DC Motors – Voltage Vs. Output Speed Vs. Torque
AB-032: DC Motors – Voltage Vs. Output Speed Vs. Torque - Precision Microdrives
AB-032
DC Motors – Voltage Vs. Output Speed Vs. Torque
The relationship between voltage, torque and output speed is a common topic of discussion between our customers and Precision Microdrives’ sales engineers.
The following article aims to discuss and elaborate on the relationship between these parameters and methods of using them together with other resources, to understand the full capabilities of our DC motors and gear motors.
Definitions of terms used in our data sheet can be found below with links generously distributed across the article for further reading.
Torque And Speed
Torque can be defined as a ‘twisting force’ that has a tendency to rotate an object about a fulcrum. In relation to DC motors and gear motors, we will typically refer to the ‘Rated Torque’ as the ‘Rated Load’ to avoid any confusion in our values. Ultimately, the two terms represent the same value – the rotational force applied to the output shaft.
When discussing ‘speed’, we are typically referring to the angular velocity of the output shaft on our DC motors and gear motors (typically in revolutions per minute). Depending on the application, this parameter will affect the rate at which a particular function is executed and it may have a significant effect on the overall performance of the device.
Why Change Torque?
The most obvious benefit of varying the torque is to maintain a constant speed when the motor’s load varies, keeping in mind the interdependent nature of speed, torque, and voltage.
Although this example may be outdated, audio cassettes are a great way of explaining how some applications need to vary the torque to match a changing load. As the cassette plays and the audiotape moves from one spindle to the other, the driving motor will experience a change in load. However, the playback must remain at a constant speed throughout – otherwise the audio pitch would be affected.
There are also instances where the motor’s load is changed dramatically between operations, rather than a slow dynamic change like the cassette example. Pulleys and lifts often experience this, the motor stops at an extremity as the load is attached or removed. Here, keeping a constant speed is not as important as the motor being able to handle a range of different torque loads as moving a heavier object requires more output torque than a light object or no load.
Each of these applications has the common theme of a varying load attached to the motor. If your application involves a fixed load, then it is likely that you will be more interested in varying the speed.
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Why Change Speed?
The ability to vary motor speed whilst maintaining a steady torque is essential to many applications for a variety of reasons.
An example of an application that requires a variable speed and steady torque is an audio CD player as it is commonly observed that the CD will rotate faster at certain points than others. This is because the information is stored in spiralled circular tracks on the disk and the length/circumference of the track is directly proportional to the amount of information stored on them. This means that the speed must be decreased as the laser is reading from the outermost tracks because there is more information per revolution. Inversely, the speed is increased as the laser reads from the innermost tracks as the spiral circumferences are smaller and therefore contain less information per revolution.
Without the ability to adjust the motor speed (with the voltage) whilst maintaining this constant torque, it would be very difficult to read and play this information at a steady rate.
This same principle can be applied to a wide variety of applications and is often critical to their successful operation. Many of our DC motors and gear motors can operate across a wide variety of speeds and loads, this allows our customers to explore the possibilities of their project and usually reach a suitable solution with a single motor.
How To Read The Typical Performance Characteristics Chart
The Typical Performance Characteristics chart appears on the front page of each of our data sheets. This graph is an extremely useful tool that illustrates the typical behaviour of an individual motor.
As we have previously discussed, many of our customers are looking for a motor or gear motor that will operate at a given speed and load. One of the best places to find a solution is our online catalogue and we can always help to recommend suitable motors and discuss customisation options. As motor speed in DC motors and gear motors is primarily dictated by the load and driving voltage, the data sheet value for ‘Rated Speed’ is taken at a ‘Rated Voltage’ and a ‘Rated Load’. This means that the data sheet values for speed are taken under controlled and specific conditions and do not represent the full capabilities of any single motor. This is where the typical performance chart is a useful tool to view a wider range of the motor capabilities.
108-106 Motor Performance Graph
The graphs for our DC motors and Gearmotors assume a fixed voltage and show how current draw, power, efficiency and motor speed are affected by a change in load. Each of the affected parameters has it’s own independent performance line and corresponding scale on the Y-axis.
The blue line on the 108-106 Typical performance Chart (above) shows the speeds at which the motor will operate between a point of no-load all the way up to its stall torque (approx. 0.725 mNm) and allows us to examine the motor performance as well as understand the relationship between speed and torque for the individual motor.
For example; if a customer requires a steady speed and torque of 1900 RPM and 0.65 mNm respectively, the data sheet “key features” section (above) would indicate that the 108-106 was not suitable as it states:
Rated load – 0.15 mNm
Rated load speed – 12,600 RPM
However, after inspecting the performance chart, at a load of 0.65 mNm on the X-axis, the blue performance line (speed), indicates on the corresponding Y-axis that the speed will be 1900RPM. The image above illustrates this and demonstrates that the 108-106 is, in fact, suitable for the customer based on their fixed speed and torque requirements. This chart can also be extended to illustrate the range of capabilities for the motor if it is to be used with a dynamic load/speed.
The Relationship Between Speed, Torque And Voltage
Now that we’ve discussed how to read the performance chart, we can look at the relationship between speed and torque. In this section, we will outline the relationship between speed and torque and explain the limits of each before considering the further effect of voltage on these parameters.
N/L speed and Stall Torque on Motor Performance Graphs
Assuming the motor is driven at a fixed voltage, there are two points that describe the peak performance of a motor at each extremity. “No-load speed” (N/L) and “stall torque”
- The stall torque represents the point at which the motor has reached its maximum operational load. At this point, the shaft will no longer rotate and the motor will be in a “stalled” condition. Please note that the motor must not be operated to stall as this will almost certainly lead to premature failure
- The no-load speed is the maximum output speed the motor will achieve at a given voltage. At this point, the motor is running freely and under no external load
Our DC motors and gear motors can operate anywhere between these limits before reaching stall. If we take a look at the blue performance line, the relationship between speed and torque is quite easily understood – the torque is inversely proportional to the motor speed – starting at the point of no-load/full speed and, as the load increases, the speed decreases proportionally until the motor stalls.
Whilst the performance chart illustrates how speed is affected when applying various loads, it does not indicate that the speed of our DC motors is also directly proportional to the voltage applied. The theory behind this principle can be found here. In short, this means that we can control the speed of a motor independently of the torque and it allows us to maintain a steady speed for a variable load and also maintain a steady torque with a varying motor speed.
This principle is employed to ensure that our CD player and cassette tape are played correctly and would probably include a closed-loop feedback system that will measure the motor speed and adjust the driving voltage to either maintain a steady speed for a variable load or provide a variable speed for a fixed load.
How We Can Change Motor Performance
There are several methods in which the performance of a motor can be customised, whether this is a highly customised solution tailored to a customer’s needs or a simple adjustment to how the motor is operated. Some common modifications are listed below:
- Windings: By modifying the number of turns in the motor coils and/or the cross-sectional area of the wire used, the terminal resistance, operating voltage/current and terminal inductance can be manipulated. This means that both the electrical and mechanical performance of a motor can be tailored to a particular specification quite easily
- Gearbox ratios: Gearboxes are an effective method of accurately altering DC motor performance using one or more gear stages. Whilst we do supply stock reduction gear motors, many of our customers like to develop their own set of gears. If you would like to experiment with your own gear chains, simple gear equations can be found in AB-024. However, we can offer custom gearboxes and stock part modifications so please feel free to contact an engineer if you would like to discuss your requirements and the options that we can offer.
- Driving voltage: This can be a simple and cost-effective way to control the performance of our motors. There are several ways in which you can adjust the driving voltage to your motor, including PWM and even dedicated driver IC’s. We have previously discussed these topics in more detail at the following links – 1 and 2
- Material selection: The materials used can significantly affect the overall performance of your gear motor. Some of the potential options here are listed below
- Gear material: A common point of failure with micro gear motors occurs at the final gear stage. This is the point at which the largest force is exerted when a load is applied to the motor. In this instance, the gear can fail long before the motor stall torque is achieved and the potential capabilities are not fully utilised. If this is the case, stronger gears can be added to the final stage(s) so that a higher torque and wider performance range can be achieved. In practice, this has been used with the 206-108 which stalls at roughly 17mNm due to gear failure. This is characterised on the characteristics chart by an abrupt halt in the torque-speed line long before stall (0 RPM) is approached. By inserting a metal gear at the final stage a torque of roughly 34mNm is achieved, doubling the torque capabilities of the motor and opening up a wider range of possibilities. This was given the part number 206-10C
- Lubricants: Ambient and operational temperature largely affect the efficiency of a gear motor and the overall performance achieved at the output shaft. Whilst the electrical efficiency of the motor can often increase at low temperatures, the gearbox efficiency and effectiveness of the lubricant can reduce such that the overall performance is lessened. A common method of reducing this effect is to use a specified cold temperature lubricant. This can increase the efficiency of the gearbox and therefore the performance of the motor at the output. This means that the temperature range listed on the datasheet is not an absolute limit and there are several ways that it can be extended. If there are any queries, Precision Microdrives engineers will be happy to help out
- Encoders: If you require greater control of your gearmotor or use it within a positioning actuator, you may require an encoder. This is a typical modification that we can offer ranging from simple tachometers for speed measurement, incremental encoders for single reference positioning, all the way to absolute encoders for exact positioning of the output shaft. These encoders can also be used in closed-loop control to maintain speed for a varying torque, vary the speed for a steady torque, or any combination of the two (examples discussed earlier in this article). Please do get in touch with a Precision Microdrives engineer if you would like further information on what we can supply
Any combination of the above can be used together to achieve a wide variety of outputs from our gear motors. So, even if you cannot find a gear motor performance chart that meets your specification, please feel free to contact our engineers as there are a variety of ways that we can look to meet your requirements.
Limitations
As with all good things, there are limitations to what can be achieved. This section aims to outline some of the associated limitations that are encountered when modifying a gear motor.
- Windings: Unfortunately, without significant modification and expense in developing a wire with a specific cross-sectional area, it is occasionally difficult to provide an exact performance at a required terminal resistance, operating voltage/current and terminal inductance. If this is the case, the requirements are often met very closely and the variations can be negligible. Dimensional limitations also apply here as very limited space is available for the windings. In practice, this means that certain mechanical/electrical performances cannot be achieved with a given motor due to the envelope available space for the windings and the required cross-sectional area/number of the winding wire. Of course, Precision Microdrives will be happy to assess the feasibility of your request, so please do not hesitate to contact us with your requirements
- Material properties: Material properties also represent a limitation when considering achievable modifications with our motors.
As discussed, cold temperature lubricants can be used to increase the performance of a material at certain temperatures, however, there are obvious physical limitations to specific materials. Notable limiting properties include the coefficient of thermal expansion, material strength, melting points and many others. If you do have any queries relating to this, please do get in touch
In this section, we have discussed a few of the obvious limitations that present themselves when modifying a gear motor. In many cases, these limitations are surpassable if less critical parameters are more flexible. So, please do enquire with our engineers to assess what can be implemented within your application.
Conclusion
Throughout this article, we have discussed some reasons why a user will vary motor speed and torque and looked at specific examples for each situation. This prompted us to consider the limits of motor speed and torque for our gear motors when driven at a fixed voltage. Here we understood that, at a fixed voltage, our motors can operate across a wide range of speeds and torques between a point of no-load (full speed) and the point of maximum load (stall).
We have also discussed how to read the “typical performance characteristics chart” to understand the full range of torque and speed capabilities of a given motor (at its rated voltage). From here we saw how the relationship between speed and torque is inversely proportional from a point of no-load to stall torque and discussed how we can adjust the driving voltage to maintain a steady speed or torque when the other variable is dynamic.
The final section of this bulletin was aimed at describing some methods of manipulating motor performance. Some of these methods are easily implemented and can be experimented with when testing, whilst others are permanent modifications that can be provided to a particular specification. If you would like to consider the options that you have for your project, please contact one of our engineers to discuss your requirements and the options that we can offer you.
Torque vs Speed
https://www.powerelectric.com/motor-resources/motors101/speed-vs-torque#:~:text=The%20output%20power%20of%20a%20motor%20sets%20the,typically%20measured%20in%20Watts%20%28W%29%20or%20horsepower%20%28hp%29.
The purpose of a rotary motor is to provide a desired rotational output speed while overcoming the various rotational loads resisting that rotational output (Torque). Speed and Torque are directly related, and are the two primary performance factors in properly selecting a motor for a specific application or use. To learn how to select the right motor to fit your application, the first step is to understand the relationship between speed, torque and output power of a motor.
Speed versus Torque
The output power of a motor sets the speed and torque performance boundaries of a motor, based on the equation:
Power (P) = Speed (n) x Torque (M)
- Power: The mechanical output power of a motor is defined as the output speed times the output torque and is typically measured in Watts (W) or horsepower (hp).
- Speed: The speed of a motor is defined as the rate at which the motor rotates. The speed of an electric motor is measured in revolutions per minute, or RPM.
- Torque: The torque output of a motor is the amount of rotational force that the motor develops. The torque of a small electric motor is commonly measured in either inch pounds (in-lbs), Newton meters (N-m) or other directly converted units of measure.
Since the rated output power of a motor is a fixed value, speed and torque are inversely related. As output speed increases, the available output torque decreases proportionately. As the output torque increases, the output speed decreases proportionately. This power, speed and torque relationship is commonly illustrated with a motor performance curve which often includes motor current draw (in Amps) and motor efficiency (in %).
Speed and Torque Considerations for Electric Motor Selection
The key to selecting the right motor for a specific function is to first understand the requirements of the application. Since most motor applications are dynamic, meaning that torque and speed requirements change within the application, it is critical to define the various operating points within the application. Knowing or calculating the speed and torque requirements at each operational point in the application will determine the overall speed and torque requirements for the appropriate motor. Motor selection can be verified by plotting the various application operational points on the selected motor’s performance curve to make sure that every speed vs. torque point falls within the appropriate zone of the curve (continuous or intermittent zones).
In many cases, application requirements will force the selection of a standard motor that is significantly oversized to insure that all operational points are covered. Applying motors that are oversized for an application results in unnecessary cost, as well as larger and heavier designs of the overall product. Fortunately, custom motor suppliers can develop motors with optimized performance curves to precisely meet application requirements. This is done by changing the electromagnetic characteristics of the motor by altering either the wire size or the number of wire turns in the winding, or both. More turns of smaller wire provides more torque and less speed where fewer turns of larger wire provides higher speed but less torque. In some applications adding gearing to the motor output provides the ideal speed versus torque relationship while keeping the cost and size of the overall solution to a minimum.
Torque - Work done and Power Transmitted
Work done
Work done is the force multiplied with the distance moved by the force - and can be expressed as
W = F s (1)
where
W = work done (J, Nm)
F = force (N)
s = distance moved by force (s)
For an angular motion
the work done can be expressed as
W = F θ r
= T θ (2)
where
W = work (Joules)
θ = angle (radians)
r = radius (m)
T = torque or moment (Nm)
Power transmitted
Power is the ratio between the work done and the time taken and can be expressed as
P = W / dt
= T θ / dt
= T ω
= 2 π n T
= 2 π (nrpm / 60) T
= 0.105 nrpm T (3)
where
P = power (Watts)
dt = time taken (s)
ω = θ / dt = 2 π n = angular velocity (rad/s)
n = speed (rev/s)
nrpm = speed (rev/min, rpm)
Note! - a machine must rotate to produce power! A machine with no rotation can deliver torque - like an electric motor - but since no distance is moved by force - no power is produced. As soon as the machine starts to rotate power is produced.
Example - required Torque to produce Power
A machine rotates with speed 3000 rev/min (rpm) and consumes 5 kW. The torque at the shaft can be calculated by modifying (3) to
T = P / 2 π n
= (5 kW) (1000 W/kW) / 2 π (3000 rev/min) / (60 sec/min)
= 15.9 Nm
Motor Power Calculations
https://www.globalspec.com/pfdetail/motors/motor-power-calculations
“What are the major criteria by which the performance of an electric motor drive can be evaluated?”
This is a question commonly asked by designers and engineers looking to specify an electric motor for an application. And the answer is pretty straightforward: motor power, speed, efficiency, and torque are vital to the performance of an electric motor for a particular application.
This article presents valuable information about sizing motors for different applications. It will cover design considerations and several calculations, including motor efficiency, torque, and motor power calculations.
Electric motor used for water pump
© [wi6995] / Adobe Stock
How do Electric Motors Work?
Electric motors are designed to convert electrical energy into mechanical energy. Despite the wide range of motor options available today, electric motors typically work based on the magnetic effect of current: an electric current flowing through a wire coil in a magnetic field creates a force that rotates the coil, thus creating torque.
To properly specify electric motors, engineers must understand and know how to calculate the following:
- Electrical Power Input of the motor
- Mechanical Power Output of the motor
- The efficiency of the motor
- Motor amperage
Motor Power Calculations (Electric Power Input)
Electric power is defined as the rate at which electrical energy is transferred by an electric circuit. The electric power input of an electric motor can be estimated using:
Where:
Pin=Electric power input (Watts or W)
I=Current (A)
V=Applied Voltage (V)
However, keep in mind that the formula above is only applicable to DC sources. Determining the electric power input for an AC source must include a power factor.
Power factor is a measure of how effectively electrical power is being used. It is the ratio of the electrical power that the motor uses (also called real power) compared to the overall amount of power supplied to the circuit (also called apparent power).
The electric power input formula becomes:
A power factor of 1 describes a perfect electric power input system (which is practically impossible to achieve).
[Learn more about electrical power generators with Engineering360]
Motor Power Calculations (Mechanical Power Output)
Mechanical power describes the output power that moves the object attached to the motor. It is simply defined as speed times torque (the rotational equivalent of a linear force).
Where:
Pout=Output power (W)
t=Torque (Nm)
w=Angular speed (rad/s)
The angular speed of the motor can be estimated if designers know the rotational speed of the motor, as shown in the equation below:
Where:
N=Rotational speed (rpm)
Motor Horsepower Calculations (Mechanical Power Output)
Mechanical power can also be defined in Horsepower (hp). For example, a mechanical power level of 1 hp is equivalent to 746 watts (W) or 0.746 kilowatts. Here is a formula for electric motor horsepower calculations:
Where:
Pout=Output power (hp)
t=Torque (lbf.ft)
N=Rotational speed (rpm)
Engineers can also utilize the electric motor horsepower calculators on manufacturers’ websites for their motor power calculation needs.
Motor Efficiency Calculator: A Key Formula for Electric Motor Performance
The efficiency of an electric motor is simply the ratio of the mechanical power output to the electric power input.
Because energy is lost (usually in the form of heat) as an electric motor converts electrical energy into mechanical energy, the mechanical power output is always less than the electric power input. What this means is that motor efficiency will always beless than 1 (or 100%); t ypical electric motor designs have efficiency ranging between70 and 96%.
Consider an electric motor with an input electrical power of 1.36 W that generates a torque of 0.0098 Nm and rotational speed of 1000 rpm. By combining the power input, power output, and efficiency form ula s, the total mechanical power output would be 1.026 W, and the motor efficiency would be 0.7544 (or 75.44%).
This means that approximately three-quarters of the electricity consumed by the motor drive is converted into useful mechanical power , while the remaining one-quarter is lost in the form of heat.
Motor Full Load Amps Calculators
Amperage (or Amps) is simply the strength of an electric current or the amount of electrical current that flows through an electric conductor in a given time.
The following sections will provide formulas for estimating amperage for single-phase and three-phase systems when mechanical power output (or motor rating) is known.
1-Phase Motor Full Load AmpsCalculations
The 1-phase motor full load amps calculations (or single-phase motor FLA calculations) are shown below:
Where:
Pout=mechanical power output (kW)
PF=Power Factor
V=Voltage (V)
Where:
Pout=Mechanical power output (measured in hp)
E=Efficiency
3-Phase Motor Amperage Calculator
The 3-phase motor amps calculations are shown below:
Where:
Pout=Mechanical power output (kW)
Where:
Pout=Mechanical power output (hp)
Engineers can also utilize the 3-phase motor amps calculators (or full load amp calculators) on motor manufacturers’ websites for amperage calculations.
Electric Motor Calculations: Manufacturers Can Help
While this article provides helpful information about electric motors, there still exist many other factors that engineers must consider when specifying electric motors for a particular application. Therefore, engineers are advised to reach out to manufacturers to discuss their application requirements.
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Motor Calculations
https://www.me.ua.edu/me360/docs/Motors/MotorCalculations.pdf
· Calculating Mechanical Power Requirements
· Torque - Speed Curves
· Numerical Calculation
· Sample Calculation
· Thermal Calculations
Calculating Mechanical Power Requirements
Physically, power is defined as the rate of doing work. For linear motion, power is the product of force multiplied by the distance per unit time. In the case of rotational motion, the analogous calculation for power is the product of torque multiplied by the rotational distance per unit time.
Where:
Prot = rotational mechanical power
M = torque
w = angular velocity
The most commonly used unit for angular velocity is rev/min (RPM). In calculating rotational power, it is necessary to convert the velocity to units of rad/sec. This is accomplished by simply multiplying the velocity in RPM by the constant (2 x p) /60:
It is important to consider the units involved when making the power calculation. A reference that provides conversion tables is very helpful for this purpose. Such a reference is used to convert the torque-speed product to units of power (Watts). Conversion factors for commonly used torque and speed units are given in the following table. These factors include the conversion from RPM to rad/sec where applicable.
For example, assume that it is necessary to determine the power required to drive a torque load of 3 oz-in at a speed of 500 RPM. The product of the torque, speed, and the appropriate conversion factor from the table is:
Calculation of power requirements is often used as a preliminary step in motor or gearmotor selection. If the mechanical power required for a given application is known, then the maximum or continuous power ratings for various motors can be examined to determine which motors are possible candidates for use in the application.
Torque - Speed Curves
One commonly used method of displaying motor characteristics graphically is the use of torque - speed curves. While the use of torque - speed curves is much more common in technical literature for larger DC machines than it is for small, ironless core devices, the technique is applicable in either case. Torque - speed curves are generated by plotting motor speed, armature current, mechanical output power, and efficiency as functions of the motor torque. The following discussion will describe the construction of a set of torque - speed curves for a typical coreless DC motor from a series of raw data measurements. Motor 1624E009S is used as an example.
Assume that we have a small motor that we know has a nominal voltage of 9 volts. With a few fundamental pieces of laboratory equipment, the torque - speed curves for the motor can be generated:
Step One (measure basic parameters):
Using a voltage supply set to 9 volts, run the motor unloaded and measure the rotational speed using a non-contacting tachometer (strobe, for instance). Measure the motor current under this no-load condition. A current probe is ideal for this measurement since it does not add resistance in series with the operating motor. Using an adjustable torque load such as a small particle brake coupled to the motor shaft, increase the torque load to the motor just to the point where stall occurs. At stall, measure the torque from the brake and the motor current. For the sake of this discussion, assume that the coupling adds no load to the motor and that the load from the brake does not include unknown frictional components. It is also useful at this point to measure the terminal resistance of the motor. Measure the resistance by contacting the mot or terminals. Then spin the motor shaft and take another measurement. The measurements should be very close in value. Continue to spin the shaft and take at least three measurements. This will ensure that the measurements were not taken at a point of minimum contact on the commutator.
Now we have measured the:
· n0= no-load speed
· I0= no-load current
· MH= stall torque
· R= terminal resistance
Step Two (plot current vs. torque and speed vs torque)
Prepare a graph with motor torque on the horizontal axis, motor speed on the left vertical axis, and motor current on the right vertical axis. Scale the axes based on the measurements in step 1. Draw a straight line from the left origin of the graph (zero torque and zero current) to the stall current on the right vertical axis (stall torque and stall current). This line represents a plot of the motor current as a function of the motor torque. The slope of this line is the proportionality constant for the relationship between motor current and motor torque (in units of current per unit torque). The reciprocal of this slope is the torque constant of the motor (in units of torque per unit current). For the resulting curves see Graph 1.
Using the relationships between motor constants discussed earlier, calculate the velocity constant of the motor from the torque constant obtained above. By multiplying the velocity constant by the nominal motor voltage, obtain the theoretical no-load speed of the motor (zero torque and no-load speed) and plot it on the left vertical axis. Draw a straight line between this point and the stall torque and zero speed point on the graph. The slope of this line is the proportionality constant for the relationship between motor speed and motor torque (in units of speed per unit torque). The slope of the line is negative, indicating that motor speed decreases with increasing torque. This value is sometimes called the regulation constant of the motor. For the resulting curves see Graph 1.
For the purpose of this discussion, it will be assumed that the motor has no internal friction. In practice, the motor friction torque is determined using the torque constant of the motor and the measured no-load current. The torque vs speed line and the torque vs current line are then started not at the left vertical axis but at an offset on the horizontal axis equal to the calculated friction torque.
Step Three (plot power vs torque and efficiency vs torque)
In most cases, two additional vertical axes are added for plotting power and efficiency as functions of torque. A second left vertical axis is usually used for efficiency and a second right vertical axis is used for power. For the sake of simplifying this discussion, efficiency vs. torque and power vs. torque will be plotted on a second graph separate from the speed vs. torque and current vs. torque plots.
Construct a table of the motor mechanical power at various points from no-load to stall torque. Since mechanical power output is simply the product of torque and speed with a correction factor for units (see section on calculating mechanical power requirements), power can be calculated using the previously plotted line for speed vs. torque. A sample table of calculations for motor 1624E009S is shown in Table 1. Each calculated point is then plotted. The resulting curve is a parabolic curve as shown in Graph 1. The maximum mechanical power occurs at approximately one-half of the stall torque. The speed at this point is approximately one-half of the noload speed.
Construct a table of the motor efficiency at various points from no-load to stall torque. The voltage applied to the motor is given, and the current at various levels of torque has been plotted. The product of the motor current and the applied voltage is the power input to the motor. At each point selected for calculation, the efficiency of the motor is the mechanical power output divided by the electrical power input. Once again, a sample table for motor 1624E009S is shown in Table 1. and a sample curve in Graph 1. Maximum efficiency occurs at about 10% of the motor stall torque.
Numerical Calculation
For an iron-less core, DC motor of relatively small size, the relationships that govern the behavior of the motor in various circumstances can be derived from physical laws and characteristics of the motors themselves. Kirchoff's voltage rule states, "The sum of the potential increases in a circuit loop must equal the sum of the potential decreases." When applied to a DC motor connected in series with a DC power source, Kirchoff's voltage rule can be expressed as "The nominal supply voltage from the power source must be equal in magnitude to the sum of the voltage drop across the resistance of the armature windings and the back EMF generated by the motor.":
Where:
Vo = Power supply (Volts)
I = Current (A)
R = Terminal Resistance (Ohms)
Ve = Back EMF (Volts)
The back EMF generated by the motor is directly proportional to the angular velocity of the motor. The proportionality constant is the back EMF constant of the motor.
Where:
w= angular velocity of the motor
Ke = back EMF constant of the motor
Therefore, by substitution:
The back EMF constant of the motor is usually specified by the motor manufacturer in volts/RPM or mV/RPM. In order to arrive at a meaningful value for the back EMF, it is necessary to specify the motor velocity in units compatible with the specified back EMF constant.
ompatible with the specified back EMF constant. The motor constant is a function of the coil design and the strength and direction of the flux lines in the air gap. Although it can be shown that the three motor constants normally specified (back EMF constant, torque constant, and velocity constant) are equal if the proper units are used, calculation is facilitated by the specification of three constants in the commonly accepted units.
The torque produced by the rotor is directly proportional to the current in the armature windings. The proportionality constant is the torque constant of the motor.
Where:
Mo = torque developed at rotor
kM = motor torque constant
Substituting this relationship:
The torque developed at the rotor is equal to the friction torque of the motor plus the resisting torque due to external mechanical loading:
Where:
Mf = motor friction torque
Ml = load torque
Assuming that a constant voltage is applied to the motor terminals, the motor velocity will be directly proportional to sum of the friction torque and the load torque. The constant of proportionality is the slope of the torque-speed curve and can be calculated by:
Where:
MH = stall torque
n0= no-load speed
An alternative approach to deriving this value is to solve for velocity, n:
Sample Calculation
Motor 1624T009S is to be operated with 9 volts applied to the motor terminals. The torque load is 0.2 ozin. Find the resulting motor speed, motor current, efficiency, and mechanical power output. From the motor data sheet, it can be seen that the no-load speed of the motor at 9 volts is 11,700 rpm. If the torque load is not coupled to the motor shaft, the motor would run at this speed.
The motor speed under load is simply the no-load speed less the reduction in speed due to the load. The proportionality constant for the relationship between motor speed and motor torque is the slope of the torque vs. speed curve, given by the motor no-load speed divided by the stall torque. In this example, the speed reduction caused by the 0.2 oz -in torque load is:
0.2 oz-in x (11,700 rpm/.634 oz-in) = -3690 rpm
The motor speed under load must then be:
11,700 rpm - 3690 rpm = 8010 rpm
The motor current under load is the sum of the no-load current and the current resulting from the load. The proportionality constant relating current to torque load is the torque constant (kM), in this case, 1.039 oz -in/A. In this case, the load torque is 0.2 oz-in, and the current resulting from the load must be:
I = 0.2 oz-in x 1 amp/1.039 oz -in = 192 mA
The total motor current must be the sum of this value and the motor no-load current. The data sheet lists the motor no-load current as 11 mA. Therefore, the total current is:
192 mA + 11 mA = 203 mA
The mechanical power output of the motor is simply the product of the motor speed and the torque load with a correction factor for units (if required). Therefore, the mechanical power output of the motor in this application is:
output power = 0.2 oz-in x 8010 rpm x .00074 = 1.18 Watts
The mechanical power input to the motor is the product of the applied voltage and the total motor current in Amps. In this application:
input power = 9 volts x .203 A = 1.82 Watts
Since efficiency is simply power out divided by power in, the efficiency in this application is:
efficiency = 1.18 Watts / 1.82 Watts = .65 = 65%
Thermal Calculations
A current I flowing through a resistance R results in a power loss as heat of I2R. In the case of a DC motor, the product of the square of the total motor current and the armature resistance is the power loss as heat in the armature windings. For example, if the total motor current was .203 A and the armature resistance 14.5 Ohms the power lost as heat in the windings is:
power loss = .203^2 x 14.5 = .59 Watts
The heat resulting from (I^2)*R losses in the coil is dissipated by conduction through motor components and airflow in the air gap. The ease with which this heat can be dissipated is a function of the motor type and construction. Motor manufacturers typically provide an indication of the motor’s ability to dissipate heat by providing thermal resistance values. Thermal resistance is a measure of the resistance to the passage of heat through a given thermal path. A large cross section aluminum plate would have a very low thermal resistance, for example, while the values for air or a vacuum would be considerably higher. In the case of DC motors, there is a thermal path from the motor windings to the motor case and a second between the motor case and the motor environment (ambient air, etc.). Some motor manufacturers specify a thermal resistance for each of the two thermal paths while others specify only the sum of the two as the total thermal resistance of the motor.
Thermal resistance values are specified in temperature increase per unit power loss. The total I2R (I^2)*R losses in the coil (the heat source) are multiplied by thermal resistances to determine the steady state armature temperature. The steady state temperature increase of the motor (T) is given by:
Where:
Tinc = temperature increase
I = current through motor windings
R = resistance of motor windings
Rth1 = thermal resistance from windings to case
Rth2 = thermal resistance case to ambient
For example, a 1624E009S motor running with a current of 0.203 Amps in the motor windings, with an armature resistance of 14.5 Ohms, a winding-to-case thermal resistance of 8 °C/Watt, and a case-to-ambient thermal resistance of 39 °C/Watt. The temperature increase of the windings is given by:
If it is assumed that the ambient air temperature is 22°C, then the final temperature of the motor windings is 50°C (22° + 28°)
It is important to be certain that the final temperature of the windings does not exceed their rated value. In the example given above, the maximum permissible winding temperature is 100°C. Since the calculated winding temperature is only 50°C, thermal damage to the motor windings will not be a problem in this application. One could use similar calculations to answer a different kind of question. For example, an application may require that a motor run at its maximum torque without being damaged by heating. To continue with the example given above, suppose it is desired to run motor 1624E009S at the maximum possible torque with an ambient air temperature of 22°C. The designer wants to know how much torque the motor can safely provide without overheating.
The data sheet for motor 1624E009S specifies a maximum winding temperature of 100°C. Since the ambient temperature is 22°C, a rotor temperature increase of 78°C is tolerable. The total thermal resistance for the motor is 47°C/Watt. By taking the reciprocal of the thermal resistance and multiplying this value by the acceptable temperature increase, the maximum power dissipation in the motor can be calculated:
Setting I2R equal to the maximum power dissipation and solving for I yields the maximum continuous current allowable in the motor windings:
The motor has a torque constant of 1.309 oz-in/A and a no-load current of 11 mA. Therefore, the maximum current available to produce useful torque is .327 Amps (.338 - .011), and the maximum usable torque available (M) is given by:
The maximum allowable current through the motor windings could be increased by decreasing the thermal resistance of the motor. The rotor-to-case thermal resistance is primarily fixed by the motor design. The case-to-ambient thermal resistance can be decreased significantly by the addition of heat sinks. Motor thermal resistances for small DC motors are usually specified with the motor suspended in free air. Therefore, there is usually some heat sinking which results from simply mounting the motor into a framework or chassis. Some manufacturers of larger DC motors specify thermal resistance with the motor mounted into a metal plate of known dimensions and material.
The preceding discussion does not take into account the change in resistance of the copper windings as a result of heating. While this change in resistance is important for larger machines, it is usually not significant for small, coreless motors and is often ignored for the sake of calculation.
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