Electrical Formula

 

To Find Single Phase Two Phase Four Wire Three Phase Direct
Current
Amps when “HP” is known HP x 746

V x %EFF x PF

HP x 746

V x %EFF x PF x 2

HP x 746

V x %EFF x PF x 1.73

HP x 746

V x %EFF

Amps when “KW”is known  KW x 1000

V x PF

KW x 1000

V x PF x 2

KW x 1000

V x PF x 1.73

KW x 1000

V

Amps when “kVA”is known kVA x 1000

V

kVA x 1000

V x 2

kVA x 1000

V x 1.73

 

Kilowatts V x A x PF

1000

V x A x PF x 2

1000

V x A x PF x 1.73

1000

V x A

1000

Kilovolt-Amps “kVA” V x A

1000

V x A x 2

1000

V x A x 1.73

1000

HP V x A x%EFFx PF

746

VxA x %EFF x PF x 2

746

VxAx%EFFxPFx1.73

746

V x A X %EFF

746

 

V=VOLTS      A=AMPS      EFF=EFFICIENCY         PF=POWER FACTOR

  1. volts x amps = watts

Same as P = VI … Power (watts) = V (volts) x I (amps)

Example: 4500 watt element, 240 volt water heater circuit, how many amps?

Amps = 4500 watt ÷ 240 volt = 18.75 amps

  1. 1000 watts = 1 kilowatt Kw

.002931 Kw needed to raise 1 pound of water 1°F

  1. Resistance Ohms = volts² ÷ watts

Example: test 4500 watt element 240 volt water heater

Correct ohm reading = 240² ÷ 4500 = 12.8 ohms

  1. Resistance Ohms = volts ÷ amps

E = IR

volts = amps x resistance (ohms)

1 horsepower = 745.6998 watts

Average efficiency of Motor Power Factor

When an actual efficiency and power factor of the motor to be controlled are not known, then the following approximation

Type Power Factor
DC motor, 35HP and less 80% to 85%
DC motor, above 35HP 85% to 90%
Synchronous Motor at 100% Power factor 92% to 95%
Three Phase Induction Motors, 25HP and less 70%
Three Phase Induction Motors,  above 25HP 80%

USEFUL ELECTRICAL FORMULAS

kVA/kW AMPERAGE CHART 80% POWER FACTOR

kVA kW 208V 220V 240V 380V 400V 440V 450V 480V 600V 2400V 3300V 4160V
6.3 5 17.5 16.5 15.2 9.6 9.1 8.3 8.1 7.6 6.1
9.4 7.5 26.1 24.7 22.6 14.3 13.6 12.3 12 11.3 9.1
12.5 10 34.7 33 30.1 19.2 18.2 16.6 16.2 15.1 12
18.7 15 52 49.5 45 28.8 27.3 24.9 24.4 22.5 18
25 20 69.5 66 60.2 38.4 36.4 33.2 32.4 30.1 24 6 4.4 3.5
31.3 25 87 82.5 75.5 48 45.5 41.5 40.5 37.8 30 7.5 5.5 4.4
37.5 30 104 99 90.3 57.6 54.6 49.8 48.7 45.2 36 9.1 6.6 5.2
50 40 139 132 120 77 73 66.5 65 60 48 12.1 8.8 7
62.5 50 173 165 152 96 91 83 81 76 61 15.1 10.9 8.7
75 60 208 198 181 115 109 99.6 97.5 91 72 18.1 13.1 10.5
93.8 75 261 247 226 143 136 123 120 113 90 22.6 16.4 13
100 80 278 264 240 154 146 133 130 120 96 24.1 17.6 13.9
125 100 347 330 301 192 182 166 162 150 120 30 21.8 17.5
156 125 433 413 375 240 228 208 204 188 150 38 27.3 22
187 150 520 495 450 288 273 249 244 225 180 45 33 26
219 175 608 577 527 335 318 289 283 264 211 53 38 31
250 200 694 660 601 384 364 332 324 301 241 60 44 35
312 250 866 825 751 480 455 415 405 376 300 75 55 43
375 300 1040 990 903 576 546 498 487 451 361 90 66 52
438 350 1220 1155 1053 672 637 581 568 527 422 105 77 61
500 400 1390 1320 1203 770 730 665 650 602 481 120 88 69
625 500 1735 1650 1504 960 910 830 810 752 602 150 109 87
750 600 2080 1980 1803 1150 1090 996 975 902 721 180 131 104
875 700 2430 2310 2104 1344 1274 1162 1136 1052 842 210 153 121
1000 800 2780 2640 2405 1540 1460 1330 1300 1203 962 241 176 139
1125 900 3120 2970 2709 1730 1640 1495 1460 1354 1082 271 197 156
1250 1000 3470 3300 3009 1920 1820 1660 1620 1504 1202 301 218 174
1563 1250 4350 4130 3765 2400 2280 2080 2040 1885 1503 376 273 218
1875 1500 5205 4950 4520 2880 2730 2490 2440 2260 1805 452 327 261
2188 1750 5280 3350 3180 2890 2830 2640 2106 528 380 304
2500 2000 6020 3840 3640 3320 3240 3015 2405 602 436 348
2812 2250 6780 4320 4095 3735 3645 3400 2710 678 491 392
3125 2500 7520 4800 4560 4160 4080 3765 3005 752 546 435
3750 3000 9040 5760 5460 4980 4880 4525 3610 904 654 522
4375 3500 10550 6700 6360 5780 5660 5285 4220 1055 760 610
5000 4000 12040 7680 7280 6640 6480 6035 4810 1204 872 695

BASIC ELECTRICAL ENGINEERING FORMULAS

BASIC ELECTRICAL CIRCUIT FORMULAS

E=Irms2R×tENERGY (dissipated on R or stored in L, C)

E=Li2/2

E=Cv2/2

Notes:

R- electrical resistance in ohms, L- inductance in henrys, C- capacitance in farads, f – frequency in hertz, t- time in seconds, π≈3.14159;

ω=2πf – angular frequency; j – imaginary unit ( j2=-1 )

Electrical Formulas

The most common used electrical formulas – Ohms Law and combinations

Common electrical units used in formulas and equations are:

  • Volt – unit of electrical potential or motive force – potential is required to send one ampere of current through one ohm of resistance
  • Ohm – unit of resistance – one ohm is the resistance offered to the passage of one ampere when impelled by one volt
  • Ampere – units of current – one ampere is the current which one volt can send through a resistance of one ohm
  • Watt – unit of electrical energy or power – one watt is the product of one ampere and one volt – one ampere of current flowing under the force of one volt gives one watt of energy
  • Volt Ampere – product of volts and amperes as shown by a voltmeter and ammeter – in direct current systems the volt ampere is the same as watts or the energy delivered – in alternating current systems – the volts and amperes may or may not be 100% synchronous – when synchronous the volt amperes equals the watts on a wattmeter – when not synchronous volt amperes exceed watts – reactive power
  • Kilovolt Ampere – one kilovolt ampere – KVA – is equal to 1000 volt amperes
  • Power Factor – ratio of watts to volt amperes

Electric Power Formulas

P = V I         (1a)

P = R I2         (1b)

P = V2/ R         (1c)

where

P = power (watts, W)

V = voltage (volts, V)

I = current (amperes, A)

R = resistance (ohms, Ω)

Electric Current Formulas

I = V / R         (2a)

I = P / V         (2b)

I = (P / R)1/2         (2c)

Electric Resistance Formulas

R = V / I         (3a)

R = V2/ P         (3b)

R = P / I2         (3c)

Electrical Potential Formulas – Ohms Law

Ohms law can be expressed as:

V = R I         (4a)

V = P / I         (4b)

V = (P R)1/2         (4c)

Example – Ohm’s law

A 12 volt battery supplies power to a resistance of 18 ohms.

I = (12 V) / (18 Ω)

0.67 (A)

Electrical Motor Formulas

Electrical Motor Efficiency

μ = 746 Php / Pinput_w         (5)

where

μ = efficiency

Php = output horsepower (hp)

Pinput_w = input electrical power (watts)

or alternatively

μ = 746 Php / (1.732 V I PF)         (5b)

Electrical Motor – Power

P3-phase = (V I PF 1.732) / 1,000         (6)

where

P3-phase = electrical power 3-phase motor (kW)

PF = power factor electrical motor

Electrical Motor – Amps

I3-phase = (746 Php) / (1.732 V μ PF)         (7)

where

I3-phase = electrical current 3-phase motor (amps)

PF = power factor electrical motor

 

Three-Phase Power Equations

Electrical 3-phase equations

Most AC power today is produced and distributed as three-phase power where three sinusoidal voltages are generated out of phase with each other. With single-phase AC power there is only one single sinusoidal voltage.

Real Power

Wapplied = 31/2 U I cos Φ

             = 31/2 U I PF        (1)

where

Wapplied = real power (W, watts)

U = voltage (V, volts)

I = current (A, amps)

PF = cos Φ = power factor (0.7 – 0.95) 

For pure resistive load: PF = cos Φ = 1

  • resistive loads converts current into other forms of energy, such as heat
  • inductive loads use magnetic fields like motors, solenoids, and relays

Example – Pure Resistive Load

For pure resistive load and power factor = 1 the real power in a 415 voltage 20 amps circuit can be calculated as

Wapplied = 31/2 415 (V) 20 (A) 1

14400 W

14.4 kW

Total Power

W = 31/2 U I         (2)

Brake Horsepower

WBHP = 31/2 U I PF μ / 746         (3)

where

WBHP = brake horse power (hp)

μ = device efficiency

Converting Ampere between Single Phase and 3 Phase

Converting amperage between single phase (120, 240 and 480 Voltage) and three phase (240 and 480 Voltage)

With single-phase AC power there is only one single sinusoidal voltage. Most AC power today is produced and distributed as three-phase power where three sinusoidal voltages are generated out of phase with each other.

Sometimes it is necessary to turn between power (VA), voltage (V) and amperage (A). The diagram and table below can be used to convert amperage between single phase and three phase equipment and vice versa.

Power
(VA)
Amperage (A)
Volts Single Phase Volts 3 Phase
Balanced Load
120 240 480 240 480
100
150
200
250
300
0.83
1.25
1.67
2.08
2.50
0.42
0.63
0.83
1.04
1.25
0.21
0.31
0.42
0.52
0.63
0.24
0.36
0.49
0.61
0.73
0.13
0.18
0.25
0.30
0.37
350
400
450
500
600
2.92
3.33
3.75
4.17
5.00
1.46
1.67
1.88
2.08
2.50
0.73
0.84
0.93
1.04
1.25
0.85
0.97
1.10
1.20
1.45
0.43
0.49
0.55
0.60
0.73
700
750
800
900
1,000
5.83
6.25
6.67
7.50
8.33
2.92
3.13
3.33
3.75
4.17
1.46
1.56
1.67
1.87
2.10
1.70
1.81
1.93
2.17
2.41
0.85
0.91
0.97
1.09
1.21
1,100
1,200
1,250
1,300
1,400
9.17
10.0
10.4
10.8
11.7
4.58
5.00
5.21
5.42
5.83
2.30
2.51
2.61
2.71
2.91
2.65
2.90
3.10
3.13
3.38
1.33
1.45
1.55
1.57
1.69
1,500
1,600
1,700
1,750
1,800
12.5
13.3
14.2
14.6
15.0
6.25
6.67
7.08
7.29
7.50
3.12
3.34
3.54
3.65
3.75
3.62
3.86
4.10
4.22
4.34
1.82
1.93
2.05
2.10
2.17
1,900
2,000
2,200
2,500
2,750
15.8
16.7
18.3
20.8
23.0
7.92
8.33
9.17
10.4
11.5
3.96
4.17
4.59
5.21
5.73
4.58
4.82
5.30
6.10
6.63
2.29
2.41
2.65
3.05
3.32
3,000
3,500
4,000
4,500
5,000
25.0
29.2
33.3
37.5
41.7
12.5
14.6
16.7
18.8
20.8
6.25
7.30
8.33
9.38
10.42
7.23
8.45
9.64
10.84
12.1
3.62
4.23
4.82
5.42
6.1
6,000
7,000
8,000
9,000
10,000
50.0
58.3
66.7
75.0
83.3
25.0
29.2
33.3
37.5
41.7
12.50
14.59
16.67
18.75
20.85
14.50
16.9
19.3
21.7
24.1
7.25
8.5
9.65
10.85
12.1

 

Asynchronous Motors – Electrical Data

Electrical motor data – nominal current, fuse, start ampere, contactor and circuit breaker of asynchronous motors

The table below can used to determine electrical data for asynchronous 380 Voltage motors.

380 Voltage 50 Hz motors have been commonly used in Europe. Note that the nominal voltage of existing 220/380 V and 240/415 V systems shall evolve toward the IEC recommended value of 230/400 V.

Rated Power Nominal current
– In – 
(A)
Directly Fused
(A)
Star – Delta Started
(A)
Star – Delta contactor
– In –
(A)
Circuit Breaker
– In –
(A)
kW HP
0.2 0.3 0.7 2 2   16
0.33 0.5 1.1 2 2   16
0.5 0.7 1.4 2 2   16
0.8 1.1 2.1 4 4   16
1.1 1.5 2.6 4 4   16
1.5 2 3.6 6 4 (16)22 16
2.2 3 5.0 10 6 (16)22 16
3 4 6.6 16 10 (16)22 16
4 5.5 8.5 20 16 (16)22 16
5.5 7.5 11.5 25 20 (16)22 16
7.5 10 15.5 35 25 (25)22 25
11 15 22.2 35 35 (40)30 40
15 20 30 50 35 (40)30 40
22 30 44 63 50 (63)/60 60
30 40 57 80 63 (63)/60 60
45 66 85 125 100 90 100
55 75 104 160 125 110 100
75 100 140 200 160 150 200
90 125 168 225 200 220 200
110 150 205 300 250 220 200
132 180 245 400 300 300 400
160 220 290 430 300 300 400
200 270 360 500 430 480 400
240 325 430 630 500 480 480

Full-voltage, single-speed motor starters

Full-voltage starters (manual and magnetic) apply full voltage directly to motor terminals.

Reduced-voltage, single-speed motor starters

Some machines or loads may require a gentle start and smooth acceleration up to full speed.

Many starters apply reduced voltage to motor windings, primary resistor, primary reactor, autotransformer and solid state. Part winding and wye-delta starters can also provide reduced-voltage starting, although technically they are not reduced-voltage starters.

Motor Protection

Motors should have protection for themselves, in the branch circuit, and in the feeder line. Protection provided by fuses and circuit breakers guards against fault conditions caused by short circuits or grounds and over currents exceeding locked-rotor values.

Electrical Units

Definition of common electrical units – like Ampere, Volt, Ohm, Siemens

Ampere – A

The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 x 10-7 Newton per meter of length.

Electric current is the same as electric quantity in movement, or quantity per unit time, expressed like

I = dq / dt

where

I = electric current (ampere, A)

dq = electric quantity (coulomb, C)

dt = time (s)

Coulomb – C

The standard unit of quantity in electrical measurements. It is the quantity of electricity conveyed in one second by the current produced by an electro-motive force of one volt acting in a circuit having a resistance of one ohm, or the quantity transferred by one ampere in one second.

Farad – F

The farad is the standard unit of capacitance. Reduced to base SI units one farad is the equivalent of one second to the fourth power ampere squared per kilogram per meter squared (sA2/kg m2).

When the voltage across a 1 F capacitor changes at a rate of one volt per second (1 V/s) a current flow of 1 Aresults. A capacitance of 1 F produces 1 V of potential difference for an electric charge of one coulomb (1 C).

In common electrical and electronic circuits units of microfarads μF (1 μF = 10-6 F) and picofarads pF (1 pF = 10-12F) are used.

Ohm – Ω

The derived SI unit of electrical resistance – the resistance between two points on a conductor when a constant potential difference of 1 volt between them produces a current of 1 ampere.

Henry – H

The Henry is the unit of inductance. Reduced to base SI units one henry is the equivalent of one kilogram meter squared per second squared per ampere squared (kg m2 s-2 A-2).

Inductance

An inductor is a passive electronic component that stores energy in the form of a magnetic field.

The standard unit of inductance is the henry abbreviated H. This is a large unit and more commonly used units are the microhenry abbreviated μH (1 μH =10-6H) and the millihenry abbreviated mH (1 mH =10-3 H). Occasionally, thenanohenry abbreviated nH (1 nH = 10-9 H) is used.

Joule – J

The unit of energy work or quantity of heat done when a force of one Newton is applied over a displacement of one meter. One joule is the equivalent of one watt of power radiated or dissipated for one second.

In imperial units the British Thermal Unit (Btu) is used to express energy. One Btu is equivalent to approximately1,055 joules.

Siemens – S

The unit of electrical conductance S = A / V

Watt

The watt is used to specify the rate at which electrical energy is dissipated, or the rate at which electromagnetic energy is radiated, absorbed, or dissipated.

The unit of power W or Joule/second

Weber – Wb

The unit of magnetic flux.

The flux that when linking a circuit of one turn, produces an Electro Motive Force – EMF – of 1 volt as it is reduced to zero at a uniform rate in one second.

  • 1 Weber is equivalent to 108 Maxwells

Tesla – T

The unit of magnetic flux density the Tesla is equal to 1 Weber per square meter of circuit area.

Volt

The Volt – V – is the Standard International (SI) unit of electric potential or electromotive force. A potential of one volt appears across a resistance of one ohm when a current of one ampere flows through that resistance.

Reduced to SI base units,

1 (V) = 1 (kg m2 / s3 A)

Electromotive Force – e.m.f

Change in electrical potential between two points

A change in electrical potential between two points in an electric circuit is called a “potential difference”. The “electromotive force (e.m.f)” provided by a source of energy – such as a generator or battery – is measured in volts.

One volt i one joule per coulomb and is defined as the difference in potential between two points in a conductor which, when carrying a current of one ampere, dissipates a power of one watt:

volts = watts / amperes = (joules/sec) / amperes = joules / ampere seconds = joules / coulombs

Resistivity, Conductivity and Temperature Coefficients for some Common Materials

Resistivity, conductivity and temperature coefficients for some common materials as silver, gold, platinum, iron and more – Including a tutorial explanation of resistivity and conductivity

The factor in the resistance which takes into account the nature of the material is the resistivity.

Resistivity is the

  • resistance of a unit cube of the material measured between the opposite faces of the cube
Material Resistivity Coefficient 2)
– ρ –
(ohm m)
Temperature
Coefficient
 2)
(per degree C, 1/oC) 
Conductivity
– σ –
(1 /Ωm)
Aluminum 2.65 x 10-8 3.8 x 10-3 3.77 x 107
Animal fat     14 x 10-2
Animal muscle     0.35
Antimony 41.8 x 10-8    
Barium (0oC) 30.2 x 10-8    
Beryllium 4.0 x 10-8    
Bismuth 115 x 10-8    
Brass – 58% Cu 5.9 x 10-8 1.5 x 10-3  
Brass – 63% Cu 7.1 x 10-8 1.5 x 10-3  
Cadmium 7.4 x 10-8    
Caesium (0oC) 18.8 x 10-8    
Calcium (0oC) 3.11 x 10-8    
Carbon (graphite)1) 3 – 60 x 10-5 -4.8 x 10-4  
Cast iron 100 x 10-8    
Cerium (0oC) 73 x 10-8    
Chromel (alloy of chromium and aluminum)   0.58 x 10-3  
Chromium 13 x 10-8    
Cobalt 9 x 10-8    
Constantan 49 x 10-8 3 x 10-5 0.20 x 107
Copper 1.724 x 10-8 4.29 x 10-3 5.95 x 107
Dysprosium (0oC) 89 x 10-8    
Erbium (0oC) 81 x 10-8    
Eureka   0.1 x 10-3  
Europium (0oC) 89 x 10-8    
Gadolium 126 x 10-8    
Gallium (1.1K) 13.6 x 10-8    
Germanium1) 1 – 500 x 10-3 -50 x 10-3  
Glass 1 – 10000 x 109   10-12
Gold 2.24 x 10-8    
Graphite 800 x 10-8 -2.0 x 10-4  
Hafnium (0.35K) 30.4 x 10-8    
Holmium (0oC) 90 x 10-8    
Indium (3.35K) 8 x 10-8    
Iridium 5.3 x 10-8    
Iron 9.71 x 10-8 6.41 x 10-3 1.03 x 107
Lanthanum (4.71K) 54 x 10-8    
Lead 20.6 x 10-8   0.45 x 107
Lithium 9.28 x 10-8    
Lutetium 54 x 10-8    
Magnesium 4.45 x 10-8    
Manganese 185 x 10-8 1.0 x 10-5  
Mercury 98.4 x 10-8 8.9 x 10-3 0.10 x 107
Mica 1 x 1013    
Mild steel 15 x 10-8 6.6 x 10-3  
Molybdenum 5.2 x 10-8    
Neodymium 61 x 10-8    
Nichrome (alloy of nickel and chromium) 100 – 150 x 10-8 0.40 x 10-3  
Nickel 6.85 x 10-8 6.41 x 10-3  
Nickeline 50 x 10-8 2.3 x 10-4  
Niobium (Columbium) 13 x 10-8    
Osmium 9 x 10-8    
Palladium 10.5 x 10-8    
Phosphorus 1 x 1012    
Platinum 10.5 x 10-8 3.93 x 10-3 0.943 x 107
Plutonium 141.4 x 10-8    
Polonium 40 x 10-8    
Potassium 7.01 x 10-8    
Praseodymium 65 x 10-8    
Promethium 50 x 10-8    
Protactinium (1.4K) 17.7 x 10-8    
Quartz (fused) 7.5 x 1017    
Rhenium (1.7K) 17.2 x 10-8    
Rhodium 4.6 x 10-8    
Rubber – hard 1 – 100 x 1013    
Rubidium 11.5 x 10-8    
Ruthenium (0.49K) 11.5 x 10-8    
Samarium 91.4 x 10-8    
Scandium 50.5 x 10-8    
Selenium 12.0 x 10-8    
Silicon1) 0.1-60 -70 x 10-3  
Silver 1.59 x 10-8 6.1 x 10-3 6.29 x 107
Sodium 4.2 x 10-8    
Soil, typical ground     10-2 – 10-4
Solder 15 x 10-8    
Stainless steel     106
Strontium 12.3 x 10-8    
Sulfur 1 x 1017    
Tantalum 12.4 x 10-8    
Terbium 113 x 10-8    
Thallium (2.37K) 15 x 10-8    
Thorium 18 x 10-8    
Thulium 67 x 10-8    
Tin 11.0 x 10-8  4.2 x 10-3  
Titanium 43 x 10-8    
Tungsten 5.65 x 10-8 4.5 x 10-3 1.79 x 107
Uranium 30 x 10-8    
Vanadium 25 x 10-8    
Water, distilled     10-4
Water, fresh     10-2
Water, salt     4
Ytterbium 27.7 x 10-8    
Yttrium 55 x 10-8    
Zinc 5.92 x 10-8  3.7 x 10-3  
Zirconium (0.55K) 38.8 x 10-8    

1) Note! – the resistivity depends strongly on the presence of impurities in the material.

2) Note! – the resistivity depends strongly on the temperature of the material. The table above is based on 20oC reference.

The electrical resistance of a wire is greater for a longer wire and less for a wire of larger cross sectional area. The resistance depend on the material of which it is made and can be expressed as:

R = ρ L / A         (1)

where

R = resistance (ohm, Ω)

ρ = resistivity coefficient (ohm m, Ω m)

L = length of wire (m)

A = cross sectional area of wire (m2)

The factor in the resistance which takes into account the nature of the material is the resistivity. Since it is temperature dependent, it can be used to calculate the resistance of a wire of given geometry at different temperatures.

The inverse of resistivity is called conductivity and can be expressed as:

σ = 1 / ρ         (2)

where

σ = conductivity (1 / Ω m)

Example – Resistance in an Aluminum Cable

Resistance of an aluminum cable with length 10 m and cross sectional area of 3 mm2 can be calculated as

R = (2.65 10-8 Ω m) (10 m) / ((3 mm2) (10-6 m2/mm2))

   = 0.09 Ω

Resistance

The electrical resistance of a circuit component or device is defined as the ratio of the voltage applied to the electric current which flows through it:

R = V / I         (3)

where

R = resistance (ohm)

V = voltage (V)

I = current (A)

Ohm’s Law

If the resistance is constant over a considerable range of voltage, then Ohm’s law,

I = V / R         (4)

can be used to predict the behavior of the material.

Temperature Coefficient of Resistance

The electrical resistance increases with temperature. An intuitive approach to temperature dependence leads one to expect a fractional change in resistance which is proportional to the temperature change:

dR / Rs = α dT         (5)

where

dR = change in resistance (ohm)

Rs = standard resistance according reference tables (ohm)

α = temperature coefficient of resistance

dT = change in temperature (K)

(5) can be modified to:

dR = α dT Rs   (5b)

Example – Resistance of a Carbon resistor when changing Temperature

A carbon resistor with resistance 1 kΩ is heated 100 oC. With a temperature coefficient -4.8 x 10-4 (1/oC) the resistance change can be calculated as

dR = (-4.8 x 10-4 1/oC) (100 oC) (1 kΩ)

    = – 0.048 (kΩ)

The resulting resistance for the resistor

R = (1 kΩ) – (0.048 kΩ)

    = 0.952 (kΩ)

    = 952 (Ω)

Power Factor Three-Phase Electrical Motors

Power Factor definition for three-phase electrical motors

The power factor of an AC electric power system is defined as the ratio of the active (true or real) power to theapparent power

where

  • Active (Real or True) Power is measured in watts (W) and is the power drawn by the electrical resistance of a system doing useful work
  • Apparent Power is measured in volt-amperes (VA) and is the voltage on an AC system multiplied by all the current that flows in it. It is the vector sum of the active and the reactive power
  • Reactive Power  is measured in volt-amperes reactive (VAR). Reactive Power is power stored in and discharged by inductive motors, transformers and solenoids

Reactive power is required for the magnetization of a motor but doesn’t perform any action. The reactive power required by inductive loads increases the amounts of apparent power – measured in kilovolt amps (kVA) – in the distribution system. Increasing of the reactive and apparent power will cause the power factor – PF – to decrease.

Power Factor

It is common to define the Power Factor – PF – as the cosine of the phase angle between voltage and current – or the “cosφ”.

PF = cos φ

where

PF = power factor

φ = phase angle between voltage and current

The power factor defined by IEEE and IEC is the ratio between the applied active (true) power – and theapparent power, and can in general be expressed as:

PF = P / S         (1)

where

PF = power factor

P = active (true or real) power (Watts)

S = apparent power (VA, volts amps)

A low power factor is the result of inductive loads such as transformers and electric motors. Unlike resistive loads creating heat by consuming kilowatts, inductive loads require a current flow to create magnetic fields to produce the desired work.

Power factor is an important measurement in electrical AC systems because

  • an overall power factor less than 1 indicates that the electricity supplier need to provide more generating capacity than actually required
  • the current waveform distortion that contributes to reduced power factor is caused by voltage waveform distortion and overheating in the neutral cables of three-phase systems

International standards such as IEC 61000-3-2 have been established to control current waveform distortion by introducing limits for the amplitude of current harmonics.

Example – Power Factor

A industrial plant draws 200 A at 400 V and the supply transformer and backup UPS is rated 200 A × 400 V = 80 kVA.

If the power factor – PF – of the loads is only 0.7 – only

80 kVA × 0.7

56 kW

of real power is consumed by the system. If the power factor is close to 1 (purely resistive circuit) the supply system with transformers, cables, switchgear and UPS could be made considerably smaller.

Any power factor less than 1 means that the circuit’s wiring has to carry more current than what would be necessary with zero reactance in the circuit to deliver the same amount of (true) power to the resistive load.

A low power factor is expensive and inefficient and some utility companies may charge additional fees when the power factor is less than 0.95. A low power factor will reduce the electrical system’s distribution capacity by increasing the current flow and causing voltage drops.

“Leading” or “Lagging” Power Factors

Power factors are usually stated as “leading” or “lagging” to show the sign of the phase angle.

  • With a purely resistive load current and voltage changes polarity in step and the power factor will be 1. Electrical energy flows in a single direction across the network in each cycle.
  • Inductive loads – transformers, motors and wound coils – consumes reactive power with current waveform lagging the voltage.
  • Capacitive loads – capacitor banks or buried cables – generates reactive power with current phase leading the voltage.

Inductive and capacitive loads stores energy in magnetic or electric fields in the devices during parts of the AC cycles. The energy is returned back to the power source during the rest of the cycles.

Power Factor for a Three-Phase Motor

The total power required by an inductive device as a motor or similar consists of

  • Active (true or real)  power (measured in kilowatts, kW)
  • Reactive power – the nonworking power caused by the magnetizing current, required to operate the device (measured in kilovars, kVAR)

The power factor for a three-phase electric motor can be expressed as:

PF = P / [(3)1/2 U I]         (2)

where

PF = power factor

P = power applied (W, watts)

U = voltage (V)

I = current (A, amps)

Typical Motor Power Factors

Power (hp) Speed (rpm) Power Factor
1/2 load 3/4 load full load
0 – 5 1800 0.72 0.82 0.84
5 – 20 1800 0.74 0.84 0.86
20 – 100 1800 0.79 0.86 0.89
100 – 300 1800 0.81 0.88 0.91
  • 1 hp = 745.7 W

Active, Apparent and Reactive Power

Apparent, True Applied and Reactive Power – kVA

Total electrical power consumption depends on real power – electrical energy consumption, and reactive power – imaginary power consumption, and can be expressed (power triangle or Pythagorean relationship).

S = (Q2 + P2)1/2       (1)

where

S = apparent power (kilovolt amps, kVA)

Q = reactive power (kilovolt amps reactive, kVAR)

P = active power (kilowatts, kW)

Apparent Power

Apparent Power is measured in volt-amperes (VA) and is the voltage on an AC system multiplied by all the current that flows in it. It is the vector sum of the active and the reactive power.

Single Phase Current

S = U I       (2a)

where

U = electric potential (V)

I = current (A)

Three Phase Current

S = 1.732 U I       (2b)

Active Power

Active (Real or True) Power is measured in watts (W) and is the power drawn by the electrical resistance of a system doing useful work.

Single Phase Current

P = U I cos φ    (3a)

where

φ = phase angle

Three Phase Current

P = 1.732 U I cos φ    (3b)

Direct Current

P = U I    (3c)

Reactive Power

Reactive inductive Power – Q – is measured in volt-amperes reactive (VAR) and is the power stored in and discharged by the inductive motors, transformers or solenoids.

Reactive power required by inductive loads increases the amount of apparent power – measured in kilovolt amps (kVA) – in the distribution system. Increasing the reactive and apparent power causes the power factor – PF – to decrease.

Single Phase Current

Q = U I sin φ    (4a)

where

φ = phase angle

Three Phase Current

Q = 1.732 U I sin φ    (4b)

Synchronous Speed of Electrical Motors

The speed at which an induction motor will operate depends on the input power frequency and the number of electrical magnetic poles in the motor

The synchronous speed for an electric induction motor is determined by the power supply frequency and the number of poles in the motor winding and can be expressed as:

ω = 2 · 60 · f / n         (1)

where

ω = pump shaft rotational speed (rev/min, rpm)

f = frequency (Hz, cycles/sec)

n = number of poles

Synchronous rotation speed at different frequencies and number of poles

Shaft rotation speed – ω – (rev/min, rpm)
Frequency
– f –
(Hz)
Number of poles – n
2 4 6 8 10 12
10 600 300 200 150 120 100
20 1200 600 400 300 240 200
30 1800 900 600 450 360 300
40 2400 1200 800 600 480 400
501) 3000 1500 1000 750 600 500
602) 3600 1800 1200 900 720 600
70 4200 2100 1400 1050 840 700
80 4800 2400 1600 1200 960 800
90 5400 2700 1800 1350 1080 900
100 6000 3000 2000 1500 1200 1000
  1. Motors designed for 50 Hz are common outside U.S
  2. Motors designed for 60 Hz are common in U.S

Variable Frequency Drive

A variable frequency drive modulates the speed of an electrical motor by changing the frequency of the power supplied.

Induction Motors – Synchronous and Full Load Speed

Synchronous and full load speed of amplitude current (AC) induction motors

Synchronous and approximate full load speed of AC electrical motors:

Speed (rpm)
Number of Poles Frequency (Hz, cycles/sec)
60 50
Synchronous Full Load Synchronous Full Load
2 3600 3500 3000 2900
4 1800 1770 1500 1450
6 1200 1170 1000 960
8 900 870 750 720
10 720 690 600 575
12 600 575 500 480
14 515 490 429 410
16 450 430 375 360
18 400 380 333 319
20 360 340 300 285
22 327 310 273 260
24 300 285 240 230
26 277 265 231 222
28 257 245 214 205
30 240 230 200 192

Frequency and Speed of Electrical Motors

The velocity of 2, 4, 6 and 8 poles electrical motors at 50 Hz and 60 Hz

The velocity at which an induction motor will operate depends on the input power frequency and the number of electrical magnetic poles in the motor.

Speed (RPM)
Poles Frequency 50 Hz Frequency 60 Hz
Synchronous Full Load Synchronous Full Load
2 3000 2850 3600 3450
4 1500 1425 1800 1725
6 1000 950 1200 1150
8 750 700 900 850
  • Slip – in synchronous electrical induction motors

Electrical Motors – Horsepower and Amps

Horsepower rating of electrical motors compared to their ampere rating

Horsepower rating compared to electric motor ampere rating – 115 VAC and 230 VAC – are indicated below:

Ampere Rating (amps)
Power Rating
(hp)
115 VAC 230 VAC
Efficiency (%)
50 70 50 70
1/4 3.2 2.3 1.6 1.2
1/3 4.3 3.1 2.2 1.5
1/2 6.5 4.6 3.2 2.3
3/4 9.7 7.0 4.9 3.5
1 13.0 9.3 6.5 4.6
1 1/2 19.5 13.9 9.7 7.0
2 25.9 18.5 13.0 9.3
5 64.9 46.3 32.4 23.2
  • 1 hp = 745.7 W

Torques in Electrical Induction Motors

Torques used to describe and classify electrical motors

Torque is the turning force through a radius with the units – Nm – in the SI-system and – lb ft – in the imperial system.

The torque developed by an asynchronous induction motor varies when the motor accelerates from full stop (or zero speed) to maximum operating speed.

Locked Rotor or Starting Torque

The Locked Rotor Torque or Starting Torque is the torque the electrical motor develop when its starts at rest or zero speed.

A high Starting Torque is more important for application or machines hard to start – as positive displacement pumps, cranes etc. A lower Starting Torque can be accepted for centrifugal fans or pumps where the start load is low or close to zero.

Pull-up Torque

The Pull-up Torque is the minimum torque developed by the electrical motor when it runs from zero to full-load speed (before it reaches the break-down torque point)

When the motor starts and begins to accelerate the torque in general decrease until it reach a low point at a certain speed – the pull-up torque – before the torque increases until it reach the highest torque at a higher speed – the break-down torque – point.

The pull-up torque may be critical for applications that needs power to go through some temporary barriers achieving the working conditions.

Break-down Torque

The Break-down Torque is the highest torque available before the torque decreases when the machine continues to accelerate to the working conditions.

Full-load (Rated) Torque or Braking Torque

The Full-load Torque is the torque required to produce the rated power of the electrical motor at full-load speed.

In imperial units the Full-load Torque can be expressed as

T =  5252 Php / nr         (1)

where

T = full-load torque (lb ft)

Php = rated horsepower

nr = rated rotational speed (rev/min, rpm)

In metric units the rated torque can be expressed as

T = 9550 PkW / nr             (2)

where

T = rated torque (Nm)

PkW = rated power (kW)

nr = rated rotational speed (rpm)

Example – Electrical Motor and Braking Torque

The torque of a 60 hp motor rotating at 1725 rpm can be expressed as:

Tfl = (60 hp) 5,252 / (1725 rpm)

182.7 lb ft

NEMA Design

NEMA (National Electrical Manufacturers Association) have classified electrical motors in four different designs where torques and starting-load inertia are important criteria.

IEC/NEMA Standard Torques (percent of full load torque)
Power (hp) 2 Pole 4 Pole
Locked Rotor Torque Pull Up Torque Break Down Torque Locked Rotor Torque Pull Up Torque Break Down Torque
3 170/160 110/110 200/230 180/215 120/150 200/250
5 160/150 110/105 200/215 170/185 120/130 200/225
7.5 150/140 100/100 200/200 160/175 110/120 200/215
10 150/135 100/100 200/200 160/165 110/115 200/200
15 – 20 140/130 100/100 200/200 150/150 110/105 200/200

Accelerating Torque

Accelerating Torque = Available Motor Torque – Load Torque

Reduced Voltage Soft Starters

Reduced Voltage Soft Starters are used to limit the start current reducing the Locked Rotor Torque or Starting Torque and are common in applications which is hard to start or must be handled with care – like positive displacement pumps, cranes, elevators and simila

Electrical Motors – Starting Devices

Direct-on-line starters, star-delta starters, frequency drives and soft starters

Commonly used starting methods for squirrel cage motors are

  • direct-on-line starters
  • star-delta starters
  • frequency drives
  • soft starters

Direct-on-Line Starters

The simplest and most common starting device is the direct-on-line starter where the equipment consists of a main contactor and a thermal or electronic overload relay.

The disadvantage of the direct-on-line method is very high starting current (6 to 10 times the rated motor currents) and high starting torque, causing

  • slipping belts, heavy wear on bearings and gear boxes
  • damaged products in the process
  • water hammers in piping systems

Star-Delta Starters

A star-delta starting device consists normally of three contactors, an overload relay and a timer for setting the time in the star-position (starting position).

The starting current is about 30 % of the direct-on-line starting device. The starting torque is about 25 % of the direct-on-line starting torque.

The stress on an application is reduced compared to the direct-on-line starting method.

Frequency Drives

With a variable frequency drive the electrical frequency to the motor is modulated between typical 0-250 Hz.

  • the rated motor torque is available at lower speed
  • the starting current is low – ranging 0.5 – 1 times the rated motor current

The frequency drive can also be used for soft stops.

Soft Starters

With soft starters thyristors are used to reduce starting voltage. With lower motor voltage

  • the starting current and starting moment can be very low compared to other methods

Stress on the application can be close to minimum compared to other methods.

Electrical Motor Efficiency

Calculating electric motor efficiency

Electrical motor efficiency is the ratio between shaft output power – and electrical input power.

Electrical Motor Efficiency when Shaft Output is measured in Watt

If power output is measured in Watt (W), efficiency can be expressed as:

ηm = Pout / Pin             (1)

where

ηm = motor efficiency

Pout = shaft power out (Watt, W)

Pin = electric power in to the motor (Watt, W)

Electrical Motor Efficiency when Shaft Output is measured in Horsepower

If power output is measured in horsepower (hp), efficiency can be expressed as:

ηm = Pout 746 / Pin            (2)

where

Pout = shaft power out (horsepower, hp)

Pin = electric power in to the motor (Watt, W)

Primary and Secondary Resistance Losses

The electrical power lost in the primary rotor and secondary stator winding resistance are also called copper losses. The copper loss varies with the load in proportion to the current squared – and can be expressed as

Pcl = R I2                 (3)

where

Pcl = stator winding – copper loss (W, watts)

R = resistance (Ω)

I = current (A, amps)

Iron Losses

These losses are the result of magnetic energy dissipated when when the motors magnetic field is applied to the stator core.

Stray Losses

Stray losses are the losses that remains after primary copper and secondary losses, iron losses and mechanical losses. The largest contribution to the stray losses is harmonic energies generated when the motor operates under load. These energies are dissipated as currents in the copper winding, harmonic flux components in the iron parts, leakage in the laminate core.

Mechanical Losses

Mechanical losses includes friction in the motor bearings and the fan for air cooling.

NEMA Design B Electrical Motors

Electrical motors constructed according NEMA Design B must meet the efficiencies below:

Power
(hp)
Minimum Nominal Efficiency1)
1 – 4 78.8
5 – 9 84.0
10 – 19 85.5
20 – 49 88.5
50 – 99 90.2
100 – 124 91.7
> 125 92.4

1) NEMA Design B, Single Speed 1200, 1800, 3600 RPM. Open Drip Proof (ODP) or Totally Enclosed Fan Cooled (TEFC) motors 1 hp and larger that operate more than 500 hours per year.

Potential Divider

Output voltage from a potential divider

The output voltage from a potential divider can be calculated as

Vout = Vin R/ (R1 + R2)          (1)

where

Vout = output voltage (V)

R = resistance (Ohms, Ω)

Vin = input voltage (V)

Example – Potential Divider

The output voltage from a potential divider with two resistors R1 = 10 Ohms and R2 = 20 Ohms, and input voltage12 V can be calculated as

Vout = (12 V) (20 Ω) / ((10 Ω) + (20 Ω))

8 (V)

Capacitors

Capacitors and capacitance – charge and unit of charge

A capacitor is a device capable to store electrical energy.

The plates of a capacitor is charged and there is an electric field between them. The capacitor will be discharged if the plates are connected together through a resistor.

Charge of a Capacitor

The charge of a capacitor can be expressed as

Q = I t         (1)

where

Q = charge (coulomb, C)

I = current (amp, A)

t = time (s)

The quantity of charge (number of electrons) is measured in the unit Coulomb – C – where

1 coulomb = 6.24 1018 electrons

The smallest charge that exists is the charge carried by an electron, equal to -1.602 10-19 coulomb.

Example – Quantity of Electricity Transferred

If a current of 5 amp flows for 2 minutes, the quantity of electricity – coulombs – can be calculated as

Q = (5 A) (2 min) (60 s/min)

600 C

Electric Field Strength

The charged plates are separated with a dielectric – an insulating medium. The electric field strength – the ratio between the potential difference or voltage and the thickness of the dielectric can be expressed as

E = V / d         (2)

where

E = electric field strength (volt/m)

V = potential difference (volt)

d = thickness of dielectric (m)

Electric Flux Density

Electric flux density is the ratio between the charge of the capacitor and the surface area of the capacitor plates and can be expressed as

D = Q / A         (3)

where

D = electric flux density (coulomb/m2)

A = surface area of the capacitor (m2)

Charge and Applied Voltage

Charge on a capacitor is proportional to the applied voltage and can be expressed as

Q = C V         (4)

where

C = constant of proportionality or capacitance (farad, F)

Capacitance

From (4) the capacitance can be expressed as

C = Q / V             (5)

One farad is defined as the capacitance of a capacitor when there is a potential difference across the plates of one volt when holding a charge of one coulomb.

It is common to use µF (10-6 F).

Absolute Permittivity

The ratio of electric flux density – D – to the electric field – E –  is called absolute permittivity – ε – of a dielectric and can be expressed as

ε = D / E         (6)

where

ε = absolute permittivity (F/m)

The absolute permittivity of free space or vacuum – also called the electric constant – ε0 = 8.85 10-12 F/m.

Relative Permittivity

Relative permittivity – also called dielectric constant – is the ratio between the flux density of the field in an actual dielectric and the flux density of the field in absolute vacuum.

The actual permittivity can be calculated by multiplying the relative permittivity by ε0.

ε = εr ε0         (7)

Energy Stored in Capacitors

The energy stored in a capacitor can be expressed as

W = 1/2 C V2         (8)

W = energy stored – or work done in establishing the electric field (joules, J)

 

 

 

 

 

Advertisements