322 lines
14 KiB
Markdown
322 lines
14 KiB
Markdown
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title = "Transconductance Amplifiers"
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draft = false
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category = "equipment"
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Tags
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: [Electronics]({{< relref "electronics.md" >}}), [Voice Coil Actuators]({{< relref "voice_coil_actuators.md" >}})
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## Description {#description}
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A Transconductance Amplifier converts the control voltage into current with a current source characteristic.
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Such a converter is called a voltage-to-current converter, also named a voltage-controlled current source or _transconductance_ amplifier.
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Such amplifier is used to control motors (e.g. voice coil, BLDC, stepper motors, ...).
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## Manufacturers {#manufacturers}
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<a id="table--tab:table-name"></a>
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<div class="table-caption">
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<span class="table-number"><a href="#table--tab:table-name">Table 1</a>:</span>
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Drivers with integrated controllers
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</div>
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| Model | Manufacturer | Linear / PWM | Axes | Interfaces | Feedback | Current Bandwidth | ASD at 1kHz [A/sqrt(Hz)] |
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|-----------------------------------------------------------------------------------------------------------------------------------|-----------------|--------------|----------------|---------------|--------------|-------------------|--------------------------|
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| [Apogee](https://prodrive-technologies.com/motion/products/servo-drives/apogee-kepler-series/) | Prodrive | PWM | 1 to 3 | +/-10V 16bits | Encoder | 7kHz | 1e-6 |
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| [S3-400/8](https://prodrive-technologies.com/motion/products/servo-drives/cygnus-series/) | Prodrive | PWM | 1 | +/-10V | Encoder | 1kHz | 1e-4 |
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| [LWM7S](https://www.maccon.co.uk/linear-servo-amplifier.html) | Macon | Linear | 1 | | Encoder/Hall | | |
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| [Automation1 XL2e](https://www.aerotech.com/product/motion-control-platforms/automation1-xl4s-high-performance-voice-coil-drive/) | Aerotech | Linear | 1 | +/-10V 16bits | Encoder/Hall | 2.5kHz | |
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| [Automation1 XL4s](https://www.aerotech.com/product/motion-control-platforms/automation1-xl4s-high-performance-voice-coil-drive/) | Aerotech | Linear | 1 (voice coil) | +/-10V 16bits | ? | | |
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| [EM-356B](https://electromen.com/en/products/item/motor-controllers/brushless-dc-motor/EM-356B) | Electromen | PWM | 1 | 0-10V | Hall | | |
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| [azbh10a4](https://www.a-m-c.com/product/azbh10a4/) | AMC | PWM | 1 | +/-10V | Hall | | |
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| [X-MCC](https://www.zaber.com/products/controllers-joysticks/X-MCC) | Zaber | ?? | 1 to 4 | | | | |
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| [TA310](https://www.trustautomation.com/products/linear-drives/ta310-linear-drive/) | TrustAutomation | Linear | 1 | +/-10V | Hall | 5kHz | |
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<a id="table--tab:table-name"></a>
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<div class="table-caption">
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<span class="table-number"><a href="#table--tab:table-name">Table 2</a>:</span>
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Pure Drivers
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</div>
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| Model | Manufacturer | Linear / PWM | Axes | Interfaces | Current Bandwidth | Max Current | ASD at 1kHz [A/sqrt(Hz)] |
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|-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------|-----------------|--------------|------------------------|------------|-------------------|-------------|--------------------------|
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| [LA300](https://varedan.com/product/analog-linear-servo-amplifiers/la-300-analog-linear-servo-amplifier/) | Varedan | Linear | 3 | +/-10V | 10kHz | 4A | |
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| [CMAu10](https://www.cedrat-technologies.com/en/products/magnetic-controllers/oem-amplifiers.html) | Cedrat | Linear | 1 | +/-10V | 5kHz | 0.5A | |
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| [TA115](https://www.trustautomation.com/products/linear-drives/ta115-linear-drive/) and [TA105](https://www.trustautomation.com/products/linear-drives/ta105-linear-drive/) | TrustAutomation | Linear | 1 | +/-10V | 5kHz | | 1e-6 |
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| [SMA6520](https://www.glentek.com/shop/?swoof=1&product_cat=linear-brushless-series&really_curr_tax=21-product_cat) | Glentek | Linear | 1 Brushless (3 phases) | +/-10V | 10kHz | | |
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| [SMA5005](https://www.glentek.com/shop/?swoof=1&product_cat=linear-brush-series&really_curr_tax=21-product_cat) | Glentek | Linear | 1 | +/-10V | 10kHz | | |
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## Required properties {#required-properties}
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Main required properties are (taken from (<a href="#citeproc_bib_item_1">Schmidt, Schitter, and Rankers 2020</a>)):
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- **Power delivery capability**
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- **Dynamic properties**
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- **Linearity**
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- **Voltage or current drive**
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- **Efficiency**
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- **Four quadrant operation**
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## Four Quadrant Operation {#four-quadrant-operation}
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The self-inductance of an electromagnetic actuator also causes another problem when the actuator is driven with a period signal, because for a sinusoidal signal the current is out of phase with the voltage.
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In the extreme case of a purely reactive load, the maximum current needs to be delivered at zero voltage, while at a quarter of the period a positive current is delivered with a negative voltage and another quarter it is just the other way around.
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In mechatronic positioning systems with a high moving mass, the real problem is caused by the kinetic energy that is involved.
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At acceleration, the motion voltage of the actuator increases in phase with the current and electric power is inserted in the system and converted into kinetic energy.
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The deceleration phase is however completely the opposite.
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While the motion voltage still has the same sign as during constant motion, the current needs to be reversed in order to reverse the energy flow.
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This means that the full amount of kinetic energy has to be absorbed by the amplifier.
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## How to size a linear drive? {#how-to-size-a-linear-drive}
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### Why it is important to properly choose a linear drive? {#why-it-is-important-to-properly-choose-a-linear-drive}
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From a TrustAutomation [white paper](https://www.trustautomation.com/resources/engineering-blog/how-to-size-a-linear-drive-for-precision-positioning-applications/):
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> The price you'll pay for the improved precision (i.e. thanks to the linear drive as compared to a PWM one) will mostly come in the form of heat.
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> Linear drive typically maintain small amounts of power inside the drive circuits, increasing heat.
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> **Excess voltage not needed by the motor is also dissipated as heat**.
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### Determine required currents and voltages {#determine-required-currents-and-voltages}
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In order to properly choose a linear amplifier, it is important to determine the voltage and torque that has to be generated.
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The required current is based on the force (resp. torque) constant \\(K\_f\\) and peak force (resp. torque).
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The required voltage is based on the back EMF constant \\(K\_u\\), peak velocity \\(v\_\text{peak}\\), peak current \\(I\_\text{peak}\\) and winding resistance \\(R\\).
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<div class="exampl">
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Consider a linear brushless motor with a force constant \\(K\_f\\) equal to 30 N/A, a BEMF constant \\(K\_u\\) equal to \\(18\\,\frac{Vrms}{m/s}\\) (i.e. \\(25\\,\frac{V}{m/s}\\)) and a electrical resistance \\(R\\) of \\(20\\,\Omega\\).
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The peak velocity \\(v\_\max\\) is 1 mm/s and the wanted applied peak force \\(F\_\text{peak}\\) is 50 N.
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The peak current required is:
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\\[ I\_\text{peak} = F\_\text{peak}/K\_f \\]
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And we obtain a peak current of 1.7 A.
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The peak voltage is:
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\\[ V\_\text{peak} = K\_u \cdot v\_\text{peak} + R \cdot I\_\text{peak} + V\_\text{margin} \\]
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With \\(V\_\text{margin}\\) of 10 V, we obtain \\(V\_\text{peak} = 45\\,V\\).
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From this simple calculation, it is possible to obtain the required capability of the amplifier.
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</div>
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### Determine safe operating area {#determine-safe-operating-area}
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There are two danger scenarios: a stalled motor and a dynamic stopping motion
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#### Stalled motor {#stalled-motor}
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Consider the voltage supply to the drive \\(V\_\text{supply}\\) and the peak current \\(V\_\text{peak}\\).
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Now suppose the motor is pushing against a hard stop, the power \\(W\_\text{drive}\\) that the drive must dissipate is equal to:
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\\[ W\_\text{drive} = I\_\text{peak} \cdot V\_\text{drive} \\]
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with:
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\\[ V\_\text{drive} = V\_\text{supply} - I\_\text{peak} R \\]
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<div class="exampl">
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For our current application, \\(V\_\text{supply} = 45\\,V\\), \\(R = 20\\,\Omega\\) and \\(I\_\text{peak} = 1.7\\,A\\) which gives:
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\\[ W\_\text{drive} = 19\\,W \\]
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Then, it should be checked that the amplifier can dissipate this amount of power.
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</div>
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#### Dynamic Stopping. {#dynamic-stopping-dot}
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With a linear drive, the kinetic energy is absorbed by the drive itself, but must be dissipated as heat.
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This energy must be added to the energy required by the drive to stop all motion.
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The kinetic energy \\(E\_K\\) is (expressed in Joules):
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\\[ E\_K = \frac{1}{2} m v\_\text{peak}^2 \\]
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with \\(m\\) the payload mass.
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During the linear deceleration phase, the power \\(W\_d\\) that has to be dissipate by the drive is:
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\\[ W\_d = \frac{E\_K}{t\_\text{dec}} \\]
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with \\(t\_\text{dec}\\) the deceleration time.
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<div class="exampl">
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Consider a mass of 5 kg with a peak velocity of 1 mm/s and a deceleration time of 0.1s, the power to be dissipated in the drive is:
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\\[ W\_d = 25\\,\mu W \\]
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which is quite negligible.
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If a velocity of 1 m/s is considered instead, we obtain \\(W\_d = 25\\,W\\).
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</div>
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### Matlab Script to size a linear drive {#matlab-script-to-size-a-linear-drive}
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```matlab
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%% Motor properties
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Kt = 28; % Force constant [N/A] or Torque constant [Nm/A]
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Ku = 28; % BEMF in [V/(m/s)] or in [V/(rad/s)]
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R = 8.5; % Winding resistance [Ohm]
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%% Motion property
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Fp = 100; % Peak force [N] or Peak torque [Nm]
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vp = 10e-3; % Peak Velocity [m/s] or peak rotation [rad/s]
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m = 5; % Mass of the payload [kg]
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td = 0.1; % Deceleration time [s]
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```
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```matlab
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%% Driver wanted properties
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V_margin = 10; % Power supply margin [V]
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Imax = Fp / Kt; % Peak current to be supplied by the driver [A]
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Vmax = vp * Ku + R * Imax + V_margin; % Peak voltage to be generated by the driver [V]
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```
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```text
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Imax = 3.6 [A], Vmax = 41 [V]
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```
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```matlab
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%% Stalled Motor
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W_stalled = Imax * (Vmax - Imax * R); % [W]
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```
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```text
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W_stalled = 37 [W]
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```
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```matlab
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%% Dynamic Stopping
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W_stop = 0.5*m*vp^2 / t_dec; % [W]
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```
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```text
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W_stop = 0.0025 [W]
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```
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## Estimation of the required current noise {#estimation-of-the-required-current-noise}
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### Voice Coil Actuator with flexible guiding {#voice-coil-actuator-with-flexible-guiding}
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```matlab
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%% Frequency vector used for the analysis
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freqs = logspace(0, 4, 1000); % [Hz]
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%% Motor properties
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Kt = 28; % Force constant [N/A]
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%% Amplifier Noise
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In = 1e-6.*ones(size(freqs)); % Current noise density [A/sqrt(Hz)]
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%% DAC Noise
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Vn = (20/2^20)^2/12*1e4*ones(size(freqs)); % DAC output noise in [V/sqrt(Hz)]
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Vn = 3e-8
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Gi = 0.2; % Amplifier Gain [A/V]
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%% Mechanical properties
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m = 200e-3; % Mobile mass [kg]
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k = 1e3; % Guiding stiffness [N/m]
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xi = 0.05; % Modal Damping
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```
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```matlab
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%% Transfer function from F [N] to x [m]
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Gx = 1/(m*s^2);
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```
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```matlab
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%% Transfer function from I [A] to x [m]
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x_asd_i = In.*abs(squeeze(freqresp(Gx*Kt, freqs, 'Hz')))';
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x_asd_v = Vn.*abs(squeeze(freqresp(Gx*Gi*Kt, freqs, 'Hz')))';
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%% Cumulative amplitude spectrum
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figure;
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tiledlayout(1, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
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nexttile();
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hold on;
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plot(freqs, sqrt(flip(-cumtrapz(flip(freqs), flip(x_asd_i.^2)))) , '-');
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plot(freqs, sqrt(flip(-cumtrapz(flip(freqs), flip(x_asd_v.^2)))) , '-');
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plot(freqs, ex , '-');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Magnitude'); xlabel('Frequency [Hz]');
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xlim([0, 1e3]);
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```
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### Approximate analytical formula {#approximate-analytical-formula}
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Parameters:
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- `Kt`: motor force constant in N/A
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- `In`: current noise density of the amplifier in \\(A/\sqrt{Hz}\\)
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- `m`: mass in kg
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- `fb`: the feedback bandwidth in Hz
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We have that the residual motion when the feedback controller is closed is approximately equal to:
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\begin{equation}
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\epsilon\_x = \sqrt{\int\_\infty^{f\_b} \left(\frac{K\_t I\_n}{m \omega^2}\right)^2 d\omega}
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\end{equation}
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\begin{equation}
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\epsilon\_x = \frac{K\_t I\_n}{m (2\pi)^2} \sqrt{\frac{1}{3 f\_b^3}}
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\end{equation}
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Therefore, this formula can be used to:
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-
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<!--listend-->
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```matlab
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%% Estimate the position stability from the current noise and system parameters
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m = 1; % [kg]
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In = 1e-6; % [A/sqrt(Hz)]
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Kt = 10; % [N/A]
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fb = 10; % [Hz]
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ex = In*Kt/m/(2*pi)^2*sqrt(1./(3*fb^3));
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```
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```text
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epsilon x = 4.6 [nm RMS]
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```
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-
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<!--listend-->
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```matlab
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%% Estimate the required current noise from the wanted position stability and the parameters of the system
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m = 1; % [kg]
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Kt = 10; % [N/A]
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fb = 50; % [Hz]
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ex = 1e-9; % [m RMS]
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In = ex*m*(2*pi)^2/Kt * sqrt(3*fb^3);
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```
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```text
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In = 2.4e-06 [A/sqrt(Hz)]
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```
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## Bibliography {#bibliography}
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<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
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<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2020. <i>The Design of High Performance Mechatronics - Third Revised Edition</i>. Ios Press.</div>
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</div>
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