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| Subtopic |
Key Concepts |
Video Focus Areas |
| 1.1 |
Significant Figures, Accuracy
& Precision |
Significant figures rules.
Accuracy vs. Precision definitions and examples. |
| 1.2 |
Error Definitions &
Propagation |
Absolute error, Relative error,
Percentage error. True error vs. Approximate error. Error propagation in
arithmetic operations. |
| 1.3 |
Round-off & Truncation
Errors |
Round-off errors (finite
representation of numbers). Truncation errors (approximating infinite
processes). Machine epsilon. |
| Subtopic |
Key Concepts |
Video Focus Areas |
| 2.1 |
Gauss Elimination Method |
Forward elimination and Back
substitution. Augmented matrix. Pivoting (partial and full) to avoid division
by zero. |
| 2.2 |
Gauss-Jordan Elimination Method |
Gauss-Jordan method (elimination
on both sides). Finding the inverse of a matrix. |
| 2.3 |
LU Decomposition Method |
LU Decomposition (Doolittle's
and Crout's methods). Solving Ax = b using LU (Ly = b, Ux = y). |
| 2.4 |
Matrix Inversion Method |
Solving Ax = b using A⁻¹.
Computational cost of matrix inversion. |
| 2.5 |
Jacobi Iterative Method |
Iterative method for solving
linear systems. Condition for convergence (diagonally dominant). |
| 2.6 |
Gauss-Seidel Iterative Method |
Gauss-Seidel method (using
updated values immediately). Faster convergence than Jacobi. |
| 2.7 |
Application: AC Ladder Circuit
using Mesh Currents |
Formulating mesh current
equations for an AC ladder circuit. Solving using iterative methods. |
| 2.8 |
Application: AC Circuit with
Double T using Nodal Analysis |
Formulating nodal equations for
a double-T circuit. Solving using Gauss-Seidel. |
| Subtopic |
Key Concepts |
Video Focus Areas |
| 3.1 |
Bisection Method |
Bracketing method. Intermediate
Value Theorem. Interval halving. Convergence (linear). |
| 3.2 |
False Position Method (Regula
Falsi) |
Bracketing method using secant
line interpolation. Faster than bisection. |
| 3.3 |
Secant Method |
Open method (no bracketing
required). Uses two initial guesses. |
| 3.4 |
Newton-Raphson Method |
Open method using derivative.
Quadratic convergence. |
| 3.5 |
Fixed Point Iteration Method |
Rewriting f(x)=0 as x = g(x).
Convergence condition |
| 3.6 |
Rate of Convergence |
Definition of convergence order.
Linear, Super-linear, Quadratic convergence. |
| 3.7 |
Applications in Electrical
Engineering |
Diode circuits (finding
operating point). Power electronics (finding switching angles). |
| Subtopic |
Key Concepts |
Video Focus Areas |
| 4.1 |
Initial Value Problem (IVP)
& Solution Methods |
Definition of IVP: dy/dx =
f(x,y), y(x₀) = y₀. Need for numerical methods. Classification of methods. |
| 4.2 |
Euler Method (One-Step) |
Euler's method: y_{n+1} = y_n +
h f(x_n, y_n). Local truncation error (O(h²)). Global error (O(h)). |
| 4.3 |
Modified Euler Method (Heun's
Method) |
Predictor-corrector method.
Euler predictor + Trapezoidal corrector. Improved accuracy (O(h²) local
error). |
| 4.4 |
Trapezoidal Rule for ODE |
Trapezoidal method: y_{n+1} =
y_n + (h/2)(f(x_n, y_n) + f(x_{n+1}, y_{n+1})). Implicit method. |
| 4.5 |
Runge-Kutta Method (RK4) |
RK4: The classic 4th-order
method. Use of 4 slopes to get a more accurate estimate. |
| 4.6 |
Application: Unsymmetrical
Voltage Doubler |
Formulating the differential
equations for the voltage doubler circuit. Solving using RK4. |
| 4.7 |
Application: Full-Wave Rectifier
with LC Filter |
Formulating the ODEs for the
rectifier-filter circuit. Solving using RK4. |
| 4.8 |
Comparison of ODE Methods &
Stability |
Stability analysis of Euler,
Heun, RK4. Stiff differential equations. |
| Subtopic |
Key Concepts |
Video Focus Areas |
| 5.1 |
Introduction to FEM |
Need for FEM. Fundamental
concept: Discretization. Applications in electrical engineering
(magnetostatics, heat transfer). |
| 5.2 |
FEM Procedure: Discretization |
Meshing the domain. Types of
elements (1D, 2D triangular, 2D quadrilateral). Nodes and elements. |
| 5.3 |
Element Governing Equations |
Deriving element equations using
the Galerkin method. Shape functions (interpolation functions). Element
stiffness matrix. |
| 5.4 |
Assembling all the Elements |
Assembly of global stiffness
matrix from element matrices. Applying boundary conditions (Dirichlet and
Neumann). |
| 5.5 |
Solving the Resulting Equations |
Solving the global system (K u =
f). Post-processing: Computing field values and derived quantities. |
| 5.6 |
Example: Magnetostatic Fields |
Applying FEM to solve
magnetostatic field problems (Poisson's equation). Potential distribution in
a capacitor. |
| Subtopic |
Key Concepts |
Video Focus Areas |
| 6.1 |
Introduction to Optimization |
Need for optimization.
Mathematical formulation: Objective function, Design variables, Constraints. |
| 6.2 |
Continuous vs. Discrete
Optimization |
Continuous variables (real
values) vs. Discrete variables (integer values). Mixed-integer programming. |
| 6.3 |
Constrained vs. Unconstrained
Optimization |
Unconstrained optimization (no
restrictions on variables). Constrained optimization (equality/inequality
constraints). |
| 6.4 |
Global vs. Local Optimization |
Local minimum vs. Global
minimum. Convex vs. Non-convex problems. |
| 6.5 |
Stochastic vs. Deterministic
Optimization |
Deterministic methods
(guaranteed convergence to local optimum). Stochastic methods (random search,
avoids local minima, e.g., Genetic Algorithms). |
| 6.6 |
Convexity & Optimality
Criteria |
Convex functions (any local
minimum is the global minimum). Optimality conditions for unconstrained
optimization (gradient = 0). |
| 6.7 |
Exhaustive Search Method |
Brute-force method. Evaluating
all possible points in the search space. |
| 6.8 |
Interval Halving & Fibonacci
Search Methods |
Efficient single-variable search
methods. Interval halving (reducing the search interval). Fibonacci search
(optimal for unimodal functions). |