ebikes.ca Hub Motor Simulator

Hub Motor Simulator

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The following program was written to help evaluate how different motor, battery, and controller combinations will perform on an electric bike. Please choose from among any of the motor and battery combinations that we carry in store, then click "Calculate" to see the torque, output power, and efficiency curves plotted as a function of the vehicle speed. See below for an explanation.

Note, the results of this simulator exaggerate the performance of the eZee and Nine Continent hub motors as the winding indctance plays a much more significant factor in these motors than it does with the lower pole count Crystalyte hubs. We have another version of the simulator model still in development that you can play with here that is more accurate.

How to Use the Simulator

The simulator above was designed to model permanent magnet motors in the context of an electric vehicle, so that you can better understand the performance and behaviour of different battery, motor, controller, and wheel combinations.

In the simple mode, you choose from the list one of the hub motors that we have modelled, the battery pack, wheel size, and motor controller current limit, then hit "Calculate". The simulator will then output 3 graphical plots against your speed (in kph or mph) on the horizontal axis.

Red Plot: The red line shows the power output of the hub motor. The power output is zero at 0 rpm, rises up to a maximum, and then falls back down to zero once the wheel is spinning at its normal unloaded speed. If the motor current is not limited by the controller in any way, then the peak power happens at exactly 50% of the motor's unloaded rpm. Otherwise, the maximum power usually coincides with the point when the controller hits the current limit.
Green Plot: This is the efficiency curve for the hub motor. It is a ratio of the mechanical power coming out of the hub motor to the electrical power going in to the controller. This ratio accounts for losses inside the controller and the hub motor, but it does not include losses due to the internal resistance of the battery. Notice though that efficiency does not necessarily mean better range or less battery usage; a fast and powerful setup running at 80% efficiency may draw more power to cover the same distance than a slower arrangement at 75% efficiency.
Blue Plot: The blue line can be configured to display either the torque output of the hub in Newton-meters, or the thrust of the wheel in pounds. Thrust naturally increases as you select smaller wheel sizes, while the torque of the motor is independent of wheel diameter. Thrust is most convenient for estimating the climbing capacity of the vehicle, and it is what most people actually mean when they talk about 'torque'. To a first order approximation, the pounds thrust needed to overcome gravity when climbing a hill is simply weight * %grade.

If you want to see the output of a setup that doesn't use the stock components, then you can choose instead to use our 'advanced settings' option. This will let you put in any custom value for your wheel diameter, your battery pack resistance, and your controller current limit.

Vbatt, Rbatt: These two terms model the battery pack. Vbatt is the open circuit battery voltage that you measure when the pack is roughly 50% charged. Rbatt determines how much this voltage sags as you draw current from the battery, and it can have a significant effect on the output curves. If you don't know the internal resistance of your battery, you can often get a good estimate from the datasheet on the cells that make up your battery. These will show the cell impedance in mOhm. Multiply that by the number of cells in your system, and then add a certain amount for the wiring and tab resistance (usually about 1-2mOhm / tab).
Controller Resistance and Current Limit: This sets the battery current limit on the motor controller, and assumes that the motor current is not regulated. The controller resistance is the combined resistance of 2 mosfets and the return trip through the controller to motor leads. It should not include the resistance of the wires from the battery pack to the controller; that quantity should be lumped into Rbatt.
Wheel Size: You can input the exact diameter of your wheel in inches.

Simulator FAQ

Help, my battery isn't listed?

We tried to include the most common battery types that our customers use in the drop down menu, but it would be quite cluttered if we listed all the possibilities. If the voltage of your battery is the same as one of the standard listed packs, then choose one that has similar impedance. For instance with lead acids you can use the 18Ah NiMH and be pretty close. Non-standard voltages will require using the "enter custom value" menu.

Why is there a sharp peak in all of the graphs?

The inflection point that you see occurs when the motor controller hits the current limit. At speeds above this point, the current from the battery gradually declines until it reaches 0 amps at the unloaded rpm (where the graph outputs intersect). At speeds slower than the inflection point, the motor controller regulates the power to the hub, restricting battery current draw to the motor controller current limit (20A, for example). In this region, from 0 rpm up to the inflection point, there is constant power input into the motor rather than a constant voltage, and so the nature of the curves is different.

How come the torque keeps going up at lower speeds even though the amps is fixed by the current limit?

That is because even though the battery current is limited by the controller, the current through the motor is not. It is the motor current, not the battery current, that determines the torque output of the hub motor. When there is no Pulse-Width-Modulation (PWM) going on in the motor controller (full throttle and moving fast enough that the battery current is under the motor controller current limit) then the amps flowing out of the battery is the same as the amps flowing through the motor. But when the controller is doing PWM, then the current through the motor is higher than the battery current by the inverse of the PWM duty cycle.

Where does it tell me how far or fast I will go?

It doesn't. This is a hub motor simulation, and not a vehicle simulator. It tells you what torque and power outputs the motor will be able to produce at a given road speed. The actual speed you will be able to achieve depends on where the combined power output of the motor and your legs match the drive power required to move the vehicle. Determining that precisely is difficult, since it relies heavily on vehicle aerodynamics, not to mention weight and % grade you are travelling on. There are some handy online calculators that have modelled the power requirements of bicycle transportation to a high degree of accuracy, such as:

http://www.kreuzotter.de/english/espeed.htm
http://www.mne.psu.edu/lamancusa/ProdDiss/Bicycle/bikecalc1.htm
http://www.machinehead-software.co.uk/powercalc_graphs.html
http://swiss2.whosting.ch/mdetting/sports/cycling.html

This information for your vehicle combined with the simulator data can provide a very accurate prediction of vehicle performance.

I want to see more info, like current, input power, etc.

We will be upgrading the simulator in the future to show numerical data at various points along the graph. In the meantime though, you can easily tell what other parameters are from the graphs that are shown. For instance, the motor current curve will look exactly like the torque curve with just a slightly higher offset in account of the cogging torque. You know that at the inflection point, the motor current is the same as the current limit you've selected, so you can use this to scale the graph and read it as amps instead of Newton Meters. Similarly, you know that at the inflection point and slower the battery current is exactly equal to the controller current limit, and above this point if falls linearly down to 0 amps, so there would not be much to gain from showing battery amps. The input power would be this scaled again by the battery voltage.

Can I license the simulator?

While we appreciate the level of interest that has been shown in the simulator by other businesses who would like to have a copy of it on their site, the answer is that no, we are currently not interested in licensing this online software. It is first and foremost a tool that we have made freely available via our website to the ebike community.

Can you give me the data, like motor constants, resistance, etc.?

The answer to this question is that we already have, it is contained within output of these curves. If you know how to make use of this raw motor data, then you will be able to extract the motor parameters from the output graphs. Hints: The motor constant is readily determined from either the unloaded RPM at a known voltage or from the torque at the known current limit, and the motor winding resistance can be extrapolated from the slope of the torque/speed curve in the linear (non current limited) region when there are no battery losses.

Simulator Details

The simulator is based on the following circuit model for a battery-powered permanent magnet motor and controller setup:

V_OC is the open circuit battery voltage.
R_batt is the first order internal resistance of the battery.
I_batt is the current draw from the battery pack.
Controller is assumed to be a single quadrant (non-regen) controller, with the indicated input and output equations.
D is the PWM duty cycle, which can range from 0 to 1. This value is generally equal to the throttle setting, but that can be over-ridden to keep I_batt at the specified current limit.
I_motor is the current through the motor windings.
R_motor is the winding resistance of the motor.
K is the motor constant, in Nm/A or V/(Rad/sec).
w is the wheel speed in radians per second.

Once this equation is solved for a given wheel speed, output torque is calculated by taking the actual torque and subracting a 2nd order approximation of the motor cogging and windage torque:

The power is calculated simply as this output torque times wheel speed:

The efficiency is calculated from the input power to the controller divided by the output power (it does not take into account the losses internal to the battery pack):

Finally, the road speed is calculated based on wheel size. The simulator also correctly shows the regen currents and negative power when the wheel is spinning faster than the unloaded voltage, so the axis go slightly in the negative region to show this.

Accuracy and Limitations

The parameter values that are choosen for the motor model are based on directly measured data that we have compiled from tests using a custom built dynamometer made for the task.

Our dynamo setup is presently limited to a maximum loading of about 5 N-m, so it can only confirm the simulation at the higher end of the speed range. When the motor is heavily loaded, it causes all of the copper wires and windings to heat up, increasing their resistance and resulting in less torque and power than the simulator may indicate. There is also a fair bit of deviation from one hub motor to the next from Crystalyte because they are all hand made. This is especially true of the cogging torque, which has an effect on the peak efficiency of the setup. The performance charactersistics listed on the crystalyte website for the 400 series motors seem to be off by about 20%. These values appear to have been generated with an earlier and weaker magnet arrangement.

The characteristics for the battery packs are also from our own independent testing, with the value for R_batt calculated from the DC impedance from 1C - 2C loading, and with V_OC chosen at the 50% state of charge. A freshly charged pack would be faster and more powerful than indicated, while a nearly flat pack would be slower.

Finally, the program does not simulate the low voltage rollback of the controller. If you have a setup with a particularly high impedance battery, like a 5Ah NiCd, it is possible that in real life it would hit the low voltage point at high loadings and have a lower power output than indicated.