{
"metadata": {
"name": "",
"signature": "sha256:ed4e87165c8ac7092d02719ed97384f22be6f2636ae0697903ffa43695fe04ea"
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Chapter 28: Series resonance and Q-factor
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example 1, page no. 492
"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"from __future__ import division\n",
"import math\n",
"import cmath\n",
"#initializing the variables:\n",
"R = 10;# in ohms\n",
"C = 40e-6;# IN fARADS\n",
"L = 0.075;# IN Henry\n",
"V = 200;# in Volts\n",
"\n",
"#calculation:\n",
" #Resonant frequency,\n",
"fr = 1/(2*math.pi*((L*C)**0.5))\n",
" #Current at resonance, I\n",
"I = V/R\n",
"\n",
"\n",
"#Results\n",
"print \"\\n\\n Result \\n\\n\"\n",
"print \"\\n (a)Resonant frequency = \",round(fr,2),\" Hz\\n\"\n",
"print \"\\n (b)Current at resonance, I is \",I,\" A\\n\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"\n",
" Result \n",
"\n",
"\n",
"\n",
" (a)Resonant frequency = 91.89 Hz\n",
"\n",
"\n",
" (b)Current at resonance, I is 20.0 A\n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example 2, page no. 493
"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"from __future__ import division\n",
"import math\n",
"import cmath\n",
"#initializing the variables:\n",
"R = 8;# in ohms\n",
"L = 0.010;# IN Henry\n",
"f = 1000;# in Hz\n",
"\n",
"#calculation:\n",
" #At resonance\n",
" #capacitance C\n",
"C = 1/(L*(2*math.pi*f)**2)\n",
"\n",
"\n",
"#Results\n",
"print \"\\n\\n Result \\n\\n\"\n",
"print \"\\n capacitance, C is \",round(C*1E6,2),\"uF\\n\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"\n",
" Result \n",
"\n",
"\n",
"\n",
" capacitance, C is 2.53 uF\n"
]
}
],
"prompt_number": 2
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example 3, page no. 493
"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"from __future__ import division\n",
"import math\n",
"import cmath\n",
"#initializing the variables:\n",
"C1 = 1000e-12;# IN fARADS\n",
"C2 = 500e-12;# IN fARADS\n",
"fr1 = 92500;# in Hz\n",
"fr2 = 127800;# in Hz\n",
"\n",
"#calculation:\n",
" #For a series R\u2013L\u2013C circuit the resonant frequency fr is given by:\n",
" #fr = 1/(2pi*(L*C)**2)\n",
"Cs = ((C1 - C2)/((fr2/fr1)**2 - 1)) - C2\n",
"L = 1/((C1 + Cs)*(2*math.pi*fr1)**2)\n",
"\n",
"\n",
"#Results\n",
"print \"\\n\\n Result \\n\\n\"\n",
"print \"\\n (a)stray capacitance, Cs is \",round(Cs*1E12,2),\"pF\\n\"\n",
"print \"\\n (b)inductance, L is \",round(L*1000,2),\"mH\\n\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"\n",
" Result \n",
"\n",
"\n",
"\n",
" (a)stray capacitance, Cs is 50.13 pF\n",
"\n",
"\n",
" (b)inductance, L is 2.82 mH\n"
]
}
],
"prompt_number": 3
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example 4, page no. 497
"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"from __future__ import division\n",
"import math\n",
"import cmath\n",
"#initializing the variables:\n",
"R = 10;# in ohms\n",
"C = 5e-6;# IN fARADS\n",
"rv = 20;#in volts\n",
"thetav = 0;# in degrees\n",
"f = 318.3;# in Hz\n",
"\n",
"#calculation:\n",
"wr = 2*math.pi*f\n",
" #The maximum voltage across the resistance occurs at resonance when the current is a maximum. \n",
" #At resonance,L = 1/c*wr**2\n",
"L = 1/(C*wr**2)\n",
" #voltage\n",
"V = rv*math.cos(thetav*math.pi/180) + 1j*rv*math.sin(thetav*math.pi/180)\n",
" #Current at resonance Ir\n",
"Ir = V/R\n",
" #p.d. across resistance, VR\n",
"VR = Ir*R\n",
" #inductive reactance, XL\n",
"XL = wr*L\n",
" #p.d. across inductance, VL\n",
"VL = Ir*(1j*XL)\n",
" #capacitive reactance, Xc\n",
"Xc = 1/(wr*C)\n",
" #p.d. across capacitor, Vc\n",
"Vc = Ir*(-1j*Xc)\n",
" #Q-factor at resonance, Qr\n",
"Qr = VL.imag/V\n",
"\n",
"\n",
"#Results\n",
"print \"\\n\\n Result \\n\\n\"\n",
"print \"\\n (a)inductance, L is \",round(L*1000,2),\"mH\\n\"\n",
"print \"\\n (b)p.d. across resistance, VR is \",VR,\" V, p.d. across inductance, VL \",round( VL.imag,2),\"j V \"\n",
"print \"and p.d. across capacitor, VC \",round(Vc.imag,2),\" V\\n\"\n",
"print \"\\n (c)Q-factor at resonance, Qr is \",round(abs(Qr),2),\" \\n\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"\n",
" Result \n",
"\n",
"\n",
"\n",
" (a)inductance, L is 50.0 mH\n",
"\n",
"\n",
" (b)p.d. across resistance, VR is (20+0j) V, p.d. across inductance, VL 200.01 j V \n",
"and p.d. across capacitor, VC -200.01 V\n",
"\n",
"\n",
" (c)Q-factor at resonance, Qr is 10.0 \n",
"\n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example 5, page no. 502
"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"from __future__ import division\n",
"import math\n",
"import cmath\n",
"#initializing the variables:\n",
"R = 80;# in ohms\n",
"C = 0.4e-6;# IN fARADS\n",
"L = 0.020;# IN Henry\n",
"Vm = 12;#in volts\n",
"\n",
"#calculation:\n",
" #Resonant frequency,\n",
"fr = 1/(2*math.pi*((L*C)**0.5))\n",
"wr = 2*math.pi*fr\n",
" #Q = wr*L/R\n",
"Q = wr*L/R\n",
"Vc = Q*Vm\n",
" #the frequency f at which VC is a maximum value,\n",
"f = fr*(1 - (1/(2*Q*Q)))**0.5\n",
" #the maximum value of the p.d. across the capacitor is given by:\n",
"Vcm = Vc/((1 - (1/(2*Q*Q)))**0.5)\n",
"\n",
"\n",
"#Results\n",
"print \"\\n\\n Result \\n\\n\"\n",
"print \"\\n (a)The resonant frequency is \",round(fr,2),\" Hz\\n\"\n",
"print \"\\n (b)the value of the p.d. across the capacitor at the resonant frequency \",round(Vc,2),\" V\\n\"\n",
"print \"\\n (c)the frequency f at which Vc is a maximum value, is \",round(f,2),\" Hz\\n\"\n",
"print \"\\n (d)the maximum value of the p.d. across the capacitor is \",round(Vcm,2),\" V\\n\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"\n",
" Result \n",
"\n",
"\n",
"\n",
" (a)The resonant frequency is 1779.41 Hz\n",
"\n",
"\n",
" (b)the value of the p.d. across the capacitor at the resonant frequency 33.54 V\n",
"\n",
"\n",
" (c)the frequency f at which Vc is a maximum value, is 1721.52 Hz\n",
"\n",
"\n",
" (d)the maximum value of the p.d. across the capacitor is 34.67 V\n"
]
}
],
"prompt_number": 5
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example 6, page no. 503
"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"from __future__ import division\n",
"import math\n",
"import cmath\n",
"#initializing the variables:\n",
"QL = 60;# Q-factor\n",
"Qc = 390;# Q-factor\n",
"\n",
"#calculation:\n",
"QT = QL*Qc/(QL + Qc)\n",
"\n",
"\n",
"#Results\n",
"print \"\\n\\n Result \\n\\n\"\n",
"print \"\\n the overall Q-factor is \",QT"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"\n",
" Result \n",
"\n",
"\n",
"\n",
" the overall Q-factor is 52.0"
]
}
],
"prompt_number": 6
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example 7, page no. 505
"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"from __future__ import division\n",
"import math\n",
"import cmath\n",
"#initializing the variables:\n",
"R = 5;# in ohms\n",
"L = 0.010;# IN Henry\n",
"fr = 10000;# in Hz\n",
"\n",
"#calculation:\n",
"wr = 2*math.pi*fr\n",
" #Q-factor at resonance is given by\n",
"Qr = wr*L/R\n",
" #Since Qr = fr/(f2 - f1),\n",
"bw = fr/Qr\n",
"\n",
"\n",
"#Results\n",
"print \"\\n\\n Result \\n\\n\"\n",
"print \"\\n bandwidth of the filter is \",round(bw,2),\" Hz\\n\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"\n",
" Result \n",
"\n",
"\n",
"\n",
" bandwidth of the filter is 79.58 Hz\n"
]
}
],
"prompt_number": 7
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example 8, page no. 507
"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"from __future__ import division\n",
"import math\n",
"import cmath\n",
"#initializing the variables:\n",
"Zr = 50;# in ohms\n",
"fr = 1200;# in Hz\n",
"Qr = 30;# Q-factor\n",
"\n",
"#calculation:\n",
" #At resonance the circuit impedance, Z\n",
"R = Zr\n",
"wr = 2*math.pi*fr\n",
" #Q-factor at resonance is given by Qr = wr*L/R, then L is\n",
"L = Qr*R/wr\n",
" #At resonance r*L = 1/(wr*C)\n",
" #capacitance, C\n",
"C = 1/(L*wr*wr)\n",
" #bandwidth,.(f2 \u2212 f1)\n",
"bw = fr/Qr\n",
" #upper half-power frequency, f2\n",
"f2 = (bw + ((bw**2) + 4*(fr**2))**0.5)/2\n",
" #lower half-power frequency, f1\n",
"f1 = f2 - bw\n",
" #At the half-power frequencies, current I\n",
" #I = 0.707*Ir\n",
" #Hence impedance\n",
"Z = (2**0.5)*R\n",
"\n",
"\n",
"#Results\n",
"print \"\\n\\n Result \\n\\n\"\n",
"print \"\\n (a)inductance, L is \",round(L*1000,2),\"mH\\n\"\n",
"print \"\\n (b)capacitance, C is \",round(C*1E9,2),\"nF\\n\"\n",
"print \"\\n (c)bandwidth is \",round(bw,2),\" Hz\\n\"\n",
"print \"\\n (d)the upper half-power frequency, f2 is \",round(f2,2),\" Hz \"\n",
"print \" and the lower half-power frequency, f1 is \",round(f1,2),\" Hz\\n\"\n",
"print \"\\n (e)impedance at the half-power frequencies is \",round(Z,2),\" ohm\\n\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"\n",
" Result \n",
"\n",
"\n",
"\n",
" (a)inductance, L is 198.94 mH\n",
"\n",
"\n",
" (b)capacitance, C is 88.42 nF\n",
"\n",
"\n",
" (c)bandwidth is 40.0 Hz\n",
"\n",
"\n",
" (d)the upper half-power frequency, f2 is 1220.17 Hz \n",
" and the lower half-power frequency, f1 is 1180.17 Hz\n",
"\n",
"\n",
" (e)impedance at the half-power frequencies is 70.71 ohm\n",
"\n"
]
}
],
"prompt_number": 2
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example 9, page no. 508
"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"from __future__ import division\n",
"import math\n",
"import cmath\n",
"#initializing the variables:\n",
"V = 0.2;# in Volts\n",
"I = 0.004;# in Amperes\n",
"fr = 3000;# in Hz\n",
"Qr = 100;# Q-factor\n",
"\n",
"#calculation:\n",
"wr = 2*math.pi*fr\n",
" #At resonance, impedance\n",
"Z = V/I\n",
" #At resonance the circuit impedance, Z\n",
"R = Z\n",
" #Q-factor at resonance is given by Qr = wr*L/R, then L is\n",
"L = Qr*R/wr\n",
" #At resonance r*L = 1/(wr*C)\n",
" #capacitance, C\n",
"C = 1/(L*wr*wr)\n",
" #Q-factor at resonance in a series circuit represents the voltage magnification Qr = Vc/V, then Vc is\n",
"Vc = Qr*V\n",
"\n",
"\n",
"#Results\n",
"print \"\\n\\n Result \\n\\n\"\n",
"print \"\\n (a)the circuit resistance is \",round(R,2),\" ohm\\n\"\n",
"print \"\\n (b)inductance, L is \",round(L*1000,2),\"mH\\n\"\n",
"print \"\\n (c)capacitance, C is \",round(C*1E9,2),\"nF\\n\"\n",
"print \"\\n (d)the voltage across the capacitor is \",round(Vc,2),\" V\\n\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"\n",
" Result \n",
"\n",
"\n",
"\n",
" (a)the circuit resistance is 50.0 ohm\n",
"\n",
"\n",
" (b)inductance, L is 265.26 mH\n",
"\n",
"\n",
" (c)capacitance, C is 10.61 nF\n",
"\n",
"\n",
" (d)the voltage across the capacitor is 20.0 V\n"
]
}
],
"prompt_number": 9
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example 10, page no. 509
"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"from __future__ import division\n",
"import math\n",
"import cmath\n",
"#initializing the variables:\n",
"R = 8.84;# in ohms\n",
"L = 0.3518;# IN Henry\n",
"C = 20e-6;# IN fARADS\n",
"\n",
"#calculation:\n",
" #Resonant frequency,\n",
"fr = 1/(2*math.pi*((L*C)**0.5))\n",
"wr = 2*math.pi*fr\n",
" #Q-factor at resonance, Q = wr*L/R\n",
"Qr = wr*L/R\n",
" #bandwidth,.(f2 \u2212 f1)\n",
"bw = fr/Qr\n",
" #the lower \u22123 dB frequency\n",
"f1 = fr - bw/2\n",
" #the upper \u22123 dB frequency\n",
"f2 = fr + bw/2\n",
"\n",
"\n",
"#Results\n",
"print \"\\n\\n Result \\n\\n\"\n",
"print \"\\n (a)Resonant frequency, fr is \",round(fr,2),\" Hz\\n\"\n",
"print \"\\n (b)Q-factor at resonance is \",round(Qr,2),\"\\n\"\n",
"print \"\\n (c)Bandwidth is \",round(bw,2),\" Hz\\n\"\n",
"print \"\\n (d)the lower -3dB frequency, f1 is \",round(f1,2),\" Hz \"\n",
"print \" and the upper -3dB frequency, f2 is \",round(f2,2),\" Hz\\n\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"\n",
" Result \n",
"\n",
"\n",
"\n",
" (a)Resonant frequency, fr is 60.0 Hz\n",
"\n",
"\n",
" (b)Q-factor at resonance is 15.0 \n",
"\n",
"\n",
" (c)Bandwidth is 4.0 Hz\n",
"\n",
"\n",
" (d)the lower -3dB frequency, f1 is 58.0 Hz \n",
" and the upper -3dB frequency, f2 is 62.0 Hz\n",
"\n"
]
}
],
"prompt_number": 3
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example 11, page no. 511
"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"from __future__ import division\n",
"import math\n",
"import cmath\n",
"#initializing the variables:\n",
"R = 15;# in ohms\n",
"L = 0.008;# IN Henry\n",
"C = 0.3e-6;# IN fARADS\n",
"rv = 7.56;#in volts\n",
"thetav = 0;# in degrees\n",
"x = 0.03;\n",
"\n",
"#calculation:\n",
" #Resonant frequency,\n",
"fr = 1/(2*math.pi*((L*C)**0.5))\n",
"wr = 2*math.pi*fr\n",
" #At resonance,\n",
"Zr = R\n",
" #voltage\n",
"V = rv*math.cos(thetav*math.pi/180) + 1j*rv*math.sin(thetav*math.pi/180)\n",
" #Current at resonance\n",
"Ir = V/Zr\n",
" #Q-factor at resonance, Q = wr*L/R\n",
"Qr = wr*L/R\n",
" #If the frequency is 3% above fr, then\n",
"de = x\n",
"I = Ir/(1 + (2*de*Qr*1j))\n",
"Z = V/I\n",
"\n",
"\n",
"#Results\n",
"print \"\\n\\n Result \\n\\n\"\n",
"print \"\\n (a)Current at resonance, Ir is \",round(abs(Ir),2),\" A\\n\"\n",
"print \"\\n (b)current flowing in the circuit when frequency 3 percent\"\n",
"print \" above the resonant frequency is \",round(I.real,2),\" + (\",round( I.imag,2),\")i A\\n\"\n",
"print \"\\n (c)impedance of the circuit when the frequency is 3 percent\"\n",
"print \" above the resonant frequency is \",round(Z.real,2),\" + (\",round(Z.imag,2),\")i A\\n\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"\n",
" Result \n",
"\n",
"\n",
"\n",
" (a)Current at resonance, Ir is 0.5 A\n",
"\n",
"\n",
" (b)current flowing in the circuit when frequency 3 percent\n",
" above the resonant frequency is 0.35 + ( -0.23 )i A\n",
"\n",
"\n",
" (c)impedance of the circuit when the frequency is 3 percent\n",
" above the resonant frequency is 15.0 + ( 9.8 )i A\n",
"\n"
]
}
],
"prompt_number": 4
}
],
"metadata": {}
}
]
}