%I_1
V_2=V_1/a;
print '(c) voltage on secondary side (in Volts)=%.f'%V_2
P_L=V_2*I_2*pf;
print '(d) power supplied to the load (in Watts)=%.f'%P_L
flux=V_1/(4.44*f*N_p);
print '(e) flux in the core (in mili-Weber)=%.2f'%(flux*10**3)

(a) a-ratio=0.2
(b) current in primary (in Amperes)=20
(c) voltage on secondary side (in Volts)=1200
(d) power supplied to the load (in Watts)=3840
(e) flux in the core (in mili-Weber)=7.21


## Example 4.3, Page 215¶

In [17]:
import math

#Variable declaration
R_1=4;#in ohms
R_2=0.04;#in ohms
X_1=12;#in ohms
X_2=0.12;#in ohms
pf=0.866;#power factor
V_p=2300;#primary voltage in volts
V_s=230;#Secondary voltage in volts
S=23000;#VA
theta=math.acos(pf);

#Calculations
I_2=complex((S*0.75/V_s)*(math.cos(theta)),math.sin(theta));#secondary current (in Amperes)
Z_2=complex(R_2,X_2);#secondary winding impedance (in ohms)
E_2=V_s+I_2*Z_2;#induced emf in secondary winding (in Volts)
a=V_p/V_s;#transformation ratio
E_1=a*E_2;#induced emf in primary winding (in Volts)
I_1=I_2/a;#current in primary winding
Z_1=complex(R_1,X_1);#primary winding impedance (in ohms)
V_1=E_1+I_1*Z_1;#source voltage
P_o=(V_s*I_2.conjugate());#output power(in Watts)
P_in=(V_1*I_1.conjugate());#input power
Eff=P_o.real/P_in.real;

#Result
print 'Efficiency (%%)=%.1f'%(Eff*100)
#answer differs due to rounding-off errors

Efficiency (%)=97.8


## Example 4.4, Page 219¶

In [50]:
import math
import cmath

#Variable declaration
#From Exa:4.3
V_2=230.;#in Volts

#Calculations
Z_1=complex(4,12);
I_s=complex(75*(math.cos(30*math.pi/180)),75*math.sin(30*math.pi/180));#in Amperes
a=10;#transformation ratio
E_1=complex(2282.87*(math.cos(2.33*math.pi/180)),2282.87*math.sin(2.33*math.pi/180));#in Volts
E_2=complex(228.287*(math.cos(2.33)*math.pi/180),228.287*math.sin(2.33*math.pi/180));#in Volts
I_p=complex(7.5*(math.cos(30*math.pi/180)),7.5*math.sin(30*math.pi/180));#in Amperes
P_o=14938.94;#in Watts
R_c1=20000;#core loss resistance on primary side
X_m1=15000;#magnetizing reactance on primary side
I_c=E_1/R_c1;#in Amperes
I_m=E_1/complex(0,X_m1);#in Amperes
I_phy=I_c+I_m;#in Amperes
I_1=I_p+I_phy;#in Amperes
V_1=E_1+Z_1*I_1;#in Volts
P_in=(V_1*I_1.conjugate());#in Watts
Eff=P_o/P_in.real;

#Result
print 'Efficiency (%%)=%.1f'%(Eff*100)

Efficiency (%)=95.5


## Example 4.5, Page 224¶

In [53]:
import math

#Variable declaration
#Values from examples 4.3 & 4.4
a = 10.
R1 = 4.     #ohms
X1 = 12.    #ohms
R2 = 0.04  #ohms
X2 = 0.12  #ohms
V2 = a*230*complex(math.cos(0*math.pi/180),math.sin(0*math.pi/180))
Ip = 7.5*complex(math.cos(30*math.pi/180),math.sin(30*math.pi/180))

#Calculations&Results
R_e1 =R1+(a**2*R2)   #ohms
X_e1 = X1+(a**2*X2)  #ohms
Z_e1 = complex(R_e1,X_e1) #ohms
V1 = V2+(Ip*Z_e1)       #V
Ie = V1/20000           #A
Im = V1/complex(0,15000)#A
I1 = Ip+Ie+Im           #A
Po = V2*complex(7.5*math.cos(-30*math.pi/180),7.5*math.sin(-30*math.pi/180))              #W
Pin = V1*complex(7.54*math.cos(-28.6*math.pi/180),7.54*math.sin(-28.6*math.pi/180))             #W
N = Po/Pin

#Result
print "Efficiency = %.2f %%"%(N.real*100)
#Incorrect solution in textbook

Efficiency = 100.23 %


## Example 4.6, Page 226¶

In [57]:
import math

#Variable declaration
S=2200;#Volt-Ampere
V_s=220;#secondary side voltage (in Volts)
V_2=V_s;
V_p=440;#primary side voltage (in Volts)
R_e1=3;#in ohms
X_e1=4;#in ohms
R_c1=2.5*1000;#in ohms
X_m1=2000;#in ohms

#Calculations
a=V_p/V_2;#transformation ratio
pf=0.707;#lagging power factor
theta=-math.acos(pf);
I_2=(S/V_2)*complex(math.cos(theta),math.sin(theta));#(in Amperes)
#Refer to equivalent circuit (fig:4.16)
I_p=I_2/a;#in Amperes
V_2p=a*V_2;
V_1=V_2p+I_p*complex(R_e1,X_e1);
I_c=V_1/R_c1;#core loss current (in Amperes)
I_m=V_1/complex(0,X_m1);
I_1=I_p+I_c+I_m;#current supplied by source (in Amperes)
P_o=V_p*I_p.conjugate();#output power (in Watts)
P_in=V_1*I_1.conjugate();#input power (in Watts)
Eff=P_o.real/P_in.real;#Efficiency
VR=(abs(V_1)-abs(V_p))/V_p;

#Results
print 'Efficiency (%%)=%.1f'%(Eff*100)
print 'voltage regulation (%%)=%.2f'%(VR*100)

Efficiency (%)=90.6
voltage regulation (%)=5.63


## Example 4.7, Page 230¶

In [60]:
import math

#Variable declaration
S=120000;#Volt-Ampere
V_p=2400;#in volts
V_s=240;#in volts
R_1=0.75;#in ohms
R_2=0.01;#in ohms
X_1=0.8;#in ohms
X_2=0.02;#in ohms
pf=0.8;#lagging
theta=-math.acos(pf);

#Calculations&Results
a=V_p/V_s;#transformation ratio
KVA=I_p_eta*V_p/1000;
print '(a) KVA rating at max efficiency =%.f'%KVA
P_cu_eta=I_p_eta**2*(R_1+a**2*R_2);#copper loss (in Watts)
P_m=P_cu_eta;#core loss
P_o=V_p*I_p_eta*pf;#power output at max efficiency
P_in=P_o+P_m+P_cu_eta;#power input at max efficiency
eta=P_o/P_in;
print '(b) max efficiency (%%)=%.f'%(eta*100)
P_in_FL=P_cu_FL+P_o_FL+P_m;
Eff=P_o_FL/P_in_FL;
print '(c) Efficiency at full load (%%)=%.1f'%(Eff*100)
R_c1=V_p**2/P_cu_eta;
print '(d) equivalent core resistance (in ohms)=%.2f'%R_c1

(a) KVA rating at max efficiency =84
(b) max efficiency (%)=94
(c) Efficiency at full load (%)=93.6
(d) equivalent core resistance (in ohms)=2686.88


## Example 4.8, Page 236¶

In [61]:
import math

#Varible declaration
#Open circuit test
Voc = 240.  #V
Ioc = 2.    #A
Poc = 120.  #W
#Short circuit test
Vsc = 150.  #V
Isc = 10.   #A
Psc = 600.  #W

#Calculations
Rcl = Voc**2/Poc  #ohms
Soc = Voc*Ioc     #apparent power under no load,ohms
Qoc = math.sqrt(Soc**2-Poc**2)  #reactive power,VAR
Xml = Voc**2/Qoc        #magnetization reactance,ohms
a = 4800./240
Rch = a**2*Rcl  #ohms
Xmh = a**2*Xml  #ohms
Reh = Psc/Isc**2
Zeh = Vsc/Isc
Xeh = math.sqrt(Zeh**2-Reh**2)
#Winding parameters to the low-voltage side
Rel = Reh/a**2
Xel = Xeh/a**2

#values for equivelent circuit
Rh = 0.5*Reh  #ohms
Xh = 0.5*Xeh  #ohms
Rl = (0.5*Reh)/a**2 #ohms
Xl = (0.5*Xeh)/a**2 #ohms

#Results
print "For the equivalent circuit"
print "The winding resistances are %d ohms and %.1f milli-ohms"%(Rh,Rl*10**3)
print "The leakage reactances are %.2f ohms and %d milli-ohms"%(Xh,Xl*10**3)

For the equivalent circuit
The winding resistances are 3 ohms and 7.5 milli-ohms
The leakage reactances are 6.87 ohms and 17 milli-ohms


## Example 4.9, Page 240¶

In [1]:
import math

#Variable declaration
#Refer to fig:4.29
#For region A
V_bA=230.;#in Volts
S_bA=46000.;#Volt-Ampere

#Calculations&Results
I_bA=S_bA/V_bA;#in Amperes
Z_bA=V_bA/I_bA;#in ohms
Z_g_pu=complex(0.023,0.092)/Z_bA;
R_L_pu=0.023/Z_bA;
X_L_pu=0.069/Z_bA;
#For region B
#Per unit parameters on high-voltage side of the step-up transformer
V_bB=2300.;#in Volts
S_bB=46000.;#Volt-Ampere
I_bB=S_bB/V_bB;#in Amperes
Z_bB=V_bB/I_bB;#in ohms
R_H_pu=2.3/Z_bB;
X_H_pu=6.9/Z_bB;
R_cH_pu1=13800./Z_bB;
X_mH_pu1=6900./Z_bB;
Z_l_pu=complex(2.07,4.14)/Z_bB;#Per-unit impedance of transmission line
#Per unit parameters on high-voltage side of the step-down transformer
X_mH_pu2=9200./Z_bB;
R_cH_pu2=11500./Z_bB;
#For region C
V_bC=115.;#in Volts
S_bC=46000.;#Volt-Ampere
I_bC=S_bC/V_bC;#in Amperes
Z_bC=V_bC/I_bC;#in ohms
R_L_pu=0.00575/Z_bC;
X_L_pu=0.01725/Z_bC;
V_L_pu=complex(1*math.cos(0*math.pi/180),1*math.sin(0*math.pi/180));
I_L_pu=complex(1*math.cos(-30*math.pi/180),1*math.sin(-30*math.pi/180));
E_l_pu=V_L_pu+complex(R_L_pu,X_L_pu)*I_L_pu;
I_l_pu=I_L_pu+E_l_pu*complex(0.01,(1./80));
E_g_pu=E_l_pu+I_l_pu*(complex(0.02,0.06)+complex(0.018,0.036)+complex(0.02,0.06));
I_g_pu=I_l_pu+E_g_pu*complex((1./120),(1./60));
V_g_pu=E_g_pu+I_g_pu*(complex((0.02+0.02),(0.08+0.06)));
V_g=V_bA*V_g_pu;
print '(a) Generator Voltage (in Volts)=%.2f'%(abs(V_g))
print 'Phase of generated voltage (in degree)=%.2f'%(math.degrees(math.atan(V_g.imag/V_g.real)))
I_g=I_bA*I_g_pu;
print '(b) Generator current (in Amperes)=%.2f'%(abs(I_g))
print 'Phase of generator current (in degree)=%.2f'%(math.degrees(math.atan(I_g.imag/I_g.real)))
P_o_pu=0.866;#rated power output at pf=0.866 lagging
P_in_pu=V_g_pu*I_g_pu.conjugate();
Eff=P_o_pu/P_in_pu.real;
print '(c) Efficiency (%%)=%.2f'%(Eff*100)
#Answers vary due to rounding-off errors

(a) Generator Voltage (in Volts)=298.97
Phase of generated voltage (in degree)=11.33
(b) Generator current (in Amperes)=199.70
Phase of generator current (in degree)=-27.80
(c) Efficiency (%)=86.02


## Example 4.10, Page 247¶

In [2]:
#Variable declaration
V_1=2400.;#in Volts
V_2=240.;#in Volts
S_o=24.*1000;#Volt-Ampere
I_1=10.;#in Amperes
I_2=100.;#in Amperes
#Refer to fig:4.31 (a)

#Calculations
V_1a=V_1+V_2;
V_2a=V_2;
a_T1=V_1a/V_2a;
a_T2=V_2a/V_1a;
a_T3=V_1a/V_1;
a_T4=V_1/V_1a;
S_oa_1=V_1a*I_1;
S_oa_2=V_1a*I_1;
S_oa_3=V_1a*I_2;
S_oa_4=V_1a*I_2;

#Results
print 'Refer to fig:4.31a'
print '(a) primary winding voltage (in Volts)=%.f'%V_1a
print '(b) secondary winding voltage (in Volts)=%.f'%V_2a
print '(c) ratio of transformation=%.f'%a_T1
print '(d) nominal rating of transformer (KVA)=%.1f'%(S_oa_1/1000)
print 'Refer to fig:4.31b'
print '(a) primary winding voltage (in Volts)=%.f'%V_2a
print '(b) secondary winding voltage (in Volts)=%.f'%V_1a
print '(c) ratio of transformation=%.3f'%a_T2
print '(d) nominal rating of transformer (KVA)=%.1f'%(S_oa_2/1000)
print 'Refer to fig:4.31c'
print '(a) primary winding voltage (in Volts)=%.f'%V_1a
print '(b) secondary winding voltage (in Volts)=%.f'%V_1
print '(c) ratio of transformation=%.1f'%a_T3
print '(d) nominal rating of transformer (KVA)=%.f'%(S_oa_3/1000)
print 'Refer to fig:4.31d'
print '(a) primary winding voltage (in Volts)=%.f'%V_1
print '(b) secondary winding voltage (in Volts)=%.f'%V_1a
print '(c) ratio of transformation=%.2f'%a_T4
print '(d) nominal rating of transformer (KVA)=%.f'%(S_oa_4/1000)

Refer to fig:4.31a
(a) primary winding voltage (in Volts)=2640
(b) secondary winding voltage (in Volts)=240
(c) ratio of transformation=11
(d) nominal rating of transformer (KVA)=26.4
Refer to fig:4.31b
(a) primary winding voltage (in Volts)=240
(b) secondary winding voltage (in Volts)=2640
(c) ratio of transformation=0.091
(d) nominal rating of transformer (KVA)=26.4
Refer to fig:4.31c
(a) primary winding voltage (in Volts)=2640
(b) secondary winding voltage (in Volts)=2400
(c) ratio of transformation=1.1
(d) nominal rating of transformer (KVA)=264
Refer to fig:4.31d
(a) primary winding voltage (in Volts)=2400
(b) secondary winding voltage (in Volts)=2640
(c) ratio of transformation=0.91
(d) nominal rating of transformer (KVA)=264


## Example 4.11, Page 249¶

In [3]:
import math

#Variable declaration
V_2a=480.;#in volts
theta=math.degrees(math.acos(pf))
a_T=120./480;#ratio of transformation of step-up transformer
a=360./120;#ratio of transformation of two-winding transformer
R_cH=8.64*1000;#in ohms
R_H=18.9;#in ohms
X_H=21.6;#in ohms
X_L=2.4;#in ohms
R_L=2.1;#in ohms

#Calculations
X_mH=6.84*1000;#in ohms
R_cL=R_cH/a**2;#equivalent core loss resistance in ohms
X_mL=X_mH/a**2;#magnetizing reactance
I_2a=(720./360)*complex(math.cos(theta*math.pi/180),math.sin(theta*math.pi/180));
I_H=I_2a;
I_pa=I_2a/a_T;
I_com=I_pa-I_2a;#current through common winding (in Amperes)
#on applying KVL to the output loop
E_L=(I_2a*complex(R_H,X_H)+V_2a-I_com*complex(R_L,X_L))/4;
V_1a=E_L+I_com*complex(R_L,X_L);
I_ca=V_1a/R_cL;#core loss current in Amperes
I_ma=V_1a/complex(0,X_mL);#magnetizing current in Amperes
I_phy_a=I_ca+I_ma;#excitation current
I_1a=I_pa+I_phy_a;
P_o=V_2a*I_2a.conjugate();
P_in=V_1a*I_1a.conjugate();
Eff=P_o.real/P_in.real;
VR=(abs(V_2anL)-V_2a)/V_2a;

#Result
print 'Efficiency (%%)=%.1f'%(Eff*100)
print 'Voltage regulation (%%)=%.2f'%(VR*100)

Efficiency (%)=80.3
Voltage regulation (%)=1.26


## Example 4.12, Page 258¶

In [4]:
import math

#Variable declaration
VA = 720

#Calculations&Results
S_3phi = 3*VA*10**-3  #VA
#For Y/Y connection
V1l = math.sqrt(3)*360  #V
V2l = math.sqrt(3)*120  #V
print "For Y/Y connection\nThe nominal ratings are %.2f kVA and 624/%.f V"%(S_3phi,V2l)

#For delta/delta connection
V1l = 300  #V
V2l = 120  #V
print "\nFor delta/delta connection\nThe nominal values of line voltages are %.2f kVA and 360/%d V"%(S_3phi,V2l)

For Y/Y connection
The nominal ratings are 2.16 kVA and 624/208 V

For delta/delta connection
The nominal values of line voltages are 2.16 kVA and 360/120 V


## Example 4.13, Page 265¶

In [5]:
import math

#Variable declaration
R_H=133.5*10**-3;#in ohms
X_H=201*10**-3;#in ohms
R_L=39.5*10**-3;#in ohms
X_L=61.5*10**-3;#in ohms
R_cL=240;#in ohms
X_mL=290;#in ohms
pf=0.8;#lagging
theta=-math.degrees(math.acos(pf));

#Calculations&Results
V_2n=138.564*complex(math.cos(0),math.sin(0));#rated load voltage for Y/Y connection
a=120/138.564;#transformation ratio
I_pA=(I_2A/a)*complex(math.cos(30*math.pi/180),math.sin(30*math.pi/180));#per phase current in primary winding
E_2n=V_2n+I_2A*complex(0.0445,0.067);#voltage induced in secondary winding
E_2L=math.sqrt(3)*E_2n*complex(math.cos(30*math.pi/180),math.sin(30*math.pi/180));
E_1n=a*E_2n*complex(math.cos(30*math.pi/180),math.sin(30*math.pi/180));#voltage induced in primary winding
I_1A=I_pA+E_1n*complex((1./240),(1./290));
print 'Line current in secondary side (in Amperes)=%.1f'%(abs(I_2A))
print 'phase angle of induced line current in secondary (in Degree)=%.2f'%(math.degrees(math.atan(I_2A.imag/I_2A.real)))
print 'Line current in primary side (in Amperes)=%.f'%(abs(I_1A))
print 'phase angle of induced line current in primary (in Degree) =%.2f'%(math.degrees(math.atan(I_1A.imag/I_1A.real)))
print 'Line voltage induced  in secondary side (in Volts)=%.1f'%(abs(E_2L))
print 'phase angle of induced line voltage in secondary (in Degree)=%.2f'%(math.degrees(math.atan(E_2L.imag/E_2L.real)))
V_1n=E_1n+I_1A*complex(R_L,X_L);
V_1L=math.sqrt(3)*V_1n*complex(math.cos(30*math.pi/180),math.sin(30*math.pi/180));
print 'Line voltage induced  in primary side (in Volts)=%.2f'%(abs(V_1L))
print 'phase angle of induced line voltage  in primary (in Degree)=%.2f'%(math.degrees(math.atan(V_1L.imag/V_1L.real)))
P_o=3*138.564*I_2A.conjugate()
P_in=3*V_1n*I_1A.conjugate()
Eff=P_o.real/P_in.real;
print 'Efficiency (%%)=%.1f'%(Eff*100)

Line current in secondary side (in Amperes)=86.6
phase angle of induced line current in secondary (in Degree)=-36.87
Line current in primary side (in Amperes)=100
phase angle of induced line current in primary (in Degree) =-6.49
Line voltage induced  in secondary side (in Volts)=251.4
phase angle of induced line voltage in secondary (in Degree)=30.92
Line voltage induced  in primary side (in Volts)=229.68
phase angle of induced line voltage  in primary (in Degree)=62.00
Efficiency (%)=92.3


## Example 4.14, Page 272¶

In [6]:
#Variable declaration
I_L=4*80/5;

#Calculations&Results
print 'Line current (in Amperes)=%.f'%I_L
V_L=110*100/1;
print 'Line voltage (in Volts)=%.f'%V_L
P=(100./1)*(80/5)*352;
print 'Power on the transmission line (in KWatts)=%.1f'%(P/1000)

Line current (in Amperes)=64
Line voltage (in Volts)=11000
Power on the transmission line (in KWatts)=563.2