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Model torque-speed characteristics of rotary piezoelectric traveling wave motor
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_{p} represents the phase capacitance. This is the electrical capacitance associated with each of the two motor phases.
The torque constant k_{t} relates the RMS current 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_{m} is the mechanical rotational speed.
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 λ to ensure that the steady-state torque-speed curve matches the values of the following user-specified parameter values:
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.
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.
Voltage at which the motor is designed to operate. The default value is 130 V.
Torque the motor delivers at the rated RMS voltage. The default value is 0.5 N*m.
Motor speed when the motor drives a load at the rated torque. The default value is 100 rpm.
Motor rotational speed when driving no load and powered at the rated voltage and driving frequency. The default value is 160 rpm.
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.
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.
Electrical capacitance associated with each of the two motor phases. The default value is 5 nF.
Rotor resistance to change in motor motion. The default value is 200 g*cm^{2}.
Rotor speed at the start of the simulation. The default value is 0 rpm.
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.
The friction that opposes rotation with a constant torque at any velocity. The default value is 1 N*m.
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).
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_{v}, see the Simscape Rotational Friction block reference page. The default value is 10 s/rad.
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:
Physical signal input value specifying the motor driving frequency in Hz.
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.
Physical signal output value that is the RMS phase current.
Physical signal output value that is the rotational speed of the rotor.
Mechanical rotational conserving port.
Mechanical rotational conserving port.