Lab 6: System Linearity TIMS
Lab 6: System Linearity TIMS
Introduction
The following lab deals with system linearity. System linearity is a characteristic of systems where they adhere to two main properties: homogeneity and additivity. A linear system will produce an output scaled by the same factor as its input (homogeneity) and will respond to combined inputs in a manner equal to the sum of its responses to each input separately (additivity). This property simplifies the analysis and understanding of systems, especially in engineering and signal processing. Linear systems can be easily understood, predicted, and modeled using a set of well-established mathematical tools. Additionally, their behavior can be extrapolated from a few observations, making them more manageable and predictable compared to non-linear systems.
Procedure
A.1 Comparator Examination
According to the provided data sheet, a comparator will transform any analog signal into a squared form and produce a standard TTL level output.
1. Set up the VCO and UTILITIES modules.
- Position the VCO module in Slot 3 and the UTILITIES module in Slot 7.
- Link the VCO's sin(t) output to the input “A” of the BUFFER AMPLIFIER and to the FREQUENCY COUNTER's input. Note: The BUFFER AMPLIFIER module can be found on the TIMS unit's lower permanent rack.
- Attach the BUFFER AMPLIFIER's output (k1A) to the COMPARATOR input of the UTILITIES module and to Scope ChA .
- Connect the CLIPPER output from the UTILITIES to Scope ChB using a green lead.
2. Launch PicoScope:
- Activate ChB, setting it to AUTO.
- Both channels should be DC-coupled.
- Adjust the time base to 500us/div.
- Enable the TRIGGER, setting it to Auto mode.
3. Using the VCO, tweak the f0 knob to get a frequency close to 1000 Hz:
- While monitoring the ChA signal on PicoScope, calibrate the BUFFER AMPLIFIER's gain k1 to reach a 1Vpp signal.
Figure 1: comparator output voltage
response to changing input peak to peak voltage. Input: grey, output:orange.
The comparator system output is linear as it begins at 0 and it expresses the properties of super position and homogeneity. In essence, superposition and homogeneity describe how linear systems consistently respond to combinations and scalings of inputs, respectively.
- Gradually raise the input voltage in steps of 1Vpp for data gathering. Remember, Vpp signifies the voltage range from the peak high to the peak low."
A.2 Exploring the Rectifier
A rectifier primarily captures the positive segment of an input signal.
1. Adjust the previous TIMS setup to focus on the rectifier's operations rather than the comparator.
- The BUFFER AMPLIFIER's output, k1A, is now linked to the rectifier input on the UTILITIES module and to Scope ChA.
- Scope ChB is joined to the UTILITIES module's rectifier output.
2. Re-implement step 3 from section A.1, setting the input signal back to a 1Vpp sinusoid at 1000Hz.
- The PicoScope configurations from step 2 should remain appropriate.
- Observe that the peak-to-peak output of the rectifier is roughly equal to the output's height, as the negative segment is omitted.
The peak-to-peak output of the rectifier is included in the graph above, it is colored orange.
3. Scrutinize the output of the rectifier.
I believe this output is still linear as it expresses the needs stated above. In a linear system, superposition is defined as expressing the output resulting from a mix of multiple inputs is the same mix of the outputs from those inputs individually. In a linear system, homogeneity is expressed as multiplying an input by a constant factor leads to the output being multiplied by that same factor. Essentially, superposition explains how linear systems react to combined inputs, while homogeneity shows how they react when inputs are scaled or magnified.
Remember: The peak-to-peak output value measures from the signal trace's highest to its lowest point.
A.3 Understanding the Multiplier
Linear systems should exhibit "additivity." Consider a linear system defined as y1(t) = Ax1(t) and y2(t) = Ax2(t). For a combined input signal x1(t) + x2(t), the output should be A(x1(t) + x2(t)). For a system that squares the input, with x(t) = Asin(t), the output becomes y(t) = A^2sin^2(t). Using the half-angle formula, this equates to ½ A^2(1-cos(2t)). Evidently, this system doesn't meet the additivity criteria, nor does it satisfy the scaling requirement, as we'll explore.
In this experiment, a sinusoid is created via the voltage controlled oscillator (VCO), its amplitude is controlled by the BUFFER AMPLIFIER, and the result is fed into both inputs of the MULTIPLIER.
1. For alignment with the upcoming steps, it's advisable to start from scratch:
- Remove all connections and shut down the PicoScope.
2. Configure the TIMS rack:
- Ensure the VCO's onboard switch is toggled to "VCO" and not "FSK" before inserting it into the TIMS rack.
- Place the MULTIPLIER module into the TIMS rack.
3. Link the VCO output to the BUFFER AMPLIFIER input and to the FREQUENCY COUNTER (using red leads):
- Calibrate the VCO knob to yield 1 kHz.
4. Attach the BUFFER AMPLIFIER's output to both inputs of the MULTIPLIER and to Scope ChA (using yellow leads).
5. Connect the MULTIPLIER's output to Scope ChB (green lead).
6. Launch the PicoScope:
- Activate Channel B.
- Ensure both channels are DC-coupled.
- Stabilize the scope's output using the Auto Trigger setting.
- Adjust the time base to display a suitable number of cycles.
7. Modify the BUFFER AMPLIFIER's k1 knob to attain a 1Vpp input (equivalent to a 0.5V input amplitude) as displayed on the PicoScope.
8. Analyze the output signal. The signal's average or DC value should equate to its amplitude. Record this amplitude value.
Table 1: Output amplitude.
-Utilize the zoom and cursor tools on the PicoScope for precise measurements.
Figure 2: reproduction of figure 1 with table 2’s values: Output amplitude.
The system is linear.
The output is half the square amplitude of the input demonstrated in the figure above.
Part B – Exploring the VCO System
A voltage-controlled oscillator (VCO) produces a sinusoidal output whose frequency is determined by its input voltage. This section delves into the VCO's output behavior under varying control inputs.
B.1 DC Control
1. To ensure alignment between the steps and your actual observations, it's recommended to start anew:
- Detach all connections and shut down the PicoScope.
2. Arrange the TIMS rack for the DC-controlled VCO examination:
- Make sure the VCO's onboard switch is toggled to "VCO" and not "FSK", and then slot this module into the TIMS rack.
- The VCO's toggle should be selected as "LO".
- Position the VCO's gain and f0 knobs approximately at their midpoints.
- For further details, refer to the VCO module specification sheet, especially the section titled "Special VCO Operation".
3. Link the VARIABLE DC output to the VCO's input and to Scope ChB. The VARIABLE DC module, found at the TIMS unit's base, allows for consistent DC voltage adjustments between -2V and 2V using its knob.
4. Attach the VCO's output to both the FREQUENCY COUNTER and Scope ChA (using yellow leads).
5. Launch the PicoScope:
- Activate Channel B.
- Ensure both channels are DC-coupled.
- Use the Auto Trigger setting to stabilize the scope's output.
- Modify the time base to display an appropriate number of cycles.
6. Calibrate the DC output to -2V, as visualized on the PicoScope.
Table 2: DC control results.
7. Progressively adjust the DC output, recording your observations.
Figure 3: DC control results.
Horizontal axis: DC value (V),
vertical axis: Signal frequency (Hz)
The system is not linear.
B.2 Frequency Control with VCO
1. Start afresh by disconnecting everything and turning off the PicoScope.
2. Set up the AUDIO OSCILLATOR and VCO modules in the TIMS rack.
3. Link the AUDIO OSCILLATOR's output to the BUFFER AMPLIFIER and F REQUENCY COUNTER, then calibrate it to 300 Hz.
4. Connect the BUFFER AMPLIFIER's output to the VCO and Scope ChB, adjusting the VCO gain as needed.
5. Attach the VCO's output to Scope ChA.
6. Activate the PicoScope, turning on Channel B, and adjust settings for optimal signal visualization. Use Single Shot Triggering to see multiple scans.
An obvious application of a system that behaves as described above is radio systems; RF.
C – Feedback System Study
Laplace transforms can represent integrals, as introduced in Signals and Systems lectures. The Laplace V2 module contains three integrators, with each integration represented as "S-1". This lab first explores the integrator's operation and then incorporates it into a feedback system.
C.1 Investigating the Integrator
1. Begin with a clean slate by disconnecting everything and powering down the PicoScope.
2. Set up the study, incorporating the AUDIO OSCILLATOR, UTILITIES, and LAPLACE V2 modules into the TIMS rack.
3. Link the AUDIO OSCILLATOR's output to the UTILITIES COMPARATOR input and the FREQUENCY COUNTER. Adjust the AUDIO OSCILLATOR to yield 1 kHz.
4. Connect the UTILITIES output to LAPLACE V2's S1in and Scope ChB.
5. Attach LAPLACE V2's S1out to Scope ChA.
6. Launch the PicoScope, adjusting settings for optimal visualization and stabilizing the output."
C.2 Exploring the Feedback System
After understanding the LAPLACE V2 integrator, let's apply it to an ARB1 signal.
1. To align the upcoming steps with actual observations, start afresh by disconnecting everything and shutting down the PicoScope.
2. Set up the TIMS rack with the LAPLACE V2 and TRIPLE ADDER V2.
3. Observe the PC MODULES CONTROLLER ARB2 signal by connecting its output to Scope ChA.
4. Launch the PicoScope to view the ARB2 signal before activating the SFP.
5. Start the S&S V2 SFP application, select the "Lab 3" experiment tab, and press the "LOAD ARB" button.
6. Now, observe the signal on the PicoScope.
7. Zoom into the transition region as showcased in Figure 18b to observe the Gibbs Phenomena.
8. Ensure the TRIPLE ADDER settings on the SFP window are correctly set up, with the appropriate slot selection and gain settings.
9. Configure the TIMS feedback circuit. This involves various connections between the ARB2, TRIPLE ADDER, and LAPLACE V2 modules, leading to a comprehensive feedback loop.
10. Re-open the PicoScope with specific settings. Figure 23 then presents the input- output relationship of the system, showing the exponential decay in the output signal when the input appears as a step function.
The time taken for the exponential curve to decrease to e-1 of its top value is the time constant which is measured to be:
78.91us
11. Examine the output of the integrator in relation to the system's input by adjusting the Scope ChB cable.
12. Modify the scope connections to concurrently observe the input and output of the integrator. Review the input to the integrator, ensuring it resembles the impulse response associated with the step response seen at the integrator's output.
Conclusion
System linearity is crucial because it ensures predictable and proportional responses to inputs. Linear systems simplify analysis and design due to their inherent properties of superposition and scalability. When a system is linear, it can be easily understood, modeled, and controlled, leading to more reliable and efficient system designs and operations. Overall, the lab was enjoyable as the steps were easy to follow. The lab was also pleasant because receiving assistance from the GTA was helpful. Students who perform this lab should gain a better understanding of system linearity.
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