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MOTOR FORMULAS AND CALCULATIONS

The formulas and calculations which appear below should be used for estimating purposes only. It is the responsibility of the customer to specify the required motor Hp, Torque and accelerating time for his application. The salesman may wish to check the customers specified values with the formulas in this section, however, if there is serious doubt concerning the customers application or if the customer requires guaranteed motor/application performance, the customer should engage an electrical engineer to do the exact calculations.

For a detailed explanation of each formula, Click on the links below to go right to it.

Rules Of Thumb Approximation
Mechanical Formulas
Temperature Conversion
High Inertia Loads
Synchronous Speed, Frequency And Number Of Poles Of AC Motors
Relation Between Horsepower, Torque, And Speed
Motor Slip
Symbols
Equivalent Inertia
Electrical Formulas
Locked Rotor Current (IL) From Nameplate Data
Effect Of Line Voltage On Locked Rotor Current (IL) (Approx.)
Basic Horsepower Calculations
Accelerating Torque
Duty Cycles

Rules Of Thumb (Approximation)
At 1800 rpm, a motor develops a 3 lb.ft. per hp
At 1200 rpm, a motor develops a 4.5 lb.ft. per hp
At 575 volts, a 3-phase motor draws 1 amp per hp
At 460 volts, a 3-phase motor draws 1.25 amp per hp
At 230 volts a 3-phase motor draws 2.5 amp per hp
At 230 volts, a single-phase motor draws 5 amp per hp
At 115 volts, a single-phase motor draws 10 amp per hp

Mechanical Formulas
Torque in lb.ft. =HP x 5250

rpm
-----HP =Torque x rpm-----rpm =120 x Frequency
math divide symbol
No. of Poles
math divide symbol
5250

Temperature Conversion
Deg C = (Deg F - 32) x 5/9
Deg F = (Deg C x 9/5) + 32

Temperature Conversion Formula

R = 1.8 K + 0.6
.K = 5/9 (R-0.6)
F = 1.8C + 32
 C = 5/9 (F-32)
R = F + 460
 .K = C + 273  

 
C = Celsius, degrees
F = Fahrenheit, degrees
 .K = Kelvin
R = Rankine, degrees
     
to C Temp. to F
-17.8
10.8
37.8
65.6
93.3
0
50
100
150
200
32.0
122.0
212.0
302.0
392.0
121.0
148.9
176.7
204.4
232.2
250
300
350
400
450
482.0
572.0
662.0
752.0
842.0
260.0
287.7
315.6
343.3
500
550
600
650
932.0
1022.0
1112.0
1202.0
to C Temp. to F
371.1
398.9
426.7
454.4
482.2
700
750
800
850
900
1292.0
1382.0
1472.0
1562.0
1652.0
510.0
537.8
565.6
593.3
621.1
950
1000
1050
1100
1150
1742.0
1832.0
1922.0
2012.0
2102.0
648.9
676.7
704.4
732.2
1200
1250
1300
1350
2192.0
2282.0
2372.0
2462.0
to C Temp. to F
760.0
787.8
815.6
843.3
872.1
1400
1450
1500
1550
1600
2552.0
2642.0
2732.0
2822.0
2912.0
899.9
927.7
955.4
983.2
1011.0
1650
1700
1750
1800
1850
3002.0
3092.0
3182.0
3272.0
3362.0
1038.8
1066.6
1094.3
1121.1
1900
1950
2000
2050
3452.0
3542.0
3632.0
3722.0

High Inertia Loads
t =WK2 x rpm
math divide symbol
308 x T av.
-----WK2 = inertia in lb.ft.2
t = accelerating time in sec.
T = Av. accelerating torque lb.ft.
T =WK2 x rpm
math divide symbol
308 x t
inertia reflected to motor = Load Inertia math left parenthesisLoad rpm
math divide symbol
Motor rpm
math right parenthesis 2

Synchronous Speed, Frequency And Number Of Poles Of AC Motors
ns =120 x f
math divide symbol
P
-----f =P x ns
math divide symbol
120
-----P =120 x f
math divide symbol
ns

Relation Between Horsepower, Torque, And Speed
HP =T x n
math divide symbol
5250
-----T =5250 HP
math divide symbol
n
-----n =5250 HP
math divide symbol
T

Motor Slip
% Slip =ns - n
math divide symbol
ns
x 100
CodekVA/HP
 
CodekVA/HP
 
CodekVA/HP CodekVA/HP
A0-3.14
 
F5.0 -5.59
 
L9.0-9.99
 
S16.0-17.99
B3.15-3.54
 
G5.6 -6.29
 
M10.0-11.19
 
T18.0-19.99
C3.55-3.99
 
H6.3 -7.09
 
N11.2-12.49
 
U20.0-22.39
D4.0 -4.49
 
I7.1 -7.99
 
P12.5-13.99
 
V22.4 & Up
E4.5 -4.99
 
K8.0 -8.99
 
R14.0-15.99
 

 

 

Symbols
I=current in amperes
E=voltage in volts
kW=power in kilowatts
kVA=apparent power in kilo-volt-amperes
HP =output power in horsepower
n=motor speed in revolutions per minute (RPM)
ns=synchronous speed in revolutions per minute (RPM)
P=number of poles
f=frequency in cycles per second (CPS)
T=torque in pound-feet
EFF=efficiency as a decimal
PF=power factor as a decimal

Equivalent Inertia

In mechanical systems, all rotating parts do not usually operate at the same speed. Thus, we need to determine the "equivalent inertia" of each moving part at a particular speed of the prime mover.

The total equivalent WK2 for a system is the sum of the WK2 of each part, referenced to prime mover speed.

The equation says:


 
WK2EQ = WK2part math left parenthesisNpart
math divide symbol
Nprime mover
math right parenthesis 2

This equation becomes a common denominator on which other calculations can be based. For variable-speed devices, inertia should be calculated first at low speed.

Let's look at a simple system which has a prime mover (PM), a reducer and a load.

WK2 = 100 lb.ft.2
 
WK2 = 900 lb.ft.2
(as seen at output shaft)

 
WK2 = 27,000 lb.ft.2
PRIME MOVER

 
3:1 GEAR REDUCER

 
LOAD

The formula states that the system WK2 equivalent is equal to the sum of WK2parts at the prime mover's RPM, or in this case:


 
WK2EQ = WK2pm + WK2Red. math left parenthesisRed. RPM
math divide symbol
PM RPM
math right parenthesis 2+ WK2Load math left parenthesisLoad RPM
math divide symbol
PM RPM
math right parenthesis 2

Note: reducer RPM = Load RPM


 
WK2EQ = WK2pm + WK2Red. math left parenthesis1
math divide symbol
3
math right parenthesis 2+ WK2Load math left parenthesis1
math divide symbol
3
math right parenthesis 2

The WK2 equivalent is equal to the WK2 of the prime mover, plus the WK2 of the load. This is equal to the WK2 of the prime mover, plus the WK2 of the reducer times (1/3)2, plus the WK2 of the load times (1/3)2.

This relationship of the reducer to the driven load is expressed by the formula given earlier:


 
WK2EQ = WK2part math left parenthesisNpart
math divide symbol
Nprime mover
math right parenthesis 2

In other words, when a part is rotating at a speed (N) different from the prime mover, the WK2EQ is equal to the WK2 of the part's speed ratio squared.

In the example, the result can be obtained as follows:

The WK2 equivalent is equal to:


 
WK2EQ = 100 lb.ft.2 + 900 lb.ft.2 math left parenthesis1
math divide symbol
3
math right parenthesis 2+ 27,000 lb.ft.2 math left parenthesis1
math divide symbol
3
math right parenthesis 2

Finally:


 
WK2EQ = lb.ft.2pm + 100 lb.ft.2Red + 3,000 lb.ft2Load

WK2EQ = 3200 lb.ft.2

The total WK2 equivalent is that WK2 seen by the prime mover at its speed.


Electrical Formulas     (For more formulas see "Formulas")
To FindAlternating Current
Single-PhaseThree-Phase
Amperes when horsepower is knownHP x 746
math divide symbol
E x Eff x pf
HP x 746
math divide symbol
1.73 x E x Eff x pf
Amperes when kilowatts are knownkW x 1000
math divide symbol
E x pf
kW x 1000
math divide symbol
1.73 x E x pf
Amperes when kVA are knownkVA x 1000
math divide symbol
E
kVA x 1000
math divide symbol
1.73 x E
KilowattsI x E x pf
math divide symbol
1000
1.73 x I x E x pf
math divide symbol
1000
kVAI x E
math divide symbol
1000
1.73 x I x E
math divide symbol
1000
Horsepower = (Output)I x E x Eff x pf
math divide symbol
746
1.73 x I x E x Eff x pff
 math divide symbol

746

I = Amperes; E = Volts; Eff = Efficiency; pf = Power Factor; kVA = Kilovolt-amperes; kW = Kilowatts


Locked Rotor Current (IL) From Nameplate Data
Three Phase: IL =577 x HP x kVA/HP
math divide symbol
E
 See: kVA/HP Chart
Single Phase: IL =1000 x HP x kVA/HP
math divide symbol
E
    
EXAMPLE: Motor nameplate indicates 10 HP, 3 Phase, 460 Volts, Code F.
IL =577 x 10 x (5.6 or 6.29)
math divide symbol
460
 
IL =70.25 or 78.9 Amperes (possible range)

Effect Of Line Voltage On Locked Rotor Current (IL) (Approx.)
IL @ ELINE = IL @ EN/P xELINE
math divide symbol
EN/P
  
EXAMPLE: Motor has a locked rotor current (inrush of 100 Amperes (IL) at the rated nameplate voltage (EN/P) of 230 volts.

What is IL with 245 volts (ELINE) applied to this motor?

IL @ 245 V. = 100 x 254V/230V

IL @ 245V. = 107 Amperes


Basic Horsepower Calculations

Horsepower is work done per unit of time. One HP equals 33,000 ft-lb of work per minute. When work is done by a source of torque (T) to produce (M) rotations about an axis, the work done is:


 
radius x 2 x rpm x lb. or 2 TM

When rotation is at the rate N rpm, the HP delivered is:


 
HP =radius x 2 x rpm x lb.
math divide symbol
33,000
=TN
math divide symbol
5,250

For vertical or hoisting motion:


 
HP =W x S
math divide symbol
33,000 x E

Where:


 
W=total weight in lbs. to be raised by motor
S=hoisting speed in feet per minute
E=overall mechanical efficiency of hoist and gearing. For purposes of estimating
E=.65 for eff. of hoist and connected gear.

For fans and blowers:


 
HP =Volume (cfm) x Head (inches of water)
math divide symbol
6356 x Mechanical Efficiency of Fan

Or


 
HP =Volume (cfm) x Pressure (lb. Per sq. ft.)
math divide symbol
3300 x Mechanical Efficiency of Fan

Or


 
HP =Volume (cfm) x Pressure (lb. Per sq. in.)
math divide symbol
229 x Mechanical Efficiency of Fan

For purpose of estimating, the eff. of a fan or blower may be assumed to be 0.65.

Note: Air Capacity (cfm) varies directly with fan speed. Developed Pressure varies with square of fan speed. Hp varies with cube of fan speed.

For pumps:


 
HP =GPM x Pressure in lb. Per sq. in. x Specific Gravity
math divide symbol
1713 x Mechanical Efficiency of Pump

Or


 
HP =GPM x Total Dynamic Head in Feet x Specific Gravity
math divide symbol
3960 x Mechanical Efficiency of Pump

 


 
where Total Dynamic Head = Static Head + Friction Head

For estimating, pump efficiency may be assumed at 0.70.


Accelerating Torque

The equivalent inertia of an adjustable speed drive indicates the energy required to keep the system running. However, starting or accelerating the system requires extra energy.

The torque required to accelerate a body is equal to the WK2 of the body, times the change in RPM, divided by 308 times the interval (in seconds) in which this acceleration takes place:


 
ACCELERATING TORQUE =WK2N (in lb.ft.)
math divide symbol
308t

Where:


 
N=Change in RPM
W=Weight in Lbs.
K=Radius of gyration
t=Time of acceleration (secs.)
WK2=Equivalent Inertia
308=Constant of proportionality

Or


 
TAcc =WK2N
math divide symbol
308t

The constant (308) is derived by transferring linear motion to angular motion, and considering acceleration due to gravity. If, for example, we have simply a prime mover and a load with no speed adjustment:

Example 1

PRIME LOADER

 
LOAD
WK2 = 200 lb.ft.2
 
WK2 = 800 lb.ft.2

The WK2EQ is determined as before:


 
WK2EQ = WK2pm + WK2Load
WK2EQ = 200 + 800
WK2EQ = 1000 ft.lb.2

If we want to accelerate this load to 1800 RPM in 1 minute, enough information is available to find the amount of torque necessary to accelerate the load.

The formula states:


 
TAcc =WK2EQN
math divide symbol
308t
or 1000 x 1800
math divide symbol
308 x 60
or 1800000
math divide symbol
18480

 
TAcc = 97.4 lb.ft.

In other words, 97.4 lb.ft. of torque must be applied to get this load turning at 1800 RPM, in 60 seconds.

Note that TAcc is an average value of accelerating torque during the speed change under consideration. If a more accurate calculation is desired, the following example may be helpful.

Example 2

The time that it takes to accelerate an induction motor from one speed to another may be found from the following equation:


 
t =WR2 x change in rpm
math divide symbol
308 x T

Where:


 
T=Average value of accelerating torque during the speed change under consideration.
t=Time the motor takes to accelerate from the initial speed to the final speed.
WR2=Flywheel effect, or moment of inertia, for the driven machinery plus the motor rotor in lb.ft.2 (WR2 of driven machinery must be referred to the motor shaft).

The Application of the above formula will now be considered by means of an example. Figure A shows the speed-torque curves of a squirrel-cage induction motor and a blower which it drives. At any speed of the blower, the difference between the torque which the motor can deliver at its shaft and the torque required by the blower is the torque available for acceleration. Reference to Figure A shows that the accelerating torque may vary greatly with speed. When the speed-torque curves for the motor and blower intersect there is no torque available for acceleration. The motor then drives the blower at constant speed and just delivers the torque required by the load.

In order to find the total time required to accelerate the motor and blower, the area between the motor speed-torque curve and the blower speed-torque curve is divided into strips, the ends of which approximate straight lines. Each strip corresponds to a speed increment which takes place within a definite time interval. The solid horizontal lines in Figure A represent the boundaries of strips; the lengths of the broken lines the average accelerating torques for the selected speed intervals. In order to calculate the total acceleration time for the motor and the direct-coupled blower it is necessary to find the time required to accelerate the motor from the beginning of one speed interval to the beginning of the next interval and add up the incremental times for all intervals to arrive at the total acceleration time. If the WR2 of the motor whose speed-torque curve is given in Figure A is 3.26 ft.lb.2 and the WR2 of the blower referred to the motor shaft is 15 ft.lb.2, the total WR2 is:


 
15 + 3.26 = 18.26 ft.lb.2,

And the total time of acceleration is:


 
WR2
math divide symbol
308
math left bracket symbolrpm1
math divide symbol
T1
+rpm2
math divide symbol
T2
+rpm3
math divide symbol
T3
+ - - - - - - - - - +rpm9
math divide symbol
T9
math right bracket symbol

Or


 
t =18.26
math divide symbol
308
math left bracket symbol150
math divide symbol
46
+150
math divide symbol
48
+300
math divide symbol
47
+300
math divide symbol
43.8
+200
math divide symbol
39.8
+200
math divide symbol
36.4
+300
math divide symbol
32.8
+100
math divide symbol
29.6
+40
math divide symbol
11
math right bracket symbol

 
t = 2.75 sec.

Figure A
Curves used to determine time required to accelerate induction motor and blower

Accelerating Torques
T1 = 46 lb.ft.T4 = 43.8 lb.ft.T7 = 32.8 lb.ft.
T2 = 48 lb.ft.T5 = 39.8 lb.ft.T8 = 29.6 lb.ft.
T3 = 47 lb.ft.T6 = 36.4 lb.ft.T9 = 11 lb.ft.


Wave Form
 


Duty Cycles

Sales Orders are often entered with a note under special features such as:

-----"Suitable for 10 starts per hour"
Or
----"Suitable for 3 reverses per minute"
Or
-----"Motor to be capable of accelerating 350 lb.ft.2"
Or
-----"Suitable for 5 starts and stops per hour"

Orders with notes such as these can not be processed for two reasons.

  1. The appropriate product group must first be consulted to see if a design is available that will perform the required duty cycle and, if not, to determine if the type of design required falls within our present product line.
  2. None of the above notes contain enough information to make the necessary duty cycle calculation. In order for a duty cycle to be checked out, the duty cycle information must include the following:
    1. Inertia reflected to the motor shaft.
    2. Torque load on the motor during all portions of the duty cycle including starts, running time, stops or reversals.
    3. Accurate timing of each portion of the cycle.
    4. Information on how each step of the cycle is accomplished. For example, a stop can be by coasting, mechanical braking, DC dynamic braking or plugging. A reversal can be accomplished by plugging, or the motor may be stopped by some means then re-started in the opposite direction.
    5. When the motor is multi-speed, the cycle for each speed must be completely defined, including the method of changing from one speed to another.
    6. Any special mechanical problems, features or limitations.

Obtaining this information and checking with the product group before the order is entered can save much time, expense and correspondence.

Duty cycle refers to the detailed description of a work cycle that repeats in a specific time period. This cycle may include frequent starts, plugging stops, reversals or stalls. These characteristics are usually involved in batch-type processes and may include tumbling barrels, certain cranes, shovels and draglines, dampers, gate- or plow-positioning drives, drawbridges, freight and personnel elevators, press-type extractors, some feeders, presses of certain types, hoists, indexers, boring machines, cinder block machines, key-seating, kneading, car-pulling, shakers (foundry or car), swaging and washing machines, and certain freight and passenger vehicles. The list is not all-inclusive. The drives for these loads must be capable of absorbing the heat generated during the duty cycles. Adequate thermal capacity would be required in slip couplings, clutches or motors to accelerate or plug-stop these drives or to withstand stalls. It is the product of the slip speed and the torque absorbed by the load per unit of time which generates heat in these drive components. All the events which occur during the duty cycle generate heat which the drive components must dissipate.

Because of the complexity of the Duty Cycle Calculations and the extensive engineering data per specific motor design and rating required for the calculations, it is necessary for customer to refer to an electrical engineer for motor sizing with a duty cycle application.

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