R C Electric Power Systems



R C Electric Power SystemsWilliam B Garner12/27/2016This document presents the basic elements of an RC electric power system and combines them into a whole system with mathematics to predict performance.Table of Contents TOC \o "1-3" \h \z \u Preface PAGEREF _Toc470684524 \h 3Introduction PAGEREF _Toc470684525 \h 3Batteries PAGEREF _Toc470684526 \h 4Characteristics PAGEREF _Toc470684527 \h 4Voltage Designations PAGEREF _Toc470684528 \h 4Capacity PAGEREF _Toc470684529 \h 5C Rating PAGEREF _Toc470684530 \h 5Internal Resistance PAGEREF _Toc470684531 \h 5Discharge Characteristics PAGEREF _Toc470684532 \h 6Battery Weight PAGEREF _Toc470684533 \h 6Connectors PAGEREF _Toc470684534 \h 7Electronic Speed Controllers PAGEREF _Toc470684535 \h 9Selecting an ESC PAGEREF _Toc470684536 \h 10Programming PAGEREF _Toc470684537 \h 10Basic Settings PAGEREF _Toc470684538 \h 10Brake Setting PAGEREF _Toc470684539 \h 10Low-Voltage Cutoff PAGEREF _Toc470684540 \h 11Cut-off Mode PAGEREF _Toc470684541 \h 11Start-up Mode PAGEREF _Toc470684542 \h 11Timing PAGEREF _Toc470684543 \h 11Pulse-Width Modulation (PWM) Switching Frequency PAGEREF _Toc470684544 \h 11Operating Mode PAGEREF _Toc470684545 \h 11One Last Thing PAGEREF _Toc470684546 \h 11Battery Eliminator Circuits (BEC) PAGEREF _Toc470684547 \h 12Types of BECs PAGEREF _Toc470684548 \h 12Safety PAGEREF _Toc470684549 \h 13Operation PAGEREF _Toc470684550 \h 13Brushless Motors PAGEREF _Toc470684551 \h 14Physical Properties PAGEREF _Toc470684552 \h 14Inrunners and Outrunners PAGEREF _Toc470684553 \h 14Physical Construction PAGEREF _Toc470684554 \h 14Physical Limitations PAGEREF _Toc470684555 \h 16Labeling PAGEREF _Toc470684556 \h 16Specifications and Their Meaning PAGEREF _Toc470684557 \h 17Electrical Parameters PAGEREF _Toc470684558 \h 17Physical Parameters PAGEREF _Toc470684559 \h 18Performance Estimating PAGEREF _Toc470684560 \h 18The Complete Power System PAGEREF _Toc470684561 \h 18Relationships PAGEREF _Toc470684562 \h 20Newton’s Method Solution PAGEREF _Toc470684563 \h 28Prediction Uncertainties PAGEREF _Toc470684564 \h 29PrefaceWhen transitioning from glow powered model airplanes to electric powered ones, there was a learning interval in trying to understand how these systems were like glow ones and how they differed. It was evident that the similarity was weak at best. Glow engines are easy to classify; basically their displacement and maximum brake horsepower at a specified rpm are all that are needed to predict their behavior. Electric motors, however, have specifications like maximum continuous current and voltages, Kv ratings, and resistance that must be considered in trying to predict their performance. While there are computer programs available to help in estimating performance or aid in selecting components, they do not explain the underlying principles behind their computations. The personal challenge was to understand this underlying structure and to develop tools that could be used in evaluating in detail the behavior of electric powered model aircraft. This document is part of a series that develops the necessary system models and the associated mathematical representations that includes performance in flight. This document examines the electric power system. Companion documents deal with propellers and model aircraft aerodynamics when powered by this type of system.IntroductionThis document attempts to demystify the performance characteristics of RC aircraft electric power systems. The subject is complex as electric motors have a number of features that interplay with each other and the load applied to them. While an attempt has been made to keep the math simple, that is not always possible if a reasonably accurate result is desired. The document divides the parts of an electric power system into four elements. Those elements are batteries, wiring and connectors, electronic speed controls or ESCs, and brushless electric motors, each described separately. The final section provides descriptions of how these elements fit together to make a complete power system. Mathematical formulas with examples are provided for the general model, followed by example cases illustrating how the formulas can be applied to answer specific questions.BatteriesHobbyists use batteries of different chemistries. Only Lithium-Polymer (LiPo) batteries are considered here. Rather than provide details in the form of narrative, the salient characteristics are provided in list form, followed by some narrative text.CharacteristicsNegative electrode:LiCoO2 or LiMn2O4Separator:Conducting Polymer electrolyte (polyethyleneoxide, PEO)Positive electrode:Lithium or carbon-Li intercalcation compoundMechanical:Flexible pouches with outer covering for strengthVoltage range per cell:2.7V (discharged) to 4.23 V (fully charged)Overcharge limit:4.235 VDischarge limit:3.0 V Going below limit will degrade charge and voltage holding under loadCell-balance:Weakest cell sets limit on pack performance. Lowest charge or voltage will go first.Voltage balance:Typically less than 8 millivoltsCharge capacity:C, in milliampere-hours (mAH)Charge rate:Amps equal to or less than one CDischarge rate:Varies from about 10C to as much as 45 C. Defined on pack.Storage:Less than 5% per month. . Discharge to ~ 50% for long periods.Safety:Will ignite when electrolyte exposed to air, burning intensely.Disposal:Drop in bucket of salt water until fully discharged - water will turn cloudy. Put in trash. Figure B1 Battery Cell Layout.Voltage DesignationsThe pack's rated voltage is determined by the number of cells placed in series. Instead of indicating the voltage, packs will sometimes indicate the number of cells in series (indicated by #S, where # is the number of cells in series). To determine the voltage of these battery pack, multiple the number of cells in series by the cell voltage (3.7v). 1S = 3.7V 2S = 7.4V 3S = 11.1V 4S = 14.8VCapacityThe battery capacity is based on two things. The first is the capacity of a single cell, and second, how many blocks are placed in parallel (indicated on some packs as #P, where # is the # of parallel blocks). The reason we refer to these in blocks is because in order to double the capacity of a pack, the total number of cells must be double. On a 1S pack (single cell in series), a cell placed in parallel will double the capacity. However, to double the capacity of a 2S pack, a 2S block must be placed in parallel with the existing block (Figure B1). Some examples: 1s2p = 2 cell (single cell voltage, twice the capacity) 2s1p = 2 cell (twice the voltage, single cell capacity) 2s3p = 6 cell (twice the voltage, three times the capacity)C Rating The final criterion for battery packs is their C rating. The C rating of the battery is the maximum continuous discharge rating of the cells and it determines how many amps a system can pull from the pack without overheating it. Discharging batteries at higher rates will raise their temperature and can lead to combustion.To actually determine how many amps a battery can supply, multiply the battery pack's Amp-Hour capacity by the C rating. An example of a 2000mAh battery pack at 10C: 2000mah / 1000 x 10C = 20 ampsBoth capacity and C rating play an important role in determining the maximum discharge rating. Typical C ratings are 20C, 25C, 30C, 35C, 45C. Make sure the battery packs have the appropriate C rating to power the system.Internal ResistanceAll batteries have internal resistance that causes a drop in output voltage as the current increases. While generally small, it can reduce the available battery power under high current conditions. The resistance is a function of the capacity of the battery and it’s C-rating. The greater the C-rating, the lower is the resistance. The following formula provides an estimate of the battery resistance for new or nearly new batteries. Used batteries can have considerably higher resistance, an indication of deteriorating capability. Rbat=.021*NAH*25CN is the number of Cells in seriesAH is the ampere-hour capacityC is the C-rating The resistance increases with the number of cells, decreases with Amp-Hour Capacity and decreases with C-rating.Discharge CharacteristicsThe output voltage of a battery depends on the current load and the state of charge. Figure B2 is a graph showing how voltage varies with relative load and state of discharge for a single cell with a capacity of 5 AH. All of the curves start at about 4.2 Volts, decrease fairly rapidly, flatten out for most of the discharge cycle, then drop sharply as the charge is nearly exhausted. As the discharge current is increased, the output voltage decreases. This decrease is due to the effects of internal resistance.Figure B2 LiPo Discharge CharacteristicBattery WeightA key characteristic of LiPo batteries is their high energy density or Watt-hours per ounce. Figure B3 plots power density as a function of C-rating, showing the trade of weight for current handling capability.914400top00Figure B3 LiPo Power DensityWhen flight duration is a criterion, low C-ratings are advisable as flight duration is a function of weight and total Watt-hour capacity. For other applications where high power is required, greater C-ratings are in order.ConnectorsThere are many kinds of connectors in use, all incompatible with one another. There are two classes of battery connectors. One class carries power to the ESC and motor. The other is used in charging to balance the cells with one another. The photo shows many of the common types of battery connectors.Figure C1 Common Battery ConnectorsThis collection of connectors is set up for battery charging; that is, they connect a charger to a battery. The most commonly used battery connectors are the Deans, EC3 and XT60 in this collection. The choice of connector depends on the particular application and compatibility with other component connectors. The first factor should be the current rating of the connector and associated wire. The second factor is the compatibility with ESCs across planes. While adapters are available, they are one more thing to misplace plus adding to the cabling losses. Table C1 lists common connector types and suggested maximum amperage ranges.Table C1 Connectors and Their Amperage RatingConnectorContinuous Amperage RatingServo/Battery Lead< 800mAJST1-5 AMini T Plug5-18 A3.5mm Bullet Connector10-35 A4mm Bullet / Banana Connector15-50 A / 0-10 ADeans T Connector20-60 A5.5mm Bullet Connector55-110 A6.5mm Bullet Connector65-150 AAnderson Power Pole15 – 45 ATable C2 lists amperage ratings for short wires, on the order of 6 to 8 inches. Table C2 Wire Gauges and Their Amperage RatingsWire GaugeContinuous Amperage Rating18 AWG10-1816 AWG18-25 A14 AWG25-40 A12 AWG40-75 A10 AWG75-120 ADual 12 AWG80-150 ADual 10 AWG150-240 AIt is possible to calculate the resistance of a pair of wires and estimate the power loss and voltage drop across it. The table lists the resistance per foot for 5 wire gauges. Suppose it is desired to calculate the losses in a pair of 18 gauge wires connecting a battery to an ESC. The wire pair is 6 inches long and carries 40 Amps of currentTable C3 Stranded copper wire resistance per footWire GaugeOhms per foot18.006116.0039914.002512.0016210.00106The total wire length is 2 X 6 inches = 12 inches since the current flows to the ESC in one wire and returns to the battery in the other. The total resistance is then 1ft x0.0061 Ohms/ft = .0061 ohms.The voltage drop is V = I x R = 40 X .0061 = 0.244 Volts. The power loss is I2 x R = 402 X 0.0061 = 9.76 Watts! The wire is way too small for this current. A 12 gauge wire would suffer a loss of 2.6 Watts and a voltage drop of 0.065 Volts. This example illustrates the desirability of keeping wires short, of appropriate gauge and properly soldered to connectors. Connectors also have some contact resistance; hence choosing a connector of adequate current handling capability is important to good high current operation.Electronic Speed ControllersAn Electronic Speed Controller, or ESC, is a device for driving an electric motor from a battery. It converts the battery voltage and current into a form that causes the motor to turn at a rate determined by the throttle setting from a receiver or other control device. An ESC usually has three sets of wires. One pair connects to the battery though a power connector. Three of the wires connect to the motor terminals via bullet-type connectors. The remaining set of three low power wires connects the ESC to the receiver or control device via a standard servo- type three-pin connector. Figure ESC1 is a photo of an ESC with the covering removed. The three wires on the left connect to the motor. The thick 2 wires on the right go to the battery. The thin three-wire lead goes to the receiver. Figure ESC1. Typical ESC with Cover RemovedThere are four integrated circuits on the board. The three identical ones are FET switches that convert battery voltage to short bursts sequentially to the motor leads. The other IC is the controller. The black cylindrical object on the right end of the board is a capacitor that suppresses noise and provides burst current capability to the switches and motor.There are two primary specifications for ESCs. One is maximum continuous current, measured in Amps that it can handle. The second is the maximum battery voltage it can manage. This voltage is sometimes listed in terms of the number of LiPo battery cells it can handle. A short term surge current is sometimes listed as well.ESCs are rated by current carrying capability. They are physically small and must dissipate heat from relatively limited surface areas. They use power FETs for switching which have low internal resistance when on. As the current rating increases, the internal resistance is reduced by using more FETs in parallel. The resistance for typical ESCs can be approximated by the following equation over the range of 10 to 60 Amperes.Resc = 4.53E-6*I^2 – 5.7E-3*I + .0179, Ohms, where I is the ESC current rating.The specifications will usually list the programmable features of the ESC and how they can be managed.Selecting an ESCSelecting an ESC is relatively easy as there are only two primary specifications of major concern. Always select a continuous current rating greater than the motor current rating as this selection will provide some protection against overcurrent conditions. The voltage rating should at least match that of the motor or battery, whichever is greater. Oversizing does no harm and does provide overload protection.The other features that are desirable are overcurrent protection (it will shut off the ESC if a dangerous overcurrent condition is detected) and a switching BEC if there is anything more than a small current load on it. Refer to the later section on BECs for more details on this selection.ProgrammingMost ESCs are programmable either by an external program card or PC, or by manipulating the transmitter throttle control in response to ESC aural signal sequences. The controls available depend upon the particular ESC, but most provide selections for each of the following functions. The information in italics that follows about settings and programming was extracted from Reference 1.Basic SettingsControllers have myriad features that can be changed to optimize your system. Some you will want to use and some you will want to disable. I’ll cover the most common elements. In most cases your controller will come with factory defaults that are suitable for 90% of user applications.Brake SettingMost controllers will let you decide if you want a brake set or not. The brake can be set to ‘on’ if you want your propeller to stop when you reduce your throttle to idle/off. This is mostly used for folding propellers, but some prefer to stop the propeller for better glide with the motor off if it’s not the folding type.There are settings for ‘soft brake’, ‘hard brake’ and ‘no brake’. The default for most controllers is no brake or soft brake.Low-Voltage CutoffThis feature ensures you don’t damage your battery pack by drawing the voltage too low. The default on most controllers is ‘auto’ and most of the time that is just fine. Various brands use different methods to determine that value, but the standard is approximately 70% to 75% of the voltage level detected when first armed. This is important because if you fly more than one flight on a pack without recharging between flights, the controller may misinterpret the number of cells and therefore allow the voltage to drop too low. A three-cell LiPo pack is fully charged at 12.6 Volts and the nominal voltage is 11.1 Volts. The ESC accurately detects the cell count and sets the cutoff at 75% or 9.0 Volts. You make a flight and land with plenty of power left. Then after a break, you reconnect the same pack without charging and the ESC detects 11.5 Volts. It now sets the cutoff at 75% of 11.5 Volts or 8.12 Volts. That is too low for a three cell pack: you’ve discharged it to 2.8 Volts per cell and they don’t like that.Some ESCs provide a means of selecting a specific per cell cutoff so that doesn’t happen.Cut-off ModeMost controllers will allow you to choose whether you want a hard or soft cutoff when it reaches its minimum voltage level. A hard cutoff means the motor is going to stop when the voltage is reached. A soft cutoff means the motor will pulse on and off to let you know it’s time to land.Start-up ModeThis mode defines how the motor starts when you open the throttle. The choices are generally ‘normal’, ‘soft start’ and ‘super soft start’ or something similar. It is self-explanatory. Normal is usually the default mode and is fine for most airplanes.TimingThis mode causes more questions than a paragraph or two can answer. The default for most ESC’s is ‘standard’ or ‘medium’. Some list the timing angle, similar to that of a car motor. Unless you want to learn motor theory and optimize for competition, you’re safe leaving this alone. The instructions will give you special notes in case you have a problem with your motor running smoothly.Pulse-Width Modulation (PWM) Switching FrequencyThis figure represents the number of times per second that a pulse is sent to the motor. Stick with the factory default unless there is a problem. In that case, a call to technical support might help.Operating ModeSome controllers give options of selecting a preprogrammed set of parameters such as ‘airplane’, ‘glider’ or ‘helicopter’.One Last ThingHow should the three wires be connected to the motor? Hook them up in any order and if the motor runs backwards, interchange any two wires. Some ESCs will let you reverse motor direction in its programming.Battery Eliminator Circuits (BEC)Types of BECsThere are two types of BECs or voltage regulators; linear and switched. There are three names for these BECs. If an ESC is labeled ‘BEC’ then it uses a linear regulator. If it is labeled ‘SBEC’ then it uses a switching regulator. The third designation is ‘UBEC’ which may be of either type, so read the description carefully to determine which type it is. Switched regulators are available as separate devices, too.The following information in italics about BECs was extracted from Reference 2.What is wrong with a linear regulator?Linear regulators are great for powering very low powered devices. They are easy to use and cheap, and therefore are very popular. However, due to the way they work, they are extremely inefficient.A linear regulator works by taking the difference between the input and output voltages, and just burning it up as waste heat. The larger the difference between the input and output voltage, the more heat is produced. In most cases, a linear regulator wastes more power stepping down the voltage than it actually ends up delivering to the target device!With typical efficiencies of 40%, and reaching as low as 14%, linear voltage regulation generates a lot of waste heat which must be dissipated with bulky and expensive heatsinks. This also means reduced battery life for your projects.How is a switching regulator better?A switching regulator works by taking small chunks of energy, bit by bit, from the input voltage source, and moving them to the output. This is accomplished with the help of an electrical switch and a controller which regulates the rate at which energy is transferred to the output (hence the term “switching regulator”).The energy losses involved in moving chunks of energy around in this way are relatively small, and the result is that a switching regulator can typically have 85% efficiency. Since their efficiency is less dependent on input voltage, they can power useful loads from higher voltage sources.Switch-mode regulators are used in devices like portable phones, video game platforms, robots, digital cameras, and your computer.Switching regulators are complex circuits to design, and as a result they aren’t very popular with hobbyists. What can switching regulators do that linear regulators can't?With high input voltages, driving loads over 200mA with a linear regulator becomes extremely impractical. Most people use a separate battery pack in these situations, so they have one battery pack for high voltage devices and one for low voltage devices. This means you have twice as many batteries to remember to charge, and twice the hassle! A switching regulator can easily power heavy loads from a high voltage, and save you from splurging on an additional battery pack.Certain kinds of switching regulators can also step up voltage. Linear regulators cannot do this; Ever.How do I tell if I need a switching regulator?As a general rule of thumb, if your linear voltage regulation solution is wasting less than 0.5 watts of power, a switching regulator would be overkill for your project. If your linear regulator is wasting several watts of power, you most certainly want to replace it with a switcher! Here is how to calculate power losses:The equation for wasted power in a linear regulator is:Power wasted = (input voltage – output voltage) * load current ESCs are sensitive to heat and high temperatures. Generally manufacturers recommend keeping the temperature below 200 degrees Fahrenheit. The heat generated by a linear BEC running at 11 Volts and one amp in a small ESC will dissipate about 6 Watts and easily raise the temperature into this range unless cooled adequately. Go for the switched version if you can.Safety ESC – motor – propeller combinations can be dangerous. When an electric motor is stalled the current surges as the motor represents a dead short when stopped. This current surge generates very high torque that is transferred to the propeller and will remain present until the current is interrupted. A hand or body piece caught in this condition has no chance of getting away unscathed. The high current may damage the ESC and the motor as well. Rule #1: Never let anyone be within the propeller arc or in front of an electric powered plane if the battery is connected, even if the transmitter or receiver has failsafe modes. These failsafe modes are only safe under certain conditions that cannot be guaranteed.Rule #2: Always have the plane anchored in some fashion when the battery is connected. An inadvertent action such as moving the throttle accidentally may cause the motor to start and the plane to lunge forward.Rule#3: When working on the plane remove the propeller unless it is needed.Rule#4: Set up a kill switch on your transmitter that disables the receiver from activating the ESC when the throttle is in any position. This provides some safety against accidental throttle movements.Rule#5: If it has one, make sure the receiver failsafe works if the transmitter is turned off while the receiver is still powered. This can happen accidentally.OperationESCs are non-linear devices that approximate linear operation by applying very short bursts of current at maximum battery voltage to the motor windings. The inductance and inertia of the motor components resists these bursts and smooths them out, approximating a continuous flow of energy. However, the flow is never completely smooth, resulting in ripples in the applied voltages and currents that degrade motor performance somewhat. The averaged voltages are not sinusoidal, approximating a sinusoid by being trapezoidal. This property also adds to the ripple in voltage and current. The net of these differences is that the motor model presented in the section on motors is not complete. The ESC generates bursts at a constant rate, usually 8,000 bursts per second. The pulse widths are varied to change the average voltage and current supplied to the motor. Short bursts correspond to low average values. As the pulse length increases, the applied average voltages and currents also increase in direct proportion. The effect of the ripples is small at small burst lengths, increases as the burst length increases until the bursts are mid-width in value, then decrease to a small value at maximum pulse width.The basic motor equations can be modified to take this variation into account. This ripple loss is included in the analysis through the inclusion of the emf current multiplier f(d) = (1+d*-d^2). This function is 1 at d = 0, increase to 1.25 at d = 0.5, and decreases to 1 at d = 1. It is empirically derived, based upon limited information on this type of ripple in motors.Bench tests with known propellers (APCE wind tunnel data) indicate that this provides a good match to measured results.Brushless MotorsThere are two types of electric motors used by RC hobbyists. The older motor is the brushed type, largely supplanted by the brushless motor. This document assumes the brushless type.Physical PropertiesInrunners and OutrunnersThere are two types of brushless motors called inrunners and outrunners. Both types have a rotating part, the rotor, and a fixed non-rotating part, the stator. The rotor carries permanent magnets while the stator carries the wire windings to which the source power is attached. Inrunners have the magnets attached around the rotating shaft with the stator on the outside fixed frame. The outrunners have the stator on the inside with the magnets attached to a rotating outer shell. Physical ConstructionFigure M1 is a photo of an outrunner showing how it is constructed. The stator is shown on the left. The stator stack is a stack of very thin steel plates with a hole in the middle that fits around the shaft, separated from the shaft by bearings. The stack is divided into stator teeth sections, each wound with multiple turns of insulated wire. The steel acts to concentrate the magnetic fields produced by the windings at the interface with the magnets mounted on the rotor, the part to the right. The rotor on the right has a shaft that fits in the stator bearings so it can rotate. Permanent magnets are mounted on the interior of the can (or cup) that is permanently attached to the shaft at one end. The magnets alternate in polarity around the perimeter. Note that each magnet has a North and South Pole, one facing the inside, the other facing outward. As a consequence, the exterior of the can has magnetic fields that may attract small metal objects such as washers, nuts and bolts.This motor has 14 magnets and 12 stator teeth, a common configuration. Generally, the more magnets and teeth the smoother the running of the motor will be.914400top00Figure M1 Outrunner Construction DetailsFigure M2 is a cross-section drawing of an inrunner motor. The magnets are attached directly to the rotating shaft. In this case the stator holding the windings and stator stack surrounds the rotor. The inrunner is an inside-out outrunner.Figure M2 Inrunner Cross-SectionThe primary physical difference between inrunners and outrunners is that the inrunner stator and rotor are enclosed in a stationary container, or can, while the outrunner exterior can rotates. The consequence is that inrunners can be mounted with clamps around the can and are free from potential rubbing of wires or other airframe parts. Out runners are mounted either from the back or front by support plates in order to let the case rotate. The rotating case must be kept clear of other objects, sometimes creating clearance problems in tight locations. Inrunners are generally long and thin while outrunners tend to be shorter and fatter. The primary heat generators in both types of motors are the copper windings and stator metal plates. The inrunner windings are attached to the outside can so they have a direct thermal path to the exterior surface. Getting rid of this heat requires adequate airflow around the can. The outrunner windings are on the interior, isolated from the exterior rotating can. The can rotates so the rotation does provide some interior air motion to provide cooling. Some designs have a small fan built into them for this purpose.The location of the rotating magnets relative to the shaft center determines the torque characteristics of a motor. Torque is defined as force acting at a distance from an axis causing potential rotation around the axis. In equation form Q = k*F*L. k is a constant, F is the force applied and L is the distance, or arm, from the force to the axis of rotation. The inrunner magnets are locate very close to the axis so the arm is short. The outrunner magnets are mounted much farther out so the arm is longer. Hence for a given amount of force the outrunner will produce more torque than the inrunner.The power transmitted to the shaft is proportional to the rotational speed. P = n * Q where n is the rotational rate and Q is the torque. Inrunners can run at much greater rates than outrunners so that they can match the power performance of outrunners but at higher rotation rates. Outrunners produce more torque at low speeds than inrunners but cannot run at as high overall speeds. Inrunners are well suited to high speed applications such as ducted fans where high fan speeds are required and thin motors are needed to fit in the limited space. They can be adapted to use with large propellers by using gears to trade motor speed for torque, although the gears add weight and reduce efficiency. Outrunners can drive large propellers directly because of their inherent torque characteristics at low rotational rates. For this reason outrunners are the choice for most propeller driven RC aircraft.Physical LimitationsThere are some constraints on motor usage that are important. Motors do not like to be hot. The magnets are susceptible to degradation and ultimate loss of magnetism if heated to too high a temperature. Magnets are usually made from alloys of Neodymium, a rare earth metal. A typical limit for safe operation is about 150 C. The external surface of a motor will be cooler than the interior, so the rule of thumb is to limit the external temperature to 100 C, or the boiling point of water. Adequate airflow for cooling is essential for limiting maximum allowable temperatures. The wires are usually coated with some form of varnish for insulation. Long duration exposure to high heat will cause the varnish to deteriorate. Hence limiting the external temperature will also limit the wiring temperature.LabelingBrushless motors generally carry external labels that describe in rudimentary terms their characteristics. In addition to some form of brand or advertisement, they generally sport the physical dimensions of the motor in millimeters and the motor’s Kv rating. (Kv rating will be explained later).The physical dimensions are of two forms. In one form the external length and diameter of the motor are listed followed by the Kv rating. In the latter form the stator length and diameter are listed followed by the Kv rating. In this case the external dimensions are greater than the listed dimensions. The only way to tell for sure is to examine drawings showing the detailed dimensions or measure them yourself. The dimensions listed on the label in Figure M3 are the external dimensions; outer diameter followed by overall length of the case. Sometimes there is another number such as the 8 on the label in the figure. This is the number of wire turns around a stator tooth. This term will be explained later.Figure M3 Typical Brushless Motor LabelSpecifications and Their MeaningThe choice of a motor for a particular application is affected by the electrical parameters of the motor. This section defines the principle electrical parameters and explains their significance.Electrical ParametersKv: The rpm that a motor turns is a direct function of the voltage applied to the motor terminals. Under no load conditions the Kv is defined as the ratio of rpm to volts at maximum rpm. Kv is given in rpm/volt. For example, a voltage of 5 volts applied to a motor with a Kv = 1000 rpm/V will rotate under no load at 5 *1000 = 5000 rpm. As a load is applied, the motor will try to maintain the rpm, slowly decreasing as the load increases. The current will increase as the load increases in order to supply the torque needed to sustain the load. There is an associated torque constant, labeled either Kq or Kt that, when multiplied by the current in amps, equals the torque. Q = Kq * I. Kq = 1355/Kv in-oz/amp.Rm:Rm is the resistive loss of the motor windings in Ohms. It is a key parameter in motor performance as the power dissipated by the resistance is equal to the current squared times the resistance. P = Rm * I2 in Watts. For example, a current of 20 amps into a resistance of 0.05 ohms will result in a power of P = .05 * 20*20 = 20Watts. If the resistance is 0.1 ohms the power rises to 40 Watts, doubling the internal heat generated. Also the power lost in heat is not available to provide output power, so minimizing resistance is generally a good thing.Imax continuous:This is the current that the motor can handle on a sustained basis without overheating. The maximum heating power allowed is the product of the maximum current squared times the resistance.Imax Burst:This is the maximum current allowable for a short period of time, typically 60 seconds. Implied, but seldom stated, is that the current is then reduced to a point that the excess heat is dissipated before another burst is made.Vmax:Maximum allowable voltage applied to the motor. It is sometimes given in terms of the number of battery cells in series, such as 3S or 6S for Lipoly batteries.P max: This is the maximum allowed input power to the motor. It can also be computed by multiplying the maximum continuous current by the maximum voltage.Io: This is the current generated under no load conditions at a specified input voltage Vo. Vo is usually specified at 10 volts. Even though there is no useful output power transferred to the shaft, some power is needed to overcome internal losses such as bearing friction, eddy current loss and air friction loss. When a varying magnetic field is applied to a magnetic material such as steel, circulating currents are generated that produce heat. These are the eddy currents. Idle current is proportional to voltage and rpm.Physical ParametersIn addition to the electrical parameters, some physical parameters are sometimes specified. The shaft diameter in millimeters is usually specified as it is needed to determine the propeller adaptor size needed. Data sheets provide the overall dimensions and mounting configuration details.As stated previously, sometimes the number of wire turns is specified. The torque produced is directly proportional to the number of wire turns while the rpm/volt, Kv, is inversely proportional. Essentially there is a trade between torque and rpm. Unless one intends to modify a motor this information is of little value as the value of Kv in a specification captures this effect. Performance EstimatingThere are math formulas available for estimating the performance of these motors under a variety of conditions. The formulas that follow have been simplified to make them easier to understand and use. As a result, however, they tend to be too optimistic, overestimating output power and rpm values and underestimating input power and current values. They do provide useful results, especially if used for comparing alternatives. The Complete Power SystemFigure PS1 describes a complete electric power system including battery, ESC and motor.Figure PS1 Complete Power System Equivalent Electric DiagramThe battery and the motor EMF are voltage generators and their volltages begin with the letter E. Loss voltages begin with the letter V. These are conventional electrical engineering formats used to distinguish between generator and load voltages.Beginning on the left, the battery no-load voltage is Ebat. The battery internal resistance is Rbat, in ohms. The battery is connected to the ESC by wires and connectors whose resistance is labeled Rcab. Next is the ESC; it is characterized by resistance Resc and throttle fraction d. “d” is the relative value of the throttle setting, ranging from 0 to 1.0. When d = 0, the motor is off; when d = 1, the motor is at full on.The voltage marked Vesc is the voltage delivered to the input of the ESC. Emot is the equivalent generator voltage applied to the motor itself taking into account the throttle setting fraction, d. Emot = d x Vesc.The motor parameters are the motor resistance Rmot, the back emf Emf in volts, the motor current, Imotor, the no load current, Inl, the no load power loss, Pnl, the motor voltage constant, Kv and the torque constant Kq. The motor outputs are the shaft power, Pshaft, the torque, Q and the revolution rate, Rpm.The terms and their definitions are:DefinitionsEbatNo load battery voltageRbatBattery internal resistance, ohmsRcabCable and connector resistance, ohmsRescESC through resistance, ohmsRmotMotor internal resistance, ohmsPnlMotor power loss due to internal non resistive causesEmfMotor back EMF, opposed to the battery voltage, voltsPshaftPower output through the motor shaftKvMotor rpm per voltIono load reference current at VoVono load reference voltageInlMotor no load current, AmpsImotMotor current transferring power to the output shaft, AmpsdFraction of full throttle setting, range 0 to 1.0EmotThe source voltage driving the motor from the ESC, VoltsRelationshipsThere are some relationships that are common to all working formulas that follow.f(d) = 1+d*(1-d)ESC loss factor, multiplied by the motor current.Vesc = Ebat –Itotal*(Rbat + Rcab + Resc)Voltage into ESCEmotor= d*VescVoltage out of ESC to motor, viewed as a generatorInl = (Io/Vo)*EmfNo load current, AmpsPnl = Inl*EmfNo load power, WattsItotal = Inl + Imot*f(d)Total current, AmpsEmf = Kv * RpmMotor back voltage, VoltsKq = 1353/KvTorque constant, in-oz. /AmpQ = Kq*ImotTorque, in-oz.Pshaft = Imot* EmfShaft power, WattsThe source voltage must equal the load voltages around a circuit for equilibrium.On the motor side of the ESC, the relationship is:Eq. #1Emotor =[ Ebat – Itotal*(Rbat + Rcab + Resc)]*dEq. #2Emotor - Emf = Itotal * Rmot Eq #3Itotal = (Emotor-Emf)/RmotThese equations can be combined in various ways, depending on what parameters are known and which are to be calculated. To illustrate some possibilities, four example cases will be presented. Table PS1 contains the assumed parameters used in the cases. The motor parameters are those of the outrunner motor pictured in the section on brushless motors.Table PS1: Assumed system parameter valuesParameterValueEbat11.1 VoltsBattery Capacity1.5 AHBattery C rating30 C (maximum current = 1.5 * 30 = 45 Amps)Rbat.042 OhmsRcab.005 OhmsResc.001 OhmsRm.107 OhmsKv1100 rpm/VoltIo1 amp @ Vo = 10 VoltsESC maximum current30 AmpsMotor maximum current18 AmpsCase1: A cautionary example. Assume that the controls are at full throttle so that the motor has maximum voltage applied. Also assume that the shaft is constrained such that it cannot turn. This case might occur if the propeller were to snag or hit an object and cause the rpm to go to zero. In this case there is no back emf (Emf=0) since a DC current does not vary and so there is no magnetic field change to generate an emf. The resistance to current flow is only that of the resistance in the total circuit from battery, through the ESC and motorEmf = rpm/Kv = 0/1100 = 0Combining Equations # 1 & #2 when d = 1 and Rpm = 0:Itotal = Ebat/(Rbat + Rcab + Resc + Rmot) = 11.1/(.04 +.005+.001+.107) = 74 Amps!Output torque Q = Kq*Itotal = 1335/1100*74 = 98.8 in-oz.!The 74 Amps is well beyond the maximum current ratings for the ESC (30 Amps) and the motor (18 Amps). The over current is so high that the ESC will overheat and burn up unless it has over current protection that shuts off the voltage. If the current continues for more than a few seconds the motor will over heat and at least damage the magnets, perhaps burn the windings. Note that the battery current limit is 45 Amps, so permanent damage may also occur in it.Also the torque is so great that it would try to turn the prop even when constrained. Certainly if the constraint were created by the propeller snagging on a piece of clothing at the least the clothing would be imperiled as would fingers if they were the cause.Case 2: A second case is to estimate the torque, rpm, shaft power and battery power by varying the current. This approach is fairly common in displaying the characteristics of a specific motor.The derivation of the appropriate equations is not given here, only the results.fd=1+d-d2) Rbce=Rbat+Rcab+RescEmot=d*(Ebat-Itotal*Rmot)Emf=Emot-Itotal*RmotRpm=Emf*KvInl=IoVo*RpmKvImot=Itotal-Inlf(d)Pshaft=Imot*EmfQ=1335Kv*ImotPnl=Inl*EmfPresistance=Itotal2*(Rbce+Rmot)Pbattery=Pshaft+Pnl+PresistanceThree graphs follow, showing the results for three values of d and with total current as the independent variable. Figure PS2 Throttle at 20%With the throttle set at 20%, the motor is running at a low throttle setting with a corresponding low rpm (Figure PS2). At a minimum load (minimum current) the rpm is near maximum for this setting and the output and input powers are small. As the current increases, rpm drops, the input power climbs and the output power flattens out. At 20 amps, the motor is not running, there is no output power while the input power is at its greatest. The difference between the input power and the output power is lost as heat in the motor, battery & cable and ESC resistances. Figure PS3 Throttle at 50%At 50% throttle, the motor is now running at half-throttle with a mid-range rpm (Figure PS3). The rpm is highest at the lowest current load, slowly decreasing as the current increases. Note that it decreases at a lower relative rate compared to the 20% setting. The output power climbs nearly linearly with current until it approaches its maximum value of 20 Amps. The maximum power lost in heat is about 50 Watts.Figure PS4 Throttle at 100%At full throttle the rpm is at its maximum value as is the available output power with load current. The rpm sags to a relatively small percentage of its low load value. The output power increases nearly linearly with load current.Figure PS5 shows the distribution of power by source at 100% throttle. The idle power is the least and actually decreases with current since it is proportional to rpm and the rpm decreases with an increase in current. The motor resistance loss is dependent only on the resistance and the current, making it independent of throttle setting.The losses due to internal battery resistance, cabling and the ESC are also dependent only on the current, not on the throttle setting. Only the motor loss is dependent upon the throttle setting.Figure PS5 Power Element by SourceCase 3: Another condition of possible interest occurs if the required motor shaft power and rpm are known and it is desired to determine the current, throttle settings and battery power. Unfortunately, there is no simple formula that allows estimation of the throttle setting needed. “d” is found from a solution to a cubic equation. Rbce=Rbat+Rcab+RescImot=Pshaft*KvRpmInl=IoVo*RpmKvEmf=Rpm/Kv0=a0+a1*d+a2*d2+a3*d3a0=Emf+Rmot*(Inl+Imot)a1=-(Ebat-Rbce*Imot+Inl-Rmot*Imot)a2=Imot*(Rbce+Rmot)a3=-Rbce*ImotIt is possible to analyze the cubic equation directly by finding its roots, but it is a messy process. The solution can also be found by using a convergence algorithm such as that described by Newton. A straight forward way of finding a solution is to evaluate the equation for successive values of d ranging from 0 to 1.0, finding when the answer becomes essentially zero. This method is illustrated in the flowing Table PS2.From the previous graph for full throttle operation assume that the shaft power required is 124 Watts at 9,653 Rpm. The graph indicates that the required current is about 15 Amps and d=1. Table PS2 lists the step by step calculations used to analyze this case.Table PS2 Case 3 SolutionEquationResultRbce = Rbat+Rcab+Resc=.042+.005+.001 = .046 OhmsImot=Pshaft*Kv/Rpm=123*1100/9600 = 14.09 ampsEmf = Rpm/Kv=9600/1100 = 8.73 voltsInl = Io/Vo*Emf= 1/10*8.73 = .873 ampsa0 =(Emf+Rmot*Inl+Rmot*Imot)=(8.73+.042*.873+.042*14.09) =9.39 voltsa1 =-(Ebat-Rbce(Imot+Inl)-Rmot*Imot)=-(11.1-0.046(14.09+.873)-.042*14.09) = -9.82 voltsa2 = Imot(Rbce-Rmot)=14.09*(.046-.042) = 0.056 voltsa3 = -Rbce*Imot=- .046*14.09 = -0.648 voltsThe equation to be solved is then:Y=9.39- 9.82*d+0.056*d2-0.648*d3 , solved for Y=0Table PS3 lists the results for the range of d = 0 to 1.0.Table PS3 Case 3 SolutionFigure PS4 Plots the results of the table.Figure PS6 Case 3 Graphic SolutionThe result is zero when d is approximately 1.0. Using this value of d the other parameters can be found.Itotal=Inl+(1+d-d2)*Imot Itotal = 0.873+ (1+1-1^2)*14.09 = 14.96 ampsEmot=Emf+Itotal*RmotEmot = 8.73+14.09*.042 = 9.32 voltsVesc=Emot/dVesc = 9.32/1 = 9.32 voltsPmotor=Pshaft*(1+d-d2)Pmotor = 123*(1+1-1^2) = 123 WattsPnl=Inl*EmfPnl = .873*8.73 = 7.62 WattsPresistance=Rbce+Rmot*Itotal2Presistance = (.046+.042)*14.96^2 = 19.7 WattsPbattery=Pmotor+Pnl+PresistancePbattery = 123+7.62+19,7 = 150.3 WattsEfficiency=Pshaft/PbatteryEfficiency = 123/150.3 = 0.82 or 82%Newton’s Method SolutionNewton’s method is an iterative algorithm that quickly converges to a result.Make an initial guess d1 for the value of dCalculate the value of the function in d.Fd=a0+a1*d1+a2*d12+a3*d3Calculate the value of the derivative of F(d1) =F’(d)F'd1=a1+2*a2*d1+3*a3*d12Evaluate:d2=d1-F(d1)F'(d1)Substitute d2 for d1 in the equation:d3=d2-F(d2)F'(d2)And repeat this process until the answer is of adequate accuracy. The following Table PS4 is an example of these computations for Case3.a0 = 9.39a1= -9.82a2 =0.056a3= -0.648Initial guess: d1 = 0.5Table PS4 Example of Applying Newton’s Method to Case 3In this case the method yields an accurate value for d on the second iteration.Case 4: Testing of propellers can be done using electric motors calibrated to the task. It is possible to test the thrust characteristics of a propeller on a homemade test stand. The measurable quantities are the thrust, the motor rpm, the total current and the throttle setting. The known parameters are the resistances of the power system, the source voltage (best if done using a regulated power supply), the idle current parameters and the motor Kv rating. What is not easy to measure is the actual shaft output power, or the propeller absorbed power as it is sometimes known. However, it is possible to get a reasonable estimate of this power using the measured data and the formulas presented before.Pshaft= RpmKv*(Itotal-Io*RpmVo*Kv)(1+d-d2) WattsThe value of d can be estimated by running the propeller at maximum throttle and recording the resulting Rpm as d =1. Any other value of d can be estimated by scaling to the measured value of Rpm to the maximum observed value.Prediction UncertaintiesThe formulas given in the foregoing section are approximations to provide a mathematically tractable set of equations. They tend to result in optimistic results. The resistance is assumed to be constant with current. However, the resistance of the wire increases with temperature, increasing the resistance loss and decreasing the voltage available to the motor. A temperature increase of 100 degrees C results in a resistance increases of about 40% for copper wire.The idle current is assumed to be a linear function of rpm. In actuality it is slightly parabolic in form with a non-zero intercept at zero rpm. Since the idle current is relatively a small part of the current except at low rpm, the effects of this difference are generally minor.The constant Kv is assumed to be constant under all conditions. This is not strictly true as it is affected by the motor timing; that is, the delay between the change in winding voltage and the reaction of the magnets to the change. It is assumed that the applied voltage is constant and that the internal alternating currents produced to excite the windings are sinusoidal. This is not the case with hobby type ESCs as they do not produce sinusoidal AC voltages and have non-linearity that degrades the performance. This characteristic is compensated for in the formulas by employing an empirically derived correction factor, d. ................
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