Introduction to Mathematics for Electronic Systems

Course Overview and Logistics

• Instructors:
  • Prirak Deep
  • Dr. Shivakrishna Dasi
• Session Schedule: Friday, Sunday, and Tuesday.
• Session Types:
  • Instructor Sessions: Focused on summarizing the weekly content.
  • TA (Teaching Assistant) Sessions: Primarily focused on problem-solving exercises.
• Eligibility and Grading Criteria:
  • Final Course Grade Eligibility: The student must attend the Final Exam.
  • Final Exam Writing Eligibility:
    • The average score of the best 5 out of the first 7 weekly assignments must be greater than or equal to $40/100$.
    • Completion of at least one of the two scheduled quizzes.
• Score Calculation Formula ($T$):
  • $T = \max(0.6 \times F + 0.3 \times \max(Qz1, Qz2), 0.45 \times F + 0.25 \times Qz1 + 0.3 \times Qz2)$
  • Where $F$ is the Final Exam score, and $Qz1, Qz2$ are the scores for Quiz 1 and Quiz 2 respectively.

Importance of Mathematics in Electronic Systems

• Role of Numbers and Relationships: Quantitative data and the functional relationships between them are essential for specifying and designing electronic systems.
• Case Study: Travel Adapter (Model: PEM-2134):
  • Input: $100-240\text{ VAC}$, $50-60\text{ Hz}$, $0.5\text{ A}$.
  • Output: $5\text{ V}$, $2.0\text{ A}$.
• Case Study: Audio System Specifications:
  • Power Output: $75\text{ Watts}$ per channel at $1\text{ kHz}$ into $8\Omega$.
  • Input Impedance: $>14\text{ k}\Omega$, line input.
  • Input Sensitivity: $250\text{ mV}$.
  • Pre-amplifier Output Gain: $+18\text{ dB}$.
  • Frequency Response: $20\text{ Hz}-20\text{ kHz} \pm 0.5\text{ dB}$.
  • Total Harmonic Distortion: $<0.05\%$ at rated output, ref. $1\text{ kHz}$.
  • Signal to Noise Ratio: $>100\text{ dB}$, A weighted, ref. rated output.
  • Channel Separation: $>60\text{ dB}$, $20\text{ Hz}-20\text{ kHz}$.
  • Mains Supply: $110/120\text{ V}$ or $220/240\text{ V}$, $50/60\text{ Hz}$.
  • Power Consumption: $<200\text{ VA}$, $8\Omega$ load, both channels driven.
  • Dimensions: $90\text{ mm (h)} \times 440\text{ mm (w)} \times 310\text{ mm (d)}$.
  • Weight: Net $7\text{ kg}$, Gross $9\text{ kg}$.
• Case Study: Elevator (Lift) Design Parameters:
  • Number of Floors: $16$.
  • Number of Shafts (Cages): $6(12)$.
  • Floor Distance: $4.5\text{ m}$.
  • Max Velocity: $2.5\text{ m/s}$.
  • Max Acceleration: $0.7\text{ m/s}^2$.
  • Jerk: $0.7\text{ m/s}^3$.
  • Cage Capacity: $20\text{ persons}$.
  • Time for Opening Door: $2.0\text{ s}$.
  • Time for Closing Door: $2.3\text{ s}$.
  • Time for Riding: $1.0\text{ s/person}$.
  • Passenger Density (Regular/Up-peak/Down-peak): $3000, 2700, 3300\text{ persons/h}$.

Mathematical Frameworks in Electronics

• Electronic Circuit Problems: The goal is to find precise relationships between voltages and currents at various points in a circuit.
• Key Mathematical Concepts:
  • Linear Equations: Used for DC (Direct Current) circuits.
  • Functions of One Variable: Voltage and current are treated as signals, which are functions of time ($t$).
  • Calculus (Differentiation and Integration): Essential because physical properties like the current through a capacitor depend on the rate of change of voltage ($I = C \frac{dV}{dt}$).
  • Differential Equations: Used for circuits containing time-varying voltages and currents.
  • Complex Numbers: Utilized to analyze AC (Alternating Current) circuits in a steady state.

Physical Quantities and Linear Relations

• Relationship Examples:
  1. Ohm's Law: $\text{Voltage} = \text{Resistance} \times \text{Current}$ ($V = RI$). In a graph of $V$ vs $I$, the slope $\tan(\theta) = R$.
  2. Newton's Second Law: $\text{Force} = \text{Mass} \times \text{Acceleration}$ ($F = ma = m \frac{dv}{dt}$). In a graph of $F$ vs $a$, the slope $\tan(\theta) = m$.
• Equation of a Straight Line:
  • Passing through origin: $y = mx$, where $m = \tan(\theta)$ is the slope.
  • General form: $y = mx + b$, where $b$ is the y-axis intercept.
• Slope Comparisons:
  • For three lines passing through the origin with slopes $a, b, c$ and angles $\theta_3 > \theta_2 > \theta_1$:
  • $a = \tan(\theta_3)$, $b = \tan(\theta_2)$, $c = \tan(\theta_1)$.
  • If $\theta_1 < \theta_2 < \theta_3$, then $\tan(\theta_1) < \tan(\theta_2) < \tan(\theta_3)$, which implies $c < b < a$.
• Identifying Non-Linear Relationships:
  • $y = 4x + 7$: Linear ($y = mx + b$).
  • $y = \frac{3}{x} + 1$: Non-linear (reciprocal relationship).
  • $y = x(x - 2) \rightarrow y = x^2 - 2x$: Non-linear (Quadratic).
  • $y = 3(x - 4)^2 \rightarrow y = 3(x^2 - 8x + 16)$: Non-linear (Quadratic).

Quadratics and Exponentials

• Quadratics:
  • General form: $y = ax^2 + bx + c$.
  • Shape: Parabolic.
  • Applications: MOSFET current-voltage characteristics, area computations, and distance calculations under constant acceleration.
• Exponentials:
  • General forms: $y = P(1 + r)^t$ or $y = k a^x$.
  • Applications: Diode current characteristics, compound interest calculations, and iterative computations.

Function Notation and Definitions

• Definition: A function $f$ maps a real number $x$ (from domain) to a unique real number $f(x)$ (in codomain).
• Notation: $f: \mathbb{R} \rightarrow \mathbb{R}$. The symbol $x \in \mathbb{R}$ means "x is a real number".
• Specifying Values:
  • Constant Function: $f(x) = c$ (same value for every $x$).
  • Linear Function: $f(x) = mx$.
  • Quadratic Function: $f(x) = ax^2 + bx + c$.
  • Exponential Function: $f(x) = P(1 + r)^x$.
• Function Composition: Functions can be combined. For example, if $f(x) = x^2$ and $h(x) = ax^2 + b$:
  • $f(h(x)) = (ax^2 + b)^2$
  • $h(f(x)) = a(x^2)^2 + b = ax^4 + b$
• Identifying Quadratic Functions:
  • $f(x) = 7x + 10$ (Linear)
  • $f(x) = 3x^2 + 9x + 19$ (Quadratic)
  • $f(x) = x^3 + x^2 + x + 1$ (Cubic / Polynomial)
  • $f(x) = (x+1)^2 - (x+1) = x^2 + 2x + 1 - x - 1 = x^2 + x$ (Quadratic)

Functions from Graphs

• Vertical Line Test: A graph represents a function if any vertical line crosses the plot at exactly one point. If it crosses at more than one point, it is a relationship but not a function.
• Range:
  • If $f(x) = 3$, the range is the set ${3} \subset \mathbb{R}$.
  • If $f(x) = x^2$, the range is the set ${y \mid y \ge 0} \subset \mathbb{R}$.

Operations with Functions

• Arithmetic Operations:
  • Addition/Subtraction: $f(x) + g(x)$ or $f(x) - g(x)$.
  • Multiplication: $f(x)g(x)$.
  • Division: $\frac{f(x)}{g(x)}$.
• Example Calculations:
  • Let $f(x) = x^2$, $g(x) = 2x + 1$, and $h(x) = 5$.
  • $f(x) + g(x) = x^2 + 2x + 1 = (x + 1)^2$.
  • $f(x)g(x) = x^2(2x + 1) = 2x^3 + x^2$.
  • $\frac{g(x)}{h(x)} = \frac{2x + 1}{5}$.

Word Problem Formulations

• Problem 1: Vaccine Manufacturing Cost:
  • $y$ = Number of vaccines per day.
  • Manufacturing cost per vaccine = $100 + y$.
  • Total manufacturing cost = $10,000$.
  • Equation: $y(100 + y) = 10,000 \rightarrow y^2 + 100y = 10,000$.
• Problem 2: Mango Distribution:
  • Let Mahesh have $x$ mangoes and Sandhya have $y$ mangoes.
  • Total mangoes: $x + y = 20 \rightarrow y = 20 - x$.
  • Mahesh loses 3 ($x - 3$) and Sandhya loses 4 ($y - 4$).
  • The product of the remaining mangoes is 42: $(x - 3)(y - 4) = 42$.
  • Substitute $y$: $(x - 3)(20 - x - 4) = 42 \rightarrow (x - 3)(16 - x) = 42$.
  • Expand: $16x - x^2 - 48 + 3x = 42 \rightarrow -x^2 + 19x - 48 = 42 \rightarrow x^2 - 19x + 90 = 0$.
  • Factor: $(x - 9)(x - 10) = 0$.
  • Solutions: $x = 9$ or $x = 10$.