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DC motor model with electrical and torque characteristics

Rotational Actuators

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 *v*_{b} in
the armature:

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

where *k*_{v} 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 *k*_{t} 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 *k*_{v} or *k*_{t} 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:

For the steady-state torque-speed relationship,

*L*has no effect.Sum the voltages around the loop and rearrange for

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

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*k*_{t}(and equivalently*k*_{v}).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*k*_{t}.

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

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

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 *i*_{noload} 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.

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.

**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.**No-load speed**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.**Rated speed (at rated load)**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.**Rated load (mechanical power)**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.

**No-load current**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.

**Rotor inertia**Resistance of the rotor to change in motor motion. The default value is

`0.01`

g*cm^{2}. 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.

The block has the following ports:

`+`

Positive electrical input

`-`

Negative electrical input

`C`

Mechanical rotational conserving port

`R`

Mechanical rotational conserving port

See the following DC motor examples:

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

Induction Motor | Servomotor | Shunt Motor | Universal Motor