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# DC Motor

DC motor model with electrical and torque characteristics

## Library

Rotational Actuators

## Description

The DC Motor block represents the electrical and torque characteristics of a DC motor using the following equivalent circuit model:

You specify the equivalent circuit parameters for this model when you set the Model parameterization parameter to By equivalent circuit parameters. The resistor R corresponds to the resistance you specify in the Armature resistance parameter. The inductor L corresponds to the inductance you specify in the Armature inductance parameter. The permanent magnets in the motor induce the following back emf vb in the armature:

${v}_{b}={k}_{v}\omega$

where kv is the Back-emf constant and ω is the angular velocity. The motor produces the following torque, which is proportional to the motor current i:

${T}_{E}={k}_{t}i$

where kt is the Torque constant. The DC Motor block assumes that there are no electromagnetic losses. This means that mechanical power is equal to the electrical power dissipated by the back emf in the armature. Equating these two terms gives:

$\begin{array}{l}{T}_{E}\omega ={v}_{b}i\\ {k}_{t}i\omega ={k}_{v}\omega i\\ {k}_{v}={k}_{t}\end{array}$

As a result, you specify either kv or kt in the block dialog box.

The torque-speed characteristic for the DC Motor block is related to the parameters in the preceding figure. When you set the Model parameterization parameter to By stall torque & no-load speed or By rated power, rated speed & no-load speed, the block solves for the equivalent circuit parameters as follows:

1. For the steady-state torque-speed relationship, L has no effect.

2. Sum the voltages around the loop and rearrange for i:

$i=\frac{V-{v}_{b}}{R}=\frac{V-{k}_{v}\omega }{R}$

3. Substitute this value of i into the equation for torque:

${T}_{E}=\frac{{k}_{t}}{R}\left(V-{k}_{v}\omega \right)$

When you set the Model parameterization parameter to By stall torque & no-load speed, the block uses the preceding equation to determine values for R and kt (and equivalently kv).

When you set the Model parameterization parameter to By rated power, rated speed & no-load speed, the block uses the rated speed and power to calculate the rated torque. The block uses the rated torque and no-load speed values in the preceding equation to determine values for R and kt.

The block models motor inertia J and damping λ for all values of the Model parameterization parameter. The resulting torque across the block is:

$T=\frac{{k}_{t}}{R}\left(V-{k}_{v}\omega \right)-J\stackrel{˙}{\omega }-\lambda \omega$

It is not always possible to measure rotor damping, and rotor damping is not always provided on a manufacturer datasheet. An alternative is to use the no-load current to infer a value for rotor damping.

For no-load, the electrically-generated mechanical torque must equal the rotor damping torque:

${k}_{t}{i}_{noload}=\lambda {\omega }_{noload}$

where inoload is the no-load current. If you select By no-load current for the Rotor damping parameterization parameter, then this equation is used in addition to the torque-speed equation to determine values for λ and the other equation coefficients.

The value for rotor damping, whether specified directly or in terms of no-load current, is taken into account when determining equivalent circuit parameters for Model parameterization options By stall torque and no-load speed and By rated power, rated speed and no-load speed.

When a positive current flows from the electrical + to - ports, a positive torque acts from the mechanical C to R ports.

### Thermal Port

The block has an optional thermal port, hidden by default. To expose the thermal port, right-click the block in your model, and then from the context menu select Simscape > Block choices > Show thermal port. This action displays the thermal port H on the block icon, and adds the Temperature Dependence and Thermal port tabs to the block dialog box.

Use the thermal port to simulate the effects of copper resistance losses that convert electrical power to heat. For more information on using thermal ports and on the Temperature Dependence and Thermal port tab parameters, see Simulating Thermal Effects in Rotational and Translational Actuators.

## Dialog Box and Parameters

### Electrical Torque Tab

Model parameterization

Select one of the following methods for block parameterization:

• By equivalent circuit parameters — Provide electrical parameters for an equivalent circuit model of the motor. This is the default method.

• By stall torque & no-load speed — Provide torque and speed parameters that the block converts to an equivalent circuit model of the motor.

• By rated power, rated speed & no-load speed — Provide power and speed parameters that the block converts to an equivalent circuit model of the motor.

Armature resistance

Resistance of the conducting portion of the motor. This parameter is only visible when you select By equivalent circuit parameters for the Model parameterization parameter. The default value is 3.9 Ω.

Armature inductance

Inductance of the conducting portion of the motor. If you do not have information about this inductance, set the value of this parameter to a small, nonzero number. The default value is 1.2e-05 H.

Define back-emf or torque constant

Indicate whether you will specify the motor's back-emf constant or torque constant. When you specify them in SI units, these constants have the same value, so you only specify one or the other in the block dialog box. This parameter is only visible when you select By equivalent circuit parameters for the Model parameterization parameter. The default value is Specify back-emf constant.

Back-emf constant

The ratio of the voltage generated by the motor to the speed at which the motor is spinning. The default value is 7.2e-05 V/rpm. This parameter is only visible when you select Specify back-emf constant for the Define back-emf or torque constant parameter.

Torque constant

The ratio of the torque generated by the motor to the current delivered to it. This parameter is only visible when you select Specify torque constant for the Define back-emf or torque constant parameter. The default value is 6.876e-04 N*m/A.

Stall torque

The amount of torque generated by the motor when the speed is approximately zero. This parameter is only visible when you select By stall torque & no-load speed for the Model parameterization parameter. The default value is 2.4e-04 N*m.

Speed of the motor when not driving a load. This parameter is only visible when you select By stall torque & no-load speed or By rated power, rated speed & no-load speed for the Model parameterization parameter. The default value is 1.91e+04 rpm.

Motor speed at the rated mechanical power level. This parameter is only visible when you select By rated power, rated speed & no-load speed for the Model parameterization parameter. The default value is 1.5e+04 rpm.

The mechanical power the motor is designed to deliver at the rated speed. This parameter is only visible when you select By rated power, rated speed & no-load speed for the Model parameterization parameter. The default value is 0.08 W.

Rated DC supply voltage

The voltage at which the motor is rated to operate. This parameter is only visible when you select By stall torque & no-load speed or By rated power, rated speed & no-load speed for the Model parameterization parameter. The default value is 1.5 V.

Rotor damping parameterization

Select one of the following methods to specify rotor damping:

• By damping value — Specify a value for rotor damping directly, by using the Rotor damping parameter on the Mechanical tab. This is the default.

• By no-load current — The block calculates rotor damping based on the values that you specify for the No-load current and DC supply voltage when measuring no-load current parameters. If you select this option, the Rotor damping parameter is not available on the Mechanical tab.

Specify the no-load current value, to be used for calculating the rotor damping. This parameter is only visible when you select By no-load current for the Rotor damping parameterization parameter. The default value is 0 A.

DC supply voltage when measuring no-load current

Specify the DC supply voltage corresponding to the no-load current value, to be used for calculating the rotor damping. This parameter is only visible when you select By no-load current for the Rotor damping parameterization parameter. The default value is 1.5 V.

### Mechanical Tab

Rotor inertia

Resistance of the rotor to change in motor motion. The default value is 0.01 g*cm2. The value can be zero.

Rotor damping

Energy dissipated by the rotor. This parameter is only visible when you select By damping value for the Rotor damping parameterization parameter on the Electrical tab. The default value is 1e-08 N*m/(rad/s). The value can be zero.

Initial rotor speed

Speed of the rotor at the start of the simulation. The default value is 0 rpm.

## Ports

The block has the following ports:

+

Positive electrical input

-

Negative electrical input

C

Mechanical rotational conserving port

R

Mechanical rotational conserving port

## Examples

See the following DC motor examples:

## References

[1] Bolton, W. Mechatronics: Electronic Control Systems in Mechanical and Electrical Engineering, 3rd edition Pearson Education, 2004.