Introduction to DC Circuits, AC Fundamentals, and Electrical Machines

Basic Electrical Definitions

• Potential Difference (Voltage): Calculated as the work done per unit charge moving against forces. Denoted by $V$, measured in Volts.
  • $V = \frac{W}{Q}$ or $V = \frac{dw}{dq}$
  • One Volt ($V$) is defined as one Joule ($J$) of energy used to pass one Coulomb ($C$) of charge.
• Electric Current: The rate of flow of electrons in a material, measured in Amperes ($A$).
  • $I = \frac{Q}{t}$ or $I = \frac{dq}{dt}$
  • One Ampere ($A$) = $1\,C/sec$.
  • $1\,C = 6.25 \times 10^{18}$ electrons.
• Power and Energy:
  • Energy is the capacity for doing work ($W$ or $E$).
  • Power ($P$) is the rate of change of energy: $P = \frac{dw}{dt} = V \times I$.
  • Measured in Watts ($W$) or $J/S$.
• Electrical Circuit: Consists of a source, load (sink), and connecting wires.
  • Closed Circuit: Current has a complete path to flow.
  • Open Circuit: Path is not closed.
  • Electrical Network: Interconnection of two or more simple circuit elements.

Network Classification and Analysis

• Network Elements:
  • Active: Capable of delivering energy (e.g., batteries, voltage sources). Average power $> 0$.
  • Passive: Absorbs energy (e.g., resistors, inductors, capacitors). Average power $< 0$.
  • Bilateral: $V-I$ relationship is identical in either direction (e.g., $R$, $L$, $C$).
  • Unilateral: $V-I$ relationship differs by direction (e.g., Vacuum diodes, rectifiers).
  • Linear / Non-Linear: Linear elements have a straight-line $V-I$ curve passing through zero.
  • Lumped / Distributed: Lumped elements are physically separable; distributed are not.
• Circuit Laws:
  • Ohm's Law: Voltage $v(t)$ is directly proportional to current $i(t)$ given constant physical states: $V = I \times R$.
  • Kirchhoff’s Current Law (KCL): Based on conservation of charge. Algebraic sum of currents at a node is zero: $\sum I_n = 0$.
  • Kirchhoff’s Voltage Law (KVL): Based on conservation of energy. Algebraic sum of potential differences in a closed loop is zero: $\sum V_n = 0$.
• Division Rules:
  • Current Division: For resistors in parallel: $I_1 = I \times \frac{R_2}{R_1 + R_2}$.
  • Voltage Division: For resistors in series: $V_1 = V \times \frac{R_1}{R_1 + R_2}$.
• Energy Sources:
  • Independent Voltage Source: Ideally maintains constant voltage regardless of current; has zero internal resistance.
  • Independent Current Source: Ideally produces constant current irrespective of voltage; has infinite internal resistance.
  • Dependent Sources: Output depends on a voltage or current elsewhere in the circuit (e.g., VCVS, CCVS, VCCS, CCCS).

Passive Circuit Elements

• Resistor ($R$): Measures opposition to electric current.
  • Instantaneous power: $p = i^2 R$.
  • Series: $R_{eq} = R_1 + R_2 + \dots + R_n$.
  • Parallel: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n}$.
• Inductor ($L$): Property of opposing changes in time-varying current. Measured in Henry ($H$).
  • $v = L \frac{di}{dt}$.
  • Magnetic flux/current ratio: $L = \frac{\phi}{I}$.
  • Series: $L_{eq} = L_1 + L_2 + \dots + L_n$.
  • Parallel: $\frac{1}{L_{eq}} = \frac{1}{L_1} + \frac{1}{L_2} + \dots + \frac{1}{L_n}$.
• Capacitor ($C$): Measures electric charge stored for a given potential. Measured in Farad ($F$).
  • $Q = CV$; Current $i = C \frac{dv}{dt}$.
  • Stored energy: $W = \frac{1}{2} CV^2$.
  • Series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots + \frac{1}{C_n}$.
  • Parallel: $C_{eq} = C_1 + C_2 + \dots + C_n$.

AC Circuit Fundamentals

• Alternating Quantity: Changes magnitude and direction at regular intervals.
• Parameters:
  • Equation: $e = E_m \sin(\omega t)$, where $\omega = 2\pi f$.
  • Frequency ($f$): Cycles per second, measured in Hertz ($Hz$). Relationship: $f = \frac{1}{T}$.
  • Average Value ($V_{av}$): For a sine wave over a half-cycle: $0.637 V_p$.
  • RMS Value ($V_{rms}$): Effective value producing same heating as DC: $V_{rms} = \frac{V_p}{\sqrt{2}} \approx 0.707 V_p$.
  • Form Factor: $\frac{RMS\,value}{Average\,value}$. (Standard sine wave = $1.11$).
  • Peak Factor: $\frac{Peak\,value}{RMS\,value}$. (Standard sine wave = $1.414$).
• Phasors and Phase Difference:
  • Leading: Current/voltage ahead of reference ($+ phase$).
  • Lagging: Current/voltage behind reference ($- phase$).
  • In phase: Phase difference is zero.
• Complex Impedance:
  • Impedance ($Z$): Total opposition to current ($Z = R + jX$).
  • Reactance ($X$): Inductive $X_L = 2\pi fL$; Capacitive $X_C = \frac{1}{2\pi fC}$.
  • Admittance ($Y$): Reciprocal of impedance ($Y = \frac{1}{Z}$, measured in mho).
• Types of Power:
  • Real Power ($P$): $VI \cos(\theta)$, measured in Watts ($W$).
  • Reactive Power ($Q$): $VI \sin(\theta)$, measured in Volt-Ampere reactive ($VAR$).
  • Apparent Power ($S$): $VI$, measured in Volt-Amperes ($VA$).
  • Power Factor: Ratio of Real to Apparent power ($\cos(\theta)$).

Electromagnetic Principles and Machines

• Foundational Rules:
  • Maxwell’s Corkscrew Rule: Relates current direction to magnetic field lines.
  • Faraday’s Laws: Induced EMF results from varying magnetic fields: $E = -N \frac{d\phi}{dt}$.
  • Fleming’s Left Hand Rule: Used for motors (Force/Magnetic Field/Current).
  • Fleming’s Right Hand Rule: Used for generators (Motion/Field/Induced Current).
  • Lenz's Law: Induced current opposes the change that produced it.
• DC Generator:
  • Components: Yoke, Poles, Field Winding, Armature, Commutator (mechanical rectifier), Brushes.
  • EMF Equation: $E_g = \frac{\phi Z N}{60} \times \frac{P}{A}$.
  • Types: Series-wound, Shunt-wound, and Compound-wound (Short/Long shunt).
• DC Motor:
  • Principle: Lorentz Force $F = BIL \sin(\theta)$.
  • Torque Equation: $\tau_a = \frac{P Z \phi I_a}{2\pi A}$ or $\tau = 9.55 \times \frac{E_b I_a}{N}\text{ Nm}$.
• Induction Motors:
  • Rotating Magnetic Field (RMF): Produced by 3-phase supply in stator windings; resultant flux $\Phi_r = 1.5 \Phi_m$ rotating at synchronous speed $N_s = \frac{120 f}{P}$.
  • Single Phase Induction Motor: Uses main and auxiliary windings; not self-starting without help.
  • Double Revolving Field Theory: Alternating flux split into two oppositely rotating fluxes of half magnitude.

Transformers

• Static Device: Changes AC voltage/current without frequency change.
• Principle: Mutual Inductance.
• Transformation Ratio ($K$): $K = \frac{V_2}{V_1} = \frac{N_2}{N_1} = \frac{E_2}{E_1} = \frac{I_1}{I_2}$.
• EMF Equation: $E = 4.44 f \phi_m N$.
• Construction: Core type (windings on limbs) or Shell type (windings on central limb).
• Losses:
  • Iron/Core Losses: Hysteresis and Eddy current losses (constant).
  • Copper Losses: $I^2 R$ losses (variable with load).
• Efficiency (\eta): $\eta = \frac{\text{Output Power}}{\text{Output Power} + \text{Losses}}$.
  • Maximum Efficiency Condition: Iron loss equals Copper loss.
• Voltage Regulation: $\frac{E_2 - V_2}{E_2} \times 100\%$.

Electrical Installations

• Switchgear (Rated up to $1\,kV$ as LT):
  • SFU (Switch Fuse Unit): Combined manual switch and fuse protection.
  • MCB (Miniature Circuit Breaker): Automatic protection against overload/short circuits (rated up to $125\,A$).
  • MCCB (Moulded Case Circuit Breaker): Higher ratings (up to $1600\,A$) with adjustable trip settings.
  • ELCB (Earth Leakage Circuit Breaker): Protects against shock by sensing leakage current imbalance.
• Wires and Cables:
  • Solid Wire: Single metal core, rigid, used for high voltage.
  • Stranded Wire: Multiple thin strands twisted together, flexible, used for low voltage.
  • Cables: Collection of two or more insulated conductors in a single covering.
• Batteries:
  • Primary Cells: Non-rechargeable (e.g., Alkaline, Zinc-Carbon, Daniel cell).
  • Secondary Cells: Rechargeable (e.g., Lead-Acid, Nickel-Cadmium, Lithium-Ion).

Questions & Discussion

• Current Calculation: If $V = 12\,V$ and $R = 24\,\Omega$, then $I = 0.5\,A$.
• Resistance Stretch: A wire with $20\,\Omega$ stretched to 8 times its length results in a new resistance of $1280\,\Omega$ (volume remains unchanged).
• Induction Motor Slip: If slip is $S$ for forward rotating flux, backward slip is $(2-S)$.
• Transformer Problem: A $100\,kVA$, $3300/240\,V$ transformer has $990$ primary turns. The EMF per turn is calculated to find secondary turns.
• Assignment Question topics: Construction of DC machines, working principles of generators/motors, EMF and torque derivations, classification of transformers and batteries, circuit breaker comparisons, and cable grading.