Lab 11: Complex Numbers Tims

 

MATLAB code:

rectangular to polar

polar to rectangular

 

 INTRODUCTION

The following labs specializes in complex numbers in TIMS which is a software used throughout the semester. Complex numbers are mathematical entities of the form a + bi, where "a" and "b" are real numbers, and "i" is the imaginary unit (defined as the square root of -1). They play a crucial role in signals and systems because they provide a convenient and powerful way to represent sinusoidal signals and manipulate their properties. The use of complex numbers simplifies mathematical operations, such as addition, subtraction, multiplication, and division, making it easier to analyze and describe the behavior of signals in both time and frequency domains.

PROCEDURE

PRELAB

To refresh the student a prelab assignment is required. Vectors converted to polar form and graphed in figure 1.

3+j4 = 5 angle 53 deg

3-j4 = 5 angle -53 deg

-3+4j = 5 angle 127 deg

-4-4j = 5 angle 233 deg

 

Figure 1: Complex numbers graphed. Note: horizontal axis = real axis.

 

LAB EXERCISES

Upon completion of the pre-lab, the TIMS machine and Picoscope are activated. This phase of the lab aims to illustrate the correlation between sine signals and complex signals. Utilizing the ARB and Triple Adder Modules of the TIMS rack, along with the SFP program, Lab 6 is loaded onto the ARB after adjusting ARB1 and ARB2 values. The resultant summed value is expressed in polar form as 1 + 𝑖, and in rectangular form as √2∠45. Figure 2 on the Picoscope displays the loaded ARB. Subsequently, the phase of ARB2 is altered from 90° to -90°. The new complex number is recorded in polar form as 1 − 𝑖, with the rectangular form described as √2∠−45.

Figure 2: Signal transient: 1+j.

Subsequently, the XY view is configured within the Picoscope to simultaneously display both the scope and XY views of the ARB sine signals. The XY plot takes the form of a circle, with the vertical distance from the center to the tip of the line representing the amplitude of the sine wave. Notably, the Vpp of the scope view equals the radius of the circle. ARB1 is then disengaged from Scope Channel A, revealing only ARB2 from Channel B, resulting in a displayed signal in the form of a line (Figure 3). This line corresponds to the polar form of the complex vector input to ARB1, 0 + 𝑖. Subsequently, Channel A is reconnected to ARB1, while Channel B is disconnected from Channel B, revealing once again a signal in the form of a straight line (Figure 4). This line corresponds to the polar form of the complex input of ARB2, 1 + 𝑖.

Figure 3: ARB1 displayed waveform.

Figure 4: ARB2 displayed waveform.

Following that, ARB2 is reconnected. The Triple Adder Module is then adjusted, setting the amplitude of ARB1 to 1 and the phase to -15°. Simultaneously, ARB2 is modified with an amplitude of 1.2 and a phase of 75°. The resulting waveforms are loaded into the Picoscope, and Figure 5 depicts the observed waveforms, hence a circle appears. These signals can be expressed as functions of time in the form 𝑥(𝑡) = 𝐴𝑐𝑜𝑠( 𝜔𝑡 + 𝜑). Specifically, for ARB1, it is written as 𝑥(𝑡) = 𝑐𝑜𝑠( 𝜔𝑜𝑡 − 15°), while for ARB2, it is expressed as 𝑥(𝑡) = 1.2𝑐𝑜𝑠( 𝜔𝑜𝑡 + 75°). The reference Triple Adder values are restored and loaded. With Channel A scope connected to ARB1 and the lead for ARB2 moved to the “f+g” output of the Triple Adder Module, the sum of the two sinusoidal waveforms is observed from Channel B, as illustrated in Figure 6.

 

Figure 5: XY plot displays a signal

Figure 6: Sine waveforms

MATLAB

MATLAB proficiently handles complex numbers using the variable 'j' or 'i,' albeit limited to rectangular form processing. To obtain the phasor representation, a function is employed, returning the angle in radians, which is subsequently converted to degrees. If there's a preference for inputting complex numbers into MATLAB using the polar form 𝑟 ∠ 𝜃, the entry method can be derived from Euler’s Identity. The relevant functions are provided in separate MATLAB files, attached to the report. 

LAB CONTINUED

Progressing with the laboratory work, the TIMS setup is restored, and the SFP settings are reverted to their reference values. Channel B of the scope is then linked to the “f+g” output of the Triple Adder Module. The settings for the ARBs undergo two variations in the SFP program, and the outcomes are observed, as depicted in Figures 7 and 8. The observed signals represent the sum of the individual signals, aligning with expectations. Both the XY and scope views exhibit a striking resemblance.

Figure 7: 180 deg phase, A=1.

Figure 8: -180 deg phase, A=1.2.

The subsequent aspect under examination in this laboratory session was the complex conjugate signals. If A can be expressed as 𝐴 = 𝑎 + 𝑗𝑏 = |𝐴|𝑒^𝑗𝛩, then the conjugate of the complex signal could be written as 𝐴^* = 𝐴 - 𝑗𝑏 = |𝐴|𝑒^−𝑗𝛩. The sum of these formulas simplifies to 2a. Building upon the setup from the previous step, this time the “Conjugate Phase” mode is toggled to “ON,” and the phase of ARB2 is set to 20°. The Picoscope is adjusted to measure the maximum amplitude of the “f+g” output of the Triple Adder Module, employing AC coupling to eliminate any DC signal. The resulting signal is observable in Figure 9. Once the original signal is observed, the phase of ARB2 is then adjusted at 30-degree intervals. The maximum amplitude values are recorded and documented in Table 2. With this table compiled, the laboratory session was concluded. Utilizing the data from Table 2, Figure 9 graphically represents the observed values.

 

Table 1: recorded channel B amplitude

 

 Figure 9: Degrees vs Max amplitude

The following figures represents information in a visual manner from the information provided.

CONCLUSION

In conclusion, the significance of complex numbers in signals and systems lies in their ability to succinctly represent and manipulate time-varying signals, especially sinusoidal waveforms, simplifying the analysis in both time and frequency domains. The TIMS software proves invaluable in enhancing our understanding of these concepts by providing a practical and interactive platform for signal processing experimentation. Through the utilization of the TIMS setup, we navigate through various scenarios, such as exploring phasor representations, observing the impact of complex conjugate signals, and analyzing the sum of signals with ease. The software allows for real-time adjustments and measurements, facilitating a hands-on learning experience that deepens our comprehension of the intricate relationships within signals and systems. Overall, the integration of complex numbers and the TIMS software not only streamlines complex signal analysis but also fosters a more intuitive understanding of the fundamental principles underlying signal processing and system behavior.