Lab 7: Laplace Matlab

 MATLAB code

 

Lab 7: Laplace Matlab 

 

Introduction 

The following lab deals with Laplace transforms and how it pertains to circuit analysis. Laplace transforms are a mathematical tool used in the field of signals and systems to analyze and manipulate complex functions, particularly those involving time-dependent signals or systems. They transform a function of time into a function of a complex variable, s, making it easier to solve differential equations and study the behavior of systems in the frequency domain. This transformation simplifies the analysis of linear time-invariant systems and facilitates the calculation of system response, transfer functions, and stability properties. It is a fundamental technique in engineering and physics for understanding the behavior of dynamic systems. Laplace transforms are invaluable in circuit analysis as they convert complex differential equations into simpler algebraic expressions, facilitating the study of electrical circuits' responses to different inputs. They enable analysis of transient and steady-state behavior, derivation of transfer functions, handling initial conditions, and exploring frequency responses. Essentially, Laplace transforms streamline the analysis and design of electrical circuits under various conditions and input signals.   

Procedure 

Section 1.1. 

Given a circuit, it is to be transformed into the Laplace domain. The instructions and procedure to reaching the final answer are detailed below. For an efficient understanding it is beneficial to follow along in the step-by-step procedure. 

 

Figure 1: Given problem with associated given circuit diagram 

The following circuit is the given diagram from the instructions in lab, reading the problem statement, the reader may understand the rest of the procedure. 

 

Figure 2: Laplace domain circuit 

The following figure represents a Laplace domain circuit which is a transformed version of figure 1. After following the initial instructions, this circuit should be the result. 

Section 1.2. 

Accurate Impedance Matrix. Its composition may vary depending on the sequence in which loops are arranged. This determination is made in reference to the circuit's manual calculation, which involves visually plotting the circuit. 

 

Figure 3: Shown work for matrix definition. 

The following figure displays shown work from using KCL to develop a matrix to be used in MATLAB. 

 

Figure 4: Resultant matrix solution. 

The following figure shows the solution of the initial instructions. These results will be utilized in the next sections and in the MATLAB code. 

Section 2. 

After initializing the symbolic variable 's' and computing the mesh analysis matrix, you can express the output voltage in MATLAB as the equation above (refer to line 45 of this code), this equation is derived through the simplification of the previously mentioned Impedance Matrix 'I'. Although the 'VOUT' equation is correct in its original form, it can be somewhat unwieldy. To improve its clarity, you can apply the 'simplify' function in MATLAB to 'VOUT' once again. This step will result in a more streamlined representation of both the numerator and denominator. Subsequently, the 'numden()' function in MATLAB can be employed to extract two variables, 'N' and 'D', housing the simplified values for the numerator (N) and denominator (D). 

Section 3. 

This segment of the MATLAB code showcases the N and D variables, making it possible to scrutinize their accuracy. When these values are displayed with four significant figures, it becomes apparent that they are exceptionally large which will find a solution. 

Section 4. 

Employing the symbolic toolbox earlier facilitated the execution of this script to some extent. Nonetheless, a challenge arises when attempting more sophisticated modeling. To address this, the polynomial obtained previously is passed through the 'sym2poly()' function in MATLAB, converting the symbolic variables 'N' and 'D' into arrays 'NP' and 'DP'. The 'sym2poly()' function extracts all numerical coefficients from the symbolic polynomials, simplifying the process of future modeling. 

Section 5. 

The subsequent phase in this procedure involves streamlining the expression derived from the previous function. To achieve this, both 'NP' and 'DP' are divided by a common factor, denoted as 'k'. This 'k' value is employed to ascertain the maximum values within 'NP' and 'DP'. Utilizing the updated 'NP' and 'DP' values, the new polynomial coefficients are computed. During testing, the value of 'NP(1)' was evaluated in the command window, revealing an exceedingly small number, nearly indistinguishable from zero. 

Section 6. 

After successfully executing all the preceding steps in the script, the next task was to utilize the 'tf()' function in MATLAB. This function is employed to generate real/complex-valued transfer function models, which represent the frequency domain characteristics of a Linear Time-Invariant (LTI) system. By utilizing this function, one can directly specify the coefficients to create a continuous-time transfer function model, assigned the variable name 'H'. Subsequently, the function enables the plotting of three distinct figures: a Bode plot, a step response, and the poles and zeros of the transfer function. It's noteworthy that in the Bode Plot, the magnitude, expressed in decibels (dB), tends to approach zero in a positive direction as the frequency increases. Additionally, the phase, measured in degrees, tends to approach zero as the frequency decreases, albeit with a negative slope 

 

Figure 5: Pole-Zero Map 

The following figure shows the pole-zero map resultant from following the above instructions. Conclusions can be made from observations. 

 

Figure 6: Bode Diagram 

The following figure displays a Bode Diagram. The Bode Diagram is of the transfer function. 

 

Figure 7: Step Response 

 The following figure displays a Step Response. The Step is of the transfer function. Proper following of the instructions as well as an understanding of Laplace Transforms. 

 

  

Conclusion  

The lab was enjoyable as it dealt with circuit analysis. The lab could be improved with more proper preperation as well as more step-by-step instruction which often made me feel very lost. As circuit analysis with laplace transforms was learned only a few days before, I did not feel adqequetly prepared. Laplace transforms are crucial in circuit analysis because they enable engineers and scientists to analyze complex electrical circuits and systems in the frequency domain. This mathematical technique simplifies the analysis of linear time-invariant (LTI) systems, making it easier to understand their behavior under different input signals and conditions. By transforming the time-domain circuit equations into the Laplace domain, engineers can use algebraic methods to solve for voltage and current responses, making it a powerful tool for designing, optimizing, and troubleshooting electronic circuits and systems.