Model torque-speed characteristics of rotary piezoelectric traveling wave motor

Rotational Actuators

The Piezo Rotary Motor block represents the torque-speed characteristics of a piezoelectric traveling wave motor. The block represents the torque-speed relationship of the motor at a level that is suitable for system-level modeling. To simulate the motor, the block uses the following models:

The motor is unpowered when the physical signal input *v* is
zero. This corresponds to applying zero RMS volts to the motor. In
this scenario, the block models the motor using the following elements:

An inertia whose value is the

**Rotor inertia**parameter value.A friction whose characteristics are determined by the parameter values in the

**Motor-Off Friction**tab.The block uses a Simscape™ Rotational Friction block to model the friction component. For detailed information about the friction model, see the Rotational Friction block reference page.

When the motor is active, Piezo Rotary Motor block represents the motor characteristics using the following equivalent circuit model.

In the preceding figure:

The AC voltage source represents the block's physical signal input of frequency

*f*and magnitude*v*.The resistor

*R*provides the main electrical and mechanical damping term.The inductor

*L*represents the rotor vibration inertia.The capacitor

*C*represents the piezo crystal stiffness.The capacitor

*C*represents the phase capacitance. This is the electrical capacitance associated with each of the two motor phases._{p}The torque constant

*k*relates the RMS current_{t}*i*to the resulting mechanical torque.The quadratic mechanical damping term,

*λω*_{m}^{2}, shapes the torque-speed curve predominantly at speeds close to maximum RPM.*ω*is the mechanical rotational speed._{m}The term $$J{\dot{\omega}}_{m}$$ represents the rotor inertia.

At model initialization, the block calculates the model parameters *R*, *L*, *C*, *k _{t}* and

**Rated torque****Rated rotational speed****No-load maximum rotational speed****Maximum torque**

These parameter values are defined for the **Rated
RMS voltage** and **Motor natural frequency** (or
rated frequency) parameter values.

The quadratic mechanical damping term produces a quadratic torque-speed curve. Piezoelectric motors torque-speed curves can typically be approximated more accurately using a quadratic function than a linear one because the torque-speed gradient becomes steeper as the motor approaches the maximum speed.

If the rotor inertia *J* is not specified
on the datasheet, you can select a value that provides a good match
to the quoted response time. The response time is often defined as
the time for the rotor to reach maximum speed when starting from rest,
under no-load conditions.

The quality factor that you specify using the **Resonance
quality factor** parameter relates to the equivalent circuit
model parameters as follows:

$$Q=\frac{1}{R}\sqrt{\frac{L}{C}}$$

This term is not usually provided on a datasheet. You can calculate its value by matching the sensitivity of torque to driving frequency.

To reverse the motor direction of operation, make the physical
signal input *v* negative.

The block has the following limitations:

When the motor is powered, the model is valid only between zero and maximum speed, for the following reasons:

Datasheets do not provide information for operation outside of normal range.

Piezoelectric motors are not designed to operate in the powered braking and generating regions.

The block behaves as follows outside the valid operating region:

Below zero speed, the model maintains a constant torque that is the zero rpm torque value. The zero rpm torque value is the

**Maximum torque**parameter value if the RMS input voltage equals the**Rated RMS voltage**parameter value, and the frequency input equals the**Motor natural frequency**parameter value.Above maximum speed, the model produces the negative torque predicted by the equivalent circuit model, but limits the absolute value of the torque to the zero-speed maximum torque.

The torque-speed characteristics are most representative when operating the model close to the rated voltage and resonant frequency.

**Motor natural frequency**Frequency at which the piezoelectric crystal naturally resonates. For most applications, set the input signal at port

`f`

to this frequency. To slow down the motor, for example in a closed-loop speed control, use a frequency slightly less than the motor natural frequency. The default value is`40`

kHz.**Rated RMS voltage**Voltage at which the motor is designed to operate. The default value is

`130`

V.**Rated torque**Torque the motor delivers at the rated RMS voltage. The default value is

`0.5`

N*m.**Rated rotational speed**Motor speed when the motor drives a load at the rated torque. The default value is

`100`

rpm.**No-load maximum rotational speed**Motor rotational speed when driving no load and powered at the rated voltage and driving frequency. The default value is

`160`

rpm.**Maximum torque**Maximum torque that the motor delivers when actively driving a load and powered at the rated voltage and frequency. The default value is

`1`

N*m.**Note:**The**Holding torque**parameter value, the load torque the motor holds when stationary, may be greater than the**Maximum torque**parameter value.**Resonance quality factor**Quality factor

*Q*that specifies how torque varies as a function of driving frequency. Increasing the quality factor results in a much more rapid decrease in torque as driving frequency is moved away from the natural frequency. The default value is`100`

.**Capacitance per phase**Electrical capacitance associated with each of the two motor phases. The default value is

`5`

nF.

**Rotor inertia**Rotor resistance to change in motor motion. The default value is

`200`

g*cm^{2}.**Initial rotor speed**Rotor speed at the start of the simulation. The default value is

`0`

rpm.

**Holding torque**The sum of the Coulomb and the static frictions. It must be greater than or equal to the

**Coulomb friction torque**parameter value. The default value is`1.5`

N*m.**Coulomb friction torque**The friction that opposes rotation with a constant torque at any velocity. The default value is

`1`

N*m.**Viscous friction coefficient**Proportionality coefficient between the friction torque and the relative angular velocity. The parameter value must be greater than or equal to zero. The default value is

`0.001`

N*m/(rad*s).**Transition approximation coefficient**The parameter sets the coefficient value that is used to approximate the transition between the static and the Coulomb frictions. For detailed information about the coefficient,

*c*, see the Simscape Rotational Friction block reference page. The default value is_{v}`10`

s/rad.**Linear region velocity threshold**The parameter sets the small vicinity near zero velocity, within which friction torque is considered to be linearly proportional to the relative velocity. MathWorks recommends that you use values in the range between

`1e-5`

and`1e-3`

rad/s. The default value is`1e-04`

rad/s.

The block has the following ports:

`f`

Physical signal input value specifying the motor driving frequency in Hz.

`v`

Physical signal input magnitude specifying the RMS supply voltage, and sign specifying the direction of rotation. If

`v`

is positive, then a positive torque acts from port C to port R.`i`

Physical signal output value that is the RMS phase current.

`wm`

Physical signal output value that is the rotational speed of the rotor.

`C`

Mechanical rotational conserving port.

`R`

Mechanical rotational conserving port.

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