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Inductance Calculations

Working Formulas and Tables

By: FREDERICK W. GROVER

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Paperback | 22 October 2009

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304 Pages

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This authoritative compilation of formulas and tables simplifies the design of inductors for electrical engineers. It features a single simple formula for virtually every type of inductor, together with tables from which essential numerical factors may be interpolated. Although compiled in the 1940s, before calculators and computers, this book provides fundamental equations that professionals and practitioners can use to produce algorithms for computer programs and spreadsheets.Starting with a survey of general principles, it explains circuits with straight filaments; parallel elements of equal length; mutual inductance of unequal parallel filaments and filaments inclined at an angle to each other; and inductance of single-layer coils on rectangular winding forms. Additional topics include the mutual inductance of coaxial circular filaments and of coaxial circular coils; self-inductance of circular coils of rectangular cross section; mutual inductance of solenoid and a coaxial circular filament and coaxial single-layer coils; single-layer coils on cylindrical winding forms; and special types of single-layer coil. An esteemed reference, this volume belongs in the library of every electrical engineer.

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Non-Fiction Science Physics Electricity Engineering & Technology Energy Technology & Engineering Electrical Engineering Science in General

Introduction	p. xiii
General Principles	p. 1
Methods of Calculating Inductances	p. 6
Basic Formulas	p. 6
Formulas for Actual Circuits and Coils	p. 9
Integration of Basic Formulas over the Cross Section of the Winding	p. 9
Taylor's Series Expansions	p. 10
Rayleigh Quadrature Formula	p. 11
Lyle Method of Equivalent Filaments	p. 12
Sectioning Principle	p. 13
Geometric Mean Distance Method	p. 14
Correction for Insulating Space	p. 15
Geometric Mean Distances	p. 17
Equal Rectangles with Corresponding Sides Parallel	p. 18
G.m.d. of Equal Parallel Rectangles, Longer Sides in Same Straight Line	p. 19
G.m.d. of Equal Parallel Rectangles	p. 20
Longer Sides Perpendicular to Line Joining Their Centers	p. 20
G.m.d. of Line of Length a from Itself	p. 21
Circular Area of Radius a from Itself	p. 21
Ellipse with Semiaxes a and b	p. 21
Rectangle, sides B and C	p. 21
Values of Constants for G.m.d. of a Rectangle	p. 22
G.m.d. of an Annulus from Itself	p. 22
Circular Area of Radius a	p. 22
G.m.d. of an Annulus	p. 23
G.m.d. of Point or Area from an Annulus	p. 23
G.m.d. of One Circular Area from Another	p. 24
General Relation among Geometric Mean Distances	p. 24
Construction of and Method of Using the Collection of Working Formulas	p. 26
Circuits Whos Elements are Straight Filaments
Parallel Elements of Equal Length	p. 31
Mutual Inductance of Two Equal Parallel Straight Filaments	p. 31
Values of Q for Use in Formula (2)	p. 32
	p. 34
Mutual Inductance of Two Equal Parallel Conductors	p. 34
	p. 35
Self-inductance of a Straight Conductor	p. 35
	p. 36
Inductance of Multiple Conductors	p. 37
Examples 4 and 5	p. 37
Inductance of Return Circuit of Parallel Conductors	p. 39
Examples 6 and 7	p. 40
Return Circuit of Two Tubular Conductors One Inside the Other	p. 41
	p. 42
Polycore Cable	p. 42
	p. 43
Inductance of Shunts	p. 43
Mutual Inductance of Unequal Parallel Filaments	p. 45
General Case	p. 45
Special Cases	p. 46
Filaments with Their Ends in 45 Common Perpendicular	p. 46
	p. 46
Mutual Inductance of Parallel Conductors of Unequal Length However Placed	p. 47
Mutual Inductance of Filaments an Angle to Each Other	p. 48
Equal Filaments Meeting at a Point	p. 48
Values of Factors in Formula (39) for the Mutual Inductance of Equal Inclined Fila-ments	p. 49
	p. 50
Unequal Filaments Meeting at a Point	p. 50
Unequal Filaments Meeting at a Point. Values of S₁ to Be Used in Formula (45)	p. 51
	p. 51
Unequal Filaments in the Same Plane, Not Meeting	p. 52
	p. 53
Mutual Inductance of Two Straight Filaments Placed in Any Desired Positions	p. 55
	p. 57
Circuits Composed of Combinations of Straight Wires	p. 59
General Formula for the Inductance of a Triangle of Round Wire	p. 59
Rectangle of Round Wire	p. 60
Regular Polygons of Round Wire	p. 60
General Formula for Calculation of the Inductance of Any Plane Figure	p. 60
Values of a for Certain Plane Figures	p. 61
Data for the Calculation of Inductance of Polygons of Round Wire	p. 62
	p. 63
Inductance of Circuits Enclosing Plane Curves	p. 65
	p. 65
Mutual Inductance of Equal, Parallel, Coaxial Polygons of Wire	p. 66
Ratios for Calculating the Mutual Inductance of Coaxial Equal Polygons	p. 67
	p. 69
Inductance of Single-layer Coils on Rectangular Winding Forms	p. 70
Formulas for Different Cases	p. 70
Coefficients, Short Rectangular Solenoid	p. 72
Values of Coefficients, Rectangular Solenoids	p. 72
	p. 73
Coils and Other Circuits Composed of Circular Elements
Mutual Inductance of Coaxial Circular Filaments	p. 77
Formulas for Filaments of Unequal Radii	p. 77
	p. 78
Values of Factor f in Formula (77)	p. 79
Auxiliary Table for Circles Very Close Together	p. 81
Auxiliary Table for Circles Very Far Apart	p. 82
Special Case. Circles of Equal Radii	p. 82
Values of f for Equal Circles Near Together	p. 83
Values of f for Equal Circles Far Apart	p. 84
	p. 85
Auxiliary Table for Equal Circles Very Near Together	p. 85
Auxiliary Table for Equal Circles Very Far Apart	p. 86
Mutual Inductance of Coaxial Circular Coils	p. 88
Mutual Inductance of Coaxial Circles of Wire	p. 88
	p. 89
Mutual Inductance of Coaxial Circular Coils of Rectangular Cross Section	p. 89
Special Case. Equal Coils of Square Cross Section	p. 90
	p. 91
Mutual Inductance of Brooks Coils	p. 91
Coupling Coefficients of Brooks Coils	p. 92
Design of Equal Coaxial Coils of Square Cross Section	p. 92
	p. 93
Design of Mutual Inductance Composed of Two Equal Brooks Coils	p. 93
Self-inductance of Circular Coils of Rectangular Cross Section	p. 94
Nomenclature	p. 94
Inductance of Circular Coils of Square Cross Sec- tion	p. 95
Values of Constant P₀′ in Formula (91) for Coils of Square Cross Section	p. 96
	p. 97
Brooks Coils	p. 97
Correction for Insulating Space	p. 98
	p. 99
Design of Brooks Coil to Obtain a Desired Inductance with a Chosen Size of Wire	p. 99
	p. 100
Design of a Brooks Coil to Obtain a Chosen Inductance and Time Constant	p. 101
	p. 105
Inductance of Circular Coil with Rectangular Cross Section of Any Desired Proportions	p. 105
Values of k for Thin, Long Coils, Formula (99)	p. 106
Values of k for Short, Thick Coils, Formula (99)	p. 107
Values of F for Disc Coils, Formula (100)	p. 108
Values of F for Thin, Long Coils, Formula (100)	p. 109
Interpolation in Tables 22, 23 and 24, 25 (Double Interpolation)	p. 110
	p. 111
Values of P for Disc Coils, Formulas (100) and (100a)	p. 113
Mutual Inductance of Solenoid and a Coaxial Circular Filament	p. 114
Basic Case. Circle in the End Plane of the Solenoid	p. 114
Values of Q₀ for Mutual Inductance Solenoid and Circle, Formula (103)	p. 115
Values of R₀ for the Mutual Inductance of Solenoid and Coaxial Circle, Formula (104)	p. 116
General Case. Circle Not in the End Plane	p. 117
	p. 117
Campbell Form of Mutual Inductance Standard	p. 119
	p. 120
Mutual Inductance of Coaxial Single-layer Coils	p. 122
Nomenclature	p. 122
General Formula	p. 123
	p. 127
Values of B_n as Function of and ², Formula (108)	p. 124
Values of B_n. Auxiliary Table for Large and ²	p. 126
	p. 127
Concentric Coaxial Coils	p. 128
	p. 133
Values of Polynomial ₆(²) as Function of ²	p. 129
Values of Polynomial ₄ (²) as Function of ²	p. 130
Values of Polynomial ₆(²) as Function of ²	p. 131
Values of Polynomial ₈(²) as Function of ²	p. 132
	p. 133
Principle of Interchange of Lengths	p. 133
	p. 134
Loosely Coupled Coils	p. 134
	p. 135
Coaxial Coils of Equal Radii	p. 137
Values of B_n for Coils of Equal Radii ( = 1)	p. 138
	p. 139
Concentric Coils of Equal Length	p. 139
	p. 140
Single-layer Coils on Cylindrical Winding Forms	p. 142
Basic Current Sheet Formulas	p. 142
Inductance of Ring Conductor	p. 143
Values of K for Short Singlelayer Coils, Formula (118)	p. 144
Values for K for Long Single-layer Coils, Formula (118)	p. 146
Correction for Insulating Space	p. 149
	p. 149
Correction Term G in Formulas (120) and (135)	p. 148
Correction for Insulating Space	p. 149
	p. 49
Correction Term H in Formulas (120) and (135)	p. 150
General Design of Single-layer Coils on Cylindrical Forms	p. 151
Design Data. Single-layer Coils, r = <$$$>	p. 152
Given Diameter, Length, and Winding Density; To Calculate the Inductance	p. 153
Design Data for Short Single-layer Coils, R =<$$$>	p. 154
	p. 154
Given Inductance, Length, and Winding Density; To Calculate the Diameter	p. 156
	p. 157
Given Inductance, Diameter, and Winding Density; To Calculate the Length	p. 157
	p. 158
Given Inductance, Diameter, and Length; To Calculate the Winding Density	p. 158
	p. 159
Given Inductance and the Shape Ratio; To Calculate Length and Diameter	p. 159
	p. 160
Inductance as a Function of the Number of Turns	p. 160
	p. 160
Most Economical Coil Shape	p. 161
	p. 162
Special Types of Single-layer Coil	p. 163
Helices of Conductor of Large Cross Section	p. 163
Helices of Round Wire	p. 163
	p. 163
Helices of Rectangular Strip	p. 164
	p. 166
Flat Spirals of Strip	p. 167
	p. 168
Toroidal Coils	p. 169
Closely Wound Single-layer Coil on a Torus	p. 169
Toroidal Coils of Rectangular Turns	p. 170
	p. 170
Single-layer Polygonal Coils	p. 170
Data for Calculations of Polygonal Single-layer Coils	p. 172
	p. 174
Series Formulas for Short Polygonal Coils	p. 175
	p. 175
Flat Spirals with Polygonal Turns	p. 176
Mutual Inductance of Circular Elements with Parallel Axes	p. 177
Mutual Inductance of Circular Filaments of Equal Radii and with Parallel Axes	p. 177
	p. 178
Values of F for Equal Circles with Parallel Axes, Formula (159)	p. 179
Angular Position for Zero Mutual Inductance of Parallel Equal Circles	p. 180
Mutual Inductance of Coplanar Circular Filaments of Equal Radii	p. 180
Constants for Equal Coplanar Circular Elements, Formulas (160) and (161)	p. 181
	p. 182
Mutual Inductance of Circular Filaments Having Parallel Axes and Unequal Radii	p. 182
Distant Circles	p. 182
Examples 65 and 66, 183	p. 184
Circles Close Together	p. 184
Graphical Solution for Circular Filaments with Parallel Axes	p. 187
	p. 186
Mutual Inductance of Eccentric Circular Coils	p. 191
Mutual Inductance of Circular Filaments Whose Axes Are Inclined to One Another	p. 193
Circular Filaments Whose Axes Intersect at the Center of One of the Coils	p. 193
	p. 194
Best Proportions for a Variometer	p. 195
Values of Constant R for Inclined Circles, Formula (168)	p. 196
Calculation in the Most General Case	p. 201
	p. 201
Mutual Inductance of Inclined Circular Filaments Whose Axes Intersect but Not at the Center of Either	p. 204
	p. 204
General Method of Treatment	p. 205
Most General Case. Inclined Circular Filaments Placed in Any Desired Position	p. 206
Mutual Inductance of Circular Coils of Small Cross Section with Inclined Axes	p. 207
Mutual Inductance of Solenoids with Inclined Axes, and Solenoids and Circular Coils with Inclined Axes	p. 209
Inclined Solenoids with Center of One on the End Face of the Other	p. 209
Concentric Solenoids with Inclined Axes	p. 210
Unsymmetrical Cases	p. 210Solenoid
	p. 212
Circuit Elements of Larger Cross Sections with Parallel Axes	p. 215
Solenoid and Circular Filament	p. 215
Values of K_n in Formulas (183) and (185)	p. 216
	p. 217
Solenoids with Parallel Axes	p. 219
	p. 221
Solenoids with Parallel Axes Having Zero Mutual Inductance	p. 224
	p. 224
Solenoid and Coil of Rectangular Cross Section with Parallel Axes	p. 224
Corrections for Coil Thickness. Coils with Parallel Axes, Formulas (188), (190), and (192)	p. 226
	p. 227
Two Coils of Rectangular Cross Sections with Parallel Axes	p. 228
	p. 231
Mutual Inductance of Disc Coils with Parallel Axes	p. 234
	p. 235
Auxiliary Tables of Functions which Appear Frequently in Inductance Formulas	p. 236
Natural Logarithms of Numbers	p. 236
or Converting Common Logarithms into Natural Loga-rithms	p. 237
Values of Zonal Harmonic Functions	p. 238
Series for Zonal Harmonics	p. 242
Differential Coefficients	p. 242
Values of Differential Coefficients of Zonal Harmonics, 244	p. 247
Formulas for the Calculation of the Magnetic Force between Coils	p. 248
Force between Two Coaxial Circular Filaments	p. 248
Maximum Value of the Force	p. 249
Values of P. Force between Coaxial Circular Filaments	p. 250
Values of q₁ (or q) for Values of k′² (or k²)	p. 251
Spacing Ratio and Force for Maxi-mum Position Coaxial Circular Filaments	p. 252
	p. 253
Force between Two Coaxial Coils of Rectangular Cross Section	p. 253
	p. 254
Direction of the Force	p. 254
Force between Solenoid and a Coaxial Circular Filament	p. 255
Center of Circle at Center of End Face of Coil	p. 255
Filament Outside the Solenoid	p. 255
Filament Inside the Solenoid	p. 256
Force between a Single-layer Coil and a Coaxial Coil of Rectangular Cross Section	p. 256
	p. 257
Force between Two Coaxial Single-layer Coils	p. 258
	p. 259
High Frequency Formulas	p. 261
General Considerations	p. 261
Straight Cylindrical Conductor	p. 264
High Frequency Resistance and Inductance of Straight Wires	p. 266
Values of x₀ for Copper Wire 1 mm. Diameter, Frequencies 1 to 100 kc	p. 267
Maximum Diameter of Conductors in Cm. for Resistance Ratio 1.01	p. 268
	p. 269
Limiting Fractional Change of Inductance with Frequency	p. 270
Isolated Tubular Conductor	p. 271
Values of Factor F₀ for Approximate Calculations Based on (215) and (227)	p. 272
	p. 273
Go-and-return Circuit of Round Wire	p. 274
	p. 276
Correction Factor for Proximity Effect Parallel Round Wires	p. 276
Go-and-return Circuit of Parallel Tubular Conductors	p. 277
Correction Factor for Proximity Effect in Parallel Tubular Conductors	p. 277
Coaxial Cable	p. 278
High Frequency Resistance of Coaxial Cable	p. 278
	p. 279
High Frequency Inductance of Coaxial Cable	p. 280
	p. 281
References	p. 283
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Inductance Calculations

Working Formulas and Tables

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