Topic:
 What supply voltage level has to be applied to the controller to operate the motor up to a maximum speed required by the application?
Situation:
The motor in use by an application is defined and the maximum speed of the motor has been defined based on the application's requirements. There is the question discussed now what supply voltage is required by the controller resp. its power stage so that the motor can reach the required maximum speed reliably even at high load or steep acceleration demands?
Solution:
The maximum reachable speed of a motor depends on the voltage applied to its windings and the "Speed constant" specified by the motor data. The supply voltage of the controller has to be higher than the motor voltage level calculated on the base of the "Speed constant".
The socalled "Nominal voltage" of a motor winding is no limiting factor. The actual motor voltage and the supply voltage of a controller must be even higher than the specified "Nominal voltage" depending on the speed requirements. The controller adjusts the voltage level of the motor (by the socalled "PWM duty cycle") according to the speed and load demands of the currently demanded working point.
There has to be taken care that the demanded speed does not exceed any of the specified mechanical limits, e.g. the "Max. speed" of the motor or the "Max. input speed" of a gear or spindle.
The following aspects have to be considered to estimate the required minimum supply voltage of a controller to fulfill the motor speed and torque requirements of an application:
Required motor voltage
The required motor voltage (i.e. the controller “Max. output voltage”) depends on the “Speed constant” of the motor defining the no load speed per volt of motor voltage. If there is fixed voltage applied to the motor, the motor speed will be reduced with increasing torque according to the “Speed / torque gradient” of the motor. Which motor voltage is required depends on the motor's operation points of an application. Operation points are characterized by the required speed and the different torque values. The highest speed and torque load typically occurs at the end of an acceleration phase. The motor voltage must be high enough to cover this extreme operation point. It can be calculated by the following formula:
Minimum required (theoretical) motor voltage
= ("Max. speed" + ("Max. load torque" * "Speed/torque gradient")) / "Speed constant"

Max. speed [rpm]:
Maximum required motor speed based on application's requirements.  Max. torque [mNm]:
Maximum required motor torque based on application's requirements, e.g. required to accelerate the load's inertia. 
Speed / torque gradient [rpm/mNm]: see motor data sheet

Speed constant [rpm/V]: see motor data sheet

Min. required motor voltage [V]:
This calculated motor voltage is a minimum value; it’s just enough to reach the extreme operation point based on the specified motor data without any reserves.
Required motor voltage in practice:
Typically the actual motor voltage should be chosen about 20% higher than the theoretically calculated value to ensure that the extreme operation point can be reached in a reliable way even taking the following two additional influencing factors into account:

Tolerance of the motor data
The specified motor data has typically a tolerance of +/10%. 
Control reserve
A controller needs some “head room” for a responsive control and should not run into physical limits when the situation is a little bit out of the ordinary.
Controller's supply voltage aspects
The data sheet and the "Hardware Reference" of a controller specifies the socalled "Max. output voltage" (or "Max. PWM duty cycle) as a percentage value of the controller's actual supply voltage. This value limits the voltage which can be output by the controller's power stage and applied to the motor winding in maximum.
 EPOS4
In case of an "EPOS4" the "Max. output voltage" is specified as 90%, i.e. the voltage applied to motor windings will be 90% of the controller's supply voltage in maximum.
Example:
EPOS4: Supply voltage = 24V
=> approx. 21.6 V max. motor voltage possible  ESCON:
In case of an "ESCON" the "Max. output voltage" is specified as 95 ... 98% depending on the concrete ESCON product type, i.e. the voltage applied to motor windings will be 95 ... 98% of the controller's supply voltage in maximum.
Example:
ESCON 50/5: Supply voltage = 24V
=> approx. 23.5 V max. motor voltage possible  Additional note: Minimum supply voltage limits
The data sheet and "Hardware Reference" of the controller specifies a minimum supply voltage level. It has to be ensured that the voltage supply does not fall below this minimum supply voltage limit during operation, e.g. due to a partly unloaded battery or overload states.  Additional note: Maximum supply voltage limits
If there is a battery present for the supply voltage, the "endofcharge" voltage has to be recognized. It has to be ensured that the "endofcharge" voltage does neither exceed the controller's "Max. absolute supply voltage" specified in its "Hardware Reference" nor the configured voltage limit of a socalled brake chopper or shunt regulator (e.g. maxon's DSR 70/30) which might be installed in the system too.
Practice?
 Step 1: Min. required voltage
The formula mentioned above is the starting point of any calculation:
Min. (theoretically) required motor voltage
= ("Max. speed" + ("Max. torque" * "Speed/torque gradient")) / "Speed constant"
One point of discussion is quite often what will be the "Max. torque" during acceleration? Although this can be calculated in theory based on friction, load's inertia, and acceleration, there are often not all these factors known or already fixed during system design. As a rule of thumb the "Max. torque" is often assumed in the range of 1 ... 3 times of the motor's specified "Nominal torque".  Step 2: Tolerance and control reserve
A "tolerance and control" reserve of 20% is added to the calculated "Min. required motor voltage".  Step 3: Controller's supply voltage
The required controller's supply voltage has to be calculated taking the specified "Max. output voltage" ratio into account.  Remark:
Power supplies with a 24V or 48V output voltage are industry standard and commonly in use. The main decision about the supply voltage of a controller is quite often very "digital" then and will just be if 24V is sufficient or then it is 48V. The concrete controller type must be selected with a matching supply voltage range and be able to provide the required motor current.
Examples:
Example 1: ECi 40
 Motor's specified data (PN: 496660):
> Speed constant: 255 rpm/V
> Speed / torque gradient: 1.6 rpm/mNm  Controller: "EPOS4 compact 50/8 CAN" (PN: 520885)
> Max. output voltage: 0.9 * supply voltage  Application's extreme operation point:
> Max. speed: 6000 rpm @ 400 mNm (during acceleration)
Assumption: Short time overload of twice motor's nominal torque.  Theoretically minimum required motor voltage:
= (6000rpm + (400mNm * 1.6rpm/mNm)) / 255rpm/V
=> Min. required motor voltage = 26 V  Taking "Tolerances and control" reserve into account:
Required motor voltage = 26V * 1.2 = 31 V  Required controller's supply voltage:
EPOS4's supply voltage = 31V / 0.9 = approx. 35 V  Power supply selection:
Typically a 48V power supply will be chosen.
Example 2: DCX 35 L
 Motor's specified data (DCX 35L, 12V Winding):
> Speed constant: 699 rpm/V
> Speed / torque gradient: 4.04 rpm/mNm  Controller: "ESCON Module 50/8" (PN: 532872)
> Max. output voltage: 0.98 * supply voltage  Application's extreme operation point:
> Max. speed: 8000 rpm @ 200 mNm (during acceleration)
Assumption: There is load with a high inertia present  Theoretically minimum required motor voltage:
= (8000rpm + (200mNm * 4.04rpm/mNm)) / 699rpm/V
=> Min. required motor voltage = 12.6 V  Taking "Tolerances and control" reserve into account:
Required motor voltage = 12.6V * 1.2 = 15 V  Required controller's supply voltage:
ESCON's supply voltage = 15V / 0.98 = approx. 15.5 V  Power supply selection:
Typically a 24V power supply will be chosen.
Cross reference to related topics:
 Meaning of maxon's specified motor data
 Meaning of maxon's specified gearhead data
 Chapters "6.3 Motor constants and diagrams" and "6.5 Motor selection" of maxon's "Formulae Handbook" > Formulae Handbook
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