{
"metadata": {
"name": "",
"signature": "sha256:5db7f11d16032beee814462a087253a6b78ac4c2f805e28898833a5c147fee39"
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"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Chapter 32: The superposition theorem
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example 1, page no. 564
"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"from __future__ import division\n",
"import math\n",
"import cmath\n",
"#initializing the variables:\n",
"rv1 = 100;# in volts\n",
"rv2 = 50;# in volts\n",
"thetav1 = 0;# in degrees\n",
"thetav2 = 90;# in degrees\n",
"r1 = 25;# in ohm\n",
"R = 20;# in ohm\n",
"r2 = 10;# in ohm\n",
"\n",
" #calculation:\n",
" #voltage\n",
"V1 = rv1*math.cos(thetav1*math.pi/180) + 1j*rv1*math.sin(thetav1*math.pi/180)\n",
"V2 = rv2*math.cos(thetav2*math.pi/180) + 1j*rv2*math.sin(thetav2*math.pi/180)\n",
" #The circuit diagram is shown in Figure 32.7. Following the above procedure:\n",
" #The network is redrawn with the 50/_90\u00b0 V source removed as shown in Figure 32.8\n",
" #Currents I1, I2 and I3 are labelled as shown in Figure 32.8.\n",
"I1 = V1/(r1 + r2*R/(R + r2))\n",
"I2 = (r2/(r2 + R))*I1\n",
"I3 = (R/(r2 + R))*I1\n",
" #The network is redrawn with the 100/_0\u00b0 V source removed as shown in Figure 32.9\n",
" #Currents I4, I5 and I6 are labelled as shown in Figure 32.9.\n",
"I4 = V2/(r2 + r1*R/(r1 + R))\n",
"I5 = (r1/(r1 + R))*I4\n",
"I6 = (R/(r1 + R))*I4\n",
" #Figure 32.10 shows Figure 32.9 superimposed on Figure 32.8, giving the currents shown.\n",
" #Current in the 20 ohm load,\n",
"I20 = I2 + I5\n",
" #Current in the 100/_0\u00b0 V source\n",
"IV1 = I1 - I6\n",
" #Current in the 50/_90\u00b0 V source\n",
"IV2 = I4 - I3\n",
"\n",
"\n",
"#Results\n",
"print \"\\n\\n Result \\n\\n\"\n",
"print \"\\n (a)current in the 20 ohm load is \",round(I20.real,2),\" + (\",round(I20.imag,2),\")i A\"\n",
"print \"\\n (b)Current in the 100/_0deg V source is \",round(IV1.real,2),\" + (\",round(IV1.imag,2),\")i A\"\n",
"print \"\\n (b)Current in the 50/_90deg V source is \",round(IV2.real,2),\" + (\",round(IV2.imag,2),\")i A\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"\n",
" Result \n",
"\n",
"\n",
"\n",
" (a)current in the 20 ohm load is 1.05 + ( 1.32 )i A\n",
"\n",
" (b)Current in the 100/_0deg V source is 3.16 + ( -1.05 )i A\n",
"\n",
" (b)Current in the 50/_90deg V source is -2.11 + ( 2.37 )i A"
]
}
],
"prompt_number": 1
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example 2, page no. 566
"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"from __future__ import division\n",
"import math\n",
"import cmath\n",
"#initializing the variables:\n",
"V1 = 12;# in volts\n",
"V2 = 20;# in volts\n",
"R1 = 5;# in ohm\n",
"R2 = 4;# in ohm\n",
"R3 = 2.5;# in ohm\n",
"R4 = 6;# in ohm\n",
"R5 = 2;# in ohm\n",
"\n",
"#calculation:\n",
" #Removing the 20 V source gives the network shown in Figure 32.12.\n",
" #Currents I1 and I2 are shown labelled in Figure 32.12\n",
"Re1 = (R4*R5/(R4 + R5)) + R3\n",
"Re2 = Re1*R2/(Re1 + R2) + R1\n",
"I1 = V1/Re2\n",
"I2 = (R2/(Re1 + R2))*I1\n",
" #Removing the 12 V source from the original network gives the network shown in Figure 32.14.\n",
" #Currents I3, I4 and I5 are shown labelled in Figure 32.14.\n",
"Re3 = (R1*R2/(R1 + R2)) + R3\n",
"Re4 = Re3*R4/(Re3 + R4) + R5\n",
"I3 = V2/Re4\n",
"I4 = (R4/(Re3 + R4))*I3\n",
"I5 = (R1/(R1 + R2))*I4\n",
" #Superimposing Figure 32.14 on Figure 32.12 shows that the current flowing in the 4 ohm resistor is given by\n",
"Ir4 = I5 - I2\n",
"\n",
"\n",
"#Results\n",
"print \"\\n\\n Result \\n\\n\"\n",
"print \"\\ncurrent in the 4 ohm resistor of the network is \",round(Ir4,2),\" A\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"\n",
" Result \n",
"\n",
"\n",
"\n",
"current in the 4 ohm resistor of the network is 0.48 A"
]
}
],
"prompt_number": 2
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example 3, page no. 567
"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"from __future__ import division\n",
"import math\n",
"import cmath\n",
"#initializing the variables:\n",
"rv1 = 30;# in volts\n",
"rv2 = 30;# in volts\n",
"thetav1 = 45;# in degrees\n",
"thetav2 = -45;# in degrees\n",
"R1 = 4;# in ohm\n",
"R2 = 4;# in ohm\n",
"R3 = 1j*3;# in ohm\n",
"R4 = -1j*10;# in ohm\n",
"\n",
" #calculation:\n",
" #voltage\n",
"V1 = rv1*math.cos(thetav1*math.pi/180) + 1j*rv1*math.sin(thetav1*math.pi/180)\n",
"V2 = rv2*math.cos(thetav2*math.pi/180) + 1j*rv2*math.sin(thetav2*math.pi/180)\n",
" #The network is redrawn with V2 removed, as shown in Figure 32.17.\n",
" #Current I1 and I2 are shown in Figure 32.17. From Figure 32.17,\n",
"Re1 = R4*(R2 + R3)/(R4 + R3 + R2)\n",
"Re2 = Re1 + R1\n",
" #current\n",
"I1 = V1/Re2\n",
"I2 = (R4/(R2 + R3 + R4))*I1\n",
" #The original network is redrawn with V1 removed, as shown in Figure 32.18\n",
" #Currents I3 and I4 are shown in Figure 32.18. From Figure 32.18,\n",
"Re3 = R1*(R2 + R3)/(R1 + R3 + R2)\n",
"Re4 = Re3 + R4\n",
"I3 = V2/Re4\n",
"I4 = (R1/(R2 + R3 + R1))*I3\n",
" #If the network of Figure 32.18 is superimposed on the network of Figure 32.17, \n",
" #it can be seen that the current in the (4+i3) ohm impedance is given by\n",
"Ir4i3 = I2 - I4\n",
"\n",
"\n",
"#Results\n",
"print \"\\n\\n Result \\n\\n\"\n",
"print \"current in (4 + i3) ohm impedance of the network is \",round(Ir4i3.real,2),\" + (\",round( Ir4i3.imag,2),\")i A\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"\n",
" Result \n",
"\n",
"\n",
"current in (4 + i3) ohm impedance of the network is 2.15 + ( 0.42 )i A\n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example 4, page no. 568
"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"from __future__ import division\n",
"import math\n",
"import cmath\n",
"#initializing the variables:\n",
"E1 = 5 + 0j;# in volts\n",
"E2 = 2 + 4j;# in volts\n",
"Z1 = 3 + 4j;# in ohm\n",
"Z2 = 2 - 5j;# in ohm\n",
"Z3 = 6 + 8j;# in ohm\n",
"\n",
"#calculation:\n",
" #The original network is redrawn with E2 removed, as shown in Figure 32.20.\n",
" #Currents I1, I2 and I3 are labelled as shown in Figure 32.20.\n",
"Ze1 = Z3*Z2/(Z3 + Z2)\n",
"Ze2 = Ze1 + Z1\n",
" #current\n",
"I1 = E1/Ze2\n",
"I2 = (Z2/(Z3 + Z2))*I1\n",
"I3 = (Z3/(Z3 + Z2))*I1\n",
" #The original network is redrawn with E1 removed, as shown in Figure 32.22\n",
" #Currents I4, I5 and I6 are shown labelled in Figure 32.22 \n",
" #with I4 flowing away from the positive terminal of the E2 source.\n",
"Ze3 = Z3*Z1/(Z3 + Z1)\n",
"Ze4 = Ze3 + Z2\n",
"I4 = E2/Ze4\n",
"I5 = (Z1/(Z3 + Z1))*I4\n",
"I6 = (Z3/(Z3 + Z1))*I4\n",
" #If the network of Figure 32.18 is superimposed on the network of Figure 32.17, \n",
" #it can be seen that the current in the (4+i3) ohm impedance is given by\n",
"i1 = I1 + I6\n",
"i2 = I3 + I4\n",
"i3 = I2 - I5\n",
" #magnitude\n",
"i1mag = abs(i1)\n",
"i2mag = abs(i2)\n",
"E1mag = abs(E1)\n",
"E2mag = abs(E2)\n",
" #phase\n",
"phi1 = cmath.phase(complex(i1.real,i1.imag))\n",
"phi2 = cmath.phase(complex(i2.real,i2.imag))\n",
" #voltage across the(6 + i8) ohm impedance\n",
"V6i8 = i3*Z3\n",
"V6i8m = abs(V6i8)\n",
" #power\n",
"P = (E1mag*i1mag*math.cos(phi1)) + (E2mag*i2mag*math.cos(phi2 - cmath.phase(complex(E2.real,E2.imag))))\n",
"\n",
"\n",
"#Results\n",
"print \"\\n\\n Result \\n\\n\"\n",
"print \"\\n(a)currents are: \\n \",round(i1.real,2),\" + (\",round( i1.imag,2),\")i A, \\n \",round(i2.real,2),\" + (\",round(i2.imag,2),\")i A \\n and \",round(i3.real,2),\" + (\",round(i3.imag,2),\")i A\"\n",
"print \"\\n(b)current in the (6 + i8) ohm resistor of the network is \",round(V6i8m,2),\" V\"\n",
"print \"\\n(c)the total active power delivered to the network is \",round(P,2),\" W\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"\n",
" Result \n",
"\n",
"\n",
"\n",
"(a)currents are: \n",
" 0.57 + ( 0.62 )i A, \n",
" 0.56 + ( 1.33 )i A \n",
" and 0.01 + ( -0.71 )i A\n",
"\n",
"(b)current in the (6 + i8) ohm resistor of the network is 7.09 V\n",
"\n",
"(c)the total active power delivered to the network is 9.29 W"
]
}
],
"prompt_number": 5
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example 5, page no. 571
"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"from __future__ import division\n",
"import math\n",
"import cmath\n",
"#initializing the variables:\n",
"rv1 = 50;# in volts\n",
"rv2 = 30;# in volts\n",
"thetav1 = 0;# in degrees\n",
"thetav2 = 90;# in degrees\n",
"R1 = 20;# in ohm\n",
"R2 = 5;# in ohm\n",
"R3 = -1j*3;# in ohm\n",
"R4 = 8;# in ohm\n",
"R5 = 8;# in ohm\n",
"\n",
"#calculation:\n",
" #voltage\n",
"V1 = rv1*math.cos(thetav1*math.pi/180) + 1j*rv1*math.sin(thetav1*math.pi/180)\n",
"V2 = rv2*math.cos(thetav2*math.pi/180) + 1j*rv2*math.sin(thetav2*math.pi/180)\n",
" #The network is redrawn with the V2 source removed, as shown in Figure 32.26.\n",
" #Currents I1 to I5 are shown labelled in Figure 32.26. \n",
" #current\n",
"Re1 = R4*R5/(R5 + R4) + R3\n",
"Re2 = Re1*R2/(R2 + Re1)\n",
"I1 = V1/(Re2 + R1)\n",
"I2 = (Re1/(R2 + Re1))*I1\n",
"I3 = (R2/(Re1 + R2))*I1\n",
"I4 = (R4/(R4 + R5))*I3\n",
"I5 = I3 - I4\n",
" #The original network is redrawn with the V1 source removed, as shown in Figure 32.27.\n",
" #Currents I6 to I10 are shown labelled in Figure 32.27\n",
"Re3 = R1*R2/(R1 + R2)\n",
"Re4 = Re3 + R3\n",
"Re5 = Re4*R4/(Re4 + R4)\n",
"Re6 = Re5 + R5\n",
"I6 = V2/Re6\n",
"I7 = (Re4/(Re4 + R4))*I6\n",
"I8 = (R4/(Re4 + R4))*I6\n",
"I9 = (R1/(R1 + R2))*I8\n",
"I10 = (R2/(R1 + R2))*I8\n",
" #current flowing in the capacitor is given by\n",
"Ic = I3 - I8\n",
" #magnitude of the current in the capacitor\n",
"Icmag = abs(Ic)\n",
"\n",
"i1 = I2 + I9\n",
"i1mag = abs(i1)\n",
" #magnitude of the p.d. across the 5 ohm resistance is given by\n",
"Vr5m = i1mag*R2\n",
" #Active power dissipated in the 20 ohm resistance is given by\n",
"i2 = I1 - I10\n",
"i2mag = abs(i2)\n",
"phii2 = cmath.phase(complex(i2.real,i2.imag))\n",
"Pr20 = R1*(i2mag)**2\n",
" #Active power developed by the V1\n",
"P1 = rv1*i2mag*math.cos(phii2)\n",
" #Active power developed by V2 source\n",
"i3 = I6 - I5\n",
"i3mag = abs(i3)\n",
"phii3 = cmath.phase(complex(i3.real,i3.imag))\n",
"P2 = rv2*i3mag*math.cos(phii3 - (thetav2*math.pi/180))\n",
" #Total power developed\n",
"P = P1 + P2\n",
"\n",
"\n",
"#Results\n",
"print \"\\n\\n Result \\n\\n\"\n",
"print \"\\n(a)the magnitude of the current flowing in the capacitor is \",round(Icmag,2),\" A\"\n",
"print \"\\n(b) the p.d. across the 5 ohm resistance is \",round(Vr5m,2),\" V\"\n",
"print \"\\n(c)the active power dissipated in the 20 ohm resistance is \",round(Pr20,0),\" W\"\n",
"print \"\\n(d)the total active power taken from the supply is \",round(P,1),\" W\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"\n",
" Result \n",
"\n",
"\n",
"\n",
"(a)the magnitude of the current flowing in the capacitor is 2.11 A\n",
"\n",
"(b) the p.d. across the 5 ohm resistance is 5.85 V\n",
"\n",
"(c)the active power dissipated in the 20 ohm resistance is 111.0 W\n",
"\n",
"(d)the total active power taken from the supply is 191.9 W"
]
}
],
"prompt_number": 1
}
],
"metadata": {}
}
]
}