{
"metadata": {
"name": "",
"signature": "sha256:f8b31f19cbf3c6b3dce4ba746cfebc233c9801395be699bc2538e86d3ab6c5f4"
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Chapter 2: Induction Motors(Additional problems)"
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Problem 1, Page 225"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Variable declaration\n",
"P = 8 #number of poles\n",
"f = 50 #Hz\n",
"\n",
"#Calculations\n",
"Ns = (120*f)/P\n",
"\n",
"#Result\n",
"print \"Synchronous speed = %d r.p.m.\"%Ns"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Synchronous speed = 750 r.p.m.\n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Problem 2, Page 225"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Variable declaration\n",
"P = 2 #assumption - number of poles\n",
"f = 50 #Hz\n",
"\n",
"#Calculations\n",
"Ns = (120*f)/P\n",
"\n",
"#Result\n",
"print \"Maximum speed = %d r.p.m.\"%Ns"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Maximum speed = 3000 r.p.m.\n"
]
}
],
"prompt_number": 2
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Problem 3, Page 225"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Variable declaration\n",
"Pa = 6 #number of poles in alternator\n",
"N = 1000 #rpm\n",
"Pi = 16 #number of poles in induction motor\n",
"S = 2.5 #slip\n",
"\n",
"#Calculations\n",
"f = (N*Pa)/120 #Hz\n",
"Ns = (120*f)/Pi #rpm\n",
"Nr = Ns - ((S*Ns)/100)\n",
"\n",
"#Results\n",
"print \"The actual motor speed is %.2f r.p.m.\"%Nr"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The actual motor speed is 365.62 r.p.m.\n"
]
}
],
"prompt_number": 3
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Problem 4, Page 225"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Variable declaration\n",
"P = 14. #no. of poles\n",
"f = 50. #Hz\n",
"N = 414. #r.p.m.\n",
"\n",
"#calculations\n",
"Ns = (120*f)/P #rpm\n",
"S = ((Ns-N)/Ns)*100\n",
"fr = (S*f)/100\n",
"\n",
"#Results\n",
"print \"Rotor frequency = %.2f Hz\"%fr\n",
"print \"Slip = %.2f %%\"%S\n",
"#Incorrect answers in the textbook"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Rotor frequency = 1.70 Hz\n",
"Slip = 3.40 %\n"
]
}
],
"prompt_number": 4
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Problem 5, Page 225"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Variable declaration\n",
"Pi = 50 #motor input - KW\n",
"S = 3./100 #slip - %\n",
"\n",
"#Calculations\n",
"Lc = Pi*S #rotor copper loss\n",
"Pm = Pi-Lc\n",
"\n",
"#Result\n",
"print \"Total mechanical power = %.1f kW\"%Pm"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Total mechanical power = 48.5 kW\n"
]
}
],
"prompt_number": 5
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Problem 6, Page 225"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"\n",
"#Variable declaration\n",
"P = 4 #no. of poles\n",
"m = 3\n",
"f = 50 #Hz\n",
"n = 85 #efficiency %\n",
"Po = 17. #useful output power kW\n",
"Ls = 900 #stator losses W\n",
"Lwf = 1100 #windage&friction losses W\n",
"Lrc = 1. #rotor copper loss kW\n",
"\n",
"#Calculations\n",
"Pi = Po/n*100 #input power\n",
"Tl = Pi-Po #total losses kW\n",
"Lc = Tl - Ls - Lwf #copper loss W\n",
"Pir = Pi-Ls #input to rotor kW\n",
"S = Lrc/Pi\n",
"Ns = 120*f/P #rpm\n",
"N = Ns-(S*Ns) #rpm\n",
"wT = 19.1*1000 #W\n",
"Td = (wT*60)/(math.pi*2*N)\n",
"Ta = (Po*1000*60)/(2*math.pi*N)\n",
"\n",
"#Results\n",
"print \"Slip = %.3f\"%S\n",
"print \"Torque developed = %.2f N-m\"%Td\n",
"print \"Torque available = %.1f N\"%Ta\n",
"#Answers vary due to rounding-off errors"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Slip = 0.050\n",
"Torque developed = 127.99 N-m\n",
"Torque available = 113.9 N\n"
]
}
],
"prompt_number": 6
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Problem 7, Page 225"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"\n",
"#Variable declaration\n",
"P = 6 #no. of poles\n",
"f = 60 #Hz\n",
"T = 70 #N-m\n",
"N = 1152. #rpm\n",
"\n",
"#Calculations\n",
"#Part a\n",
"Pi = (2*math.pi*1200*T)/60/1000 #input to rotor W\n",
"Ns = (120.*f)/P #rpm\n",
"S = (Ns-N)/Ns\n",
"\n",
"#Part b\n",
"Lrc = S*Pi*1000 #W\n",
"Td = (2*math.pi*N*T)/60\n",
"\n",
"#Part c\n",
"Pm = Td #W\n",
"\n",
"#Results\n",
"print \"Total input to the rotor = %.1f kW\"%Pi\n",
"print \"Rotor copper loss = %d W\"%Lrc\n",
"print \"Mechanical power developed = %.2f W\"%Pm\n",
"#answers vary due to rounding-off errors"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Total input to the rotor = 8.8 kW\n",
"Rotor copper loss = 351 W\n",
"Mechanical power developed = 8444.60 W\n"
]
}
],
"prompt_number": 7
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Problem 8, Page 226"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"\n",
"#Variable declaration\n",
"Vs = 100 #voltage between two slip rings V\n",
"R2 = 0.2 #ohms/phase\n",
"Xo = 1 #ohm/phase\n",
"S = 0.04 #slip\n",
"\n",
"#Calculations\n",
"Eo = Vs/math.sqrt(3)\n",
"I2 = (S*Eo)/math.sqrt(R2**2+(S*Xo)**2)\n",
"#During maximum torque,\n",
"Sm = R2/Xo \n",
"Ir = (Sm*Eo)/math.sqrt(S**2+(S*Xo)**2)\n",
"\n",
"#Results\n",
"print \"Rotor current when slip is 4%% is %.1f A\"%I2\n",
"print \"Slip during maximum torque is %.1f\"%Sm\n",
"print \"Rotor current during maximum torque is %.1f A\"%Ir #incorrect answer in the textbook"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Rotor current when slip is 4% is 11.3 A\n",
"Slip during maximum torque is 0.2\n",
"Rotor current during maximum torque is 204.1 A\n"
]
}
],
"prompt_number": 8
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Problem 9, Page 226"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Variable declaration\n",
"S = 1 #slip at the time of starting\n",
"X2 = 1 #ohms/pase\n",
"X1 = 0.1 #internal rotor resistance ohms/phase\n",
"\n",
"#Calculations\n",
"#Total resistance required for maximum torque is 1 Ohm\n",
"Re = X2-X1\n",
"\n",
"#Result\n",
"print \"External resistance needed to be added is %.1f ohms/phase\"%Re"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"External resistance needed to be added is 0.9 ohms/phase\n"
]
}
],
"prompt_number": 9
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Problem 10, Page 226"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import sympy\n",
"from sympy.solvers import solve\n",
"from sympy import Symbol\n",
"\n",
"#Variable declaration\n",
"T = 162.8 #maximum torque Nw-m\n",
"N = 1365 #speed rpm\n",
"f = 50 #Hz\n",
"P = 4 #no. of poles\n",
"R2 = 0.2 #ohms/phase\n",
"\n",
"#Calculations\n",
"Ns = 120*f/P\n",
"S = Ns-N/Ns #slip at maximum torque\n",
"X2 = R2/S #ohms\n",
"Th = T/2 #half of maximum torque\n",
"R2 = Symbol('R2')\n",
"x = solve((Th*R2**2-18.1*40*R2+400),R2) #solving the equation\n",
"\n",
"#Results\n",
"print \"Required resistance = \",round(x[0],2),\"ohms\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Required resistance = 0.59 ohms\n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Problem 11, Page 226"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Variable declaration\n",
"N = 290. #speed of motor at full load rpm\n",
"f = 50 #Hz\n",
"\n",
"#Calculations\n",
"P = 20 #since motors running at full load\n",
"Ns= 120*f/P\n",
"S1 = (Ns-N)/Ns*100\n",
"#since S is directly proportinal to R2,\n",
"#when R2 is doubled, S is also doubled\n",
"S = 2*S1\n",
"\n",
"#Results\n",
"print \"The number of poles are %d\"%P\n",
"print \"Slip = %.1f %%\"%S1\n",
"print \"Slip if the rotor resistance is doubled = %.1f %%\"%S"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The number of poles are 20\n",
"Slip = 3.3 %\n",
"Slip if the rotor resistance is doubled = 6.7 %\n"
]
}
],
"prompt_number": 13
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Problem 12, Page 226"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Variable declarartion\n",
"S = 2.5/100 #slip\n",
"#Isc=4*If\n",
"\n",
"#Calculations\n",
"'''\n",
"Ts Isc ^2\n",
"-- = --- * slip\n",
"Tf If\n",
"'''\n",
"Ts_by_Tf = 4**2*S*100\n",
"\n",
"#Result\n",
"print \"Starting torque is %d %% of full load torque\"%Ts_by_Tf"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Starting torque is 40 % of full load torque\n"
]
}
],
"prompt_number": 14
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Problem 13, Page 226"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"\n",
"#Variable declaration\n",
"f = 50 #Hz\n",
"P = 6 #no. of poles\n",
"S = 1 #slip\n",
"Eo = 62.8 #phase voltage V\n",
"m = 3.\n",
"Xo = 6 #rotor reactance ohms\n",
"R2 = 1 #resistance ohms\n",
"\n",
"#Calculations\n",
"Ns = 120*f/P #rpm\n",
"ws = 2*math.pi*Ns/60 #rad/sec\n",
"K = m/ws\n",
"Ts = (K*S*Eo**2*R2)/((R2**2)+(S*Xo)**2) #at starting\n",
"#For maximum torque\n",
"R2 = 1 #ohm\n",
"X2 = 6 #ohms\n",
"S = R2/X2\n",
"Tmax = (K*Eo**2)/(2*X2)\n",
"\n",
"#Result\n",
"print \"The starting torque is %.3f N-m\"%Ts\n",
"print \"The maximum torque develped is %.1f N-m\"%Tmax\n",
"#Incorrect soltion in the textbook"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The starting torque is 3.054 N-m\n",
"The maximum torque develped is 9.4 N-m\n"
]
}
],
"prompt_number": 15
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Problem 14, Page 226"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"\n",
"#Variable declaration\n",
"Vl = 200. #line voltage of stator V\n",
"t = 0.67 #turn ratio\n",
"Ns = 1500. #rpm\n",
"R2 = 0.1 #resistance ohms\n",
"S = 0.04 #slip\n",
"X = 0.9 #reactance ohms\n",
"m = 3. #number of phases\n",
"\n",
"#Calculations\n",
"Vp = Vl/math.sqrt(3) #phase voltage of stator V\n",
"E = Vp*t\n",
"ws = (2*math.pi*Ns)/60\n",
"K = m/ws\n",
"T = (K*S*E**2*R2)/(R2**2+(S*X)**2)\n",
"\n",
"#Result\n",
"print \"Total torque = %d N-m\"%T\n",
"#Incorrect answer in the textbook"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Total torque = 40 N-m\n"
]
}
],
"prompt_number": 16
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Problem 15, Page 226"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Variable declaration\n",
"P = 8 #number of pos\n",
"f = 50. #Hz\n",
"S = 0.04 #slip\n",
"Po = 15*1000 #oput power W\n",
"R2 = 0.2 #rotor resistance Ohms\n",
"X = 1.5 #rotor reactance Ohms\n",
"N = 400 #ohms\n",
"\n",
"#Calculations\n",
"Ns = (120*f)/P #rpm\n",
"S_dash = (Ns-N)/Ns\n",
"R2_dash = (R2*S_dash)/S #ohms\n",
"R = R2_dash-R2 #resistance to be added\n",
"Pi = Po/(1-S) #W\n",
"Lc = S_dash*Pi #rotor copper loss W\n",
"P = Pi - Lc #ouput power W\n",
"\n",
"#Results\n",
"print \"The external resistance to be connected per phase is %.2f ohms\"%R\n",
"print \"Total rotor copper loss = %d W\"%Lc\n",
"print \"Ouput power of motor = %d W\"%P\n",
"\n",
"#Incorrect answers in the textbook"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The external resistance to be connected per phase is 2.13 ohms\n",
"Total rotor copper loss = 7291 W\n",
"Ouput power of motor = 8333 W\n"
]
}
],
"prompt_number": 17
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Problem 16, Page 227"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Variable declaration\n",
"n = 0.8 #efficiency \n",
"O = 20*735.5 #load/output\n",
"\n",
"#Calculations\n",
"i = O/n #input W\n",
"Tl = i - O #total losses W\n",
"#since total losses is a sum of rotor copper loss+stator copper loss+iron loss+mechanical loss\n",
"K = (Tl*3)/10 #rotor copper loss W\n",
"Ri = O+(K/3)+K #input to the rotor W \n",
"S = K/Ri #slip\n",
"\n",
"#Result \n",
"print \"Slip = %.3f\"%S"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Slip = 0.068\n"
]
}
],
"prompt_number": 18
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Problem 17, Page 227"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"\n",
"#Variable declaration\n",
"Pi = 20 #input power KW\n",
"N = 960. #speed rpm\n",
"f = 50 #Hz\n",
"P = 6 #number of poles\n",
"R2 = 1./3 #resistance ohms\n",
"\n",
"#Calculations\n",
"Ns = 1000.\n",
"S = (Ns-N)/Ns #slip\n",
"Lrc = S*Pi*1000 #rotor copper loss W\n",
"Lrc_ph = Lrc/3 #rotor copper loss per phase\n",
"I2 = math.sqrt(Lrc_ph/R2)\n",
"\n",
"#Result\n",
"print \"Rotor current per phase = %.1f A\"%I2"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Rotor current per phase = 28.3 A\n"
]
}
],
"prompt_number": 19
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Problem 18, Page 227"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Variable declaration\n",
"P = 4 #number of poles \n",
"f = 50. #Hz\n",
"N = 645. #speed rpm\n",
"R2 = 0.04 #rotor resistance ohms\n",
"\n",
"#Calculations\n",
"Ns = (120*f)/P #rpm\n",
"S = (Ns-N)/Ns #slip\n",
"X2 = R2/S #rotor reactance ohms\n",
"a = R2/X2\n",
"St_by_mt = (2*a)/(a*2+1) #starting torque/maximum torque\n",
"S = 0.03\n",
"Flt_by_mt = (2*a*S)/(a**2+S**2) #full load torque/maximum torque\n",
"\n",
"#Results\n",
"print \"Maximum torque at starting is %d %%\"%(St_by_mt*100)\n",
"print \"Maximum torque when slip is 3%% is %d %%\"%(Flt_by_mt*100)\n",
"\n",
"#Value of Ns ic wrongly calculated in the textbook. Hence the answers vary"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Maximum torque at starting is 53 %\n",
"Maximum torque when slip is 3% is 10 %\n"
]
}
],
"prompt_number": 20
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Problem 20, Page 227"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Variable declaration\n",
"R2 = 0.5 #rotor resistance ohms\n",
"X2 = 5 #rotor reactance ohms\n",
"Tm_by_tfl = 2.5 #max torque/full load torque\n",
"\n",
"#Calculations\n",
"#Tfl = Tm/2.5 = KE^2/2.5*2*X2\n",
"#Part a\n",
"Tst_by_Tfl_a = (25*R2)/(R2**2+X2**2)\n",
"\n",
"#Part b\n",
"Tst_by_Tfl_b = (25*R2)/(3*(R2**2+X2**2))\n",
"\n",
"#Part c\n",
"#Rotor voltage at starting = 0.75E2\n",
"Tst_by_Tfl_c = (0.75**2*25*R2)/(R2**2+X2**2)\n",
"\n",
"#Results\n",
"print \"The ratio of starting torque to full load torque for the given cases are as below:\"\n",
"print \"Part a: %.2f\"%Tst_by_Tfl_a\n",
"print \"Part a: %.2f\"%Tst_by_Tfl_b\n",
"print \"Part a: %.2f\"%Tst_by_Tfl_c"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The ratio of starting torque to full load torque for the given cases are as below:\n",
"Part a: 0.50\n",
"Part a: 0.17\n",
"Part a: 0.28\n"
]
}
],
"prompt_number": 21
}
],
"metadata": {}
}
]
}