--- title: "Engineering Math, Power & Programming Articles: PID, Laplace, Matrices, Python | Ajarn Mod" url: https://modsengineeringmath.com/en/blog/ site: "Mod's Engineering Mathematics (อาจารย์มด)" language: en updated: 2026-10-25 --- > English articles from Ajarn Mod in six categories: sequences and series, vectors and linear algebra, the Laplace transform, control and PID, power and motors, and programming in the AI era. Markdown copy of https://modsengineeringmath.com/en/blog/ for AI assistants and search tools. Math is LaTeX ($…$ and $$…$$); interactive labs and figures are only on the web page. # Engineering Math, Power & Programming Articles: PID, Laplace, Matrices, Python | Ajarn Mod THภาษาไทยENEnglish // Articles · 24 in English · 6 categories Each short article answers one question that students and engineers search for again and again: what the Laplace transform is, how to invert it, how to correct a low power factor, or which programming language to start with in the AI era. Read one through and you can do the work yourself. The complete PID tuning guide is the long one, a full PID Academy lesson. Every article links to the full lessons, interactive labs and reference formulas in the library. Pick a category below. ✎All 24ΣSequences & Series 7[A]Vectors & Linear Algebra 7ℒLaplace Transform 3PIDControl Systems & PID 23φPower Systems & Motors 4Programming & AI 1 ## Latest articles Python · AI Learning to Program in the AI Era: Why Start with Python AI writes code in any language, but if you cannot read code you cannot tell whether it is right. Why Python is the best first language, six core… 25 Oct 2026Read the article → Linear algebra Cramer's Rule: How to Use It, Why It Works, and When Not to Use It Cramer's rule explained: replace a column by the constants and divide determinants, 2×2 and 3×3 examples, one unknown, a two-line proof, the Δ = 0… 25 Oct 2026Read the article → Linear algebra 12 Determinant Properties That Appear on Every Exam: det(kA), det(A⁻¹), det(adj A) and the Traps The 12 determinant properties used in entrance exams and engineering mathematics: row operations, det(kA) = kⁿ det A, products, inverses and the… 25 Oct 2026Read the article → ## Articles by category ## ΣSequences & Series 7 articles · Open the category → Short articles that answer one question about sequences and series at a time, from the geometric series formula and a flowchart for choosing a convergence test to the Taylor and Laurent series used in engineering. Each has worked numbers and links to the full lessons. Sequences & series How to Choose a Convergence Test: an 8-Step Flowchart with an Example at Every Step Pick the right convergence test the first time: an 8-step flowchart (n-th term, geometric, p-series, telescoping, limit comparison, ratio, root,… 24 Oct 2026Read the article → Sequences & series What Is a Laurent Series and How Is It Different from a Taylor Series? (Explained with Pictures) Laurent series vs Taylor series: negative powers, the principal part, the annulus of convergence, types of singularity, residues and the link to… 24 Oct 2026Read the article → Sequences & series How to Find the Radius of Convergence: 3 Methods, Endpoint Checks and the Complex-Plane Reason Three ways to find the radius of convergence of a power series (ratio test, root test, nearest complex singularity), plus the endpoint checks for… 24 Oct 2026Read the article → Sequences & series What Do You Learn in Sequences and Series? A Roadmap from Sequences to Laurent Series The whole sequences and series chapter on one page: what each topic answers, its key formula and typical exam questions, from sequences and… 24 Oct 2026Read the article → Sequences & series Geometric Series: Formula, Examples and How to Find the Sum Geometric series formula for the sum of n terms and the infinite sum a₁/(1 − r) when |r| < 1, with the derivation, step-by-step examples, common… 17 Oct 2026Read the article → Sequences & series Mathematical Induction Examples with Step-by-Step Solutions, Including Strong Induction 14 worked proof-by-induction examples: writing the induction statement P(n), sums, divisibility, inequalities, recurrences and strong induction,… 17 Oct 2026Read the article → Sequences & series What Is a Taylor Series, What Is It Used For, with Examples What a Taylor series is, the Maclaurin series table to memorise, and 8 worked examples: approximation, expanding about other points, how many… 17 Oct 2026Read the article → ## [A]Vectors & Linear Algebra 7 articles · Open the category → Vectors and matrices are the common language of circuits, forces and control systems. The articles in this category answer the questions people search for most: how to find a determinant and an inverse, what rank tells you, and which method to use for a large linear system. Linear algebra Cramer's Rule: How to Use It, Why It Works, and When Not to Use It Cramer's rule explained: replace a column by the constants and divide determinants, 2×2 and 3×3 examples, one unknown, a two-line proof, the Δ = 0… 25 Oct 2026Read the article → Linear algebra 12 Determinant Properties That Appear on Every Exam: det(kA), det(A⁻¹), det(adj A) and the Traps The 12 determinant properties used in entrance exams and engineering mathematics: row operations, det(kA) = kⁿ det A, products, inverses and the… 25 Oct 2026Read the article → Numerical methods Solving Linear Systems Numerically: Gauss, LU, Jacobi or Gauss–Seidel — Which Should You Use? Numerical methods for linear systems compared: Gaussian elimination with pivoting, LU, Cholesky, Jacobi, Gauss–Seidel, SOR, convergence and the… 25 Oct 2026Read the article → Linear algebra Learning Matrices and Determinants: A 7-Step Roadmap from High School and Entrance Exams to Engineering Mathematics A roadmap for matrices, determinants, rank, Cramer's rule and numerical methods: what to learn first, how far entrance exams go, what university… 25 Oct 2026Read the article → Linear algebra What Is the Rank of a Matrix? How to Find It and What It Tells You About a Linear System The rank of a matrix explained visually: finding it by row reduction and by minors, using rank to decide whether Ax = b has one, infinitely many or… 25 Oct 2026Read the article → Linear algebra How to Find the Determinant and Inverse Matrix (2×2 and 3×3) with Examples Learn to find 2×2 and 3×3 determinants by cofactors and the inverse matrix with the adjugate formula, with numeric examples checked by multiplying… 16 Oct 2026Read the article → Vectors Dot Product vs Cross Product: Formulas, Meaning and Examples Compare the dot product (scalar product) and cross product (vector product): formulas, geometric meaning, calculation from components, and how they… 16 Oct 2026Read the article → ## ℒLaplace Transform 3 articles · Open the category → The Laplace transform turns the differential equations of circuits and control systems into algebra. This category starts with what the transform means and the table of pairs, and goes on to inverting it with partial fractions. Laplace How to Find the Inverse Laplace Transform: 3 Partial-Fraction Cases with Examples Invert the Laplace transform with partial fractions for distinct, repeated and complex poles, with checked examples and what to do when the… 16 Oct 2026Read the article → Laplace Laplace Transform Table: The Most Commonly Used Pairs The 15 most common Laplace transform pairs and a table of properties (time shift, frequency shift, derivative, integral, convolution, initial and… 16 Oct 2026Read the article → Laplace What Is the Laplace Transform and What Is It Used For? The Laplace transform turns differential equations into algebraic equations. See the definition, the 3-step method, a worked y′ + 3y = 6 example… 16 Oct 2026Read the article → ## PIDControl Systems & PID 2 articles · Open the category → A PID controller sits inside almost every machine. This category explains what the P, I and D terms do and teaches real tuning step by step on a checked plant model. Control systems PID Tuning: The Complete Guide from Step Test to Ziegler–Nichols, SIMC and Anti-Windup A full PID tuning course in one article: identify the process with a step test, tune with Ziegler–Nichols, Cohen–Coon, SIMC or relay, compare the… 16 Oct 2026Read the article → Control systems What Is a PID Controller? How Kp, Ki and Kd Work and How to Tune Them A PID controller adds proportional, integral and derivative action on the error. See what Kp, Ki and Kd do, the step-response table, a… 16 Oct 2026Read the article → ## 3φPower Systems & Motors 4 articles · Open the category → Electrical work in factories and buildings, from power factor and the penalty on the bill to choosing a capacitor and connecting and starting three-phase motors. Every article uses the numbers of a 230/400 V 50 Hz system. Motors · research How Many Seconds Should a Star–Delta Timer Be Set To? Numerical Experiments on a Squirrel-Cage Induction Motor: Changeover Time, Dead Time, Current and Energy How many seconds for a star–delta timer? Research simulating a 7.5 kW squirrel-cage motor: t = c·τm, a c table by load, dead time, KM3 phase order,… 19 Oct 2026Read the article → 3-phase motors 3-Phase Motor Delta (Δ) Connection: Remember U1–W2, V1–U2, W1–V2 and Choose from the Nameplate How to connect a 3-phase motor in delta: links U1-W2, V1-U2, W1-V2, what U1 V1 W1 U2 V2 W2 mean and which wire colours go where, line vs phase… 17 Oct 2026Read the article → Power factor How to Calculate the Size of a Power Factor Correction Capacitor (kvar) with an Example The formula Qc = P(tan φ1 − tan φ2) with a 100 kW, 400 V load corrected from PF 0.75 to 0.95 giving 55.3 kvar, microfarads per phase, a multiplier… 16 Oct 2026Read the article → Power factor What Is Power Factor? The cos φ Formula and the Relationship between kW, kVA and kvar Power factor is the ratio of real power to apparent power. See the P Q S power triangle, a worked current calculation at PF 0.8 versus 1.0, lagging… 16 Oct 2026Read the article → ## Programming & AI 1 article · Open the category → Now that AI can write code in any language, the skills that still matter are thinking in steps, reading code and checking results against engineering knowledge. This category starts with Python and connects it to circuits, numerical computing and PLCs. Python · AI Learning to Program in the AI Era: Why Start with Python AI writes code in any language, but if you cannot read code you cannot tell whether it is right. Why Python is the best first language, six core… 25 Oct 2026Read the article → // Keep reading ## Want to go deeper than an article? An article is a shortcut for one specific question. For the full background, derivations and labs where you can change the values, open the library. The lessons are written in Thai with English technical terms, and the formulas are the same as in these articles. The motor control and PLC courses are also available in full English editions. English lessons → Thai library Laplace transform (Thai) Vectors (Thai) Power factor (Thai) Linear algebra (Thai) Control systems (Thai)