Principles of Electrical Engineering and DC Circuit Analysis

Course Overview: Principles of Electrical Engineering and DC Circuits

• Course Code: BAEEE101
• Department: School of Electrical Engineering (SELECT) at Vellore Institute of Technology (VIT).

Module 1: Principles of Electrical Engineering: Circuits and Power Conversion Equipment

• Topics Covered:
  • Electric circuit components.
  • Mesh current analysis and Node voltage analysis.
  • Thevenin's and Superposition theorems.
  • Single-phase AC circuits: RL, RC, RLC.
  • Power and Energy Calculations and Power Factor.
  • Basics of Electrical Safety and Earthing.
  • Introduction to electromechanical energy conversion.
  • Operation of Electrical Machines: DC Motor, Induction motors, BLDC Motor, and single-phase Transformers.
  • Concepts of Power Electronics and Industrial Applications of Electrical Drives (Qualitative Analysis).

Module 2: Foundations of Electronics and Communication Systems: Devices, Circuits, and Systems

• Topics Covered:
  • Characteristics of PN Junction Diodes, Zener Diodes, BJT, and MOSFET.
  • Rectifiers and Voltage Regulators.
  • Introduction to Operational Amplifiers.
  • Electromagnetic Spectrum and Elements of Communication Systems.
  • Overview of cellular communication.
  • Fundamentals of Satellite Communication and Radar.

Advantages of Electricity in Daily Life

• Why use Electricity?
  • Possibility of bulk production and transportability over long distances.
  • Operation is silent.
  • Ease of control.
  • Transfer is very fast.
  • Easily convertible to other forms of energy.
  • Possibility of storage.

Fundamental Electrical Quantities

• Electric Charge ($Q$ or $q$):
  • A property of subatomic particles like protons and electrons.
  • Measured in Coulombs ($C$).
  • It forms the basis for electricity and explains the electrical behavior of materials.
  • It is the most elementary quantity in an electric circuit.
• Electric Current ($I$ or $i$):
  • Defined as the directed flow of electric charge under the influence of an electric field.
• Voltage ($V$):
  • Also known as potential difference.
  • Defined as the electric pressure that makes charges flow in a conductor.
  • Technically, the work or energy required to move a unit charge from one point to another.
• Electric Power ($P$):
  • The time rate of expanding or absorbing energy in a circuit.
  • Equivalently, the rate of doing work in an electric circuit.

Classification of Electrical Networks

• Electrical Network: An interconnection of electrical elements such as resistors ($R$), inductors ($L$), capacitors ($C$), and sources.
• Electric Circuit: An interconnection of elements that provides at least one closed path, allowing a return path for current. All circuits are networks, but not all networks are circuits.
• Importance of Classification:
  • Simplifies analysis and problem-solving.
  • Assists in choosing appropriate methods like KCL/KVL, Thevenin/Norton, or Superposition.
  • Essential for modeling real systems in power, electronics, and control.
• Categorization:
  • Active vs. Passive:
    • Active: Contains energy sources (independent or dependent) and can deliver power. Examples include batteries with resistors, transistors, and Operational Amplifiers.
    • Passive: Contains only $R$, $L$, and $C$ elements. These absorb or store energy but do not generate it. Example: an RLC network without sources.
  • Linear vs. Non-linear:
    • Linear: The Voltage-Current ($V-I$) relation is linear, and parameters remain constant regardless of applied voltage or current. Follows the principles of superposition and homogeneity. Examples: Ideal resistors, inductors, and capacitors.
    • Non-linear: The $V-I$ relation is not linear; parameters vary with voltage, current, or temperature. Examples: Diodes, transistors, and filament lamps.
  • Bilateral vs. Unilateral:
    • Bilateral: Behavior and characteristics (impedance) remain the same regardless of the direction of current or voltage. Examples: $R$, $L$, and $C$ elements.
    • Unilateral: Behavior depends on the direction of current or voltage. Example: A diode circuit which conducts primarily in one direction.
  • Time-invariant vs. Time-variant:
    • Time-invariant: Element values do not change over time, making analysis easier.
    • Time-variant: Element values change over time. Examples: Resistance changing with a sensor ($R(t)$), switching circuits, or variable capacitors.

Fundamental Circuit Laws

• Ohm's Law: States that the voltage $V$ across a resistor is directly proportional to the current $I$ flowing through it.
  • Equation: $V = IR$
• Kirchhoff’s Current Law (KCL): The algebraic sum of currents going away from or coming towards a node is zero.
  • Convention: Current coming toward a node is considered positive, and current leaving is negative (or vice-versa).
• Kirchhoff’s Voltage Law (KVL): The algebraic sum of voltages equals zero for any closed path (loop) in an electrical circuit.

Circuit Topology Definitions

• Node: An equipotential surface where two or more circuit elements are joined. Example nodes listed in a diagram: $\{1, 2, 3, 4\}$.
• Junction: A point in a network where three or more circuit elements are joined. Example junctions: $\{2, 4\}$.
• Loop: Any closed path of a network. Example loops: Loop 1 $\{1, 2, 4, 1\}$, Loop 2 $\{2, 3, 4, 2\}$, Loop 3 $\{1, 2, 3, 4, 1\}$.
• Mesh: The most elementary form of a loop; it cannot be divided into any other smaller loops. Example meshes: Mesh 1 $\{1, 2, 4, 1\}$, Mesh 2 $\{2, 3, 4, 2\}$.

Electrical Sources

• Independent Sources: Provide a specified voltage or current that is completely independent of other circuit elements.
  • Representations: Time-varying voltage source symbols and continuous DC voltage source symbols.
• Dependent (Controlled) Sources: The value depends on another voltage or current elsewhere in the circuit. These are used to model transistors, op-amps, and amplifiers.

Types of Dependent Sources

1. VCVS (Voltage Controlled Voltage Source):
   • Output: Voltage Source.
   • Controlled by: Voltage.
   • Equation: $V_1 = A v_1$
   • Parameter: $A = \text{Voltage gain}$ (unitless).
2. VCCS (Voltage Controlled Current Source):
   • Output: Current Source.
   • Controlled by: Voltage.
   • Equation: $i_1 = g v_1$
   • Parameter: $g = \text{transconductance}$ (Unit: Siemens ($S$) or Amperes/Volt ($A/V$)).
3. CCVS (Current Controlled Voltage Source):
   • Output: Voltage Source.
   • Controlled by: Current.
   • Equation: $v_0 = r i_1$
   • Parameter: $r = \text{trans-resistance}$ (Unit: Ohms ($\Omega$) or Volts/Ampere ($V/A$)).
4. CCCS (Current Controlled Current Source):
   • Output: Current Source.
   • Controlled by: Current.
   • Equation: $i_x = \beta i_1$
   • Parameter: $\beta = \text{Current Gain}$ (unitless).

Resistance Network Simplification

• Series Resistance: The equivalent resistance ($R_{eq}$) is the sum of individual resistances.
  • $R_{eq} = R_1 + R_2 + \dots + R_n$
• Parallel Resistance: The reciprocal of equivalent resistance is the sum of the reciprocals of individual resistances.
  • $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n}$
• Equivalent Resistance Concept: A single resistance that can replace an entire circuit or network between two specific terminals.

Delta-Star ($\Delta-Y$) Transformations

• Delta (Δ) to Star (Y) Conversion:

  • Used when three resistors ($R_{AB}$, $R_{BC}$, $R_{CA}$) form a closed mesh.
  • Formulas for star arms ($R_A, R_B, R_C$):
    • $R_A = \frac{R_{AB} R_{CA}}{R_{AB} + R_{BC} + R_{CA}}$
    • $R_B = \frac{R_{BC} R_{AB}}{R_{AB} + R_{BC} + R_{CA}}$
    • $R_C = \frac{R_{CA} R_{BC}}{R_{AB} + R_{BC} + R_{CA}}$
  • Memory Rule: Any arm of a star connection is equal to the product of two adjacent Δ arms divided by the sum of all Δ arms.

• Star (Y) to Delta (Δ) Conversion:

  • Converts resistors $R_A$, $R_B$, and $R_C$ connected to a common neutral node into a closed mesh.
  • Formulas for delta arms:
    • $R_{AB} = R_A + R_B + \frac{R_A R_B}{R_C}$
    • $R_{BC} = R_B + R_C + \frac{R_B R_C}{R_A}$
    • $R_{CA} = R_C + R_A + \frac{R_C R_A}{R_B}$

Numerical Examples for Resistance

• Y to Δ conversion example:
  • Input Star: $R_1 = 7.5\,\Omega$, $R_2 = 5\,\Omega$, $R_3 = 3\,\Omega$.
  • Calculations:
    • $R_a = 7.5 + 5 + \frac{7.5 \times 5}{3} = 25\,\Omega$
    • $R_b = 7.5 + 3 + \frac{7.5 \times 3}{5} = 15\,\Omega$
    • $R_c = 5 + 3 + \frac{5 \times 3}{7.5} = 10\,\Omega$
• Δ to Y conversion example:
  • Input Delta: $R_a = 15\,\Omega$, $R_b = 10\,\Omega$, $R_c = 25\,\Omega$.
  • Calculations:
    • $R_1 = \frac{10 \times 25}{50} = 5\,\Omega$
    • $R_2 = \frac{25 \times 15}{50} = 7.5\,\Omega$
    • $R_3 = \frac{15 \times 10}{50} = 3\,\Omega$

Voltage and Current Division Rules

• Voltage Division (Series Circuits):

  • The total applied voltage is distributed among resistors.
  • For two resistors $R_1$ and $R_2$ in series with voltage $V$:
    • $V_{R1} = V \times \frac{R_1}{R_1 + R_2}$
    • $V_{R2} = V \times \frac{R_2}{R_1 + R_2}$
  • Example with 3 resistors ($10\,\Omega$, $20\,\Omega$, $30\,\Omega$) and $60\,V$ source:
    • $V_{10\Omega} = 60 \times \frac{10}{10 + 20 + 30} = 10\,V$
    • $V_{20\Omega} = 60 \times \frac{20}{10 + 20 + 30} = 20\,V$
    • $V_{30\Omega} = 60 \times \frac{30}{10 + 20 + 30} = 30\,V$

• Current Division (Parallel Circuits):

  • Total current ($I_T$) splits between parallel branches.
  • For two resistors $R_1$ and $R_2$ in parallel:
    • $I_1 = I_T \times \frac{R_2}{R_1 + R_2}$
    • $I_2 = I_T \times \frac{R_1}{R_1 + R_2}$
  • Relationship: $I_T = I_1 + I_2$
  • Example with $R_1=10\,\Omega$, $R_2=20\,\Omega$, and $V=50\,V$:
    • $R_{eq} = \frac{10 \times 20}{10 + 20} = 6.67\,\Omega$
    • $I_T = \frac{50}{6.67} = 7.5\,A$
    • $I_1 = 7.5 \times \frac{20}{30} = 5\,A$
    • $I_2 = 2.5\,A$

Source Transformation

• Voltage to Current Source:
  • A voltage source $V$ with series resistance $R_{se}$ converts to a current source $I = \frac{V}{R_{se}}$ with parallel internal resistance equal to $R_{se}$.
• Current to Voltage Source:
  • A current source $I$ with parallel resistance $R_{sh}$ converts to a voltage source $V = I \times R_{sh}$ with series internal resistance equal to $R_{sh}$.
• Polarity Conventions:
  • The direction of equivalent current is from the negative to the positive terminal internal to the source.
  • When converting current to voltage, the positive terminal is at the head of the arrow and the negative at the tail.

Nodal Analysis

• Definition: Nodal analysis focuses on identifying voltages at nodes where three or more branches are connected.
• Procedure:
  1. Identify the total number of nodes ($N$). The number of equations required is $N-1$.
  2. Mark branch currents.
  3. Write node equations using Kirchhoff’s Current Law (KCL).
  4. Arrange equations in matrix form.
  5. Solve the system using Cramer’s rule.

Nodal Analysis Example

• Setup: 2-node circuit ($N=2$, so 1 equation). Source voltage $10\,V$, branches with resistors $4\,\Omega$, $2\,\Omega$, and $4\,\Omega$.
• Equation at Node 1:
  • $I_1 = I_2 + I_3$
  • $\frac{10 - V_1}{4} = \frac{V_1}{2} + \frac{V_1}{4}$
  • Result: $V_1 = 2.5\,V$
• Currents:
  • $I_1 = \frac{10 - 2.5}{4} = 1.875\,A$
  • $I_2 = \frac{2.5}{2} = 1.25\,A$
  • $I_3 = \frac{2.5}{4} = 0.625\,A$

Mesh Analysis

• Definition: Mesh analysis involves closed paths that do not contain any other loops within them.
• Procedure:
  1. Identify the number of meshes.
  2. Mark each mesh current (usually in a clockwise direction).
  3. Apply Kirchhoff’s Voltage Law (KVL) to each mesh to write equations.
  4. Convert the equations into a matrix format.
  5. Use Cramer’s rule to solve for mesh currents.

Mesh Analysis Example

• Setup: 3-mesh circuit to find power in an $18\,\Omega$ resistor.
  • Mesh 1: $12 I_1 - 6 I_2 = 120$
  • Mesh 2: $-6 I_1 + 34.5 I_2 - 24 I_3 = 0$
  • Mesh 3: $-21 I_2 + 42 I_3 = 0$
• Solving:
  • Determinant (\Delta): $10584$
  • Determinant for Mesh 3 ($\Delta_3$): $15120$
  • $I_3 = \frac{15120}{10584} = 1.428\,A$
• Power Calculation ($P = I^2 R$):
  • $P = 1.4282^2 \times 18 = 36.7\,W$

Network Theorems

• Superposition Theorem:
  • In a linear network with two or more sources, the response in any element is the algebraic sum of responses caused by each source acting alone.
  • To act "alone," other ideal voltage sources are replaced by short circuits, and ideal current sources are replaced by open circuits.
  • Applicability: Only valid for linear systems. It does not apply to power calculations because power is a nonlinear function ($P \propto I^2$).
• Thevenin’s Theorem:
  • Any two-terminal linear network containing voltage/current sources and resistances can be replaced by an equivalent circuit.
  • The equivalent circuit consists of a single voltage source ($V_{th}$) in series with a single resistance ($R_{th}$).
  • $V_{th}$ is the open-circuit voltage across the terminals.
  • $R_{th}$ is the equivalent resistance measured between the terminals when all internal energy sources are replaced by their internal resistances (voltage sources shorted, current sources opened).