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Principles of Electrical Measurement. . . . . . . . . . . . . . . . . . . . . . . . 261
Principles of Oscilloscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
Electrical Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
Voltage Ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
Resistance Ratio Bridges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Electricity Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
Inductance Measurement.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
Geometric Mean Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
Values for Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Mutual Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Self Inductance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285, 298
10Electrical Measurement
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Principles of Electrical Measurement
Resistance
where = resistanceV = voltageAT = ampere turnsT = turns
Ampere Turns
Amperes
where A = amperes
Direct Current
The Universal (Arytron) Shunt
For 0 to 10 mA, use:
whereRsh1 = 0.111 ohm shuntRsh2 = 1.11 ohm shuntRsh3 = 11.1 ohm shuntRm = 100
For 10.01 to 100 mA, use:
For 100.01 mA to 1 amp, use:
999 13 1 2( ) ( )R R R Rsh m sh sh= + +
99 12 3 1( ) ( )R R R Rsh sh m sh+ = +
0 009 0 0011 2 3. ( ) . ( )R R R Rsh sh sh m+ + =
AAT
=
ATV
T=
( )
=VAT
T( )
Chapter 10/Electrical Measurement 261
Rm = 100
Rm = 100
Rm = 100
9/1/
99/ 1/
999/1/
100 mA Configuration
1 amp Configuration
The Universal (Arytron) Shunt
Rsh1
Rsh2
Rsh2 Rsh1
Rsh3
Rsh3
Rsh3
Rsh2
Rsh1
M
M
M
10 mA Configuration
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Ohms Law for Direct Current
P = power in wattsI = current in amperesE = electromotive force in voltsR = resistance in ohms
Two resistances in parallel combination:
Any number of resistances inparallel combination:
For calculating capacitance inseries combinations, substituteC for R in the above equations.
Ohms Law for AlternatingCurrent
whereZ = impedance in ohmsXL = inductive reactance in ohmsXc = capacitive reactance in
ohmsL = inductance henrysC = capacitance in faradsf = frequency in cycles per
second2 f = 377 for 60 cps
fLC CX
XLL
XL f L
Xf C
LXL
f f C
Cf X
c
c
= = =
=
=
= =
=
12
12 2
2
12
21
2
12
2
( )
cc
c
c
f L
Z R X R XL X
Z R when XL X
=
= + = + = =
1
2 2
2 2 2 2
( )
( )
1 1 1 1
1 2req R R Rn= + +
reqR RR R
= +
1 2
1 2
262 ISA Handbook of Measurement Equations and Tables
P
E
I
R
P
Z
I
E
IR
EI EI
IZ
ER
2
EP
2
EP
2
EZ
2
EI
ER
EZ
I R2I R2PE
PE
P
I2
P
I2PI
PI
PR
PR
PZ
PZ
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Determining Required ShuntResistance
whereRsh = shunt resistorIm = full-scale deflection currentRm = dc resistance of meterIsh = current to be shunted
dc Voltmeters
Determining the Total ResistanceRequired to Drop Full-scale Volt-age at fsd Current
whereRt = required resistance dropMr = desired meter rangeIm = full-scale deflection currentRm = dc resistance of meter
Meter Sensitivity
whereMs = meter sensitivityV = voltsIm = full-scale deflection current
Series Voltmeters
Determining the Value of a Multiple Resistor
whereRv = multiple resistor valueV = full-scale voltage fordesired rangeIm = full-scale deflection currentRm = meter resistance
dc Bridges
Balance for a Wheatstone Bridge
whereRx = unknown resistanceRa and Rb = ratio armsRs = variable standard resistancewhen
Ra = Rb bridge is balancedand Rx = Rs
RRR
Rxa
bs=
RVI
Rvm
m=
MV
IM ohms Vs
ms= =
1/
RMI
Rtr
mm=
RI R
Ishm m
sh=
Chapter 10/Electrical Measurement 263
Null
Current for Bridge Mathematics
Rx Rs
RbRala
lx ls
lb
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Principles of Oscilloscopes
Alternating Current Waveforms
*0.9 for full-wave rectification.*0.45 for half-wave rectification.**1.11 for full-wave rectification.**2.22 for half-wave rectification.
Factors Used for Sinusoidal Wave Shape
Given Average r.m.s Peak Peak to Peak
Average 1.0 1.11** 2.22**
1.57 3.14
r.m.s. 0.90* 0.45*
1.0 1.414 2.828
Peak 0.637 0.707 1.0 2.00
Peak to Peak 0.318 0.3541 0.500 1.0
264 ISA Handbook of Measurement Equations and Tables
+1.0
+0.707
+0.636
0
-0.636
-0.707
-1.0
Time
Am
plit
ud
e
Period
0 90 180 270 360avg.
avg.
r.m.s
r.m.s
Peak
Peak
Peakto
peak
A Sinusoidal Wave Form
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Electrical Power
Determining the Gain or Loss ofPower in Decibels
wherePo = power outPi = power in
dBPPo
i= 10log
Conversion Tables, PowerRatios to Decibel (dB) Values
(cont.)
PowerRatioLoss
10 logRatio - db +
PowerRatioGain
0.3981 4.0 2.512
0.3162 5.0 3.162
0.2512 6.0 3.981
0.1995 7.0 5.012
0.1585 8.0 6.310
0.1259 9.0 7.943
0.1000 10.0 10.00
0.0794 11.0 12.59
0.0631 12.0 15.85
0.0501 13.0 19.95
0.0399 14.0 25.12
0.0316 15.0 31.62
0.0251 16.0 39.81
0.0199 17.0 50.12
0.0159 18.0 63.10
0.01259 19.0 79.43
0.0100 20.0 100.0
0.0010 30.0 103
10-4 40.0 104
10-5 50.0 105
10-6 60.0 106
10-7 70.0 107
10-8 80.0 108
10-9 90.0 109
Conversion Tables, PowerRatios to Decibel (dB) Values
PowerRatioLoss
10 logRatio - db +
PowerRatioGain
1.000 0.0 1.000
0.9772 0.1 1.023
0.9550 0.2 1.047
0.9333 0.3 1.072
0.9120 0.4 1.096
0.8913 0.5 1.122
0.8710 0.6 1.148
0.8511 0.7 1.175
0.8318 0.8 1.202
0.8128 0.9 1.230
0.7943 1.0 1.259
0.6310 2.0 1.585
0.5012 3.0 1.995
Chapter 10/Electrical Measurement 265
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Determining Voltage or CurrentGain (dB) when Input and Out-put Are Not Equal
whereV = voltageI = impedanceR = resistance
Determining Voltage or CurrentLoss (dB) when Input and Out-put Are Not Equal
dBV or I input R output
V or I output R input= 20log
dBV or I output R input
V or I input R output= 20log
Voltage/Current Ratio Tables(cont.)
Voltage/CurrentRatioGain
Decibels Voltage/CurrentRatioLoss
1.585 4.0 0.6310
1.788 5.0 0.5623
1.995 6.0 0.5012
2.239 7.0 0.4467
2.512 8.0 0.3981
3.162 10.0 0.3162
3.548 11.0 0.2818
3.981 12.0 0.2515
4.467 13.0 0.2293
5.012 14.0 0.1995
5.632 15.0 0.1778
6.310 16.0 0.1585
7.079 17.0 0.1413
7.943 18.0 0.1259
8.913 19.0 0.1122
10.00 20.0 0.1000
31.62 30.0 0.0316
102 40.0 10-2
316.23 50.0 0.000316
103 60.0 10-3
3.16 x 103 70.0 3.162 x 10-4
104 80.0 10-4
3.16 x 104 90.0 3.162 x 10-5
105 100.0 10-5
Voltage/Current Ratio Tables
Voltage/CurrentRatioGain
Decibels Voltage/CurrentRatioLoss
1.000 0.0 1.000
1.012 0.1 0.9886
1.023 0.2 0.9772
1.035 0.3 0.9661
1.047 0.4 0.9550
1.059 0.5 0.9441
1.072 0.6 0.9333
1.084 0.7 0.9226
1.096 0.8 0.9120
1.109 0.9 0.9016
1.122 1.0 0.8913
1.259 2.0 0.7943
1.413 3.0 0.7079
266 ISA Handbook of Measurement Equations and Tables
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Resistance Ratio Bridges
Measuring Inductance
and
whereLx = reactive componentRx = resistive component
Measuring Capacitance
and
whereCx = reactive componentRx = resistive component
RRR
Rxa
bs=
CRR
Cxa
bs=
RRR
Rxa
bs=
LRR
Lxa
bs=
Chapter 10/Electrical Measurement 267
detector
Lx
Rx
Ra
Rb Ls
RsRs = standard resistor
Ls = standard inductor
unknown inductor(resistance + inductance)
Resistance Ratio Bridge to Measure Inductance
Cx
Rx
Ra
RbCs
Rs Rs = standard resistorCs = standard capacitor
unknown capacitance(reactive and resistive component)
Resistance Ratio Bridge to Measure Capacitance
detector
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Measuring Capacitance, Wien Bridge
Measuring Capacitance, Schering Bridge
and
Measuring Inductance, MaxwellBridge
and
RRR
Rxb
sa=
L R R Cx b a s=
R RCCx s
b
s=
C CRRx s
b
s=
CRR
RR
Cxs
xs=
2
1
268 ISA Handbook of Measurement Equations and Tables
R2
Rs
Rs
Rs
RbRx
Lx
Ra
Rb
Rx
Cx
Cs
Cs
Cb
Cs
Cx
Rx
R1
R1 = 2 R2
detector
detector
Wien Bridge
Schering Bridge
Maxwell Bridge
detector
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Measuring Inductance, HayBridge Q Ratio Greater than 10
and
Measuring Inductance, HayBridge Q Ratio Less than 10
and
whereQ = reactive/resistive ratio
Measuring Inductance, OwensBridge
and
Measuring Wattage
Average Power in a Cycle
whereP = powerE = sinusoidal voltageI = current = phase angle that current lagsbehind voltage
r.m.s. Values of Voltage and Current
and
IIm=
2
EEm=
2
P E I= cos
RCC
Rxa
sa=
L R R Cx b s a=
RR R
R Qxb a
s x= +
( )11
LR R C
Q
xb a s
x
=
+
11
2
RRR
Rxb
sa=
L R R Cx b a s=
Chapter 10/Electrical Measurement 269
detector
Lx
Rx
Rb
Rs
CsRa
Hay Bridge
Rs
Cs
Lx
Rx
La
Ca
detector
Owens Bridge
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Conversion Tables for Electricity
To Convert from To Multiply by:
Amp/hr Coulomb 3600
Btu Calorie 251.996
Btu ft-lb force 778.169
Btu Horsepower-hr 0.000393015
Btu Kilocalorie 0.251996
Btu Kg-meter force 107.586
Btu Kw-hr 0.000293071
Btu/hr Btu/min 0.01666667
Btu/hr Btu/sec 0.000277778
Btu/hr Calorie/sec 0.0699988
Btu/hr Horsepower 0.000393015
Btu/hr Watt 0.293071
Btu/min Calorie/sec 4.19993
Btu/min Horsepower 0.0235809
Btu/min Watt 17.5843
Btu/min-ft2 Watt/m2 189.273
Btu/lb Calorie/gm 0.555556
Btu/lb Watt-hr/Kg 0.64611
Btu/sec Horsepower 1.41485
Btu/sec Kw 1.055056
Btu/sec-ft2 Kw-m2 11.3565
Btu/ft2 Watt-hr/m2 3.15459
Calorie Btu 0.00396832
Calorie ft-lb force 3.08803
Calorie Horsepower-hr 0.00000155961
270 ISA Handbook of Measurement Equations and Tables
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Conversion Tables for Electricity (cont.)
To Convert from To Multiply by:
Calorie Kg-force-m 0.426935
Calorie Kw-hr 0.000001163
Calorie Watt-hr 0.001163
Calorie/C Btu/F 0.0022046
Calorie/gm Btu/lb 1.8
Calorie/min Watt 0.06978
Calorie/sec Watt 4.1868
Calorie/sec-cm2 Kw/m2 41.868
Chu (C heat unit) Btu 1.8
Chu (C heat unit) Calorie 453.592
clo C-m2/watt 0.155
Coulomb amp-sec 1.0
Decibel Neper 0.115129255
Erg Watt-hr 2.777778 x 10-11
Erg/cm2-sec Watt/cm3 0.001
ft-lb force Btu 0.00128507
ft-lb force Calorie 0.323832
ft-lb force Horsepower-hr 5.05051 x 10-7
ft-lb force Watt-hr 0.000376616
ft-lb force/min Horsepower 0.000030303
ft-lb force/min Watt 0.022597
ft-lb force/sec Horsepower 0.00181818
ft-lb force/sec Watt 1.355818
Horsepower Btu/hr 2544.43
Horsepower Btu/min 42.4072
Chapter 10/Electrical Measurement 271
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Conversion Tables for Electricity (cont.)
To Convert from To Multiply by:
Horsepower Btu/sec 0.706787
Horsepower ft-lb force/hr 1980000.0
Horsepower ft-lb force/min 33000.0
Horsepower ft-lb force/sec 550.0
Horsepower Kilocalorie/hr 641.186
Horsepower Kilocalorie/min 10.6864
Horsepower Kilocalorie/sec 0.178107
Horsepower Kg-force-m/sec 76.0402
Horsepower Kw 0.74570
Horsepower/hr Btu 2544.43
Horsepower/hr ft-lb force 1980000.0
Horsepower/hr Kilocalorie 641.186
Horsepower/hr Kw-hr 0.74570
Kilocalorie/hr Watt 1.163
Kilocalorie/hr-m2 Watt/m2 1.163
Kilocalorie/Kg Btu/lb 1.8
Kilocalorie/min ft-lb force/sec 51.4671
Kilocalorie/min Horsepower 0.0935765
Kilocalorie/min Watt 69.78
Kilocalorie/sec Kw 4.1868
Kw Btu/hr 3412.14
Kw Btu/min 56.8690
Kw Btu/sec 0.947817
Kw ft-lb force/hr 2655220.0
Kw ft-lb force/min 44253.7
272 ISA Handbook of Measurement Equations and Tables
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Conversion Tables for Electricity (cont.)
To Convert from To Multiply by:
Kw ft-lb force/sec 737.562
Kw Horsepower 1.34102
Kw Kilocalorie/hr 859.845
Kw Kilocalorie/min 14.3308
Kw Kilocalorie/sec 0.0238846
Kw Kg force-m/hr 367098.0
Kw Kg force-m/min 6118.3
Kw Kg force-m/sec 101.972
Kw-hr Btu 3412.14
Kw-hr ft-lb force 2655220.0
Kw-hr horsepower-hr 1.34102
Kw-hr Kilocalorie 859.845
Kw-hr Kg-force-m 367098.0
Kw-hr/lb Btu/lb 3412.14
Kw-hr/lb Kilocalorie/kg 1895.63
Kw-hr/Kg Btu/lb 1547.72
Megajoule Kw-hr 0.2777778
Neper Decibel 8.68589
Ohm/ft Ohm/m 3.28084
Ohm-cm Ohm-m 0.01
Pond Gram-force 1.0
Statohm Ohm 8.987552 x 1011
Statvolt Volt 299.7925
Volt/in Volt/m 39.37008
Volt-sec Weber 1.0
Chapter 10/Electrical Measurement 273
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Conversion Tables for Electricity (cont.)
To Convert from To Multiply by:
Watt Btu/hr 3.41214
Watt Btu/min 0.056869
Watt Calorie/min 14.3308
Watt Calorie/sec 0.238846
Watt Erg/sec 10000000.0
Watt ft-lb-force/min 44.2537
Watt ft-lb-force/sec 0.737562
Watt Horsepower 0.00134102
Watt Joule/sec 1.0
Watt Kilocalorie/hr 0.859845
Watt Kg-force-m/sec 0.101972
Watt/in2 Btu/hr-ft2 491.348
Watt/in2 Kilocalorie/hr-m2 1332.76
Watt/in2 Watt/m2 1550.003
Watt/m2 Kilocalorie/hr-m2 0.859845
Watt-hr Btu 3.41214
Watt-hr Calorie 859.845
Watt-hr ft-lb force 2655.22
Watt-hr Horsepower-hr 0.00134102
Watt-hr Joule 3600.0
Watt-hr Kg-force-m 367.098
Watt-sec Erg 10000000.0
Watt-sec Joule 1.0
Watt-sec Newton-m 1.0
274 ISA Handbook of Measurement Equations and Tables
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Inductance Measurement
The most direct method of calcu-lating inductances is based on thedefinition of flux linkages perampere. To calculate flux link-ages, it is necessary to write theexpression for the magneticinduction at any point of the field,and then to integrate this expres-sion over the space occupied bythe flux that is linked to the ele-ment in question.
Biot-Savart Law of MagneticField Intensity
wheredH = magnetic field densityi = currentds = length of circuit elementr = radius vector = angle between ds and theradius vector
Mutual Inductance of Two Conductors
Values of loge in the equation:loge R = loge p + loge k
(Longer sides of rectangles insame straight line.)
See Tables on next page for val-ues.
=
=cp
Bc
,1
dHi ds
r= 2 sin
Chapter 10/Electrical Measurement 275
d
ds
r
B B
c c
p
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Geometric Mean Distances
In calculating the mutual inductance of two conductors whose crosssectional dimensions are small compared with their distance apart, weassume that the mutual inductance is the same as the mutual induc-tance of the filaments along their axes, and use the appropriate basicformula for filaments to calculate mutual inductance. For conductorswhose cross section is too large to justify this assumption, it is neces-sary to average the mutual inductances of all the filaments of which theconductors consist. That is, the basic formula for the mutual inductanceis to be integrated over the cross sections of the conductors.Values of logc k in equation:
Geometric Mean Distance of Equal Parallel Rectangles,Longer Sides of Rectangle in Same Straight Line
1 = 0
.02 .04 .06 .08 1.0
0.05 -0.0002 -0.0002 -0.0002 -0.0001 -0.0001 +0.0000
0.10 -0.0008 -0.0008 -0.0007 -0.0005 -0.0003 +0.0000
0.15 -0.0019 -0.0018 -0.0016 -0.0012 -0.0006 +0.0000
0.20 -0.0034 -0.0032 -0.0028 -0.0021 -0.0012 +0.0000
0.25 -0.0053 -0.0051 -0.0044 -0.0034 -0.0019 +0.0000
0.30 -0.0076 -0.0073 -0.0064 -0.0048 -0.0027 +0.0001
0.35 -0.0105 -0.0100 -0.0087 -0.0066 -0.0036 +0.0002
0.40 -0.0138 -0.0132 -0.0115 -0.0086 -0.0047 +0.0002
0.45 -0.0176 -0.0169 -0.0146 -0.0110 -0.0059 +0.0003
0.50 -0.0220 -0.0210 -0.0182 -0.0136 -0.0073 +0.0005
0.55 -0.0269 -0.0257 -0.0222 -0.0164 -0.0087 +0.0007
0.60 -0.0325 -0.0310 -0.0267 -0.0196 -0.0103 +0.0010
0.65 -0.0388 -0.0369 -0.0316 -0.0231 -0.0120 +0.0014
0.70 -0.0458 -0.0435 -0.0370 -0.0269 -0.0137 +0.0019
0.75 -0.0536 -0.0509 -0.0431 -0.0310 -0.0156 +0.0023
0.80 -0.0625 -0.0591 -0.0470 -0.0354 -0.0176 +0.0031
0.85 -0.0725 -0.0683 -0.0569 -0.0401 -0.0195 +0.0037
0.90 -0.0839 -0.0786 -0.0648 -0.0451 -0.0216 +0.00046
0.95 -0.0973 -0.0903 -0.0734 -0.0504 -0.0236 +0.0056
1.00 -0.1137 -0.1037 -0.0828 -0.0561 -0.0258 +0.0065
276 ISA Handbook of Measurement Equations and Tables
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(Longer sides of the rectangle per-pendicular to lines joining theircenters.)
Geometric Mean Distances ofEqual Parallel Rectangles (con-cluded)
BBp
cB
= =,
log log loge c cR p k= +
Geometric Mean Distance of Equal Parallel Rectangles,Longer Sides of the Rectangle Perpendicular to Centers
B = 0 0.2 0.4 0.6 0.8 1.0
0.1 0.0008 0.0008 0.0007 0.0005 0.0003 0.0000
0.2 0.0033 0.0032 0.0028 0.0021 0.0012 0.0000
0.3 0.0074 0.0071 0.0062 0.0048 0.0027 0.0001
0.4 0.0129 0.0124 0.0109 0.0084 0.0050 0.0003
0.5 0.0199 0.0191 0.0169 0.0131 0.0077 0.0005
0.6 0.0281 0.0271 0.0240 0.0185 0.0111 0.0011
0.7 0.0374 0.0361 0.0320 0.0251 0.0155 0.0019
0.8 0.0477 0.0461 0.0411 0.0321 0.0200 0.0031
0.9 0.0589 0.0569 0.0506 0.0404 0.0254 0.0046
1.0 0.0708 0.0685 0.0614 0.0492 0.0313 0.0065
0.9 0.0847 0.0821 0.0738 0.0596 0.0382
0.8 0.1031 0.0999 0.0903 0.0745 0.0485
0.7 0.1277 0.1240 0.1125 0.0925
0.6 0.1618 0.1573 0.1436 0.1194
0.5 0.2107 0.2053 0.1886
0.4 0.2843 0.2776 0.2567
0.3 0.4024 0.3942
0.2 0.6132 0.6021
0.1 1.0787
Chapter 10/Electrical Measurement 277
c c
p
B
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For accurate interpolation in the case of broad rectangles, near together(1/B small and D small), write:
loge R = loge B + loge K'
Values for logeK'
1/B = 0 0.1 0.2 0.3 0.4 0.5
0.00 -1.5000
0.05 -1.3542
0.10 -1.2239 -1.2278
0.15 -1.1052 -1.1084
0.20 -0.9962 -0.9989 -1.0073
0.25 -0.8953 -0.8977 -0.9049
0.30 -0.8015 -0.8037 -0.8098 -0.8208
0.35 -0.7140 -0.7159 -0.7215 -0.7311
0.40 -0.6321 -0.6337 -0.6387 -0.6472 -0.6596
0.45 -0.5550 -0.5565 -0.5610 -0.5687 -0.5797
0.50 -0.4825 -0.4838 -0.4879 -0.4948 -0.5046 -0.5178
278 ISA Handbook of Measurement Equations and Tables
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Values of Constants for the Geometric Mean Distance of a Rectangle
Sides of the rectangle are B and c. The geometric mean distance R isgiven by:loge R = loge (B + c) - 1.5 + loge e.
R = K (B + c), loge K = - 1.5 + loge e
Geometric Mean Distance of a Line of Length (a) from Itself
or
Circular Area of Radius (a) from Itself
or
Ellipse with Semiaxes (a) and (b)
log loge eRa b
=+
2
14
R a= 0 7788.
log loge eR a= 14
R a= 0 22313.
log loge eR a= 32
Values for Constants K, logee
B/c or c/B K loge e B/c or c/B K loge e
0.00 0.22313 0.0000 0.50 0.22360 0.00211
0.025 0.22333 0.00089 0.55 0.22358 0.00203
0.05 0.22346 0.00146 0.60 0.22357 0.00197
0.10 0.22360 0.00210 0.65 0.22356 0.00192
0.15 0.22366 0.00239 0.70 0.22355 0.00187
0.20 0.22369 0.00249 0.75 0.22354 0.00184
0.25 0.22369 0.00249 0.80 0.22353 0.00181
0.30 0.22368 0.00244 0.85 0.22353 0.00179
0.35 0.22366 0.00236 0.90 0.22353 0.00178
0.40 0.22364 0.00228 0.95 0.223525 0.00177
0.45 0.22362 0.00219 1.00 0.223525 0.00177
Chapter 10/Electrical Measurement 279
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-
Geometric Mean Distance of anAnnulus from Itself
Geometric Mean Distance of aPoint or Area from an Annulus
loglog log
ee eR
p p p p
p p=
12
1 22
2
12
22
12
log log loge p eR = 1
Values for Geometric Mean Distance of an Annulus
p2/p1 loge d1 d2
0.00 0.2500 -12
0.05 0.2488 -36 -24
0.10 0.2452 -57 -21
0.15 0.2395 -75 -18
0.20 0.2320 -92 -16
0.25 0.2228 -105 -14
0.30 0.2123 -116 -12
0.35 0.2007 -127 -10
0.40 0.1880 -135 -8
0.45 0.1745 -142 -7
0.50 0.1603 -144 -6
0.55 0.1456 -147 -5
0.60 0.1304 -152 -4
0.65 0.1148 -156 -3
0.70 0.0989 -159 -3
0.75 0.0827 -162 -2
0.80 0.0663 -163 -1
0.85 0.0499 -164 -1
0.90 0.0333 -165 -1
0.95 0.0167 -166 -1
1.00 0.0000 -167
280 ISA Handbook of Measurement Equations and Tables
A area
point
p 1
p2
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Inductance of Parallel Elements of Equal Length
Mutual Inductance of Two Equal Parallel Straight Filaments
or
M = 0.002lQ
M lld
l
d
d
l
dle
= + +
+ +
0 002 1 12
2
2
2. log
Values for Q, d/ld/l Q d1
0.050 2.7382 -903
0.055 2.6479 -822
0.060 2.5657 -752
0.065 2.4905 -693
0.070 2.4212 -642
0.075 2.3570 -597
0.080 2.2973 -558
0.085 2.2415 -524
0.090 2.2189 -493
0.095 2.1398 -466
0.100 2.0932 -440
0.105 2.0492 -418
0.110 2.0074 -397
0.115 1.9677 -379
0.120 1.9298 -361
0.125 1.9837 -345
0.130 1.8592 -330
0.135 1.8262 -318
0.140 1.7944 -305
0.145 1.7639 -293
0.150 1.7346 -281
Chapter 10/Electrical Measurement 281
p
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-
Values for Q, d/l (cont.)
d/l Q d1
0.155 1.7065 -271
0.160 1.6794 -262
0.165 1.6532 -253
0.170 1.6279 -244
0.175 1.6035 -236
0.180 1.5799 -228
0.185 1.5571 -222
0.190 1.5349 -215
0.195 1.5134 -208
0.200 1.4926 -398
0.210 1.4528 -376
0.220 1.4152 -355
0.230 1.3797 -337
0.240 1.3460 -321
0.250 1.3139 -305
282 ISA Handbook of Measurement Equations and Tables
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-
Values for Q, d/l (cont.)
d/l Q d1 d/l Q d1
0.260 1.2834 -290 0.520 0.8016 -227
0.270 1.2544 -277 0.540 0.7789 -215
0.280 1.2267 -265 0.560 0.7574 -204
0.290 1.2002 -253 0.580 0.7370 -194
0.300 1.1749 -243 0.600 0.7176 -184
0.310 1.1506 -233 0.620 0.6992 -175
0.320 1.1273 -224 0.640 0.6817 -167
0.330 1.1049 -214 0.660 0.6650 -160
0.340 1.0835 -207 0.680 0.6490 -152
0.350 1.0627 -199 0.700 0.6338 -145
0.360 1.0429 -192 0.720 0.6193 -139
0.370 1.0238 -186 0.740 0.6054 -134
0.380 1.0052 -178 0.760 0.5920 -128
0.390 0.9874 -172 0.780 0.5792 -122
0.400 0.9702 -166 0.800 0.5670 -118
0.410 0.9536 -161 0.820 0.5552 -113
0.420 0.9375 -156 0.840 0.5439 -109
0.430 0.9219 -151 0.860 0.5330 -105
0.440 0.9068 -146 0.880 0.5225 -101
0.450 0.8922 -141 0.900 0.5124 -97
0.460 0.8781 -137 0.920 0.5027 -93
0.470 0.8644 -133 0.940 0.4934 -90
0.480 0.8511 -130 0.960 0.4843 -87
0.490 0.8381 -125 0.980 0.4756 -84
0.500 0.8256 -240 1.000 0.4672 -81
Chapter 10/Electrical Measurement 283
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Values for Q, l/d
l/d Q d1 l/d Q d1
1.00 0.4672 -84 0.50 0.2451 -94
0.98 0.4588 -83 0.48 0.2357 -95
0.96 0.4505 -84 0.46 0.2262 -96
0.94 0.4421 -85 0.44 0.2166 -95
0.92 0.4336 -85 0.42 0.2071 -96
0.90 0.4251 -85 0.40 0.1975 -97
0.88 0.4166 -86 0.38 0.1878 -97
0.86 0.4080 -87 0.36 0.1781 -97
0.84 0.3993 -87 0.34 0.1684 -97
0.82 0.3906 -87 0.32 0.1587 -98
0.80 0.3819 -88 0.30 0.1489 -98
0.78 0.3731 -88 0.28 0.1391 -98
0.76 0.3643 -89 0.26 0.1293 -99
0.74 0.3554 -90 0.24 0.1194 -98
0.72 0.3464 -90 0.22 0.1096 -99
0.70 0.3374 -90 0.20 0.0977 -99
0.68 0.3284 -91 0.18 0.0898 -100
0.66 0.3193 -91 0.16 0.0798 -99
0.64 0.3102 -92 0.14 0.0699 -100
0.62 0.3011 -93 0.12 0.0599 -99
0.60 0.2918 -92 0.10 0.0500 -100
0.58 0.2826 -93 0.08 0.0400 -100
0.56 0.2733 -93 0.06 0.0300 -100
0.54 0.2640 -94 0.04 0.0200 -100
0.52 0.2546 -95 0.02 0.0100 -100
284 ISA Handbook of Measurement Equations and Tables
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Mutual Inductance of Two EqualParallel Conductors
Self-Inductance of a StraightConductorGeneral Formula
wherer = geometric mean distance1= arithmetic mean distance ofthe points of the cross section
For a Round Wire, Radius p
For a Round Magnetic Wire
where = permeability
For Rectangular Wire, Sides Band C
whereB and C = see table, Values ofconstants for Geometric MeanDistance for Rectangles
For Elliptical Wire
where = semiaxes of the ellipse
Inductance of Multiple Conductors
Two Equal Parallel Wires, Sepa-rated by Distance (d) betweenCenters
Three Equal Parallel Wires, at theCorners of an Equilateral Triangleof Side (d)
wherer = geometric mean distance ofcircular area of radius (p)
Inductance of a Return Circuit ofParallel Conductors
Equal Round Wires of Radius (p)
Equal Permeable Round Wires
L ldp
dle
= +
0 0044
. log
L ldp
dle
= +
0 00414
. log
L ll
rde=
0 002
212 1 3. log ( ) /
L ll
pde=
0 002
2 78
. log
L ll
e= +
0 002
20 05685. log .
L ll
B Cee e= +
+
0 0022 1
2. log log
L ll
pe= +
0 0022
14
. log
L ll
pe=
0 0022 3
4. log
L ll
r le= +
0 0022
1 1. log
M ll
dk
dl
d
le e= +
0 0022
11
4
2
2. log log
Chapter 10/Electrical Measurement 285
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-
Return Circuit of Two Tubular Conductors, One Inside the Other
whereloge 1 and loge 3 = values from table, geometric mean distance ofan annulus
Return Circuit of Polycore Cable
Mutual Inductance of Unequal Parallel Filaments
General Formula
where = l + m = l = m
M hd
hd
hd
hd
d=
+ + 0 001 1 1 1 1 2 2. sin sin sin sin
+ + + + +
2 2 2 2 2 2d d d
L lpa
PP
pp
ppe e
= +
0 002
2
1
1
2
1
2
2
1
21
2. log log
+ + +
1 14
1n
an ne e
log log
L lpp
PP
pp
e= +
0 0022
1
1
3
2
1
2
2
1
2. log loog log loge e epp
1
21 31 + +
286 ISA Handbook of Measurement Equations and Tables
pa
p1
p 2
m
p
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Mutual Inductance of Filaments Inclined at an Angle
Equal Filaments Meeting at aPoint
Mutual Inductance between Filaments
or
M lS= 0 001.
M l hl
l R=
+0 004 1
1. cos tan
R l12 22 1= ( cos )
Value of Factor S (cont.)
cos S d1
-0.05 -0.0867 -867
-0.10 -0.1707 -840
-0.15 -0.2523 -815
-0.20 -0.3316 -793
-0.25 -0.4088 -772
-0.30 -0.4840 -752
-0.35 -0.5574 -734
-0.40 -0.6290 -716
-0.45 -0.6991 -701
-0.50 -0.7677 -686
-0.55 -0.8348 -671
-0.60 -0.9006 -658
-0.65 -0.9651 -645
-0.70 -1.0284 -633
-0.75 -1.0906 -622
-0.80 -1.1517 -611
-0.85 -1.2118 -601
-0.90 -1.2709 -591
-0.95 -1.3290 -581
-1.00 -1.3862 -572
Values of Factor S
cos S d1
0.95 3.7830 -7236
0.90 3.0594 -4462
0.85 2.6132 -3316
0.80 2.2816 -2679
0.75 2.0137 -2274
0.70 1.7863 -1991
0.65 1.5872 -1780
0.60 1.4092 -1618
0.55 1.2474 -1488
0.50 1.0986 -1382
0.45 0.9604 -1294
0.40 0.8310 -1218
0.35 0.7092 -1154
0.30 0.5938 -1097
0.25 0.4841 -1048
0.20 0.3793 -1003
0.15 0.2789 -964
0.10 0.1825 -929
0.05 0.0896 -896
0.00 0.0000 -867
Chapter 10/Electrical Measurement 287
R1
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-
Unequal Filaments Meeting at a Point
or M l S= 0 001 1 1.
M l hm
l Rm h
lm R
=+
+ +
0 002 111 1
11
1
1. cos tan tan
Values for S1, Unequal Filaments Meeting at a Point
cos ml
1
11= 0.8 0.6 0.4 0.2
0.95 3.7830 3.3406 2.7622 2.0473 1.1776
0.90 2.0594 2.7095 2.2597 1.6957 0.9918
0.85 2.6132 2.3178 1.9422 1.4690 0.8688
0.80 2.2816 2.0256 1.7028 1.2950 0.7727
0.75 2.0137 1.7889 1.5073 1.1513 0.6917
0.70 1.7863 1.5876 1.3402 1.0272 0.6209
0.65 1.5872 1.4113 1.1931 0.9172 0.5572
0.60 1.4092 1.2534 1.0609 0.8177 0.4991
0.55 1.2474 1.1098 0.9404 0.7264 0.4452
0.50 1.0986 0.9776 0.8291 0.6417 0.3947
0.40 0.8310 0.7398 0.6283 0.4880 0.3020
0.30 0.5938 0.5288 0.4496 0.3501 0.2179
0.20 0.3793 0.3378 0.2876 0.2244 0.1404
0.10 0.1825 0.1626 0.1385 0.1083 0.0680
0.00 0.0000 0.0000 0.0000 0.0000 0.0000
-0.10 -0.1707 -0.1522 -0.1298 -0.1018 -0.0644
-0.20 -0.3316 -0.2956 -0.2523 -0.1982 -0.1257
-0.30 -0.4840 -0.4314 -0.3684 -0.2898 -0.1844
-0.40 -0.6290 -0.5608 -0.4791 -0.3772 -0.2406
-0.50 -0.7677 -0.6845 -0.5850 -0.4611 -0.2948
-0.60 -0.9006 -0.8031 -0.6865 -0.5416 -0.3470
-0.70 -1.0284 -0.9172 -0.7844 -0.6194 -0.3976
-0.80 -1.1517 -1.0272 -0.8788 -0.6944 -0.4467
288 ISA Handbook of Measurement Equations and Tables
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Unequal Filaments in the SamePlane, Not Meeting
Equations Connecting the Two Systems
where
Equation for Mutual Inductance
Ml h
mR R
v m hl
R Rh
mR
21
1 2
1
1 4
1
cos( )tan
( )tan tan
= ++
+ ++
33 4
1
2 3
+
+
R
v hl
R Rtan
R l v m l v m
R l v v l
12 2 2
22 2 2
2
2
= + + + + +
= + + +
( ) ( ) ( )( )cos
( ) ( )cos
RR v v
R v m v m
32 2 2
42 2 2
2
2
= +
= + + +
cos
( ) ( )cos
vm l R R m R R l
l m=
+
2
4
242
32
22
22
32 2
2 2 4
( ) (
=
+
l m R R l R R m
l m
2
4
222
32 2 2
42
32 2
2 2 4
( ) ( )
2 42
32
22
12= + R R R R
22
cos
=lm
Chapter 10/Electrical Measurement 289
b
m
C
a
BA
d
p
p
R3
R4
R1
R 2
m
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Mutual Inductance of Two Filaments Placed in Any Desired Position
where
Circuits Composed of Combinations of Straight Wires
Equation for the Inductance of a Triangle of Round Wire
where
a,b,c = sides of the triangle
V a b a c b c a b c2 2 2 2 2 2 2 4 4 42= + + ( )
L aa
bb
cc
b c hc b a
e e e= + +
+ 0 0022 2 2 1
2 2. log log log ( )sin
22
12
22
12 2
V
a ba b c
Va c h
a c
++
++ ( )sin h ( )sin
bbV
a b c
a b c
2
4
+ +
+ + +
( )
( )
=+ + +
+
tancos ( )( )sin
sin
tancos (
12 2
1
12
d l v mdR
d
+
=+
l vdR
d vdR
) sinsin
tancos sin
sin
2
2
12 2
3
+ +
tancos ( )sin
sin1
2 2
4
d v mdR
Ml h
mR R
v m hl
R R
0 0012
2
1
1 2
1
1 4
. cos( )tan
( )tan ta
= ++
+ ++
nn
tansin
hm
R R
v hR R
d
+
+
1
3 4
1
2 32
1
290 ISA Handbook of Measurement Equations and Tables
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Equation for the Inductance of a Rectangle of Round Wire
Regular Polygons of Round Wire
Equilateral Triangle
Square
Pentagon
Hexagon
Octagon
Equation for the Calculation of Inductance of Any Plane Figure
wherel = perimeter of the figure
L ll
e= +
0 002
24
. log
L ss
e= + +
0 016 0 21198 4
. log .
L ss
e= +
0 012 0 15152 4
. log .
L ss
e= +
0 010 0 40914 4
. log .
L ss
e= +
0 008 0 77401 4
. log .
L ss
e= +
0 006 1 40546 4
. log .
L aa b
a b a hab
b hbae e
= + + +
0 0042 2
2 2 2 1 1. log log sin sin
+ + +
24
( ) ( )a b a b
Chapter 10/Electrical Measurement 291
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-
Values for (alpha) for Certain Plane Figures
Rectangles
0.05 4.494
0.10 3.905
0.15 3.589
0.20 3.404
0.25 3.270
0.30 3.172
0.40 3.041
0.50 2.962
0.60 2.913
0.70 2.882
0.80 2.865
0.90 2.856
1.00 2.854
Isosceles Triangles
5 4.884
10 4.152
20 3.690
30 3.424
40 3.284
50 3.217
60 3.197
70 3.214
80 3.260
90 3.331
100 3.426
110 3.546
120 3.696
130 3.875
140 4.105
150 4.399
160 4.813
170 7.514
292 ISA Handbook of Measurement Equations and Tables
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Mutual Inductance of Equal,Parallel, Coaxial Polygons ofWire
s = length of the side of thepolygon.
d = distance between theirplanes.
Squares
Equilateral Triangles
Hexagons
Ms
F= 62
2 6ad
sd
=
Ms
F= 32
2 3ad
sd
=
Ms
F= 42
2 4ad
sd
=
Chapter 10/Electrical Measurement 293
Regular Polygons
N
3 3.197
4 2.854
5 2.712
6 2.636
7 2.591
8 2.561
9 2.542
10 2.529
11 2.519
12 2.513
13 2.506
14 2.500
15 2.495
16 2.492
17 2.489
18 2.486
19 2.484
20 2.482
21 2.481
22 2.480
23 2.478
24 2.477
2.452
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Values for (F) in Coaxial Equal Polygons, d/s
d/s Triangles F Diff. Squares F Diff. Hexagon F Diff.
0.00 1.0000 1.000 1.000
0.05 0.7245 -2755 0.8642 -1358 0.9449 -551
0.10 0.6640 -605 0.8362 -280 0.9350 -99
0.15 0.6217 -423 0.8165 -197 0.9283 -67
0.20 0.5890 -327 0.8007 -158 0.9231 -52
0.25 0.5624 -266 0.7875 -132 0.9188 -43
0.30 0.5402 -222 0.7760 -115 0.9150 -38
0.35 0.5215 -187 0.7658 -102 0.9117 -33
0.40 0.5054 -161 0.7565 -93 0.9087 -30
0.45 0.4914 -140 0.7480 -85 0.9057 -30
0.50 0.4792 -122 0.7402 -78 0.9029 -28
0.55 0.4686 -106 0.7329 -73 0.9003 -26
0.60 0.4592 -94 0.7262 -67 0.8078 -25
0.65 0.4507 -85 0.7200 -62 0.8054 -24
0.70 0.4437 -70 0.7140 -60 0.8031 -23
0.75 0.4372 -65 0.7085 -55 0.8906 -25
0.80 0.4314 -58 0.7035 -50 0.8884 -22
0.85 0.4263 -51 0.6988 -47 0.8863 -21
0.90 0.4216 -47 0.6941 -47 0.8843 -20
0.95 0.4175 -41 0.6899 -42 0.8823 -20
1.00 0.4138 -37 0.6861 -38 0.8802 -21
294 ISA Handbook of Measurement Equations and Tables
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Coaxial Triangles
Coaxial Squares
Coaxial Hexagons
M ssd
ds
d
s
d
se= + +
0 012 0 15152 0 3954 0 1160 0 052
2
2
4
4. log . . . . ....
M ssd
ds
d
s
d
se= +
0 008 0 7740 0 0429 0 1092
2
4
4. log . . . ....
M ssd
ds
d
s
d
se= + +
0 006 1 4055 2 209
11
12
203
864
2
2
4
4. log . . ....
Values for (F) in Coaxial Equal Polygons, s/d
s/d Triangles F Diff. Squares F Diff. Hexagon F Diff.
1.00 0.4138 0.6861 0.8802
0.90 0.4066 -72 0.6783 -78 0.8761 -41
0.80 0.3996 -70 0.6701 -82 0.8713 -48
0.70 0.3930 -66 0.6613 -88 0.8656 -57
0.60 0.3866 -64 0.6525 -88 0.8592 -64
0.50 0.3808 -58 0.6439 -86 0.8518 -74
0.40 0.3757 -51 0.6362 -77 0.8440 -78
0.30 0.3714 -43 0.6289 -73 0.8364 -76
0.20 0.3682 -32 0.6221 -68 0.8297 -67
0.10 0.3662 -20 0.6182 -39 0.8243 -54
0.00 0.3655 -7 0.6169 -13 0.8225 -18
Chapter 10/Electrical Measurement 295
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Inductance of Single-Layer Coils on Rectangular Winding Forms
where
g a a2 2 12= +
L Naab
ba
hab
ba
hab
a
b
ba
= +
0 008
12
12
12
12 11
1 1 1 12
2. sin sin11
1
12
2
1 1
1
1
1
12
12
sin
sin sin
ha
ba
b
ab
haa
ab
ha
+
11 1 1
22
2
2
1
2
2
2
221
13
1 11
2aaa
bg
b
baa
g
b
g
b+
+
+ +
tan
+ +
13
13
1 11
2
12
1
2
1
2
2
2
2baa
baa
a
b
a
b
b22
1
12
212
21
3 313
231
21
1
2
16aa
a
b
a
b
baa
g a a
b
+
296 ISA Handbook of Measurement Equations and Tables
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Coefficients, Short Rectangle Solenoid
11
11 = +
k
Chapter 10/Electrical Measurement 297
1 1 2 3 5 7
1.00 0.4622 0.6366 0.2122 -0.0046 0.0046 -0.0382
0.95 0.4574 0.6534 0.2234 -0.0046 0.0053
0.90 0.4512 0.6720 0.2358 -0.0046 0.0064 -0.0525
0.85 0.4448 0.6928 0.2496 -0.0042 0.0080
0.80 0.4364 0.7162 0.2653 -0.0031 0.0103 -0.0838
0.75 0.4260 0.7427 0.2829 -0.0010 0.0141
0.70 0.4132 0.7730 0.3032 0.0026 0.0198 -0.1564
0.65 0.3971 0.8080 0.3265 0.0085 0.0291
0.60 0.3767 0.8488 0.3537 0.0179 0.0432 -0.3372
0.55 0.3500 0.8970 0.3858 0.0331 0.0711
0.50 0.3151 0.9549 0.4244 0.0578 0.1183 -0.7855
0.40 0.1836 1.1141 0.5305 0.1679 0.3898 -2.4030
0.30 -0.0314 1.3359 0.7074 0.5433 2.0517 -7.850
0.20 -0.6409 1.9099 1.0610 2.3230 14.5070 15.51
0.10 -3.2309 3.5014 2.1220 22.5480 497.360 14282.0
HB electric chap10.qxd 3/2/2006 10:29 AM Page 297
-
Self-Inductance of CircularCoils of Rectangular Cross-Section
Nomenclaturea = mean radius of turnsb = axial dimension of the
cross-sectionc = radial dimension of the
cross-sectionN = total number of turnsnb = number of turns per layernc = number of layerspb = distance between centers
of adjacent turns in thelayer
pc = distance between centersof corresponding wires inconsecutive layers
For Closely Wound Coils:
wherepb = pc = diameter of the covered wire
b n
c n
Nbc
b
c
==
=
b n p
c n p
N n n
b b
c c
b c
===
298 ISA Handbook of Measurement Equations and Tables
c
b
a
HB electric chap10.qxd 3/2/2006 10:29 AM Page 298
Front MatterTable of Contents10. Electrical Measurement10.1 Principles of Electrical Measurement10.2 Principles of Oscilloscopes10.3 Electrical Power10.4 Voltage Ratios10.5 Resistance Ratio Bridges10.6 Electricity Conversions10.7 Inductance Measurement10.8 Geometric Mean Distances10.9 Values for Q10.10 Mutual Inductance10.11 Self InductanceIndex
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