Fundamental of Electrical Engineering

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This section covers Ohm’s law, the basic electrical quantities, conducting and insulating materials, series/parallel circuit reduction, star-delta conversion, Kirchhoff’s laws, and the fundamental classifications used to describe electric circuits and their elements.

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Ohm’s Law & Basic Quantities Quantity Definition / Relation Voltage (V) Work done per unit charge in moving charge between two points; measured in volts Current (I) Rate of flow of charge; measured in amperes, I = dQ/dt Ohm’s law V = IR, for a conductor at constant temperature — the resistance R is constant regardless of V or I Power (P) Rate of doing work/energy transfer;

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P = VI = I²R = V²/R, measured in watts Energy (W) Power consumed over time;

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W = Pt, measured in joules (or kWh for electrical billing) Conducting & Insulating Materials Conducting materials (e.g. copper, aluminium, silver) have abundant free electrons and very low resistivity, allowing current to flow easily.

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Insulating materials (e.g. rubber, PVC, mica, porcelain) have very few free electrons and extremely high resistivity, blocking current flow and providing electrical isolation/safety.

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Series & Parallel Circuits, Star-Delta Conversion Configuration Key Property Series circuit Same current flows through all elements; total resistance R = R₁+R₂+...+Rₙ; voltages add up Parallel circuit Same voltage across all elements; total conductance adds (1/R = 1/R₁+1/R₂+...); currents add up Star (Y) to Delta (Δ) conversion Converts a 3-terminal star network to an equivalent delta network (and vice versa) for simplifying network analysis; each delta resistance is derived from products/sums of the star resistances (and vice versa) Kirchhoff’s Laws Kirchhoff’s Current Law (KCL): the algebraic sum of currents entering a node equals the sum leaving it (conservation of charge).

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Kirchhoff’s Voltage Law (KVL): the algebraic sum of voltage drops and rises around any closed loop equals zero (conservation of energy).

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Circuit Classifications Classification Description Linear circuit Circuit elements obey a straight-line V-I relationship (e.g. ideal resistors); superposition applies Non-linear circuit Contains elements (e.g. diodes, transistors) whose V-I relationship is not proportional; superposition does not directly apply Bilateral circuit Circuit whose behavior (e.g. resistance) is the same regardless of the direction of current flow (e.g. a resistor) Unilateral circuit Circuit whose behavior changes with the direction of current flow (e.g. a diode) Active circuit Contains one or more energy sources (voltage/current sources) capable of delivering power Passive circuit Contains only passive elements (R, L, C) that can only absorb or store energy, not generate it

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This section covers the four classical network theorems used to simplify and analyze linear electrical circuits: superposition, Thevenin’s, Norton’s, and the maximum power transfer theorem.

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Superposition Theorem In a linear circuit with multiple independent sources, the response (current or voltage) in any element equals the algebraic sum of the responses caused by each source acting alone, with all other independent sources deactivated (voltage sources shorted, current sources opened).

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Applies only to linear circuits, and only to voltage/current (not power, which is a non-linear/quadratic function of these quantities).

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Thevenin’s & Norton’s Theorems Theorem Statement Thevenin’s theorem Any linear two-terminal network can be replaced by an equivalent voltage source Vᴛₕ in series with a resistance Rᴛₕ, where Vᴛₕ is the open-circuit voltage at the terminals and Rᴛₕ is the resistance seen from the terminals with all independent sources deactivated Norton’s theorem Any linear two-terminal network can be replaced by an equivalent current source Iᴚ in parallel with a resistance Rᴚ, where Iᴚ is the short-circuit current at the terminals and Rᴚ = Rᴛₕ Thevenin-Norton equivalence The two equivalents are related by the source transformation Vᴛₕ = Iᴚ × Rᴚ (Rᴚ=Rᴛₕ), so either can be derived directly from the other Maximum Power Transfer Theorem Maximum power is transferred from a source (represented by its Thevenin equivalent) to a load when the load resistance Rₗ equals the source’s Thevenin resistance Rᴛₕ (Rₗ=Rᴛₕ) for a purely resistive load, at which the transfer efficiency is 50%.

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For AC circuits with complex source impedance Zᴛₕ, maximum power transfer occurs when the load impedance is the complex conjugate of the source impedance (Zₗ=Zᴛₕ*).

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This section covers the principle of generation of alternating voltage/current, the standard equations and waveform description of AC quantities, and the average, peak, and rms values used to characterize them.

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Generation of Alternating Voltage/Current An alternating voltage/current is generated by rotating a coil in a uniform magnetic field (or rotating a magnetic field past a stationary coil), inducing an EMF via Faraday’s law that varies sinusoidally with time as the flux linkage changes.

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The instantaneous value is given by v(t) = Vₘsin(ωt), where Vₘ is the peak (maximum) value and ω=2πf is the angular frequency (f = supply frequency in Hz).

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Waveform Description Term Meaning Cycle One complete set of positive and negative values of an alternating quantity Time period (T) Time taken to complete one cycle;

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T = 1/f Frequency (f) Number of cycles completed per second, measured in Hz Phase/phase angle The fraction of a cycle (in degrees/radians) that has elapsed from a reference point, or the angular displacement between two alternating quantities Average, Peak & RMS Values Value Definition (Sinusoidal Wave) Peak value (Vₘ) The maximum instantaneous value reached during a cycle Average value Mean of instantaneous values over a half-cycle (full-cycle average of a pure sinusoid is zero); for a sinusoid, average = 0.637 × Vₘ (=2Vₘ/π) RMS (root-mean- square) value The DC-equivalent value that produces the same heating effect; for a sinusoid, RMS = 0.707 × Vₘ (=Vₘ/√2) Form factor Ratio of RMS value to average value; for a pure sinusoid, form factor ≈ 1.11 Peak (crest) factor Ratio of peak value to RMS value; for a pure sinusoid, crest factor ≈ 1.414 (√2)

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This section covers the steady-state and transient behavior of R-L, R-C, and R-L-C circuits when subjected to a sudden change (e.g. switching), including the concept of time constant and the different damping conditions of second-order (R-L-C) circuits.

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Steady State vs Transient Response The transient response is the temporary behavior of a circuit immediately following a sudden change (switching on/off, a step input), before the circuit settles.

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The steady-state response is the behavior of the circuit after all transients have decayed and the circuit has settled into its final, stable pattern (constant for DC, or a fixed-amplitude sinusoid for AC).

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First-Order Circuits (R-L, R-C) Circuit Time Constant & Behavior R-L circuit Time constant τ=L/R; current i(t) rises/decays exponentially toward its final value, reaching ≈63.2% of the total change in one time constant R-C circuit Time constant τ=RC; capacitor voltage rises/decays exponentially, reaching ≈63.2% of the total change in one time constant Second-Order Circuits (R-L-C) Damping Condition Description Overdamped Damping factor > natural frequency; response decays slowly to steady state without oscillation, via two distinct real exponential terms Critically damped Damping factor = natural frequency; response returns to steady state in the shortest time possible without oscillating Underdamped Damping factor < natural frequency; response oscillates with exponentially decaying amplitude before settling Undamped Zero resistance/damping (ideal, no losses); response oscillates continuously at the natural frequency without decay

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This section covers the power triangle (active, reactive, and apparent power), power factor, and the phenomenon of resonance in series and parallel AC circuits containing resistance, inductance, and capacitance.

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Active, Reactive & Apparent Power Power Type Definition Active (real) power, P The actual power consumed/converted to useful work, P = VIcosφ (watts, W) Reactive power, Q Power that oscillates between source and reactive (L/C) elements without net consumption, Q = VIsinφ (volt-amperes reactive, VAR) Apparent power, S The product of RMS voltage and current, S = VI (volt-amperes, VA);

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S² = P²+Q² Power Factor Power factor (cosφ) = P/S = the cosine of the phase angle between voltage and current; indicates how effectively apparent power is converted to active (useful) power.

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Lagging power factor occurs with inductive loads (current lags voltage); leading power factor occurs with capacitive loads (current leads voltage); unity power factor (cosφ=1) occurs with purely resistive loads.

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Low power factor increases current for a given real power, increasing I²R losses and requiring larger conductors/equipment; power factor correction (typically adding capacitors to offset inductive lagging loads) improves it.

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Resonance Resonance Type Condition & Effect Series resonance Occurs when Xₗ=X₊ (inductive reactance = capacitive reactance); circuit impedance is minimum (purely resistive), current is maximum for a given applied voltage Parallel resonance Occurs when the inductive and capacitive branch currents/susceptances cancel; circuit impedance is maximum, line current drawn from the source is minimum Resonant frequency, f₀ f₀ = 1/(2π√(LC)), the frequency at which Xₗ=X₊

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This final section of the chapter covers the star and delta connections of a three-phase system, the relationship between line and phase quantities, the neutral concept, the effect of unbalanced load, and active/reactive power in three-phase circuits.

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Star & Delta Connections Connection Line-Phase Relationship Star (Y) connection Line voltage Vₗ = √3 × phase voltage Vₚₕ (line voltages are 30° ahead of corresponding phase voltages); line current Iₗ = phase current Iₚₕ; a neutral point is available Delta (Δ) connection Line voltage Vₗ = phase voltage Vₚₕ; line current Iₗ = √3 × phase current Iₚₕ; no neutral point is available Neutral Concept The neutral is the common point of a star-connected system (ideally at zero/reference potential for a balanced load), providing a return path for unbalanced current and enabling both line and phase (line-to- neutral) voltages to be used.

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For a perfectly balanced three-phase load, the neutral carries zero current; any imbalance between phases causes current to flow in the neutral conductor.

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Effect of Unbalanced Load An unbalanced load (unequal impedance or loading per phase) causes unequal phase currents, a resulting neutral current (in a star system with neutral), and can cause the neutral point voltage to shift (neutral displacement) in a system without a solidly grounded/connected neutral — leading to unequal phase voltages across the loads.

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Three-Phase Power Power Formula (Balanced 3-Phase System) Active power P = √3 VₗIₗcosφ = 3VₚₕIₚₕcosφ Reactive power Q = √3 VₗIₗsinφ = 3VₚₕIₚₕsinφ Apparent power S = √3 VₗIₗ = 3VₚₕIₚₕ