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diff --git a/Fundamentals_of_Electrical_Drives/Chapter6.ipynb b/Fundamentals_of_Electrical_Drives/Chapter6.ipynb new file mode 100755 index 00000000..8ac325e6 --- /dev/null +++ b/Fundamentals_of_Electrical_Drives/Chapter6.ipynb @@ -0,0 +1,1817 @@ +{ + "metadata": { + "name": "" + }, + "nbformat": 3, + "nbformat_minor": 0, + "worksheets": [ + { + "cells": [ + { + "cell_type": "heading", + "level": 1, + "metadata": {}, + "source": [ + "Chapter 6: Induction Motor Drives" + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example No:6.1,Page No:147" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from __future__ import division\n", + "import cmath\n", + "\n", + "#variable declaration\n", + "#ratings of the Y-connected induction motor \n", + "f=50 # frequency in HZ\n", + "Vl=440 #line voltage in V\n", + "P=6 # number of poles\n", + "N=950 #speed in rpm\n", + "\n", + "#parameters referred to the stator\n", + "Xr_=1.2 # rotor winding reactance in ohm\n", + "Rr_=0.4 # resistance of the rotor windings in ohm\n", + "Rs=0.5 # resistance of the stator windings in ohm\n", + "Xs=Xr_ # stator winding reactance in ohm\n", + "Xm=50 # no load reactance\n", + " \n", + "#calculation\n", + "Ns=120*f/P #synchronous speed in rpm\n", + "s=(Ns-N)/Ns #full load slip\n", + "x=math.sqrt((Rs+Rr_/s)**2+(Xs+Xr_)**2) #total impedance\n", + "Ir_=(Vl/math.sqrt(3))/x #full load rotor current\n", + "angle=-math.atan((Xs+Xr_)/(Rs+Rr_/s)) #angle in radian\n", + "Ir_=cmath.rect(Ir_,angle) #full load rotor current in rectangular form\n", + "Im=Vl/math.sqrt(3)/Xm*(-1j) #magnetizing current\n", + "Is=Ir_+Im #full load current\n", + "\n", + "Zf=Rs+Xs*1j+1j*Xm*(Rr_/s+1j*Xr_)/(Rr_/s+1j*(Xr_+Xm))\n", + "Zb=Rs+Xs*1j+1j*Xm*(Rr_/(2-s)+1j*Xr_)/(Rr_/(2-s)+1j*(Xr_+Xm))\n", + "Z=Zf+Zb\n", + "I=(Vl/math.sqrt(3))/abs(Z) #motor current\n", + "Wms=2*math.pi*Ns/60\n", + "\n", + "#torque due to positive sequence\n", + "Tp=(1/Wms)*(3*I**2*Xm**2*Rr_/s)/((Rr_/s)**2+(Xr_+Xm)**2)\n", + "#torque due to negative sequence\n", + "Tn=-(1/Wms)*(3*I**2*Xm**2*Rr_/(2-s))/((Rr_/(2-s))**2+(Xr_+Xm)**2)\n", + "T=Tp+Tn #net torque\n", + "Wm=Wms*(1-s) #rated speed in in rad/sec\n", + "Tl=0.0123*Wm**2 #required torque of the load\n", + "\n", + "#results\n", + "print\"Full load motor current Is:\",round(abs(Is),1),round(math.degrees(cmath.phase(Is))),\"\u00b0\",\"A\"\n", + "print\"Tp:\",round(Tp,2),\"N-m\"\n", + "print\"Tn:\",round(Tn,3),\"N-m\"\n", + "print\"\\nSince I:\",round(I,2),\"A and N:\",N,\"rpm\"\n", + "print\"And I:\",round(I,2),\"A <\",\"Is:\",round(abs(Is),2),\"A, the motor will run safely\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "Full load motor current Is: 30.5 -25.0 \u00b0 A\n", + "Tp: 125.61 N-m\n", + "Tn: -3.299 N-m\n", + "\n", + "Since I: 24.26 A and N: 950 rpm\n", + "And I: 24.26 A < Is: 30.54 A, the motor will run safely\n" + ] + } + ], + "prompt_number": 197 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example No:6.2,Page No:156" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from __future__ import division\n", + "import cmath\n", + "\n", + "#variable declaration\n", + "#ratings of the Delta connected Induction motor\n", + "f=50 #frequency in HZ\n", + "Vl=2200 #line voltage in V\n", + "P=8 #number of poles\n", + "N=735 #rated speed in rpm\n", + "#parameters referred to the stator\n", + "Xr_=0.55 # rotor winding reactance in ohm\n", + "Xs=0.45 # stator winding reactance in ohm\n", + "Rr_=0.1 # resistance of the rotor windings in ohm\n", + "Rs=0.075 # resistance of the stator windings in ohm\n", + "\n", + "#calculation\n", + "Ns=120*f/P #synchronous speed in rpm\n", + "s=(Ns-N)/Ns #full load slip\n", + "x=math.sqrt((Rs+Rr_/s)**2+(Xs+Xr_)**2) #total impedance\n", + "Ip=(Vl)/x #full load phase current\n", + "Il=math.sqrt(3)*Ip #full load line current\n", + "Wms=2*math.pi*Ns/60 \n", + "Tl=(1/Wms)*(3*Ip**2*Rr_/s) #full load torque\n", + "\n", + "#(i)if the motor is started by star-delta switching\n", + "y=math.sqrt((Rs+Rr_)**2+(Xs+Xr_)**2)\n", + "Ist=(Vl/math.sqrt(3))/y #Maximum line current during starting\n", + "Tst=(1/Wms)*(3*Ist**2*Rr_) #Starting torque\n", + "ratio1=Tst/Tl #ratio of starting torque to full load torque\n", + "z=Rs+math.sqrt(Rs**2+(Xs+Xr_)**2)\n", + "Tmax=3/(2*Wms)*(Vl/math.sqrt(3))**2/z #maximum torque\n", + "ratio2=Tmax/Tl #ratio of maximum torque to full load torque\n", + "\n", + "#(ii) If the motor is started using auto transformer\n", + "y=math.sqrt((Rs+Rr_)**2+(Xs+Xr_)**2)\n", + "Ist1=Vl*math.sqrt(3)/y #starting current direct online\n", + "aT=math.sqrt(2*Il/Ist1) #transofrmation ratio\n", + "Ilst=2*Il/aT #starting motor line current\n", + "Ipst=Ilst/math.sqrt(3) #starting motor phase current\n", + "Tst1=(1/Wms)*(3*Ipst**2*Rr_) #starting torque\n", + "\n", + "#(iii) If motor is started using part winding method\n", + "Rs_=2*Rs\n", + "Xs_=2*Xs\n", + "y=math.sqrt((Rs_+Rr_)**2+(Xs_+Xr_)**2)\n", + "Ist2=(Vl*math.sqrt(3))/y #starting line current \n", + "Ip=Ist2/math.sqrt(3) #starting phase current\n", + "Tst2=(1/Wms)*(3*Ip**2*Rr_) #starting torque\n", + "\n", + "#(iv) motor is started using series reactors in line\n", + "Rs_=Rs/3 ; Rr_=Rr_/3\n", + "Xs_=Xs/3 ; Xr_=Xr_/3\n", + "Il=2*Il #line current at start\n", + "x=(Vl/math.sqrt(3))**2/(Il**2) #x=(Rs_+Rr_)**2+(Xs_+Xr_+Xe)**2\n", + "y=x-(Rs_+Rr_)**2 #y=(Xs_+Xr_+Xe)**2\n", + "z=math.sqrt(y) #z=(Xs_+Xr_+Xe)\n", + "Xe=z-Xs_-Xr_\n", + "\n", + "\n", + "#results\n", + "print\"(i)Maximum value of line current during starting Ist:\",round(Ist),\"A\"\n", + "print\" Ratio of starting torque to full load torque :\",round(ratio1,3)\n", + "print\" Ratio of maximum torque to full load torque :\",round(ratio2,2)\n", + "print\"\\n(ii)transofrmation ratio aT:\",round(aT,3)\n", + "print\" starting torque :\",round(Tst1),\"N-m\"\n", + "#answer for the starting torque in the book is wrong due to accuracy\n", + "print\"\\n(iii)maixmum line current during starting :\",round(Ist2),\"A\"\n", + "print\" Starting torque :\",round(Tst2),\"N-m\"\n", + "#answer for the starting torque in the book is wrong due to accuracy\n", + "print\"\\n(iv)value of the reactor Xe:\",round(Xe,3),\"ohm\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "(i)Maximum value of line current during starting Ist: 1251.0 A\n", + " Ratio of starting torque to full load torque : 0.173\n", + " Ratio of maximum torque to full load torque : 0.83\n", + "\n", + "(ii)transofrmation ratio aT: 0.627\n", + " starting torque : 7041.0 N-m\n", + "\n", + "(iii)maixmum line current during starting : 2590.0 A\n", + " Starting torque : 8539.0 N-m\n", + "\n", + "(iv)value of the reactor Xe: 0.527 ohm\n" + ] + } + ], + "prompt_number": 199 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example No:6.3,Page No:159" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from __future__ import division\n", + "import cmath\n", + "\n", + "#variable declaration\n", + "#ratings of the star connected Induction motor\n", + "f=50 # frequency in HZ\n", + "Vl=400 # line voltage in V\n", + "P=6 # number of poles\n", + "#parameters referred to the stator\n", + "Xr_=2 # rotor winding reactance in ohm\n", + "Xs=Xr_ # stator winding reactance in ohm\n", + "Rr_=1 # resistance of the rotor windings in ohm\n", + "Rs=Rr_ # resistance of the stator windings in ohm\n", + "\n", + "#calculation\n", + "Ns=120*f/P #synchronous speed in rpm\n", + "Wms=2*math.pi*Ns/60 \n", + "\n", + "#(i)\n", + "Sm=-Rr_/math.sqrt(Rs**2+(Xs+Xr_)**2) #slip at maximum torque\n", + "x=math.sqrt((Rs+Rr_/Sm)**2+(Xs+Xr_)**2)\n", + "Ir_=(Vl/math.sqrt(3))/x #current at maximum torque \n", + "Tmax=(1/Wms)*3*Ir_**2*Rr_/Sm #maximum torque\n", + "N=(1-Sm)*Ns #range of speed\n", + "\n", + "#(ii)an overhauling torque of 100Nm\n", + "Tl=100 #overhauling torque in Nm\n", + "# Tl=(3/Wms)*(Vl**2*Rr_/s)/y\n", + "# where y=(Rs+Rr_/s)**2+(Xs+Xr_)**2\n", + "a=(1/Wms)*(Vl**2*Rr_)/(-Tl) #a=s*(Rs+Rr_/s)**2+(Xs+Xr_)**2\n", + "a = 17\n", + "b = 17.3\n", + "c = 1\n", + "# calculate the discriminant\n", + "d = (b**2) - (4*a*c)\n", + "# find two solutions\n", + "s1 = (-b-cmath.sqrt(d))/(2*a)\n", + "s2 = (-b+cmath.sqrt(d))/(2*a)\n", + "\n", + "N2=(1-s2)*Ns #motor speed and we neglect s1 \n", + "\n", + "#slight difference in the answer due to accuracy\n", + "\n", + "#(iii)when a capacitive reactance of 2 ohm is inserted in each phase stator\n", + "Xc=2 #reactance of the capacitor\n", + "Sm=-Rr_/math.sqrt(Rs**2+(Xs+Xr_-Xc)**2) #slip at maximum torque \n", + "x=math.sqrt((Rs+Rr_/Sm)**2+(Xs+Xr_-Xc)**2)\n", + "Ir_=(Vl/math.sqrt(3))/x #current at maximum torque \n", + "Tmax_=(1/Wms)*3*Ir_**2*Rr_/Sm #maximum overhauling torque with capacitor\n", + "ratio=Tmax_/Tmax\n", + "\n", + "\n", + "#results\n", + "print\"(i)Maximum overhauling torque that the motor can hold is:\",round(abs(Tmax),1),\"N-m\"\n", + "print\" Range of speed is from \",Ns,\"to\",round(N),\"rpm\"\n", + "print\"\\n(ii)Now s*(1+1/s)**2+16s=\",a\n", + "print\" Or 17s**s+17.3s+1=0\"\n", + "print\"The solution for s are \",round(s1.real,3),\"and\",round(s2.real,3)\n", + "print\"\\nTherefore Motor speed is: \",round(abs(N2)),\"rpm\"\n", + "print\"Note :There is a slight difference in the answer due to the decimal place\"\n", + "print\"\\n(iii) Ratio of maximum torque with capacitor and to maximum torque without capacitor is:\",round(ratio,2)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "(i)Maximum overhauling torque that the motor can hold is: 244.6 N-m\n", + " Range of speed is from 1000.0 to 1243.0 rpm\n", + "\n", + "(ii)Now s*(1+1/s)**2+16s= 17\n", + " Or 17s**s+17.3s+1=0\n", + "The solution for s are -0.956 and -0.062\n", + "\n", + "Therefore Motor speed is: 1062.0 rpm\n", + "Note :There is a slight difference in the answer due to the decimal place\n", + "\n", + "(iii) Ratio of maximum torque with capacitor and to maximum torque without capacitor is: 2.53\n" + ] + } + ], + "prompt_number": 202 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example No:6.4,Page No:162\n" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from __future__ import division\n", + "import cmath\n", + "\n", + "#variable declaration\n", + "#ratings of the motor is same as that of Ex-6.3\n", + "f=50 # frequency in HZ\n", + "Vl=400 #line voltage in V\n", + "P=6 # number of poles\n", + "#parameters referred to the stator\n", + "Xr_=2 # rotor winding reactance in ohm\n", + "Xs=Xr_ # stator winding reactance in ohm\n", + "Rr_=1 # resistance of the rotor windings in ohm\n", + "Rs=Rr_ # resistance of the stator windings in ohm\n", + "N=950 #full load speed in rpm\n", + "SR=2.3 #stator to rotor turns ratio\n", + "\n", + "#calculation\n", + "Ns=120*f/P #synchronous speed in rpm\n", + "Wms=2*math.pi*Ns/60 \n", + "s=(Ns-N)/Ns #full load slip\n", + "x=math.sqrt((Rs+Rr_/s)**2+(Xs+Xr_)**2)\n", + "Irf_=(Vl/math.sqrt(3))/x #full load current\n", + "Tf=(1/Wms)*3*Irf_**2*Rr_/s #full load torque\n", + "\n", + "#(i)initial braking current and torque\n", + "S=2-s #during plugging at 950rpm\n", + "y=math.sqrt((Rs+Rr_/S)**2+(Xs+Xr_)**2)\n", + "Ir_=(Vl/math.sqrt(3))/y #initial braking current\n", + "ratio1=Ir_/Irf_\n", + "T=(1/Wms)*3*Ir_**2*Rr_/S #initial braking torque\n", + "ratio2=T/Tf\n", + "\n", + "#(ii)when an external resistance is connected such \n", + "#that maximum braking current is 1.5 times the full load current\n", + "Ir_=1.5*Irf_\n", + "x=(Vl/math.sqrt(3))/Ir_ #x=math.sqrt((Rs+(Rr_+Re_)/S)**2+(Xs+Xr_)**2)\n", + "y=x**2 #y=(Rs+(Rr_+Re_)/S)**2+(Xs+Xr_)**2\n", + "z=y-(Xs+Xr_)**2 #z=(Rs+(Rr_+Re_)/S)**2\n", + "a=math.sqrt(z) #a=(Rs+(Rr_+Re_)/S)\n", + "b=(a-Rs)*S #b=(Rr_+Re_)\n", + "Re_=b-Rs\n", + "Re=Re_/SR**2\n", + "T=(1/Wms)*3*Ir_**2*(Rr_+Re_)/S #initial braking torque\n", + "ratio=T/Tf\n", + "\n", + "\n", + "#results\n", + "print\"(i)Ratio of initial braking current to full load current is:\",round(ratio1)\n", + "print\" Ratio of initial braking torque to full load torque is:\",round(ratio2,2)\n", + "print\"\\n(ii)Ratio of initial braking torque to full load torque when the resistance is added is:\",round(ratio,3)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "(i)Ratio of initial braking current to full load current is: 5.0\n", + " Ratio of initial braking torque to full load torque is: 0.64\n", + "\n", + "(ii)Ratio of initial braking torque to full load torque when the resistance is added is: 1.426\n" + ] + } + ], + "prompt_number": 203 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example No:6.5,Page No:165" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from __future__ import division\n", + "import cmath\n", + "\n", + "#variable declaration\n", + "#ratings of the star connected Induction motor\n", + "f=50 # frequency in HZ\n", + "Vl=440 # line voltage in V\n", + "P=6 # number of poles\n", + "Vp=Vl/math.sqrt(3) #phase voltage\n", + "#parameters referred to the stator\n", + "Xr_=1.2 # rotor winding reactance in ohm\n", + "Xs=Xr_ # stator winding reactance in ohm\n", + "Rr_=0.4 # resistance of the rotor windings in ohm\n", + "Rs=0.5 # resistance of the stator windings in ohm\n", + "Xm=50 # no load reactance\n", + "N=950 # full load speed\n", + "Sm=2 # slip at maximum torque\n", + "\n", + "#calculation\n", + "Rr_=Sm*math.sqrt(Rs**2+(Xs+Xr_)**2) #Since Sm=Rr_/math.sqrt(Rs**2+(Xs+Xr_)**2)\n", + "Ns=120*f/P #synchronous speed in rpm\n", + "Wms=2*math.pi*Ns/60 \n", + "s=(Ns-N)/Ns #slip at 950 rpm\n", + "\n", + "x=1j*Xm*(Rr_/s+1j*Xr_)\n", + "y=Rr_/s+1j*(Xr_+Xm)\n", + "Zp=Rs+1j*Xs+x/y\n", + "Ip=Vp/math.sqrt(3)/Zp #In the book the value of Ip is wrong which leads to other subsequent wrong answers in the book\n", + "Irp_=Ip*(1j*Xm)/(Rr_/s+1j*(Xr_+Xm))\n", + "Tp=(1/Wms)*3*abs(Irp_)**2*Rr_/s\n", + "x=1j*Xm*(Rr_/(2-s)+1j*Xr_)\n", + "y=Rr_/(2-s)+1j*(Xr_+Xm)\n", + "Zn=Rs+1j*Xs+x/y\n", + "In=Vp/math.sqrt(3)/Zn\n", + "Irn_=In*(1j*Xm)/(Rr_/(2-s)+1j*(Xr_+Xm))\n", + "Tn=-(1/Wms)*3*abs(Irn_)**2*Rr_/(2-s) #In the book the value of In is wrong\n", + "\n", + "T=Tp-Tn\n", + "I=abs(Ip)+abs(In)\n", + "Rr_=0.4 # from the parameters of the motor referred to the stator\n", + "x=math.sqrt((Rs+Rr_/s)**2+(Xs+Xr_)**2)\n", + "If=(Vl/math.sqrt(3))/x #full load current\n", + "Tf=(1/Wms)*3*If**2*Rr_/s #full load torque\n", + "\n", + "ratio1=I/If\n", + "ratio2=abs(T)/Tf\n", + "\n", + "#results\n", + "print\"Ratio of braking current to full load current is:\",round(ratio1,3)\n", + "print\"Ratio of braking torque to full load torque is:\",round(ratio2,3)\n", + "#answer in the book is wrong\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "Ratio of braking current to full load current is: 1.448\n", + "Ratio of braking torque to full load torque is: 0.565\n" + ] + } + ], + "prompt_number": 204 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example No:6.6,Page No:171" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from __future__ import division\n", + "from array import array\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "import matplotlib.cbook as cbook\n", + "%matplotlib inline\n", + "\n", + "#variable declaration\n", + "#ratings of the star connected Induction motor which operates under dynamic braking\n", + "f=50 # frequency in HZ\n", + "P=4 # number of poles\n", + "#parameters referred to the stator\n", + "Xr_=3.01 # rotor winding reactance in ohm\n", + "Xs=Xr_ # stator winding reactance in ohm\n", + "Rr_=4.575 # resistance of the rotor windings in ohm\n", + "Rs=1.9 # resistance of the stator windings in ohm\n", + "J=0.1 # moment of inertia of the motor load system in kg-m2\n", + "Id=12 # given DC current\n", + "\n", + "N=1500 #given asynchronous speed\n", + "#magnetization chacrateristic at the given asynchronous speed\n", + "Im=[0.13,0.37,0.6,0.9,1.2,1.7,2.24,2.9,3.9,4.9,6,8,9,9.5] #magnetization current\n", + "E=[12.8,32,53.8,80,106,142,173,200,227,246,260,280,288,292] #back emf\n", + "\n", + "#calculation\n", + "torque=[]\n", + "speed=[]\n", + "temp=[]\n", + "Is=math.sqrt(2/3)*Id #value of stator current for two lead connection\n", + "Wms=2*math.pi*N/60\n", + "for i in range (1,14,1):\n", + " x=(Is**2-Im[i]**2)/(1+2*Xr_*Im[i]/E[i]) #x=Ir_**2\n", + " Ir_=math.sqrt(x) #required rotor current\n", + " y=(E[i]/Ir_)**2-Xr_**2\n", + " S=Rr_/math.sqrt(y) #required slip\n", + " N=S*Ns #required speed\n", + " T=(3/Wms)*(Ir_)**2*Rr_/S #required torque\n", + " speed.append(round(N))\n", + " torque.append(round(T,1))\n", + " temp.append(round(T,1))\n", + "print\"Hence the magnetizatioin curve is\"\n", + "print\"Speed :\",speed,\"rpm\"\n", + "for i in range(0,13,1):\n", + " torque[i]=-1*torque[i]\n", + "print\"Braking torque is :\",torque,\"N-m\"\n", + "\n", + "#results\n", + "#plot of of torque vs speed\n", + "plt.figure(1)\n", + "plt.plot(torque,speed)\n", + "plt.xlabel('Torque, N-m') \n", + "plt.ylabel('Speed, rpm')\n", + "plt.title('Torque vs Speed')\n", + "plt.grid(True)\n", + "plt.tight_layout()\n", + "plt.show()\n", + "\n", + "#plot of Wm vs J/T\n", + "inertia_over_torque=[]\n", + "for i in range(2,13,1):\n", + " J_T=1000*J/temp[i]\n", + " inertia_over_torque.append(round(J_T,4))\n", + "print\"J/t :\",inertia_over_torque \n", + "\n", + "Wm=[1,4,8,12,16,20,25,55,95,125,160] \n", + "#the values of Wm are taken for the angular frequency with maximum value of Wms=50*pi rad/s\n", + "plt.figure(2)\n", + "plt.plot(Wm,inertia_over_torque)\n", + "plt.xlabel(r'Angular speed $\\omega_m$') \n", + "plt.ylabel(' $J/T,1*10^-2$')\n", + "plt.title(r'$J/T vs \\omega_m$')\n", + "plt.grid(True)\n", + "x=[6.5,6.5]\n", + "y=[2,4.5]\n", + "plt.plot(x,y,'blue')\n", + "plt.annotate('A',xy=(8,2), xytext=(8,2.1), arrowprops=dict(facecolor='black', shrink=0),)\n", + "plt.annotate('B',xy=(8,4.5), xytext=(8,4.6), arrowprops=dict(facecolor='black', shrink=0),)\n", + "plt.annotate('C',xy=(80,3.4), xytext=(80,3.5), arrowprops=dict(facecolor='black', shrink=0),)\n", + "plt.annotate('D',xy=(157,8.3), xytext=(157,8.4), arrowprops=dict(facecolor='black', shrink=0),)\n", + "plt.annotate('E',xy=(157,2), xytext=(157,2.1), arrowprops=dict(facecolor='black', shrink=0),)\n", + "plt.show()\n", + "\n", + "print\"Hence from the plot the area ABCDEA between the curve and the speed axis for speed change \"\n", + "print\"for synchronous to 0.02 times synchrnous speed is the stopping time is which equal to: 9.36 sec\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "Hence the magnetizatioin curve is\n", + "Speed : [2975.0, 949.0, 578.0, 421.0, 306.0, 246.0, 207.0, 174.0, 150.0, 128.0, 86.0, 56.0, 34.0] rpm\n", + "Braking torque is : [-2.6, -8.3, -13.5, -18.4, -24.8, -30.0, -34.0, -36.9, -37.5, -35.9, -27.9, -19.5, -12.2] N-m\n" + ] + }, + { + "metadata": {}, + "output_type": "display_data", + "png": 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+ "text": [ + "<matplotlib.figure.Figure at 0x7f444007ee50>" + ] + }, + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "Hence from the plot the area ABCDEA between the curve and the speed axis for speed change \n", + "for synchronous to 0.02 times synchrnous speed is the stopping time is which equal to: 9.36 sec\n" + ] + } + ], + "prompt_number": 205 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example No:6.7,Page No:176" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from __future__ import division\n", + "\n", + "#variable declaration\n", + "#ratings of the star connected Induction motor\n", + "f=50 # frequency in HZ\n", + "Vl=2200 # line voltage in V\n", + "P=6 # number of poles\n", + "#parameters referred to the stator\n", + "Xr_=0.5 # rotor winding reactance in ohm\n", + "Xs=Xr_ # stator winding reactance in ohm\n", + "Rr_=0.12 # resistance of the rotor windings in ohm\n", + "Rs=0.075 # resistance of the stator windings in ohm\n", + "J=100 # combine inertia of motor and load in kg-m2\n", + "\n", + "#calculation\n", + "#(i) during starting of the motor\n", + "Sm=Rr_/math.sqrt(Rs**2+(Xs+Xr_)**2) #slip at maximum torque\n", + "Wms=4*math.pi*f/P #angular frequency\n", + "x=Rs+math.sqrt(Rs**2+(Xs+Xr_)**2)\n", + "Tmax=(3/2/Wms)*(Vl/math.sqrt(3))**2/x #maximum torque\n", + "tm=J*Wms/Tmax #mechanical time constant of the motor\n", + "ts=tm*(1/4/Sm+1.5*Sm) #time taken during starting\n", + "Es=1/2*J*Wms**2*(1+Rs/Rr_) #energy disspated during starting\n", + "\n", + "#(ii) when the motor is stopped by plugging method\n", + "tb=tm*(0.345*Sm+0.75/Sm) #time required to stop by plugging\n", + "Eb=3/2*J*Wms**2*(1+Rs/Rr_) #energy disspated during braking \n", + "\n", + "#(iii)required resistance to be inserted during plugging\n", + "tb1=1.027*tm #minimum value of stopping time during braking\n", + "x=1.47*(Xs+Xr_) #x=Rr_+Re\n", + "Re=x-Rr_ #Re is the required external resistance to be connected\n", + "Ee=3/2*J*Wms**2*(Re/(Re+Rr_)) #energy disspated in the external resistor\n", + "Eb1=Eb-Ee #total energy disspated during braking \n", + "\n", + "\n", + "#results\n", + "print\"(i)Time taken during starting is ts:\",round(ts,4),\"s\"\n", + "print\" Energy disspated during starting is Es:\",round(Es/1000),\"kilo-watt-sec\"\n", + "print\"\\n(ii)Time taken to stop by plugging is tb:\",round(tb,2),\"s\"\n", + "print\" Energy disspated during braking is Eb:\",round(Eb/1000),\"kilo-watt-sec\"\n", + "print\"\\n(iii)Minimum Time taken to stop by plugging is tb:\",round(tb1,2),\"s\"\n", + "print\" Required external resistance to be connected is Re:\",round(Re,2),\"ohm\"\n", + "print\" Total Energy dissipated during braking is Eb:\",round(Eb1/1000,2),\"kilo-watt-sec\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "(i)Time taken during starting is ts: 1.108 s\n", + " Energy disspated during starting is Es: 891.0 kilo-watt-sec\n", + "\n", + "(ii)Time taken to stop by plugging is tb: 3.08 s\n", + " Energy disspated during braking is Eb: 2673.0 kilo-watt-sec\n", + "\n", + "(iii)Minimum Time taken to stop by plugging is tb: 0.5 s\n", + " Required external resistance to be connected is Re: 1.35 ohm\n", + " Total Energy dissipated during braking is Eb: 1162.36 kilo-watt-sec\n" + ] + } + ], + "prompt_number": 206 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example No:6.8,Page No:184" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from __future__ import division\n", + "import cmath\n", + "from sympy import *\n", + "from array import array\n", + "import numpy as np\n", + "\n", + "#variable declaration\n", + "#ratings of the delta connected Induction motor\n", + "f=50 # frequency in HZ\n", + "Vl=400 # line voltage in V\n", + "P=4 # number of poles\n", + "Pm=2.8*1000 # rated mechanical power developed\n", + "N=1370 # rated speed in rpm\n", + "#parameters referred to the stator\n", + "Xr_=5 # rotor winding reactance in ohm\n", + "Xs=Xr_ # stator winding reactance in ohm\n", + "Rr_=5 # resistance of the rotor windings in ohm\n", + "Rs=2 # resistance of the stator windings in ohm\n", + "Xm=80 # no load reactance\n", + "\n", + "#calculation\n", + "Ns=120*f/P #synchronous speed in rpm\n", + "Wms=2*math.pi*Ns/60 #synchronous speed in rad/s\n", + "s=(Ns-N)/Ns #full load slip\n", + "x=(Rs+Rr_/s)**2+(Xs+Xr_)**2 #total impedance\n", + "T=(3/Wms)*(Vl**2*Rr_/s)/x #full load torque\n", + "Tl=T\n", + "K=Tl/(1-s)**2 #since Tl=K*(1-s)**2\n", + "\n", + "#(i) when the motor is running at 1200 rpm\n", + "N1=1200 #speed in rpm\n", + "s1=(Ns-N1)/Ns #slip at the given speed N1\n", + "Tl=K*(1-s1)**2 #torque at the given speed N1\n", + "\n", + "y=(Rs+Rr_/s1)**2+(Xs+Xr_)**2 #total impedance\n", + "a=Tl*(Wms/3)*y*(s1/Rr_) #Since T=(3/Wms)*(Vl**2*Rr_/s)/x and a=V**2\n", + "V=math.sqrt(a) #required voltage at the given speed N1\n", + "Ir_=V/((Rs+Rr_/s1)+1j*(Xs+Xr_))#rotor current\n", + "Im=V/(1j*Xm) #magnetizing current\n", + "Is=Ir_+Im #total current\n", + "Il=abs(Is)*math.sqrt(3) #line current\n", + "\n", + "#(ii)when the terminal voltage is 300 V\n", + "V1=300 #terminal voltage in V\n", + "s = Symbol('s')\n", + "x=(Rs+Rr_/s)**2+(Xs+Xr_)**2 \n", + "T=(3/Wms)*(V1**2*Rr_/s)/x\n", + "\n", + "#Now we have to solve for the value of slip 's' from the given equation 104s**4- 188s**3 + 89s**2 - 179s + 25=0\"\n", + "coeff = [104,-188,89,-179,25] #coeffcient of the polynomial equation \n", + "s=np.roots(coeff) #roots of the polynomial equation\n", + "\n", + "T=K*(1-s[3].real)**2 #torque at the given terminal voltage of 300 V\n", + "N=Ns*(1-s[3].real) #speed at the given terminal voltage of 300 V\n", + "Ir_=V1/((Rs+Rr_/s[3].real)+1j*(Xs+Xr_))#rotor current\n", + "Im=V1/(1j*Xm) #magnetizing current\n", + "Is=Ir_+Im #total current\n", + "Il1=abs(Is)*math.sqrt(3) #line current\n", + "\n", + "\n", + "#results\n", + "print\"(i)Required torque is Tl:\",round(Tl,1),\"N-m\"\n", + "print\" Required motor terminal voltage is V:\",round(V,1),\"V\"\n", + "print\" Required line current is Il:\",round(Il,2),\"A\"\n", + "print\"\\n(ii)The roots of the polynomial equation are \"\n", + "print\" s1:\",round(s[0].real,3),\"s2:\",round(s[1].real,3),\"s3:\",round(s[2].real,3),\"s4:\",round(s[3].real,3)\n", + "print\" Hence Only s4:\",round(s[3].real,3),\"is valid\"\n", + "print\"\\nRequired torque is Tl:\",round(T,2),\"N-m\"\n", + "print\"Required speed is N:\",round(N),\"rpm\"\n", + "print\"Required line current is Il:\",round(Il1,2),\"A\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "(i)Required torque is Tl: 36.9 N-m\n", + " Required motor terminal voltage is V: 253.2 V\n", + " Required line current is Il: 17.89 A\n", + "\n", + "(ii)The roots of the polynomial equation are \n", + " s1: 1.818 s2: -0.079 s3: -0.079 s4: 0.147\n", + " Hence Only s4: 0.147 is valid\n", + "\n", + "Required torque is Tl: 41.94 N-m\n", + "Required speed is N: 1279.0 rpm\n", + "Required line current is Il: 16.88 A\n" + ] + } + ], + "prompt_number": 207 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example No:6.9,Page No:199" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from __future__ import division\n", + "from array import array\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "%matplotlib inline\n", + "\n", + "#variable declaration\n", + "#ratings of the star connected squirrel Induction motor\n", + "f=50 # frequency in HZ\n", + "Vl=400 # line voltage in V\n", + "P=4 # number of poles\n", + "N=1370 # rated speed\n", + "\n", + "#the frequency variation is from 10 Hz to 50 Hz\n", + "fmin=10 \n", + "fmax=50\n", + "#parameters referred to the stator\n", + "Xr_=3.5 # rotor winding reactance in ohm\n", + "Xs=Xr_ # stator winding reactance in ohm\n", + "Rr_=3 # resistance of the rotor windings in ohm\n", + "Rs=2 # resistance of the stator windings in ohm\n", + "\n", + "#calculation\n", + "Ns=120*f/P #synchronous speed\n", + "N1=Ns-N #increase in speed from no load to full torque rpm\n", + "Wms=2*math.pi*Ns/60\n", + "s=(Ns-N)/Ns #full load slip\n", + "#(i)to obtain the plot between the breakdown torque and the frequency\n", + "K=0.1\n", + "k=[]\n", + "frequency=[]\n", + "torque=[]\n", + "for i in range (0,9):\n", + " K=K+0.1\n", + " f1=K*f\n", + " x=Rs/K+math.sqrt((Rs/K)**2+(Xs+Xr_)**2)\n", + " Tmax=(3/2/Wms)*(Vl/math.sqrt(3))**2/x\n", + " k.append(round(K,1))\n", + " frequency.append(round(f1))\n", + " torque.append(round(Tmax,2))\n", + "print\"K :\",k\n", + "print\"f :\",frequency,\"Hz\"\n", + "print\"Tmax :\",torque,\"N-m\"\n", + "\n", + "#plotting the values of Tmax vs f \n", + "plt.figure(1)\n", + "plt.plot(frequency,torque,'y')\n", + "plt.xlabel('$f$,Hz')\n", + "plt.ylabel('$Tmax$,N-m')\n", + "plt.grid(True)\n", + "plt.title('Torque vs frequency characteristic')\n", + "plt.show()\n", + "\n", + "#(ii) to obtain the starting torque and current at rated frequency and voltage\n", + "x=(Rs+Rr_)**2+(Xs+Xr_)**2\n", + "Tst=(3/Wms)*(Vl/math.sqrt(3))**2*Rr_/x #starting torque at 50 Hz frequency\n", + "Ist=(Vl/math.sqrt(3))/math.sqrt(x) #starting current at 50 Hz frequency\n", + "\n", + "K=fmin/fmax #minimum is available at 10 Hz\n", + "y=((Rs+Rr_)/K)**2+(Xs+Xr_)**2\n", + "Tst_=(3/Wms)*(Vl/math.sqrt(3))**2*Rr_/K/y #starting torque at 10 Hz frequency\n", + "Ist_=(Vl/math.sqrt(3))/math.sqrt(y) #starting current at 10 Hz frequency\n", + "\n", + "ratio1=Tst_/Tst #ratio of starting torque to the rated starting torque\n", + "ratio2=Ist_/Ist #ratio of starting current to the rated starting current\n", + "\n", + "\n", + "#results\n", + "print\"\\n(i)Hence from the plot we can see that for a constant (V/f) ratio breakdown torque decreases with frequency\"\n", + "print\"\\n(ii)Hence the required ratio of starting torque to the rated starting torque is :\",round(ratio1,3)\n", + "print\" Hence the required ratio of starting current to the rated starting current is :\",round(ratio2,2)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "K : [0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0]\n", + "f : [10.0, 15.0, 20.0, 25.0, 30.0, 35.0, 40.0, 45.0, 50.0] Hz\n", + "Tmax : [22.93, 31.18, 37.44, 42.22, 45.94, 48.89, 51.27, 53.24, 54.88] N-m\n" + ] + }, + { + "metadata": {}, + "output_type": "display_data", + "png": 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9/XCEhaXTF9UQYkTUQ+CQGOYVhZqRMS0uX/4Qp08/g169PoJa/ZwoioFQx/NO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\n1zuCLgZimfuknNyinNwSS05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CIYQQAFQQCCGE3EQFgRBCCAAqCIQQQm6igkAIIQQAFQRC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\nB9dt23sqOyvAAAAAAElFTkSuQmCC\n", + "text": [ + "<matplotlib.figure.Figure at 0x7f443fef8950>" + ] + }, + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "\n", + "(i)Hence from the plot we can see that for a constant (V/f) ratio breakdown torque decreases with frequency\n", + "\n", + "(ii)Hence the required ratio of starting torque to the rated starting torque is : 0.549\n", + " Hence the required ratio of starting current to the rated starting current is : 0.33\n" + ] + } + ], + "prompt_number": 208 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example No:6.10,Page No:201" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from __future__ import division\n", + "from array import array\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "%matplotlib inline\n", + "\n", + "#variable declaration\n", + "#ratings of the star connected squirrel Induction motor is same as that of Ex-6.9\n", + "f=50 # frequency in HZ\n", + "Vl=400 # line voltage in V\n", + "P=4 # number of poles\n", + "N=1370 # rated speed\n", + "\n", + "#the frequency variation is from 5 Hz to 50 Hz\n", + "fmin=5 \n", + "fmax=50\n", + "#parameters referred to the stator\n", + "Xr_=3.5 # rotor winding reactance in ohm\n", + "Xs=Xr_ # stator winding reactance in ohm\n", + "Rr_=3 # resistance of the rotor windings in ohm\n", + "Rs=2 # resistance of the stator windings in ohm\n", + "\n", + "#calculation\n", + "Ns=120*f/P #synchronous speed\n", + "N1=Ns-N #increase in speed from no load to full torque rpm\n", + "Wms=2*math.pi*Ns/60\n", + "s=(Ns-N)/Ns #full load slip\n", + "Tmax=torque[8] #maximum torque as obtain from Ex-6.9\n", + "#to obtain the plot between the voltage and the frequency\n", + "K=0\n", + "k=[]\n", + "frequency=[]\n", + "line_voltage=[]\n", + "for i in range (0,10):\n", + " K=K+0.1\n", + " f1=K*f\n", + " x=2*K*Wms*Tmax/3\n", + " y=Rs+math.sqrt((Rs)**2+(K*(Xs+Xr_))**2)\n", + " Vl_square=3*x*y\n", + " Vl=math.sqrt(Vl_square)\n", + " k.append(round(K,1))\n", + " frequency.append(round(f1))\n", + " line_voltage.append(round(Vl,1))\n", + "print\"K :\",k\n", + "print\"f :\",frequency,\"Hz\"\n", + "print\"Vl :\",line_voltage,\"V\"\n", + "#plotting the values of line voltage Vl vs f \n", + "plt.figure(1)\n", + "plt.plot(frequency,line_voltage,'b')\n", + "plt.xlabel('$f$,Hz')\n", + "plt.ylabel('Line voltae,volts')\n", + "plt.grid(True)\n", + "plt.title('Line voltage vs frequency characteristic')\n", + "#for constant V/f ratio\n", + "x=[0,10,20,30,40,50]\n", + "y=[0,80,160,240,320,400]\n", + "plt.plot(x,y,'--')\n", + "plt.annotate('Constant V/f ratiuo', xy=(21, 160), xytext=(30, 160),arrowprops=dict(facecolor='black', shrink=0),)\n", + "plt.show()\n", + "\n", + "print\"\\nHence for a constant breakdown torque at all frequencies,\"\n", + "print\"V/f ratio has to be progressively increased with increase in frequency\"\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "K : [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0]\n", + "f : [5.0, 10.0, 15.0, 20.0, 25.0, 30.0, 35.0, 40.0, 45.0, 50.0] Hz\n", + "Vl : [84.3, 123.8, 159.2, 193.7, 228.0, 262.3, 296.7, 331.1, 365.5, 400.0] V\n" + ] + }, + { + "metadata": {}, + "output_type": "display_data", + "png": 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wIbB2LfDCC+Xfl9dyNh08t5IJ4/pUNc6Fmq5ykZoqDEYbOlSYQvvSpYoVDKdP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\nTkSuQmCC\n", + "text": [ + "<matplotlib.figure.Figure at 0x7f443fefdb90>" + ] + }, + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "\n", + "Hence for a constant breakdown torque at all frequencies,\n", + "V/f ratio has to be progressively increased with increase in frequency\n" + ] + } + ], + "prompt_number": 209 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example No:6.11,Page No:202" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from __future__ import division\n", + "\n", + "#variable declaration\n", + "#ratings of the star connected squirrel Induction motor is same as that of Ex-6.9\n", + "f=50 # frequency in HZ\n", + "Vl=400 # line voltage in V\n", + "P=4 # number of poles\n", + "N=1370 # rated speed\n", + "#parameters referred to the stator\n", + "Xr_=3.5 # rotor winding reactance in ohm\n", + "Xs=Xr_ # stator winding reactance in ohm\n", + "Rr_=3 # resistance of the rotor windings in ohm\n", + "Rs=2 # resistance of the stator windings in ohm\n", + "\n", + "#calculation\n", + "Ns=120*f/P #synchronous speed\n", + "N1=Ns-N #increase in speed from no load to full torque rpm\n", + "Wms=2*math.pi*Ns/60 #synchronous speed\n", + "s=(Ns-N)/Ns #full load slip\n", + "D=Ns-N #drop in speed from no load to full load torque at 50 Hz\n", + "\n", + "#(i)when the frequency is 30 Hz and 80% of full load torque\n", + "f1=30 #given frequency in Hz\n", + "d=D*0.8 #drop in speed from no load to 80% full load torque\n", + "Ns1=120*f1/P #synchronous speed at the given frequency f1=30 Hz\n", + "N1=Ns1-d #required motor speed \n", + "\n", + "#(ii)when the speed is 1000 rpm for a full load torque\n", + "N2=1000 #given speed in rpm\n", + "Ns2=N2+D #synchronous speed\n", + "f2=P*Ns2/120 #required frequency\n", + "\n", + "#when the speed is 1100 rpm and the frequency is 40 Hz\n", + "N3=1100 #given speed in rpm\n", + "f3=40 #given frequency in Hz\n", + "Ns3=120*f3/P #synchronous speed at the given frequency f1=40 Hz\n", + "D1=Ns3-N3 #drop in speed from no load to N1=1100 rpm\n", + "x=(Rs+Rr_/s)**2+(Xs+Xr_)**2\n", + "Tf=(3/Wms)*(Vl/math.sqrt(3))**2*(Rr_/s)/x #full load torque\n", + "T1=D1/D*Tf #required torque \n", + "\n", + "\n", + "#results\n", + "print\"(i)Hence the required motor speed is :\",round(N1),\"rpm\"\n", + "print\"\\n(ii)Hence the required frequency is :\",round(f2,2),\"Hz\"\n", + "print\"\\n(iii)Hence the required torque is :\",round(T1,2),\"N-m\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "(i)Hence the required motor speed is : 796.0 rpm\n", + "\n", + "(ii)Hence the required frequency is : 37.67 Hz\n", + "\n", + "(iii)Hence the required torque is : 19.52 N-m\n" + ] + } + ], + "prompt_number": 210 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example No:6.12,Page No:204" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from __future__ import division\n", + "\n", + "#variable declaration\n", + "#ratings of the star connected Induction motor is same as that of Ex-6.9\n", + "f=50 # frequency in HZ\n", + "Vl=400 #line voltage in V\n", + "P=4 # number of poles\n", + "N=1370 #rated speed\n", + "\n", + "#parameters referred to the stator\n", + "Xr_=3.5 # rotor winding reactance in ohm\n", + "Xs=Xr_ # stator winding reactance in ohm\n", + "Rr_=3 # resistance of the rotor windings in ohm\n", + "Rs=2 # resistance of the stator windings in ohm\n", + "\n", + "#calculation\n", + "Ns=120*f/P #synchronous speed\n", + "N1=Ns-N #increase in speed from no load to full torque rpm\n", + "#(i)when f1=30Hz and 80% of full load\n", + "Ns=120*f/P\n", + "f1=30 #frequency \n", + "N2=0.8*N1 #increase in speed from no load to 80% of full torque rpm\n", + "Ns1=f1/f*Ns\n", + "N=Ns1+N2 #machine speed\n", + "\n", + "#(ii)at a speed of 1000rpm\n", + "N2=1000 #given speed in rpm\n", + "N3=N2-N1 #synchronous speed\n", + "f3=P*N3/120 #required frequency\n", + "\n", + "#(iii)when frequency is 40Hz and speed is 1300 rpm\n", + "f4=40 #frequency in hz\n", + "N2=1300 #speed in rpm\n", + "Ns=120*f4/P #required synchronous speed in rpm\n", + "N4=N2-Ns #increase in speed from no load speed in rpm\n", + "Tf=25.37 #full load torque as calculated in Ex-6.11\n", + "Tm=-N4/N1*Tf #motor torque\n", + "\n", + "#(iv) when the motor is under dynamic braking\n", + "\n", + "\n", + "#results\n", + "print\"(i)Required speed is :\",N,\"rpm\"\n", + "print\"\\n(ii)required frequency is:\",f3,\"Hz\"\n", + "print\"\\n(iii)Required motor torque :\",round(Tm,2),\"N-m\"\n", + "print\"\\n(iv)The value of the frequency,speed and motor torque calculated in (i),(ii) and(iii)\"\n", + "print\" will be the same when the motor is operated under dynamic braking\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "(i)Required speed is : 1004.0 rpm\n", + "\n", + "(ii)required frequency is: 29.0 Hz\n", + "\n", + "(iii)Required motor torque : -19.52 N-m\n", + "\n", + "(iv)The value of the frequency,speed and motor torque calculated in (i),(ii) and(iii)\n", + " will be the same when the motor is operated under dynamic braking\n" + ] + } + ], + "prompt_number": 211 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example No:6.13,Page No:204" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from __future__ import division\n", + "\n", + "#variable declaration\n", + "#ratings of the star connected Induction motor is same as that of Ex-6.9\n", + "f=50 # frequency in HZ\n", + "Vl=400 #line voltage in V\n", + "P=4 # number of poles\n", + "N=1370 #rated speed\n", + "#parameters referred to the stator\n", + "Xr_=3.5 # rotor winding reactance in ohm\n", + "Xs=Xr_ # stator winding reactance in ohm\n", + "Rr_=3 # resistance of the rotor windings in ohm\n", + "Rs=2 # resistance of the stator windings in ohm\n", + "\n", + "#calculation\n", + "Wms=4*math.pi*f/P\n", + "f1=60 #frequency in Hz during speed control of the motor\n", + "K=f1/f #the value of K at 60Hz \n", + "x=Rs+math.sqrt(Rs**2+K**2*(Xs+Xr_)**2)\n", + "Tmax_=3/(2*K*Wms)*(Vl/math.sqrt(3))**2/x #torque at 60 Hz frequency\n", + "z=Rs+math.sqrt(Rs**2+(Xs+Xr_)**2)\n", + "Tmax=3/(2*Wms)*(Vl/math.sqrt(3))**2/z #maximum torque\n", + "ratio=Tmax_/Tmax #ratio\n", + "\n", + "#results\n", + "print\"Ratio of Motor breakdown torque at 60Hz to rated torque at 50Hz is:\",round(ratio,3)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "Ratio of Motor breakdown torque at 60Hz to rated torque at 50Hz is: 0.727\n" + ] + } + ], + "prompt_number": 212 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example No:6.14,Page No:209" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from __future__ import division\n", + "\n", + "print\"When operating at a frequency K times rated frequency f then\"\n", + "print\"Im**2=[((Rr_/Ksf)**2+(2*pi*Lr_)**2)/((Rr_/Ksf)**2+(2*pi*Lm+2*Pi*Lr_)**2)]*Is**2----(1)\"\n", + "print\"Since Im is constant for constant flux therefore\"\n", + "print\"K*s*f=constant--------(2)\"\n", + "print\"K*Wms*s=constant-------(3) which is the slip speed\"\n", + "print\"s*K=constant----------(4)\"\n", + "print\"Thereofre for a frequency K*f\"\n", + "print\"T=(3/K/Wms)*[(Is*K*Xm)**2*(Rr_/s)/((Rr_/s)**2+K**2*(Xm+Xr_)**2]\"\n", + "print\"T=(3/K/Wms*s)*[(Is*Xm)**2*(Rr_)/((Rr_/s/K)**2+(Xm+Xr_)**2]-------(5)\"\n", + "print\"\\nHence for a given slip speed (K*Wms*s),K*s is constant and from (1) for a given K*s*f and constant flux\"\n", + "print\"operation Is is fixed. Now from (5) T is also fixed. Thus, motor develps a constant torque and draws a\"\n", + "print\"constant current from the inverter at all frequencies for a given slip speed\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "When operating at a frequency K times rated frequency f then\n", + "Im**2=[((Rr_/Ksf)**2+(2*pi*Lr_)**2)/((Rr_/Ksf)**2+(2*pi*Lm+2*Pi*Lr_)**2)]*Is**2----(1)\n", + "Since Im is constant for constant flux therefore\n", + "K*s*f=constant--------(2)\n", + "K*Wms*s=constant-------(3) which is the slip speed\n", + "s*K=constant----------(4)\n", + "Thereofre for a frequency K*f\n", + "T=(3/K/Wms)*[(Is*K*Xm)**2*(Rr_/s)/((Rr_/s)**2+K**2*(Xm+Xr_)**2]\n", + "T=(3/K/Wms*s)*[(Is*Xm)**2*(Rr_)/((Rr_/s/K)**2+(Xm+Xr_)**2]-------(5)\n", + "\n", + "Hence for a given slip speed (K*Wms*s),K*s is constant and from (1) for a given K*s*f and constant flux\n", + "operation Is is fixed. Now from (5) T is also fixed. Thus, motor develps a constant torque and draws a\n", + "constant current from the inverter at all frequencies for a given slip speed\n" + ] + } + ], + "prompt_number": 213 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example No:6.15,Page No:210" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from __future__ import division\n", + "\n", + "#variable declaration\n", + "#ratings of the star connected Induction motor \n", + "f=50 # frequency in HZ\n", + "Vl=400 #line voltage in V\n", + "P=4 # number of poles\n", + "N=1370 #rated speed\n", + "#parameters referred to the stator\n", + "Xr_=3.5 # rotor winding reactance in ohm\n", + "Xs=Xr_ # stator winding reactance in ohm\n", + "Rr_=3 # resistance of the rotor windings in ohm\n", + "Rs=2 # resistance of the stator windings in ohm\n", + "Xm=55 # magnetizing reactance in ohm\n", + "\n", + "#calculation\n", + "Ns=120*f/P #synchronous speed in rpm\n", + "s=(Ns-N)/Ns #full load slip\n", + "x=1j*Xm*(Rr_/s+1j*Xr_)\n", + "y=Rr_/s+1j*(Xr_+Xm)\n", + "Z=Rs+1j*Xs+x/y #total motor impedance\n", + "Isf=(Vl/math.sqrt(3))/abs(Z) #full load stator current\n", + "Irf_=Isf*(1j*Xm)/(Rr_/s+1j*(Xr_+Xm)) #full load rotor current\n", + "Tf=(3/Wms)*abs(Irf_)**2*Rr_/s #full load torque\n", + "N1=Ns-N #full load slip speed\n", + "#(i) when the motor is operating at 30Hz\n", + "f1=30 #given frequency in Hz\n", + "#at rated slep speedvalue of torque and stator current is same as the rated value\n", + "T=Tf \n", + "Is=abs(Isf) \n", + "Ns1=f1/f*Ns #synchronous at f1=30Hz\n", + "N2=Ns1-N1 #required motor speed at 30Hz\n", + "\n", + "#(ii)at a speed of 1200 rpm\n", + "N3=1200 #speed in rpm\n", + "Ns1=N3+N1 #required synchronous speed\n", + "f1=Ns1/Ns*f #required frequency at N2=1200rpm\n", + "\n", + "#(iii)when speed-torque curves are assumed to be straight lines at 30Hz at half the rated motor torque\n", + "f2=30 #frequency in Hz\n", + "N1_=N1/2 #slip at half the rated torque\n", + "Ns1=f2/f*Ns #synchronous at f1=30Hz\n", + "N4=Ns1-N1_ #required motor speed\n", + "\n", + "#(iv) When the motor is operating at 45hz and braking torque is equal to rated torque\n", + "f3=45 #given frequency in Hz\n", + "N1_=-N1 #slip speed when braking at rated torque\n", + "Ns1=f3/f*Ns #synchronous at f1=45Hz\n", + "N5=Ns1-N1_ #required motor speed\n", + "\n", + "\n", + "#results\n", + "print\"(i)At 30Hz the required value of Torque is T:\",round(T,2),\"N-m \"\n", + "print\" Stator current is Is:\",round(Is,4),\"A\"\n", + "print\" Motor speed is :\",N2,\"rpm\"\n", + "print\"\\n(ii)Required inverter frequency is :\",round(f1,2),\"Hz\"\n", + "print\"\\n(iii)Required motor speed at 30Hz is:\",N4,\"rpm\"\n", + "print\"\\n(iv)Required motor speed at 45Hz is:\",N5,\"rpm\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "(i)At 30Hz the required value of Torque is T: 22.71 N-m \n", + " Stator current is Is: 7.2442 A\n", + " Motor speed is : 770.0 rpm\n", + "\n", + "(ii)Required inverter frequency is : 44.33 Hz\n", + "\n", + "(iii)Required motor speed at 30Hz is: 835.0 rpm\n", + "\n", + "(iv)Required motor speed at 45Hz is: 1480.0 rpm\n" + ] + } + ], + "prompt_number": 214 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example No:6.16,Page No:215" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "import cmath\n", + "from __future__ import division\n", + "\n", + "#variable declaration\n", + "#ratings of the Delta connected slipring Induction motor\n", + "f=50 # frequency in HZ\n", + "Vl=400 #line voltage in V\n", + "P=6 # number of poles\n", + "SR=2.2 #ratio of stator to rotor\n", + "#parameters referred to the stator\n", + "Xr_=1 # rotor winding reactance in ohm \n", + "Rr_=0.2 # resistance of the rotor windings in ohm\n", + "s=0.04 # given slip when motor runs at full load\n", + "\n", + "#calculation\n", + "Ns=120*f/P #synchronous speed\n", + "Wms=2*math.pi*Ns/60\n", + "x=(Rr_/s)**2+Xr_**2\n", + "Tf=(3/Wms)*(Vl)**2*(Rr_/s)/x #full load torque\n", + "K=Tf/(Ns*(1-s))**2\n", + "N=850 #speed of the motor in rpm\n", + "Tl=K*N**2 #torque at the given speed N\n", + "s=(Ns-N)/Ns #slip at the given speed N\n", + "y=Tl*(Wms/3)/Vl**2 #y=X/(X**2+Xr_**2) and X=(Re+Rr_)/s\n", + "print\"The torque at the given speed of 850rpm is:\",round(Tl),\"N-m\"\n", + "print\"With a slip of s:\",s\n", + "print\"To find the external resistance connected the given quadratic equation is X**2-6.633X+1=0\"\n", + "print\"With X=(Re-Rr_)/s where Re is the required external resistance\" \n", + "a = 1\n", + "b = -1/y\n", + "c = 1\n", + "# calculate the discriminant\n", + "d = (b**2) - (4*a*c)\n", + "# find two solutions\n", + "X1 = (-b-cmath.sqrt(d))/(2*a)\n", + "X2 = (-b+cmath.sqrt(d))/(2*a)\n", + "\n", + "#results\n", + "print\"The solution for X are \",round(X1.real,4),\"and\",round(X2.real,4)\n", + "Re1=X1*s-Rr_\n", + "Re2=X2*s-Rr_\n", + "\n", + "if (Re1.real>0) :\n", + " print\"\\nThe number Re1:\",round(abs(Re1),3),\"ohm is feasible:\"\n", + " R=Re1.real/SR**2\n", + " print\"Rotor referred value of the external resistance is:\",round(R,3),\"ohm\"\n", + "\n", + "if (Re2.real>0) :\n", + " print\"\\nThen Re2:\",round(abs(Re2),3),\"ohm is feasible\" \n", + " R=Re2.real/SR**2\n", + " print\"Hence Rotor referred value of the external resistance is:\",round(R,3),\"ohm\" " + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "The torque at the given speed of 850rpm is: 691.0 N-m\n", + "With a slip of s: 0.15\n", + "To find the external resistance connected the given quadratic equation is X**2-6.633X+1=0\n", + "With X=(Re-Rr_)/s where Re is the required external resistance\n", + "The solution for X are 0.1544 and 6.4786\n", + "\n", + "Then Re2: 0.772 ohm is feasible\n", + "Hence Rotor referred value of the external resistance is: 0.159 ohm\n" + ] + } + ], + "prompt_number": 217 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example No:6.17,Page No:217" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from __future__ import division\n", + "from sympy import *\n", + "\n", + "#variable declaration\n", + "#ratings of the star connected Induction motor \n", + "f=50 # frequency in HZ\n", + "Vl=440 #line voltage in V\n", + "P=6 # number of poles\n", + "Ns=120*f/P #synchronous speed\n", + "#parameters referred to the stator\n", + "Xr_=1.2 # rotor winding reactance in ohm\n", + "Xs=Xr_ # stator winding reactance in ohm\n", + "Rr_=0.4 # resistance of the rotor windings in ohm\n", + "Rs=0.5 # resistance of the stator windings in ohm\n", + "Xm=50 # magnetizing reatance\n", + "a=3.5 # stator to rotor turns ratio\n", + "delta=0 # duty ratio at the given breakdown torque\n", + "Sm=1 # slip at standstill\n", + "#calculation\n", + "#slip at maximum torque without an external resistance is Sm=Rr_/math.sqrt(Rs**2+(Xs+Xr_)**2) \n", + "#when an external resistanc Re referred to the stator is connected \n", + "x=Sm*math.sqrt(Rs**2+(Xs+Xr_)**2) #x=Re+Rr_\n", + "Re=x-Rr_\n", + "y=0.5*a**2*(1-delta) # y=0.5*a**2*R*(1-delta) #y=Re\n", + "R=Re/y\n", + "\n", + "N = Symbol('N')\n", + "Sm=(Ns-N)/Ns\n", + "c=(x*Sm-Rr_)/(0.5*a**2*R) #c=(1-delta)\n", + "delta=1-c #given duty ratio\n", + "\n", + "#results\n", + "print\"variation of the duty ratio is\",round(delta*1000/N,3),\"*N*10**(-3)\"\n", + "print\"Hence the duty ratio must change linearly with speed N\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "variation of the duty ratio is 1.195 *N*10**(-3)\n", + "Hence the duty ratio must change linearly with speed N\n" + ] + } + ], + "prompt_number": 218 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example No:6.18,Page No:223" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from __future__ import division\n", + "import cmath\n", + "\n", + "#variable declaration\n", + "#ratings of the star connected Induction motor \n", + "f=50 # frequency in HZ\n", + "Vl=440 # line voltage in V\n", + "P=6 # number of poles\n", + "N=970 # rated speed\n", + "n=2 # ratio of stator to rotor\n", + "Sm=0.25 # it is given the speed range is 25% below the synchronous speed which is proportional to the Slip\n", + "#parameters referred to the stator\n", + "Xr_=0.4 # rotor winding reactance in ohm\n", + "Xs=0.3 # stator winding reactance in ohm\n", + "Rr_=0.08 # resistance of the rotor windings in ohm\n", + "Rs=0.1 # resistance of the stator windings in ohm\n", + "alpha=165 # maximum value of the firing angle in degress\n", + "\n", + "#calculation\n", + "Ns=120*f/P # synchronous speed\n", + "Wms=2*math.pi*Ns/60\n", + "#(i) transformer turns ratio\n", + "a=-Sm/math.cos(math.radians(alpha)) #since Sm=a*math.cos(alpha) \n", + "m=n/a #since a=n/m where m is the transformer ratio\n", + "\n", + "#(ii)When speed is 780 rpm and firing angle is 140 degrees\n", + "N1=780 #given speed\n", + "alpha1=140 #given firing angle\n", + "s1=(Ns-N1)/Ns #slip at the given speed N1\n", + "Vd1=(3*math.sqrt(6)/math.pi)*s1*(Vl/math.sqrt(3))/n\n", + "Vd2=(3*math.sqrt(6)/math.pi)*(Vl/math.sqrt(3))/m*math.cos(math.radians(alpha1))\n", + "Rs_=Rs*(1/n)**2 #stator resistance referred to the rotor\n", + "Rr=Rr_*(1/n)**2 #rotor resistance referred to the rotor\n", + "Rd=0.01 #equivalent resistance of the DC link inductor\n", + "Id=(Vd1+Vd2)/(2*(s1*Rs_+Rr)+Rd)\n", + "T1=abs(Vd2)*Id/s1/Wms #required torque\n", + "\n", + "#(iii)when speed is 800rpm and firing angle is half the rated motor torque\n", + "N1=800 #given speed\n", + "s=(Ns-N)/Ns #rated slip\n", + "x=(Rs+Rr_/s)**2+(Xs+Xr_)**2\n", + "Trated=(3/Wms)*(Vl/math.sqrt(3))**2*(Rr_/s)/x #rated torque\n", + "T_half=Trated/2 #half rated torque\n", + "s1=(Ns-N1)/Ns #given slip at speed N1=800rpm\n", + "Vd1=(3*math.sqrt(6)/math.pi)*s1*(Vl/math.sqrt(3))/n\n", + "cos_alpha1 = Symbol('cos_alpha1')\n", + "Vd2=(3*math.sqrt(6)/math.pi)*(Vl/math.sqrt(3))/m*cos_alpha1\n", + "Id=(Vd1+Vd2)/(2*(s1*Rs_+Rr)+Rd)\n", + "T=abs(Vd2)*Id/s1/Wms #required torque\n", + "#let cos_alpha1=-X\n", + "X=-cos_alpha1\n", + "#since the given torque is half of the rated value\n", + "#To find the find the firing angle we assumed cos(alpha1)=-X\n", + "#The given quadratic equation is X**2-0.772X+0.06425=0\n", + "a = 1\n", + "b = -0.772\n", + "c = 0.06425\n", + "# calculate the discriminant\n", + "d = (b**2) - (4*a*c)\n", + "# find two solutions\n", + "X1 = (-b-cmath.sqrt(d))/(2*a)\n", + "X2 = (-b+cmath.sqrt(d))/(2*a)\n", + "alpha1=-math.acos(X2.real) #since cos(alpha1)=-X where alpha1 is radians\n", + "alpha1=math.degrees(alpha1) #angle in degrees\n", + "alpha1=180+alpha1 #required firing angle\n", + "\n", + "\n", + "#results\n", + "print\"(i)Transformer ratio is: \",round(m,3)\n", + "print\"\\n(ii)Required torque is :\",round(T1),\"N-m\"\n", + "print\" There is a slight difference in the answer for the torque due to accuracy i.e more number of decimal place\"\n", + "print\"\\n(iii)The half rated torque at the given speed of\",N1,\"rpm is:\",round(T_half,2),\"N-m\"\n", + "print\" With a slip of s:\",s1\n", + "print\"The solution for X are \",round(X1.real,4),\"and\",round(X2.real,4)\n", + "print\"For X1:\",round(X1.real,4),\" the motor is unstable hence we discard this value and we use X2:\",round(X2.real,4)\n", + "print\"\\nHence the required firing angle is :\",round(alpha1,1),\"\u00b0\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "(i)Transformer ratio is: 7.727\n", + "\n", + "(ii)Required torque is : 271.0 N-m\n", + " There is a slight difference in the answer for the torque due to accuracy i.e more number of decimal place\n", + "\n", + "(iii)The half rated torque at the given speed of 800 rpm is: 302.66 N-m\n", + " With a slip of s: 0.2\n", + "The solution for X are 0.0949 and 0.6771\n", + "For X1: 0.0949 the motor is unstable hence we discard this value and we use X2: 0.6771\n", + "\n", + "Hence the required firing angle is : 132.6 \u00b0\n" + ] + } + ], + "prompt_number": 219 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example No:6.19,Page No:225" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from __future__ import division\n", + "import cmath\n", + "\n", + "#variable declaration\n", + "#ratings of the star connected Induction motor is same as that of Ex-6.17\n", + "f=50 # frequency in HZ\n", + "Vs=440 # line voltage in V\n", + "P=4 # number of poles\n", + "#parameters referred to the stator\n", + "Xr_=1.2 # rotor winding reactance in ohm\n", + "Xs=Xr_ # stator winding reactance in ohm\n", + "Rr_=0.4 # resistance of the rotor windings in ohm\n", + "Rs=0.5 # resistance of the stator windings in ohm\n", + "Xm=50 # magnetizing reatance\n", + "a=3.5 # stator to rotor turns ratio\n", + "\n", + "#calculation\n", + "Ns=120*f/P # synchronous speed in rpm\n", + "Wms=2*math.pi*Ns/60 # synchronous speed in rad/s\n", + "#(i)when motor speed is 1200rpm with a voltage of 15+0j V\n", + "V=cmath.rect(15,0)\n", + "N1=1200 #speed in rpm\n", + "Vr_=a*V #rotor voltage\n", + "s1=(Ns-N1)/Ns #slip at the given speed N1=1200 rpm\n", + "Z=Rs+Rr_/s1+1j*(Xs+Xr_) #total impedance\n", + "Ir_=(Vs/math.sqrt(3)-Vr_/s1)/Z #rotor current\n", + "phi_r=cmath.phase(Vr_)-cmath.phase(Ir_) #angle between Vr_ and Ir_\n", + "Pr=3*(abs(Ir_))**2*Rr_ #rotor copper loss\n", + "P1=3*abs(Vr_)*abs(Ir_)*math.cos(phi_r) #power absorbed by Vr_\n", + "Pg=(Pr+P1)/s1 #gross power \n", + "T=Pg/Wms #required motor torque\n", + "\n", + "#(ii)when motor speed is 1200rpm with a unity power factor \n", + "N1=1200 #speed in rpm\n", + "Ir_=cmath.rect(abs(Ir_),0) #machine is operating at unity power factor\n", + "x=Ir_*Z #x=(Vs-Vr_/s1)*phi_r where phi_r is the angle between Vr_ and Ir_\n", + "\n", + "#x=a+b\n", + "d=(Vs/math.sqrt(3)-Vr_/s1*math.cos(phi_r))**2\n", + "e=(Vr_/s1*math.sin(phi_r))**2\n", + "f=x/(d+e)\n", + "theta=cmath.phase(f) #required angle in radian\n", + "\n", + "#Now we should solve for the quadratice equation for the rotor current\n", + "# 0.9*Ir_**2 + 50.8*Ir_ + 90.12 = 0\n", + "a1 = 0.9\n", + "b1 = 50.8\n", + "c1 = 90.12\n", + "# calculate the discriminant\n", + "d = (b1**2) - (4*a1*c1)\n", + "# find two solutions\n", + "Ir_1 = (-b1-cmath.sqrt(d))/(2*a1)\n", + "Ir_2 = (-b1+cmath.sqrt(d))/(2*a1)\n", + "\n", + "Ir_=Ir_2 #Ir_2 is chosen because for Ir_1 the motor is unstable\n", + "Vr_sin_phi_r=abs(Ir_)/2.083\n", + "Vr_cos_phi_r=s1*(Vs/math.sqrt(3)+2.5*Vr_sin_phi_r)\n", + "Vr_=Vr_cos_phi_r+1j*Vr_sin_phi_r #total rotor voltage referred to the stator\n", + "Vr_=Vr_/a #total rotor voltage referred to the rotor\n", + "phase=round(math.degrees(cmath.phase(Vr_))) #required phase of the voltage in degrees\n", + "\n", + "#results\n", + "print\"(i)The torque is :\",round(T,2),\"N-m and since it is negative the motor is operating in regenerative braking \"\n", + "print\"\\n(ii)Now theta \u03b8:\",round(math.degrees(theta),2),\"\u25e6\"\n", + "print\" The solution for Ir_ are \",round(Ir_1.real,3),\"and\",round(Ir_2.real,3)\n", + "print\" We choose Ir_:\",round(Ir_2.real,3),\"A since higher value corresponds to unstable region\"\n", + "print\"\\n Hence the required voltage magnitude is Vr:\",round(abs(Vr_),2),\"V\",\";phase:\",phase,\"\u25e6\"\n", + "print\"\\n Note :There is a slight difference in the answers due to accuracy i.e more number of decimal place\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "(i)The torque is : -8.61 N-m and since it is negative the motor is operating in regenerative braking \n", + "\n", + "(ii)Now theta \u03b8: 43.83 \u25e6\n", + " The solution for Ir_ are -54.611 and -1.834\n", + " We choose Ir_: -1.834 A since higher value corresponds to unstable region\n", + "\n", + " Hence the required voltage magnitude is Vr: 14.64 V ;phase: 1.0 \u25e6\n", + "\n", + " Note :There is a slight difference in the answers due to accuracy i.e more number of decimal place\n" + ] + } + ], + "prompt_number": 220 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Example No:6.20,Page No:234" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "from __future__ import division\n", + "import cmath\n", + "\n", + "#variable declaration\n", + "#ratings of the single phase Induction motor \n", + "f=50 # frequency in HZ\n", + "Vs=220 # supply voltage in V\n", + "P=4 # number of poles\n", + "N=1425 # rated speed in rpm\n", + "#parameters referred to the stator\n", + "Xr_=6 # rotor winding reactance in ohm\n", + "Xs=Xr_ # stator winding reactance in ohm\n", + "Rr_=5 # resistance of the rotor windings in ohm\n", + "Rs=2 # resistance of the stator windings in ohm\n", + "Xm=60 # magnetizing reatance\n", + "\n", + "#calculation\n", + "N1=1200 #when the motor is operating at the given speed in rpm\n", + "Ns=120*f/P # synchronous speed\n", + "Wms=2*math.pi*Ns/60\n", + "s=(Ns-N)/Ns #rated slip\n", + "\n", + "Zf=1j*(Xm)*(Rr_/s+1j*Xr_)/2/(Rr_/s+1j*(Xr_+Xm))\n", + "Rf=Zf.real ; Xf=Zf.imag\n", + "Zb=1j*(Xm)*(Rr_/(2-s)+1j*Xr_)/2/(Rr_/(2-s)+1j*(Xr_+Xm))\n", + "Rb=Zb.real ; Xb=Zb.imag\n", + "Zs=Rs+1j*Xs\n", + "Z=Zs+Zf+Zb\n", + "Is=(Vs)/Z\n", + "T=(abs(Is))**2/Wms*(Rf-Rb)\n", + "Tl=T\n", + "K=Tl/N**2\n", + "\n", + "#therefore for a speed of of N1=1200 rpm we get \n", + "Tl=K*N1**2 #required load torque for the given speed N1\n", + "s1=(Ns-N1)/Ns # slip for the given speed N1\n", + "\n", + "Zf=1j*(Xm)*(Rr_/s1+1j*Xr_)/2/(Rr_/s1+1j*(Xm))\n", + "Rf=Zf.real ; Xf=Zf.imag\n", + "Zb=1j*(Xm)*(Rr_/(2-s1)+1j*Xr_)/2/(Rr_/(2-s1)+1j*(Xr_+Xm))\n", + "Rb=Zb.real ; Xb=Zb.imag\n", + "x=(Wms*Tl)/(Rf-Rb) #since Tl=(abs(Is))**2/Wms*(Rf-Rb) and x=Is**2\n", + "Is=math.sqrt(x)\n", + "Z=Zs+Zf+Zb\n", + "V=Is*abs(Z)\n", + "\n", + "#results\n", + "print\"Hence the motor terminal voltage at the speed of\",N1,\"rpm is : \",round(V,1),\"V\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "Hence the motor terminal voltage at the speed of 1200 rpm is : 127.9 V\n" + ] + } + ], + "prompt_number": 221 + } + ], + "metadata": {} + } + ] +}
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