Instrumentation and control

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

HB electric chap10.qxd 3/2/2006 10:29 AM Page 259

Principles of Electrical Measurement

Resistance

whereΩ = 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

=Ω

ATV

T=Ω

( )

Ω =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


Ohm’s 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 in

parallel combination:

For calculating capacitance inseries combinations, substituteC for R in the above equations.

Ohm’s Law for AlternatingCurrent

whereZ = impedance in ohmsXL = inductive reactance in ohmsXc = capacitive reactance in

ohmsL = inductance henrysC = capacitance in faradsf = frequency in cycles per

second2π 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

π π ππ

π

π π

π

( )

cc

c

c

f L

Z R X R XL X

Z R when XL X

=

= + = + −= =

1

2 2

2 2 2 2

( )

( )

π

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

E

R

2

E

P

2

E

P

2

E

Z

2

E

I

E

RE

ZI R2

I R2 P

E

P

E

P

I2

P

I2

P

I

P

I

PR

P

R

PZ

P

Z


Determining Required ShuntResistance

whereRsh = shunt resistorIm = full-scale deflection currentRm = dc resistance of meterIsh = current to be shunted

dc Voltmeters

Determining the Total Resistance

Required 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=


Null

Current for Bridge Mathematics

Rx Rs

RbRa

la

lx ls

lb


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


+1.0

+0.707

+0.636

0

-0.636

-0.707

-1.0

Time

Am

pli

tud

e

Period

0˚ 90˚ 180˚ 270˚ 360˚avg.

avg.

r.m.s

r.m.s

Peak

Peak

Peakto

peak

A Sinusoidal Wave Form


Electrical Power

Determining the Gain or Loss of

Power 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



Determining Voltage or Current

Gain (dB) when Input and Out-

put Are Not Equal

whereV = voltageI = impedanceR = resistance

Determining Voltage or Current

Loss (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



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=


detector

Lx

Rx

Ra

Rb Ls

Rs

Rs = standard resistor

Ls = standard inductor

unknown inductor

(resistance + inductance)

Resistance Ratio Bridge to Measure Inductance

Cx

Rx

Ra

Rb

Cs

RsRs = standard resistor

Cs = standard capacitor

unknown capacitance

(reactive and resistive component)

Resistance Ratio Bridge to Measure Capacitance

detector


Measuring Capacitance,

Wien Bridge

Measuring Capacitance,

Schering Bridge

and

Measuring Inductance, Maxwell

Bridge

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


R2

Rs

Rs

Rs

Rb

Rx

Lx

Ra

Rb

Rx

Cx

Cs

Cs

Cb

Cs

Cx

Rx

R1

R1 = 2 R2

detector

detector

Wien Bridge

Schering Bridge

Maxwell Bridge

detector


Measuring Inductance, Hay

Bridge Q Ratio Greater than 10

and

Measuring Inductance, Hay

Bridge Q Ratio Less than 10

and

whereQ = reactive/resistive ratio

Measuring Inductance, Owens

Bridge

and

Measuring Wattage

Average Power in a Cycle

whereP = powerE = sinusoidal voltageI = currentφ = phase angle that current lagsbehind voltage

r.m.s. Values of Voltage and

Current

and

IIm=

2

EEm=

2

P E I= cosφ

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=


detector

Lx

Rx

Rb

Rs

CsRa

Hay Bridge

Rs

Cs

Lx

Rx

La

Ca

detector

Owens Bridge


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



Conversion Tables for Electricity (cont.)


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





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





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





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



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 Magnetic

Field Intensity

wheredH = magnetic field densityi = currentds = length of circuit elementr = radius vector θ = 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.

γ =∆

=cp

Bc

,1

dHi ds

r= 2 sinθ


d

dsχ

θ r

B B

c c

p


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

γ 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



(Longer sides of the rectangle per-pendicular to lines joining theircenters.)

Geometric Mean Distances of

Equal 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


c c

p

B


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



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



Geometric Mean Distance of an

Annulus from Itself

Geometric Mean Distance of a

Point 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 ζ

Values for Geometric Mean Distance of an Annulus

p2/p1 logeζ 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


A area

point

p 1

p2


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


ι

p


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



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



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



Mutual Inductance of Two Equal

Parallel Conductors

Self-Inductance of a Straight

Conductor

General Formula

wherer = geometric mean distanceζ1= arithmetic mean distance ofthe points of the cross section

For a Round Wire, Radius p

For a Round Magnetic Wire

where µ = permeability

For Rectangular Wire, Sides Band C

whereB and C = see table, Values ofconstants for Geometric MeanDistance for Rectangles

For Elliptical Wire

where α = β 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 of

Parallel Conductors

Equal Round Wires of Radius (p)

Equal Permeable Round Wires

L ldp

dle= + −

0 0044

. logµ

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 .

α β

L ll

B Cee e=

++ −

0 0022 1

2. log log

L ll

pe= − +

0 0022

14

. logµ

L ll

pe= −

0 0022 3

4. log

L ll

r le= − +

0 0022

1 1. logζ

M ll

dk

dl

d

le e= − − + −

0 0022

11

4

2

2. log log



Return Circuit of Two Tubular Conductors, One Inside the Other

whereloge ζ1 and loge ζ3 = values from table, geometric mean distance ofan annulus

Return Circuit of Polycore Cable

Mutual Inductance of Unequal Parallel Filaments

General Formula

whereα = l + m − ζβ = l − ζγ = m − ζ

M hd

hd

hd

hd

d= −

− + − +− − − −0 001 1 1 1 1 2 2. sin sin sin sinαα

ββ

γγ

ζζ

α

+ + + + − +

β γ ζ2 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 elog logρ ξ

L lpp

PP

pp

e= +

−

×0 002

2

1

1

3

2

1

2

2

1

2. log loog log loge e epp

1

21 31− + +

ζ ζ


p

a

p1

p 2

ι

ζ m

p


Mutual Inductance of Filaments Inclined at an Angle

Equal Filaments Meeting at a

Point

Mutual Inductance between

Filaments

or

M lS= 0 001.

M l hl

l R=

+−0 004 1

1. cos tanε

R l12 22 1= −( cos )ε

Value of Factor S (cont.)

cos εε 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 εε 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


ι

ι

εR1


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 tanε

Values for S1, Unequal Filaments Meeting at a Point

cos ε 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



Unequal Filaments in the Same

Plane, 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

εµ

µ

= ++

+ ++

−

−

− −

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

µ µ ε

µ µ ε

RR v v

R v m v m

32 2 2

42 2 2

2

2

= + −

= + + − +

µ µ ε

µ µ ε

cos

( ) ( )cos

vm l R R m R R l

l m=

− − + − − −

2

4

242

32

22

22

32 2

2 2 4

( ) (α

α

µα

α=

− − + − − −

l m R R l R R m

l m

2

4

222

32 2 2

42

32 2

2 2 4

( ) ( )

α242

32

22

12= − + −R R R R

22

cos εα

=lm


b

m

C

a

BA

d

p

p

ν

ε

ι

ν

µ

ι

R3

R4

R1

R 2

∈

m


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

ρ ρ ρ

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

− + +

+ + +

( )

( )µ

ωε µ ε

ε

ε

=+ + +

−+

−

−

tancos ( )( )sin

sin

tancos (

12 2

1

12

d l v mdR

d µµ εε

ε µ εε

+

=+

−

l vdR

d vdR

) sinsin

tancos sin

sin

2

2

12 2

3

−+ +

−tancos ( )sin

sin1

2 2

4

d v mdRε µ ε

ε

Ml h

mR R

v m hl

R R

0 0012

2

1

1 2

1

1 4

. cos( )tan

( )tan ta

εµ

µ

= ++

+ ++

−

−

− nn

tansin

hm

R R

v hR R

d

−

−

+

−+

−

Ω

1

3 4

1

2 32

1ε



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

. logρ

αµ

L ss

e= + +

0 016 0 21198

4. log .

ρµ

L ss

e= − +

0 012 0 15152

4. log .

ρµ

L ss

e= − +

0 010 0 40914

4. log .

ρµ

L ss

e= − +

0 008 0 77401

4. log .

ρµ

L ss

e= − +

0 006 1 40546

4. log .

ρµ

L aa b

a b a hab

b hbae e= + + +

− −

− −0 004

2 22 2 2 1 1. log log sin sin

ρ ρ

− + + +

24

( ) ( )a b a bµ



Values for αα (alpha) for Certain Plane Figures

Rectanglesβ α

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

ε α

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



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= ∫62π

2 6ad

sd

=

π

Ms

F= ∫32π

2 3ad

sd

=

π

Ms

F= ∫42π

2 4ad

sd

=

π


Regular Polygons

N α

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


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



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



Inductance of Single-Layer Coils on Rectangular Winding Forms

where

g a a2 212= +

L Naab

ba

hab

ba

hab

a

b

ba

= + − −

− −0 008

12

12

12

12 1

1

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+ −

+

+ +

−π

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− −

+



Coefficients, Short Rectangle Solenoid

βπ11

11′ = +

k


κ β1’ β1 β2 β3 β5 β7

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


Self-Inductance of CircularCoils of Rectangular Cross-Section

Nomenclature

a = 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 = pcδ = diameter of the covered wire

b n

c n

Nbc

b

c

==

=

δδ

δ

b n p

c n p

N n n

b b

c c

b c

===


c

b

a


Date post:	25-May-2015
Category:	Engineering
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