Moku PID Controller User Manual

Moku PID Controller

وضاحتیں

• بند لوپ bandwidth: >100 kHz
• خصوصیات: Real-time configurable feedback controllers
• درخواستیں: Suitable for temperature and laser frequency stabilization
• اضافی Features: Embedded Oscilloscope and Data Logger

تعارف

The Moku PID (Proportional-Integral-Derivative) Controller features real-time configurable feedback controllers with a closed-loop bandwidth of >100 kHz. This enables each controller to be used in applications requiring both low and high feedback bandwidths, such as temperature and laser frequency stabilization. The PID Controller also comes withan embedded Oscilloscope and Data Logger to observe short- and long-term behavior of the controller. Below, we provide a guide to the underlying architecture of the instrument. We also include a general exampفوری آغاز گائیڈ میں اور ایک چھوٹی سی تعداد میں گہرائی سابقamples to showcase different ways to use the Moku’s PID Controller. These user manuals are tailored to the graphical interfaces available on macOS, Windows, iPadOS, and visionOS. If you’d prefer to automate your application, you can use Moku API; available for Python, MATLAB, LabVIEW, and more. Refer to the API Reference to get started. AI-powered help is available to aid both workflows. AI help is built into the Moku application, and provides fast, intelligent answers to your questions, whether you’re configuring instruments or troubleshooting setups. It draws from Moku manuals, the Liquid Instruments Knowledge Base, and more, so you can skip the datasheets and get straight to the solution.

مین مینو سے AI مدد تک رسائی حاصل کریں۔

Figure 1. PID Controller user interface showing the instrument block diagram (top), embedded Oscilloscope panel (bottom), and the Oscilloscope settings panels (bottom right)

For more information on the specifications for each Moku device, please refer to our Product Documentation, where you can find the Specifications and the PID Controller Datasheets.

فوری آغاز گائیڈ

Here we outline how to set up the Moku PID Controller and highlight a typical use case for the instrument. In this example, we incorporate the PID Controller into a feedback system. The measured signal is provided as Input 1, with a reference signal provided as Input 2. The output is sent to the actuator in the feedback system from Output1. In this case, the PID Controller is used as a simple proportional-integral (PI) controller, with no derivative term.

• مرحلہ 1: Configure the analog front-end settings for the signal inputs
  Set the analog front-end settings for the input. In this case, both Input1 and Input2 have a 50 Ω input impedance, 0 dB attenuation, and use DC coupling.
• مرحلہ 2: Configure the Control matrix
  اس میں سابقample, the matrix is chosen to be [1,-1;0,0]. This indicates the matrix takes the difference between the two inputs, the sensed and reference signal, and then gives it to the controller.
• مرحلہ 3: Configure the input/output offset
  Depending on the control loop settings, it is sometimes desirable to introduce a DC offset in the error signal calculation. For example, if the error signal at Input 1 has a DC offset of 10 mV, setting the input offset to –10 mV would compensate for it. Similar adjustments can be made by adding output offsets after the controller block.
• مرحلہ 4: جلد کو ترتیب دیںtage حدود
  In addition to the offsets, the user can also put voltage limits on each of the output ports. These limits ensure that excessive voltages are not applied to any component in the control system. For this example, the offsets are set to 0 with no limits on the output port.
• مرحلہ 5: Configure the PID Controller
  Now configure the response by selecting the PID block. Doing so opens an interactive window that displays the PID response as a function of frequency. The behavior of the PID Controller can then be changed by enabling/disabling the different terms and putting in the gain value for each term. This can be done by dragging the markers on the interactive graph and changing them as desired. For this example, the Derivative and Double Integrator are disabled with only the Integrator and Proportional gain active. The Proportional gain is at 0 dB, with the Integrator crossover frequency at 1 kHz.
  نوٹ: This step can be repeated multiple times to change the PID Controller behavior as required.
• مرحلہ 6: Observe signals on the Oscilloscope
  After the PID Controller is set, probe points can be used to observe the signals. Enable the probe points before the controller and at the controller output. Clicking on these probe points opens up the embedded Oscilloscope menu and displays the signal at that point in the chain. Please see the Oscilloscope manual for more details on its operation.
• مرحلہ 7: Enable the outputs.
  Once the Oscilloscope is set up to observe the signals, the output can be enabled. Click on the output icon to select between Off, 0 dB gain, and 14 dB gain. For this example، 0 dB سب سے چھوٹی رینج کے طور پر منتخب کیا گیا ہے۔

Figure 3. Using the embedded Oscilloscope to monitor signals before and after the controller.

• مرحلہ 8: Updating the PID Controller
  With the output enabled, the feedback system becomes closed. The embedded Oscilloscope is useful to observe the error and control signal. Using these probe points to monitor changes, the PID Controller can be tuned to optimize loop performance or maximize noise suppression.
  نوٹ: other Moku instruments, such as the Phasemeter and Time & Frequency Analyzer, can offer additional metrics to help quantify performance.

Figure 4. Tuning the PID Controller gains by observing the signals on the Oscilloscope.

آپریشن کا اصول

Moku’s PID Controller instrument provides an easy-to-use interface for tuning proportional, integral, and derivative gains in a feedback loop. The PID is implemented by cascading two PID controllers to yield the final output. This architecture enables features such as a double integrator or multiple-section frequency response in Advanced mode. The basic control structure is shown in the block diagram below.

Figure 5. Block diagram of the Moku PID Controller.

Both PIDA and PIDB have an identical structure. The behavior of the PID controller can be encapsulated by the time domain expression as

c t = Kpe t + KI∫ e t dt + KD dx t

Using a Laplace transform, this can be converted to the frequency domain as

C s = KPE s + KIE s s + KDE s s

PID controllers are commonly used in feedback systems as they are easy to use and implement. Conceptually, each path contributes a correction to the measured error between the input and the reference signal. The proportional term applies a correction based on the current error but cannot eliminate steady-state error. The integral term addresses this by accumulating the error signal over time, which helps stability by driving the steady-state error toward zero. To further improve performance, the derivative term responds to the rate of change of the error, which dampens rapid fluctuations that the proportional and integral terms might otherwise amplify. In practice, the PI configuration is widely used, as it offers low steady-state error while being simple to implement. The Moku PID Controller also provides the ability to set saturation on the integrator and derivative terms. These saturation levels allow the systems to have a finite gain at very low and very high frequencies. Limiting the integrator gain at low frequencies prevents long-term noise accumulation that could otherwise drive the system to its voltage limits. Similarly, setting saturation limits can avoid infinite gain for high-frequency noise in differentiators and thereby improve performance. While saturation limits improve stability and help during tuning, setting them too low can restrict the controller’s ability to correct errors, leading to poor steady-state performance. Please refer to the six-part app series for a deeper understanding of feedback systems and PID controllers.

• حصہ 1: Frequency-domain control: defining a transfer function
• حصہ 2: Feedback control: constructing feedback control loops
• حصہ 3: Stability and delays: assessing stability in feedback control loops
• حصہ 4: Loop shaping: frequency domain tuning
• حصہ 5: Understanding actuator saturation in control systems
• حصہ 6: PID Controllers: Frequency-Domain Models & Applications

آلے کا استعمال کرتے ہوئے

سگنل ان پٹ
The analog frontend settings for each input channel of the PID Controller can be individually configured. Click the icon to configure the input settings for the signal input.

Figure 6. Configuration of analog inputs on the PID Controller.

• AC اور DC ان پٹ کپلنگ کے درمیان منتخب کریں۔
• Select between 50 Ω and 1 MΩ input impedance (hardware dependent).
• Select an input attention.

کنٹرول میٹرکس

The control matrix combines, rescales, and redistributes the input signal to the two independent PID Controllers. The output vector is the product of the control matrix multiplied by the input vector.

شکل 7۔ بلاک ڈایاگرام اور پاتھ اسکیمیٹک میں میٹرکس کو کنٹرول کریں۔

جہاں Path1 = a × In1 + b × In2 اور Path2 = c × In1 + d × In2۔

The value of each element in the control matrix can be set between -20 to +20. The gain can be incremented by 0.1 when the absolute value is less than 10 and by 1 when the absolute value is between 10 and 20. Thus t,, he matrix can be used to add or subtract two input signals to instead utilize a differential or common mode input for the PID Controller..

پی آئی ڈی کنٹرولر
Each channel is equipped with an independent PID Controller, positioned after the Control matrix that combines inputs from a pair of channels. This configuration allows precise control over each channel’s feedback path following signal blending. If more than two channels are available, you can access the other channels by clicking the arrow at the top. Each Control matrix feeds two PID blocks, each of which, in tu, is connected to an output. The signal path is shown as a block diagram in the PID instrument. To configure the PID gains, the PID block can be selected and then operated either in Basic or Advanced Mode.

Figure 8. Accessing multiple PIDs on Moku: Pro.

بنیادی موڈ

The Basic mode of the PID Controller provides a simplistic way to change the PID gains.

Figure 9. Interface to access the Basic mode of the PID block.

1. Enable/Disable button for the corresponding gain parameter.
2. Field to observe or type in the numbers for each gain parameter.
3. Corresponding interactive PID response plot.
4. Markers on the plot indicate the enabled gain parameters.
5. Toggle between magnitude and phase graphs.
6. Increase/Decrease the Overall gain of the PID Controller.
7. Toggle between Basic and Advanced mode.
8. Close the PID block.

The gain fields of the different parameters are described below

Table 1. Parameters of the PID block

Quick PID configuration
In the Basic mode of the PID controller, users can change the Proportional, Integrator, and Differentiator without the need to open the block, as shown in the screenshot.

Figure 10. Accessing quick control on the PID block.

1. Enable/Disable button for Proportional (P), Integrator(I) and Derivative (D).
2. Field to observe and/or type in the numbers for each gain parameter-

اعلی درجے کی وضع
The Advanced mode in the PID Controller provides us with the flexibility to manually adjust the gain settings of the PID Controller. The user can access each gain parameter from two PID cascaded blocks – Section A and Section B. The combined response of the two sections is shown in the PID response plot.

Figure 11. Accessing the interface for Advanced mode on the PID Block.

1. Enable/Disable button to select the corresponding Section. Disabling any Section would ensure only the other Section is active. Disabling both Sections would result in a pass-through/ signal relay logic.
2. Enable/Disable the corresponding gain parameter in each Section.
3. Field to observe or type in the numbers for each gain parameter in dB or Hz.
4. Corresponding PID response plot.
5. Toggle between magnitude and phase graphs.
6. Close the PID block.

The gains of the different parameters are shown below

Table 2. Different Parameters of the PID section

نوٹ: Double integrators can be implemented in the Advanced mode by cascading the cascade of integrators in Section A and Section B.

Controller path settings
Other block diagram elements in the PID Controller include switches to enable/disable the signal in the processing path, offsets that can be applied to the input signal or the control signal, and applying voltage limits on the output channels.

Figure 12. PID Controller path settings.

1. Type in the Input offset before the Controller.
2. Open/close the input switch from the input signal to the Controller.
3. Open/close the output switch from the Controller to the output.
4. Type in the Output offset before it is generated as output.
5. Enable/Disable the Voltagای محدود کرنے والا۔
6. Type in the high and low voltage حدود.
7. Enable/Disable the output and set the output gain (if applicable).

آفسیٹس
A DC offset can be applied to the signal both before and after the controller. Input offsets can be added or subtracted from the measured process variable before it is fed to the PID block. These are used to correct for any sensor calibration errors or to handle known deviations from the error point. Output offsets are added to the output of the PID block before it’s sent to the actuator or system. These offsets are used to maintain operation in the system around a known nominal value, or when the actuator needs a default bias to operate.

سوئچ کرتا ہے۔
The switches can be used to engage or disengage the control loop. When the switches are open, the input switch feeds zeros to the controller while the output switch gives zeros to the output. Upon clicking the input switch and closing it, the input signal is again fed to the controller. Similarly, upon clicking the output switch, the controller signal is passed to the output signal path. Every time the switches are opened and closed, the Integrator and Differentiator registers in the PID Controller are cleared.

والیومtage حدود
والیومtage limits can be applied before the signals are generated from the output ports. These limits ensure the output is maintained at these voltage levels whenever the signal crosses the specified threshold. For example, consider a system that only works with positive voltages. An input offset would be useful to generate a zero-crossing error signal with an output offset to return it to a positive level. The voltage limits would be useful to ensure that the minimum voltage is always greater than zero.

ڈیٹا کا مشاہدہ

ایمبیڈڈ آسیلوسکوپ

شکل 13۔ پروب پوائنٹ سگنلز viewایمبیڈڈ آسیلوسکوپ میں ایڈ۔

ڈیٹا لاگ ان 

Figure 14. Embedded Data Logger in PID Controller.

The embedded Data Logger can stream over a network or save data to the onboard storage of our Moku. For details, refer to the Data Logger user manual. More streaming information is in our API Reference.

ڈیٹا برآمد کرنا
شیئر آئیکن پر کلک کرکے ڈیٹا ایکسپورٹ کریں۔ کسی بھی فعال پروب پوائنٹس کو لائیو ڈیٹا ایکسپورٹ یا لاگنگ میں پکڑا جائے گا۔ براہ راست اور لاگ شدہ ڈیٹا برآمد کرنے کے لیے بالترتیب ایمبیڈڈ اوسیلوسکوپ یا ڈیٹا لاگر کھولیں۔

لائیو ڈیٹا 

شکل 15۔ ڈیٹا ایکسپورٹ کرنے والا صارف انٹرفیس اور سیٹنگز۔

To save live data

1. Select the type of data to export
   • Traces Saves the trace data for all visible signal traces, in either a CSV or MATLAB format.
   • Screenshots: ve the app window as an image, in either a PNG or JPG format.
   • Settings saves the current instrument settings to a TXT file.
   • Measurements saves the active measurement values in either a CSV or MATLAB format.
   • High-res data ,the full memory depth of statistic values for all visible channels, in LI, CSV, HDF5, MAT, or NPY format.
2. ایکسپورٹ فارمیٹ منتخب کریں۔
3. منتخب کریں۔ Filename prefix for your export. This is defaulted to “MokuPIDControllerData” and can be changed to any fileحروف عددی حروف اور انڈر سکور کا نام۔ ایک ٹائمسٹamp اور ڈیٹا فارمیٹ کو اس بات کو یقینی بنانے کے لیے سابقہ ​​میں شامل کر دیا جائے گا۔ filename is unique. For example: “MokuPIDControllerData_YYYYMMDD_HHMMSS_Traces.csv”
4. کسی بھی متن کی بنیاد پر محفوظ کیے جانے کے لیے اضافی تبصرے درج کریں۔ file ہیڈر
5. اپنے مقامی کمپیوٹر پر برآمد کی منزل کا انتخاب کریں۔ اگر "میرا files" یا "Share" کو منتخب کیا جاتا ہے، جب ایکسپورٹ بٹن پر کلک کیا جاتا ہے تو درست مقام کا انتخاب کیا جاتا ہے۔ My کا استعمال کرتے ہوئے ایک ساتھ برآمد کی متعدد اقسام کو برآمد کیا جا سکتا ہے۔ Files اور شیئر کریں، لیکن ایک وقت میں کلپ بورڈ پر صرف ایک برآمدی قسم برآمد کی جا سکتی ہے۔
6. ڈیٹا ایکسپورٹ کریں، یا
7. ایکسپورٹ کیے بغیر ایکسپورٹ ڈیٹا ونڈو کو بند کریں۔

لاگ ان ڈیٹا

تصویر 16۔ File exporting user interface and settings.

لاگ ان ڈیٹا کو بچانے کے لیے:

1. سبھی کو منتخب کریں۔ files logged to the device’s memory to download or convert.
2. منتخب کردہ کو حذف کریں۔ file/s
3. Browse and select file/s ڈاؤن لوڈ یا تبدیل کرنے کے لیے۔
4. Select an optional file تبادلوں کی شکل
5. Select a location to export your selected files سے
6. ڈیٹا ایکسپورٹ کریں۔
7. ایکسپورٹ کیے بغیر ایکسپورٹ ڈیٹا ونڈو کو بند کریں۔

Examples

Using PID in a feedback system
The Moku PID Controller can be directly incorporated into different feedback systems. A simple example involves using a PID controller to control the flow of fluid in a tank.

Figure 17. Block diagram of the water tank system.

Consider a simple block diagram of a tank system. The tank uses two valves to control the inflow and outflow of a fluid into the tank. A sensor is used to measure the fluid level in the tank and is given to the Moku as a voltage signal. The Moku PID Controller would then produce a signal to control the valves.

• مرحلہ 1: Configure the analog front-end settings for the signal inputs
  Set the analog front-end settings for the input. In this case, both inputs have a 50 Ω input impedance to match the source, -20 dB attenuation, and use DC coupling.
• مرحلہ 2: Configure the Control matrix
  Configure the Control matrix to take Input1 in control path 1, and Input1 in control path 2. As the same water level information is required for both systems, both control paths would use the same information. The matrix will take the values [1, 0; 1, 0].
• مرحلہ 3: Configure the input and output offsets
  The input offsets provide the reference set point. Depending on the valve, the height can be translated to a voltage using a scaling factor. This can then be used to generate the reference DC offset and thus create an error signal. Since the valves operate in unipolar mode, the output offsets need to ensure the signal is positive at all times. This can be reinforced by enabling the voltage limits to have a minimum of 0 V.

Figure 18. PID Controller interface for implementing feedback in the tank system.

• مرحلہ 4: Configure the PID block
  The PID Controller can be set to the desired configuration for operation. The optimal values can be analytically calculated by doing an open-loop analysis on the tank system. Alternatively, the control loop can be enabled at very low gains and slowly increase them until it becomes unstable.
• مرحلہ 5: Enable the outputs
  Once the PID blocks are configured, the outputs can be enabled. These outputs would be used to control the valve operation.
• مرحلہ 6: Observe the controller inputs and outputs
  Put probes on the input channels and at the outputs of the PID Controller.

اضافی ٹولز

مین مینو
The main menu can be accessed by clicking the icon in the top-left corner.

AI Help… Opens a window to chat to an AI trained to provide Moku-specific help (Ctrl/Cmd+F1)
میرے آلات returns to the device selection screen
سوئچ کریں۔ to another instrument
محفوظ کریں/یاد کرنے کی ترتیبات

• Save current instrument state (Ctrl/Cmd+S)
• Load last saved instrument state (Ctrl/Cmd+O)
• Show the current instrument settings, with the option to export the settings.

آلہ ری سیٹ کریں۔ to its default state (Ctrl/Cmd+R)
Sync Instrument slots in Multi-Instrument Mode*
بیرونی 10 MHz clock selection determines whether the internal 10 MHz clock is used.
گھڑی ملاوٹ کی ترتیب opens the clock blending configuration pop-up *
بجلی کی فراہمی access panel*
File مینیجر access tool
File تبدیل کرناr access tool
ترجیحات access tool
If available, use the current settings or device.

HelpThe 

• مائع آلات website opens in the default browser
• Shortcuts list (Ctrl/Cmd+H)
• Manual Open the user manual in your default browser (F1)
• Report an issue to the Liquid Instruments team
• Privacy Policy opens in the default browser
• Export diagnostics exports a diagnostics file you can send to the Liquid Instruments team for support.
• About the Show app version, check for updates or licence information

File کنورٹر 

دی File converter can be accessed from the main menu. The File converter converts a Moku binary (.li) format on the local computer to either .csv, .mat, .hdf5, or .npy format. The converted file اصل فولڈر میں اسی فولڈر میں محفوظ ہے۔ file.

تصویر 20۔ File Converter user interface.

بدلنا a file

1. منتخب کریں a file قسم
2. کھولنا a file (Ctrl/Cmd+O) یا فولڈر (Ctrl/Cmd+Shift+O) یا ڈریگ اور ڈراپ File کنورٹر کو تبدیل کرنے کے لئے file.

ترجیحات اور ترتیبات

The preferences panel can be accessed via the Main Menu. In here, you can reassign the color representations for each channel, switch between light and dark mode, etc. Throughout the manual, the default colors are used to present instrument features.

شکل 21. ڈیسک ٹاپ (a) اور آئی پیڈ (b) ایپ کے لیے ترجیحات اور ترتیبات۔

1. Change the App theme between dark and light mode.
2. Choose if a warning opens before closing any instrument windows.
3. Tap to change the color associated with the input channels.
4. Tap to change the color associated with the output channels.
5. Tap to change the color associated with the math channel.
6. Select if instruments open with the last used setting  r default values each time.
7. Clear all auto-saved settings and reset them to their defaults.
8. محفوظ کریں اور ترتیبات کو لاگو کریں۔
9. Reset all application preferences to their default state.
10. Notify when a new version of the app is available. Your device must be connected to the internet to check for updates.
11. دائروں کے ساتھ اسکرین پر ٹچ پوائنٹس کی نشاندہی کریں۔ یہ مظاہروں کے لیے مفید ہو سکتا ہے۔
12. Open information about the installed Moku application and license.

بیرونی حوالہ گھڑی

Your Moku may support the use of an external reference clock, which allows Moku to synchronize with multiple Moku devices, other lab equipment, lock to a more stable timing reference, or integrate with laboratory standards. The reference clock input and output are on the rear panel of the device. Each external reference option is hardware-dependent. Review آپ کے موکو کے لیے دستیاب بیرونی حوالہ جات کے اختیارات۔

حوالہ ان پٹ: Accepts a clock signal from an external source, such as another Moku, a laboratory frequency standard, or an atomic reference (for example، ایک روبیڈیم گھڑی یا GPS نظم و ضبط والا آسکیلیٹر)۔

حوالہ آؤٹ پٹ: Supplies the Moku internal reference clock to other equipment that requires synchronization.

If your signal is lost or is out of frequency, your Moku will revert to using its own internal clock until the reference signal returns. If this occurs, check that the source is enabled and that the correct impedance, amplitude, tolerance, frequency, and modulation are attached to the reference. Check the required specifications in the device specsheets. When the reference returns within range, sthe status changes to “validating” and then “valid” once the lock is re-established.

10 میگاہرٹز بیرونی حوالہ

To use the 10 MHz external reference function, ensure “always use internal” is disabled in the Moku application, found in the main menu under “External 10 MHz clock”. Then, when an external signal is applied to your Moku reference input and your Moku has locked to it, a pop-up will show in the app. On some devices, the external reference information will be shown in the LED status as well. More information can be found in your Moku Quick Start Guide.

شکل 22۔ "ہمیشہ اندرونی استعمال کریں" حوالہ کے ساتھ موکو مین مینو کو غیر فعال اور بیرونی حوالہ استعمال کرنا۔

گھڑی ملاوٹ کی ترتیب

If available, Moku blends up to four clock sources simultaneously for more accurate phase, frequency, and interval measurements across all time scales. A low-phase-noise Voltage-Controlled Crystal Oscillator (VCXO) is blended with a 1 ppb Oven-Controlled Crystal Oscillator (OCXO) for optimal wide-band phase noise and stability, which can be blended further with an external frequency reference and GPS disciplining to synchronize Moku with your lab and UTC. The VCXO and OCXO will always be used for the clock generation signal. The external and 1 pps references are optional and can be enabled or disabled in the “Clock blending configuration…” settings from the main menu. The loop bands are adjusted based on the different possible k source configurations, shown in Figure 23, where the frequencies of the bands represent where each oscillator’s phase noise dominates. Read how the clock blending works on Mok:  DD e lta for more details.

شکل 23. بیرونی 10 میگاہرٹز فریکوئنسی حوالہ اور GNSS فعال کے ساتھ موکو کلاک بلینڈنگ کنفیگریشن ڈائیلاگ۔

1. VCXO jitter reference is always used for clock generation, handling high-frequency jitter with the lowest noise.
2. OCXO jitter reference is always used for clock generation, ensuring moderate-term stability.
3. بیرونی 10/100 MHz فریکوئنسی حوالہ مقامی آسکیلیٹر میں بڑھنے کو درست کرنے کے لیے "10 MHz" یا "100 MHz" بیرونی حوالہ استعمال کرتا ہے، یہ نوٹ کرتے ہوئے کہ آپ کے Moku کو 10 MHz اور 100 MHz سورس کے درمیان ہر تبدیلی کے بعد دوبارہ شروع کرنا ہوگا۔
4.  1 پی پی ایس ہم وقت سازی کا حوالہ UTC کے ساتھ مطابقت پذیر ہونے اور مقامی آسکیلیٹر میں درست بڑھنے کے لیے ایک "بیرونی" یا "GNSS" حوالہ استعمال کرتا ہے۔ تخمینہ شدہ گھڑی کا استحکام اس بات کا ایک پیمانہ ہے کہ حوالہ کی کارکردگی مقامی OCXO/VCXO ٹائم بیس (جیسا کہ فی الحال ملایا گیا ہے اور، اگر فعال ہے، بیرونی 10/100 میگاہرٹز بیرونی حوالہ سے چلایا جاتا ہے) کے مقابلے میں کتنا انحراف ہوتا ہے۔

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