Concept of Basic Electrical and Electronics Engineering

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This section covers the fundamental electrical quantities, Ohm’s law, conducting and insulating materials, series-parallel circuits, star-delta transformation, Kirchhoff’s laws, and the classification of circuit elements — the foundation of all circuit analysis.

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Basic Quantities & Ohm’s Law Quantity Unit & Symbol Charge (Q) Coulomb (C) Current (I) Ampere (A) = Coulomb/second Voltage (V) Volt (V) = Joule/Coulomb Resistance (R) Ohm (Ω) Power (P) Watt (W) Energy (E) Joule (J) or kWh Ohm’s Law:

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V = I × R (at constant temperature; voltage across a conductor is directly proportional to the current through it).

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P = V × I = I²R = V²/R.

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E = P × t (commercially billed in kWh;

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Conductors (copper, aluminium, silver) have abundant free electrons → very low resistivity; insulators (rubber, glass, mica, PVC) have tightly bound electrons → extremely high resistivity.

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Series vs Parallel Circuits Series Parallel Same CURRENT through all elements Same VOLTAGE across all elements Rₜₒₜₐₗ = R1 + R2 + … 1/Rₜₒₜₐₗ = 1/R1 + 1/R2 + … Voltage divides Current divides Rₜₒₜₐₗ increases Rₜₒₜₐₗ decreases (less than smallest R) Star-Delta (Y-Δ) Transformation Used to simplify bridge/ladder networks that are neither purely series nor purely parallel.

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Rₐ = (R_ab × R_ca) / (R_ab + R_bc + R_ca), and similarly (cyclically) for R_b and R_c.

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R_ab = (Rₐ R_b + R_b R_c + R_c Rₐ) / R_c, and similarly (cyclically) for R_bc and R_ca.

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For three equal resistances:

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R_star = R_delta ÷ 3, and R_delta = 3 × R_star.

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Kirchhoff’s Laws & Circuit Types KCL (Kirchhoff’s Current Law): sum of currents entering a node = sum leaving (conservation of charge).

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KVL (Kirchhoff’s Voltage Law): algebraic sum of voltages around any closed loop = 0 (conservation of energy).

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Linear circuit — obeys superposition (V ∝ I); non-linear — does not (e.g., diode).

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Active circuit contains an energy source (battery/source); passive has none (only R, L, C).

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Bilateral element works identically in both directions (resistor); unilateral works in one direction only (diode).

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Network theorems simplify the analysis of complex linear circuits.

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This section also covers impedance of R-L, R-C and R-L-C circuits, series/parallel resonance, and active & reactive power.

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Key Network Theorems Theorem Statement Superposition In a linear circuit with multiple sources, total response = sum of responses due to each source acting alone (others replaced by their internal impedance).

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Thevenin’s Any linear two-terminal network can be replaced by a single voltage source (Vth) in series with a resistance (Rth).

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Norton’s Any linear two-terminal network can be replaced by a single current source (IN) in parallel with a resistance (RN).

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Maximum Power Transfer Maximum power is delivered to the load when R_load = R_source (Rth); for AC, Z_load = Zₜₕ* (complex conjugate).

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Impedance of R-L, R-C, R-L-C Circuits R-L series:

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Z = R + jXₗ, |Z| = √(R²+Xₗ²), φ = tan¹⁻(Xₗ/R) — current LAGS voltage.

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Z = R − jX_C, |Z| = √(R²+X_C²), φ = tan¹⁻(X_C/R) — current LEADS voltage.

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Z = R + j(Xₗ − X_C).

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If Xₗ > X_C → inductive (lagging); if Xₗ < X_C → capacitive (leading); if Xₗ = X_C → purely resistive (resonance).

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Inductive reactance Xₗ = 2πfL (rises with frequency);

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Capacitive reactance X_C = 1/(2πfC) (falls with frequency).

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Series & Parallel Resonance Series Resonance Parallel Resonance Occurs when Xₗ = X_C Occurs when susceptances cancel (tank circuit) Impedance is MINIMUM (= R) Impedance (dynamic resistance) is MAXIMUM (= L/(RC)) Current is MAXIMUM Line current is MINIMUM f0 = 1/(2π√(LC)) f0 ≈ 1/(2π√(LC)) (approx., for high-Q coil) Quality factor:

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Q = ωL/R = 1/(ωRC) = (1/R)√(L/C); higher Q → sharper (narrower) resonance curve.

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Active & Reactive Power Active (real) power:

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P = VI cosφ (watts) — power actually consumed/dissipated.

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Q = VI sinφ (VAR) — power oscillating between source and reactive elements.

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Power factor = cosφ = P/S.

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Alternating current (AC) periodically reverses direction, typically following a sinusoidal waveform.

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This section covers AC generation, waveform values, and the three-phase system.

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Generation of Alternating EMF AC is generated by electromagnetic induction (a coil rotating within a magnetic field), based on Faraday’s law.

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Instantaneous voltage: v = Vₘ sin(ωt), where Vₘ = peak value, ω = 2πf (angular frequency in rad/s).

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Frequency (f) = number of cycles per second (Hz);

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Nepal’s standard supply frequency is 50 Hz.

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Key Waveform Values Value Formula (sinusoid) Peak (Vm) Maximum amplitude of the waveform RMS (effective) V_rms = Vm/√2 = 0.707·Vm Average (half cycle) V_avg = 2Vm/π = 0.637·Vm Average (full cycle) Zero (positive and negative half-cycles cancel) Form factor RMS / Average = 1.11 Peak factor (crest factor) Peak / RMS = 1.414 (√2) RMS value is the equivalent DC value that produces the same heating effect; it is the value normally quoted on nameplates and meters (e.g., 230 V mains).

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Three-Phase System A three-phase system has three sinusoidal EMFs of equal magnitude, displaced 120° from one another; far more efficient than single-phase for bulk power generation and transmission.

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Star (Y) connection:

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V_line = √3 × Vₚₕₐₛₑ;

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Delta (Δ) connection:

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I_line = √3 × Iₚₕₐₛₑ.

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Star connection is generally preferred for transmission because it needs lower phase (insulation) voltage for a given line voltage, and provides a neutral for unbalanced/single-phase loads.

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This section covers the semiconductor diode and its characteristics, BJT configuration and biasing with small/large-signal modeling, and the working principle and applications of the MOSFET and CMOS.

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Semiconductor Diode A diode is a PN-junction device that conducts current predominantly in one direction (forward bias) and blocks it in the other (reverse bias).

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V-I characteristic: negligible current until the forward voltage exceeds the cut-in (knee) voltage (≈0.7V for silicon, ≈0.3V for germanium), after which current rises sharply; in reverse bias, only a small leakage/saturation current flows until breakdown voltage is reached.

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BJT — Configuration & Biasing Configuration Key Feature Common Emitter (CE) High voltage & current gain, moderate input/output impedance; most widely used for amplification Common Base (CB) High voltage gain, current gain <1, low input impedance, high output impedance; used for high-frequency applications Common Collector (CC) Voltage gain ≈1 (emitter follower), high current gain, high input impedance, low output impedance; used for impedance matching/buffering Biasing establishes a stable DC operating (Q) point so the transistor amplifies without distortion; common schemes include fixed bias, collector-to-base bias, and voltage-divider (self) bias — voltage-divider bias gives the best stability against β variation and temperature.

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Small-signal model: a linearized AC-equivalent circuit (hybrid-π or h-parameter model) valid for small signal excursions around the Q-point, used to analyze gain/impedance.

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Large-signal model: accounts for the full non-linear V-I characteristics, needed when signal swings are large (e.g. power amplifiers, switching applications).

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MOSFET & CMOS Device Working Principle MOSFET Voltage-controlled device; gate voltage creates/controls a conducting channel between source and drain through an insulated gate (very high input impedance, negligible gate current) Enhancement-mode MOSFET Channel does not exist at V_GS=0; must apply V_GS beyond threshold voltage Vₜ to create (enhance) the channel Depletion-mode MOSFET Channel exists at V_GS=0; applying gate voltage can further enhance or deplete (reduce) the channel CMOS Complementary MOSFET technology combining an NMOS and PMOS transistor pair; near- zero static power dissipation, the basis of virtually all modern digital ICs

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This section covers the basic principle of oscillation (the Barkhausen criterion) and the main categories of sinusoidal and non-sinusoidal signal generators:

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RC, LC, and crystal oscillators, and general waveform generators.

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Basic Principle of Oscillation An oscillator is a circuit that generates a periodic waveform without an external AC input, using positive feedback from an amplifier and a frequency-selective feedback network.

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Barkhausen criterion for sustained oscillation: the loop gain magnitude must equal 1 (|Aβ|=1) and the total phase shift around the loop must be 0° (or a multiple of 360°).

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Oscillator Types Oscillator Frequency-Determining Element & Typical Use RC oscillator (e.g.

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Wien bridge, phase-shift) Uses resistor-capacitor networks for phase shift/frequency selection; suited to audio- frequency range (low to moderate frequencies) LC oscillator (e.g.

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Hartley, Colpitts) Uses an inductor-capacitor tank circuit for frequency determination; suited to radio- frequency (RF) range Crystal oscillator Uses the piezoelectric resonance of a quartz crystal for exceptionally stable, precise frequency generation; used where high frequency stability is essential (e.g. clocks, communication systems) Hartley oscillator:

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LC oscillator using a tapped inductor (or two inductors) with a common capacitor in the feedback network.

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Colpitts oscillator:

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LC oscillator using a tapped capacitor (two capacitors in series) with a single inductor in the feedback network.

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Crystal oscillators offer very high frequency stability and a very high quality factor (Q) compared to RC/LC oscillators, due to the mechanical resonance properties of quartz.

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Waveform Generators Waveform generators produce standard non-sinusoidal waveforms — square, triangular, sawtooth, and pulse — typically built from comparators, integrators, and switching circuits (e.g. astable/monostable multivibrators, 555 timer circuits).

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A function generator is a practical instrument capable of producing multiple waveform shapes (sine, square, triangular) over a range of adjustable frequencies.

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This section covers the classification of amplifier output stages (Class A/B/AB), biasing considerations, power BJTs, transformer-coupled push-pull stages, tuned amplifiers, and operational amplifiers.

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Classification of Output Stages Class Conduction Angle & Efficiency Class A Transistor conducts for the full 360° cycle; low efficiency (max ≈25–50%) but lowest distortion Class B Transistor conducts for half the cycle (180°); higher efficiency (max ≈78.5%) but suffers crossover distortion in push-pull form Class AB Conducts slightly more than 180° (a compromise between A and B); reduces crossover distortion of Class B while retaining better efficiency than Class A Biasing the Class AB Stage A small quiescent (bias) current is set in each output transistor (using diode or V_BE-multiplier biasing) so both devices remain slightly ON near the zero-crossing, eliminating the crossover distortion seen in pure Class B operation.

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Power BJTs & Push-Pull Stages Power BJTs are designed to handle large currents/voltages and dissipate significant heat; they require adequate heat-sinking and derating to avoid thermal runaway.

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Transformer-coupled push-pull stage: two transistors alternately amplify positive and negative half- cycles, combined via a center-tapped output transformer; provides good power efficiency and eliminates DC component in the output, though transformers add bulk/cost and bandwidth limitations.

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Tuned Amplifiers A tuned amplifier uses an LC resonant (tank) circuit as its load, providing high gain over a narrow band of frequencies centered at the resonant frequency — used in RF and IF amplification stages of radio receivers.

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Operational Amplifiers (Op-Amps) An ideal op-amp has infinite open-loop gain, infinite input impedance, zero output impedance, and infinite bandwidth.

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Virtual short concept: in a negative-feedback configuration, the op-amp's two input terminals are assumed to be at the same voltage (though no current flows between them).

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Common op-amp configurations: inverting amplifier, non-inverting amplifier, voltage follower (buffer), summing amplifier, differentiator, and integrator.

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When a circuit is switched, the currents and voltages in inductors and capacitors cannot change instantaneously — the circuit passes through a transient before settling into steady state.

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This section covers the transient and steady-state analysis of R-L, R-C and R-L-C circuits.

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Transient vs Steady State In an inductor, current cannot change instantaneously (iₗ is continuous); in a capacitor, voltage cannot change instantaneously (v_C is continuous).

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At DC steady state (after the transient has died out): an inductor behaves as a short circuit (di/dt = 0, so Vₗ = 0); a capacitor behaves as an open circuit (dv/dt = 0, so I_C = 0).

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R-L Circuit Transient (DC Switching) Current growth (switching onto DC source V): i(t) = (V/R)[1 − e^(−t/τ)], where time constant τ = L/R.

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Current decay (source removed, circuit shorted): i(t) = I₀ · e^(−t/τ).

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After about 5τ, the transient is considered practically complete (<1% of the initial value remaining).

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R-C Circuit Transient Charging: v_C(t) = V[1 − e^(−t/τ)], where time constant τ = RC.

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Discharging: v_C(t) = V₀ · e^(−t/τ).

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Charging current: i(t) = (V/R)·e^(−t/τ) — maximum at t = 0, decaying to zero as the capacitor charges.

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R-L-C Circuit — Damping Characteristic equation of a series RLC circuit:

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Ls² + Rs + 1/C = 0, giving roots s₁,₂ = −α ± √(α² − ω₀²).

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Damping factor (neper frequency): α = R/2L.

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Undamped natural frequency: ω₀ = 1/√(LC).

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Damping ratio: ζ = α/ω₀ = (R/2)√(C/L).

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Condition Response Type α > ω₀ Overdamped — two distinct real roots; decays without oscillation α = ω₀ Critically damped — repeated real root; fastest non-oscillatory decay α < ω₀ Underdamped — complex conjugate roots; oscillatory, decaying (damped sinusoid)

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This section covers network (transfer) functions, poles and zeros, driving-point analysis of one-port networks, and the four standard parameter sets used to characterize linear two-port networks.

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Transfer Functions, Poles & Zeros A network (transfer) function H(s) = Output(s)/Input(s), expressed in the Laplace domain as a ratio of polynomials N(s)/D(s).

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Zeros: values of s that make H(s) = 0 (roots of the numerator).

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Poles: values of s that make H(s) → ∞ (roots of the denominator).

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Pole location and response: poles in the left-half s-plane → stable, decaying response; poles on the jω- axis → sustained oscillation (marginally stable); poles in the right-half plane → unstable, growing response.

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One-Port Passive Networks A one-port (two-terminal) network is fully characterized by a single driving-point impedance Z(s) = V(s)/I(s), or equivalently its admittance Y(s) = 1/Z(s).

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For a physically realizable passive network, Z(s) must be a positive real function (see Section 1.6).

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Two-Port Network Parameters Parameter Set Defining Equations Z (open-circuit impedance) V1 = Z11·I1 + Z12·I2 V2 = Z21·I1 + Z22·I2 (found with the opposite port open) Y (short-circuit admittance) I1 = Y11·V1 + Y12·V2 I2 = Y21·V1 + Y22·V2 (found with the opposite port shorted) ABCD (transmission) V1 = A·V2 − B·I2 I1 = C·V2 − D·I2 (relates input to output; ideal for cascading) h (hybrid) V1 = h11·I1 + h12·V2 I2 = h21·I1 + h22·V2 (mixed variables; standard for transistor models) Reciprocity (passive network, no dependent sources):

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AD − BC = 1; h12 = −h21.

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Z11 = Z22 (also Y11 = Y22, A = D) when the network looks identical from either port.

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Interconnection of Two-Port Networks Connection Combination Rule Series Add the Z-parameter matrices Parallel Add the Y-parameter matrices Cascade Multiply the ABCD matrices, in signal order

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Network synthesis is the reverse of network analysis: given a prescribed function, realize a physical network that produces it.

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This section covers Hurwitz polynomials, positive real functions, Foster and Cauer realizations, and ladder-network synthesis.

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Hurwitz Polynomials A polynomial P(s) is Hurwitz if all of its roots lie in the left half of the s-plane (real parts ≤ 0), with any roots on the jω-axis being simple (non-repeated).

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Necessary (not sufficient) conditions: all powers of s must be present with real, positive coefficients (for a strictly Hurwitz polynomial).

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Hurwitz polynomials appear as the denominator of any stable, physically realizable network function.

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Positive Real Functions (PRF) A function F(s) is positive real if:

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(1) F(s) is real for every real s, and (2) Re[F(s)] ≥ 0 whenever Re[s] ≥ 0 (equivalently, no poles/zeros in the right-half plane and Re[F(jω)] ≥ 0 for all ω).

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Physical significance: the driving-point impedance or admittance of any passive RLC network is always a positive real function — this is the necessary and sufficient condition for a one-port function to be realizable using only passive R, L, C elements.

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Foster & Cauer Forms (LC Network Synthesis) Form Method Resulting Structure Foster I Partial-fraction expansion of Z(s) Series combination of parallel L‖C tank circuits Foster II Partial-fraction expansion of Y(s) Parallel combination of series L-C branches Cauer I Continued-fraction expansion, removing the pole at s = ∞ first Ladder network starting with a series element Cauer II Continued-fraction expansion, removing the pole at s = 0 first Ladder network starting with a shunt element LC network: poles and zeros of Z(s) lie only on the jω-axis and must alternate strictly between poles and zeros.

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RC driving-point impedance: poles and zeros lie only on the negative real axis and alternate, with the critical frequency nearest the origin always a pole.

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RL driving-point impedance: poles and zeros lie only on the negative real axis and alternate, with the critical frequency nearest the origin always a zero.

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Two-Port Synthesis & Ladder Networks Resistively terminated (singly or doubly terminated) LC two-port networks are synthesized to realize a prescribed transfer/attenuation characteristic while working between resistive source and load — the standard approach for filter design.

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A ladder network consists of series and shunt elements alternating along the signal path; both Cauer forms correspond directly to ladder structures.

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Ladder realizations are preferred in practice because they have low sensitivity to component tolerances compared with other (e.g., lattice) topologies.