KCL: The algebraic sum of currents entering any node equals zero — based on conservation of charge.
KVL: The algebraic sum of all voltages around any closed loop equals zero — based on conservation of energy. They hold in AC because at any instant, charge is still conserved and energy is still conserved. At very high frequencies, distributed capacitance between wires can cause apparent violation — this is a lumped-circuit assumption boundary.

With AC: Current flows because capacitor reactance Xc = 1/(2πfC) allows current to pass. The bulb glows.
With DC: After initial transient, capacitor blocks steady-state DC completely. Current drops to zero and the bulb does NOT glow (after brief flash during charging).

Peak Value (Vm): Maximum amplitude of the sinusoidal waveform.
RMS Value: The equivalent DC value that delivers the same average power to a resistive load. For 230 V AC supply, peak voltage = 230 × √2 ≈ 325 V.
We use RMS because it directly represents heating effect — a practical measure for power calculations: P = V²_rms / R.

No, power factor (PF) cannot exceed 1. PF = cos φ where φ is the phase angle between voltage and current. Since cos φ ranges from −1 to +1, and PF is always taken as a magnitude, 0 ≤ PF ≤ 1.

Unity PF (PF = 1): Voltage and current are perfectly in phase. All power drawn is real/active power — no reactive power. This occurs for pure resistive loads.

Any linear network with voltage/current sources and resistors can be replaced by a single Thevenin voltage (Vth) in series with a Thevenin resistance (Rth) at any two terminals. Steps:
1. Remove the load. Find open-circuit voltage → V_th.
2. Kill all independent sources (short voltage sources, open current sources).
3. Find resistance looking into terminals → R_th.
4. Draw equivalent: V_th in series with R_th.

Both are rated for the same voltage (230 V). Using P = V²/R → R = V²/P. The 100 W bulb has lower resistance. Lower resistance allows more current to flow at the same voltage, hence more power dissipation and brighter light. The 60 W bulb has higher resistance — restricts current — dissipates less power.

Resonance occurs when inductive reactance equals capacitive reactance: XL = XC. At resonance:
• Series RLC: Impedance is minimum (= R), current is maximum, PF = 1.
• Parallel RLC: Impedance is maximum, current is minimum, circuit acts as open for AC source.
• Voltage across L and C can be much larger than supply voltage (voltage magnification).

For two identical batteries (EMF = E, internal resistance = r) in parallel: Why? Parallel connection doesn’t add voltage — it increases current capacity and reduces internal resistance. Each battery shares the load current equally, halving the effective internal resistance.

Impedance (Z): Total opposition to AC current flow. It is the complex ratio of voltage to current phasor. Admittance (Y): Reciprocal of impedance. Ease with which AC flows. G = conductance, B = susceptance. Admittance is more convenient for parallel circuit analysis.

Skin effect is the tendency of AC current to flow predominantly at the outer surface (skin) of a conductor rather than uniformly through its cross-section, increasing the effective resistance. As frequency increases → skin depth decreases → current confined to thin surface layer → effective cross-section shrinks → resistance increases.

At 50 Hz, skin depth in copper ≈ 9.35 mm (not significant for standard wires). At MHz range, skin depth is micrometres — critical for RF cables and hollow conductors (waveguides).

In a linear network with multiple independent sources, the total response (voltage or current) equals the algebraic sum of responses due to each source acting alone (others killed).

Cannot be applied when:
• Circuit contains non-linear elements (diodes, transistors in non-linear region).
• For power calculations (P = I²R is non-linear — power is NOT superimposable).
• Circuits with dependent sources (keep dependent sources active while killing independents).

Ideal Voltage Source: Must maintain constant terminal voltage regardless of load current. If internal resistance existed, voltage would drop (V = E − I·r). So r = 0 ensures V_terminal = E always.

Ideal Current Source: Must deliver constant current regardless of load resistance. For constant I, the source must have infinite impedance — otherwise current would divert internally. R_internal = ∞ forces all current into the external circuit.

Time constant τ is the time required for a capacitor to charge to 63.2% of its final value (or discharge to 36.8%).

After 5τ, capacitor is considered fully charged (99.3%). It represents the speed of response of the circuit to a step input.

Reactive Power (Q): Power alternately stored and released by reactive elements (L and C). Measured in VAR (volt-ampere reactive). It does no net useful work — energy oscillates between source and reactive element. However, it is essential for creating magnetic fields in motors, transformers, and maintaining voltage levels in power systems. Without reactive power, inductive loads cannot operate.

Ideal voltage source short-circuited: Infinite current flows (V/0 = ∞). This is physically impossible and destroys real sources. In circuit analysis, we avoid this invalid condition.

Ideal current source open-circuited: Infinite voltage develops across the open terminals (I × ∞ = ∞). Again physically impossible — in reality, arcing or insulation breakdown would occur.

These represent the boundary violations in ideal source models.

Duality means two circuits have the same mathematical form with variables interchanged. Dual pairs: Voltage↔Current, R↔G (conductance), L↔C, KVL↔KCL, Series↔Parallel, Mesh↔Node analysis.

Mutual Inductance (M): When changing current in one coil induces EMF in a neighbouring coil via magnetic flux linkage. k = 1: All flux of coil 1 links coil 2 (ideal transformer). k = 0: No coupling. For air-core transformers: k ≈ 0.4–0.9. For iron-core: k ≈ 0.95–0.99.

Advantages of 3-phase:
• Constant power delivery (no pulsating power as in single-phase).
• 25% less copper needed for same power transmitted (more economical).
• Motors are self-starting with rotating magnetic field — no starting auxiliary needed.
• Higher power density per conductor.
• Neutral wire can be eliminated for balanced loads.

Star (Y): One end of each phase joined at neutral point. 4-wire system possible. Lower voltage per phase — safer insulation requirement. Used for loads requiring neutral (single-phase taps).

Delta (Δ): All three phases form a closed loop. No neutral. Higher phase current — used where high-torque starting required. Common for motors and generators.

Star-Delta starter uses this conversion to reduce starting current of induction motors.

Faraday’s Law: EMF induced is proportional to rate of change of magnetic flux linkage. The negative sign embodies Lenz’s Law: the induced EMF always opposes the change in flux that caused it (conservation of energy). If opposing force didn’t exist, energy would be created from nothing.

Lenz’s Law Significance: Explains back-EMF in motors, braking effect in eddy current brakes, behaviour of transformers under load, and why inductors resist changes in current.

A transformer works on mutual electromagnetic induction. AC in primary winding creates alternating flux in the iron core, which links the secondary winding and induces EMF (Faraday’s Law). Why not DC? DC produces constant (non-changing) flux. Since e = −N(dΦ/dt), zero rate of change means zero induced EMF in secondary. Also, a DC source would cause primary to draw excessive current (only resistance limits it — no back-EMF from reactance), potentially burning the primary winding.

Slip = 0: Rotor runs at synchronous speed. No relative motion between rotating field and rotor → no induced EMF → no rotor current → no torque. Impossible at load (theoretical only).
Slip = 1: Rotor is stationary (Nr = 0). Starting condition — maximum rotor current and flux cutting rate. Torque exists but motor hasn’t started yet. High starting current drawn.
Normal operating slip: 2–8% for squirrel-cage induction motors.

At start, the stator creates a rapidly rotating magnetic field (at synchronous speed). The rotor (with DC excitation) is stationary. The stator field tries to drag the rotor, but reverses direction 50 times per second — resulting in zero average torque (alternating push-pull cancels). The rotor cannot accelerate fast enough to catch the rotating field.

Starting methods:
1. Damper winding (amortisseur): Short-circuited bars on rotor face act like squirrel cage — provide induction motor starting torque. Motor is started as induction motor then excited and pulled into synchronism.
2. Starting with a variable frequency drive (VFD): Slowly ramp up frequency from 0 Hz to 50 Hz — rotor can follow.
3. Use of a small starting motor (pony motor).

Core-Type: Windings surround the core. Core is like a hollow rectangle. Windings on two limbs. Better cooling (windings exposed), easier insulation, used for high-voltage applications.

Shell-Type: Core surrounds the windings. Figure-8 cross-section. Both primary and secondary on central limb — good leakage flux minimisation. Better mechanical support for windings — used for high-current, low-voltage applications and audio transformers.

This is a dangerous condition! With field disconnected, magnetic flux drops to near zero (only residual magnetism remains). To maintain back-EMF = V − I·Ra, the motor must accelerate dramatically. The motor races (runaway condition) — speed becomes dangerously high. Centrifugal forces can mechanically destroy the armature (“armature fly-apart”). This is especially dangerous for series motors running at no load.

When the armature rotates in a magnetic field, it generates an EMF that opposes the supply voltage — this is back-EMF (Eb). Significance:
• Limits armature current — acts as self-regulating mechanism.
• At no load, Eb ≈ V (very little current drawn).
• Under heavy load, Eb drops → Ia increases → more torque.
• Back-EMF is the motor’s way of converting electrical energy to mechanical energy.

Types of losses:
1. Copper losses (I²R): Ohmic losses in winding resistance — reduced by using larger conductors, low-resistance materials, better insulation.
2. Core/Iron losses: Hysteresis (use grain-oriented silicon steel) + Eddy current (use laminated core).
3. Mechanical losses: Friction (bearings), windage (cooling fan) — reduced by good bearings, aerodynamic design.
4. Stray load losses: Due to leakage flux, non-uniform current distribution.

Crawling: Due to harmonic components (particularly 7th harmonic) in flux waveform, motor may achieve a stable running speed at 1/7th of synchronous speed (about 1/7 × Ns). Motor “crawls” at low speed instead of accelerating to rated speed. Caused by slot harmonic interaction. Remedied by skewing rotor slots.

Cogging (Magnetic Locking): Rotor teeth lock with stator teeth due to magnetic attraction when number of rotor slots equals stator slots. Motor refuses to start. Remedied by: using a different number of rotor and stator slots, or skewing rotor bars.

Squirrel Cage: Rotor has short-circuited aluminium/copper bars — simple, rugged, low cost. Low starting torque, high starting current. Cannot add external resistance. Used for pumps, fans, compressors.

Slip-Ring (Wound Rotor): Rotor has 3-phase windings brought out via slip rings. External resistance can be added to rotor circuit → high starting torque, reduced starting current. Speed can be controlled. Used for cranes, elevators, mills requiring high starting torque.

Essential conditions for parallel operation of transformers:
1. Same voltage ratio (turns ratio) — else circulating current flows at no load.
2. Same percentage impedance — for proportional load sharing.
3. Same polarity — to avoid short circuit.
4. Same phase sequence (for 3-phase) — to avoid circulating currents.
5. Same phase displacement (same vector group, e.g., Dyn11 + Dyn11).

DC Shunt Motor: Flux is approximately constant (field winding connected directly across supply). Speed is nearly constant from no-load to full load. Good speed regulation — used for constant-speed drives (lathes, fans, pumps).

DC Series Motor: Field winding in series with armature → Φ ∝ Ia. As load increases, current & flux both increase → torque increases rapidly (T ∝ Ia²). Speed decreases sharply with load. Extremely high starting torque. Used for traction (trains), cranes, elevators.

An auto-transformer has a single winding with a tapping point. Part of the winding is common to both primary and secondary — power transfer is via both conduction AND magnetic induction.

Advantages: Less copper, smaller size and weight, better voltage regulation, lower leakage reactance, higher efficiency for same rating.

Limitations: No electrical isolation between primary and secondary (safety hazard). Cannot be used where isolation is required. High short-circuit current if fault occurs.

The commutator is a mechanical rectifier in DC machines. In a generator, it converts the alternating EMF generated in armature conductors into unidirectional (DC) output. In a motor, it reverses current in armature coils at the right moment to maintain continuous torque in one direction.

Without commutator, output would be AC (alternating) — the machine would behave as an AC machine. Brushes and commutator are the major maintenance points in DC machines — subject to sparking, wear, and require periodic replacement.

When load current flows through armature, it creates its own magnetic field that distorts and weakens the main field flux — this effect is armature reaction. Effects: flux weakening (reduces torque/EMF), flux distortion (shifts magnetic neutral axis), sparking at brushes.

Compensation methods:
1. Brush shift (advance or retard by geometric neutral axis shift — temporary fix).
2. Interpoles (commutating poles between main poles): Cancel armature reaction in commutation zone.
3. Compensating winding: Embedded in pole faces, connected in series with armature — best method for large machines.

The Buchholz relay is a gas-actuated protection device used in oil-immersed transformers to detect internal faults. It is installed in the pipe connecting transformer tank to the conservator tank.

Working: Any internal fault (arc, insulation deterioration) decomposes transformer oil and generates gas. Gas bubbles rise and collect in the Buchholz relay. Minor faults: float activates alarm. Severe faults: oil surge actuates trip coil, disconnecting the transformer.
It detects: inter-turn faults, core faults, partial discharge, oil level drop.

Per-unit (pu) value = Actual quantity / Base quantity.
Typically: Base VA (MVA), Base Voltage (kV) are chosen; Base current and Base impedance are derived. Advantages:
• Eliminates the need to carry actual transformers ratios in calculations (transformer appears as series impedance).
• Simplifies analysis of interconnected systems at different voltage levels.
• Typical per-unit values of machines fall in similar ranges — easy comparison.
• Reduces computational complexity in power flow, fault analysis.

Fuse: Sacrificial element — melts and destroys itself on overcurrent. Single use. Fast acting. Low cost. No moving parts. Used for small loads, branch circuit protection.

Circuit Breaker (CB): Reusable switching device — trips on fault and can be reset. Can interrupt at specific current values. Adjustable trip settings. Suitable for high voltages and fault currents. Used in distribution panels, transmission line protection.

CB Types: Air CB (low voltage), Oil CB, SF6 CB (high voltage), Vacuum CB (medium voltage — maintenance-free, used in DRDO/ISRO applications).

Differential protection compares the current entering the protected equipment with the current leaving it. Under normal conditions, (scaled) input current ≈ output current → differential current ≈ 0 → no trip. During an internal fault, current difference becomes significant → relay trips. Advantages: Highly sensitive to internal faults, immune to external faults (through faults), fast operation, selective (only trips faulted equipment).

Challenge: Magnetising inrush current (at switch-on) has large differential component but no fault — solved using harmonic restraint (2nd harmonic blocking).

Ferranti Effect: In long transmission lines under no-load or light load conditions, the receiving-end voltage is higher than the sending-end voltage. Caused by the charging current of the line’s distributed capacitance flowing through the line inductance, creating a voltage rise.

Occurs prominently in: Long EHV cables (underground), long overhead lines at no load (>300 km), long HVDC cables. Mitigation: Shunt reactors at receiving end or along the line to absorb the leading reactive current and suppress the voltage rise.

Earthing (Grounding) connects metallic parts of electrical installation to earth at zero potential. Purposes:
1. Safety: Provides a low-resistance path for fault current → trips protective devices → prevents electric shock.
2. Voltage stabilization: Limits overvoltages (lightning, switching surges).
3. EMI reduction: Shields against electromagnetic interference.
4. Lightning protection: Dissipates lightning energy to earth safely.

Types: Solid earthing (neutral directly connected), resistance earthing, reactance earthing, isolated neutral.

BJT (Bipolar Junction Transistor): Current-controlled device. Base current controls collector current. Uses both holes and electrons (bipolar). Low input impedance. Better for high-speed, high-current applications. Faster switching at moderate voltages.

FET (Field Effect Transistor): Voltage-controlled device. Gate voltage controls drain current. High input impedance (MOSFET gate draws virtually zero current). Lower noise. Better for signal amplification, analog switches, op-amps. MOSFET dominates digital IC design. Choose BJT: High-current linear amplifiers, audio power stages, high-speed ECL logic.
Choose MOSFET: Digital logic, power electronics, low-power circuits, RF switching.

Ideal Op-Amp Properties:
• Infinite open-loop gain (A = ∞)
• Infinite input impedance (Zin = ∞ → Iin = 0)
• Zero output impedance (Zo = 0)
• Infinite bandwidth (BW = ∞)
• Zero offset voltage, zero noise
• CMRR = ∞ (perfectly rejects common-mode signals)

Virtual Ground (Virtual Short): In a closed-loop inverting/non-inverting op-amp with negative feedback, since gain is infinite, the differential input voltage must be essentially zero (V+ ≈ V−). The inverting input is not actually connected to ground — but it appears to be at ground potential → “virtual ground.”

CMRR (Common Mode Rejection Ratio): Ratio of differential-mode gain to common-mode gain. A high CMRR means the op-amp amplifies the differential signal strongly but rejects signals common to both inputs (noise, interference). For a 741 op-amp: CMRR ≈ 90 dB typically.

Why important? In sensor applications (ECG, instrumentation amplifiers, strain gauges), the signal is small and differential, while noise (50 Hz hum, electromagnetic interference) appears equally on both inputs as common-mode. High CMRR ensures only the real signal is amplified.

Regular PN Diode (Reverse Bias): Very small leakage current flows. If reverse voltage exceeds breakdown voltage, catastrophic (destructive) breakdown occurs — device damaged.

Zener Diode: Heavily doped. Undergoes controlled, reversible breakdown at a precise voltage (Zener voltage, Vz). After breakdown, voltage remains nearly constant despite large changes in current. This is its operating point — used intentionally. Uses: Voltage reference, voltage regulator, overvoltage protection, clipping circuits.

A PLL is a feedback control system that locks the phase of its output oscillator to the phase of an input signal. Components: Phase Detector (PD), Low Pass Filter (LPF), Voltage Controlled Oscillator (VCO).

The PD compares input phase with VCO output phase → generates error voltage. LPF smooths the error → drives VCO until output frequency/phase matches input (locked state). Applications: Frequency synthesis (radio tuners), FM demodulation, clock recovery in communication, motor speed control, GPS receivers, ISRO satellite transponders, carrier recovery in modems.

Gain-Bandwidth Product (GBW or GBP) is a constant for a given op-amp. As gain increases, bandwidth decreases proportionally. So if GBW = 1 MHz and gain is configured to 100 (40 dB), the usable bandwidth is only 10 kHz. At unity gain (Av = 1), bandwidth = 1 MHz (full GBW).

Linear Power Supply: Uses a pass transistor in its active (linear) region to drop excess voltage. Simple, low noise, very stable. Low efficiency (40–60% typical — excess energy dissipated as heat). Heavy (large transformer). Good for analog, audio, RF circuits.

Switching Power Supply (SMPS): Transistor switches ON/OFF at high frequency (100 kHz–MHz) — inductor/capacitor filter output. High efficiency (85–95%). Compact, lightweight. Generates electromagnetic interference (EMI). Used almost everywhere: computers, phones, adapters, DRDO radar PSUs.

A comparator compares input with a single reference threshold. If input > reference → high output; else low. Very sensitive to noise — noisy input near threshold causes rapid output toggling (multiple transitions).

A Schmitt Trigger uses hysteresis — it has two different thresholds: upper trigger point (UTP) and lower trigger point (LTP). Output switches HIGH only when input exceeds UTP, and switches LOW only when input falls below LTP. This hysteresis makes the Schmitt trigger immune to noise — it won’t oscillate from small noise around the threshold. Used in: signal conditioning, wave shaping, touch switches, oscillators.

A rectifier converts AC to pulsating DC by allowing current flow in one direction only. Half-Wave: Uses 1 diode. Only positive half-cycle used. Wasteful, high ripple, poor efficiency (40.6%). Simple, cheap.

Full-Wave (Bridge): Uses 4 diodes. Both half-cycles used. Lower ripple, higher efficiency (81.2%), better regulation. Ripple frequency = 2×supply frequency (100 Hz for 50 Hz input).

A standard differential amplifier has low and unequal input impedance — loading the sensor source and having poor CMRR.

An Instrumentation Amplifier (In-Amp) uses three op-amps: two input buffer stages (very high input impedance, no loading) feeding a third difference stage. Gain is set by a single resistor. Properties: Very high input impedance (>10 GΩ), excellent CMRR (>80 dB), precise adjustable gain, very low offset and noise.

A flip-flop is a bistable multivibrator — a 1-bit memory element that stores one of two states.

SR (Set-Reset): S=1,R=0 → Q=1; S=0,R=1 → Q=0; S=R=1 → forbidden (undefined) state.
D (Data): Q follows D input at clock edge. Eliminates forbidden state. Used in registers, buffers.
JK: Like SR but J=K=1 toggles output. Universal FF — can be configured as SR, D, or T.
T (Toggle): T=1 → output toggles on each clock; T=0 → holds. Used in counters.

Setup Time (tsu): Minimum time data must be stable BEFORE the active clock edge for reliable capture.
Hold Time (th): Minimum time data must remain stable AFTER the active clock edge.

Violation: If setup or hold time is violated, the flip-flop enters a metastable state — an unstable equilibrium where the output is neither valid HIGH nor valid LOW. It will eventually resolve, but unpredictably. This can cause system malfunction — timing errors, spurious operations, or system crash.

Mitigation: Synchronizers, margin in timing closure, slower clock, careful PCB layout to minimise propagation delays.

SRAM (Static RAM): Uses 6 transistor bistable latch per bit. Holds data as long as powered (no refresh). Fast (5–25 ns), expensive, high power, low density. Used for cache memory (L1, L2, L3).

DRAM (Dynamic RAM): Uses 1 transistor + 1 capacitor per bit. Data stored as charge on capacitor. Capacitor leaks charge → must be refreshed periodically (every 64 ms typically).
Slower, cheaper, lower power per bit, high density. Used for main memory (DDR4, DDR5 RAM).

FPGA (Field Programmable Gate Array): Reconfigurable hardware — contains configurable logic blocks (CLBs), interconnects, and I/O cells that can be programmed after manufacturing using HDL (VHDL/Verilog). Programmed via bitstream loaded from flash. Re-programmable anytime.

ASIC (Application-Specific IC): Custom chip fixed at fabrication — cannot be changed. Much faster, lower power, lower cost at volume. High NRE (Non-Recurring Engineering) cost for design & fabrication (crores of rupees).

Microcontroller (MCU): Fixed processor executing software — sequential, flexible via programming but fundamentally software-based. Cannot do true parallel hardware operations.

2’s complement is the standard method for representing signed integers in binary. To get 2’s complement: invert all bits (1’s complement) then add 1. Why important?
• Subtraction becomes addition (A − B = A + (−B) in 2’s complement) → single hardware adder handles both add and subtract.
• Single representation for zero (unlike 1’s complement which has +0 and −0).
• Range for 8 bits: −128 to +127 (2^7 negative values, 2^7−1 positive).
• Hardware simplicity — all modern processors, microcontrollers use 2’s complement arithmetic.

The transfer function G(s) is the ratio of the Laplace transform of output Y(s) to input X(s), with all initial conditions zero. It carries: natural frequencies (poles), zeros, gain, stability information (pole locations in s-plane), steady-state gain (G(0)), frequency response (Bode plot via G(jω)).

Poles: roots of denominator — determine natural modes and stability.
Zeros: roots of numerator — shape the response, create notches in frequency response.

The Routh-Hurwitz criterion determines stability of a system without computing actual poles. A system is stable if all elements in the first column of the Routh array are positive (same sign).

For characteristic polynomial: a₀s³ + a₁s² + a₂s + a₃ = 0 Number of sign changes in first column = number of RHP poles.
If any element is zero or negative → system is unstable.

Phase Margin (PM): How much additional phase lag the system can tolerate before becoming unstable. PM = 180° + ∠G(jω) at the gain crossover frequency (where |G(jω)| = 0 dB). Gain Margin (GM): How much gain can be increased before instability. GM = 1/|G(jω_pc)| at phase crossover frequency (where phase = −180°). In dB: GM = −20log|G(jω_pc)|.

Stable if: PM > 0° and GM > 1 (GM > 0 dB). Good design: PM ≈ 45°–60°, GM ≥ 6 dB — provides robustness to plant uncertainty.

Proportional (P): Output proportional to error. Reduces rise time and steady-state error. Large Kp → oscillation, instability.
Integral (I): Eliminates steady-state error (accumulates past error). Slows response, can cause windup. Used for type-1 systems that need zero steady-state error for step input.
Derivative (D): Anticipates future error (rate of change). Improves stability, reduces overshoot. Sensitive to noise — usually filtered in practice.

PID tuning methods: Ziegler-Nichols, Cohen-Coon, IMC, model-based, auto-tune.

Steady-state error (e_ss) is the difference between desired output and actual output as t→∞. Determined by system type (number of open-loop poles at origin). Error constants: Kp (position), Kv (velocity), Ka (acceleration) determine the magnitude of steady-state error.
Adding integral term (adding pole at origin) increases system type by 1 — eliminates steady-state error for that input class.

Nyquist Theorem: A bandlimited signal with maximum frequency f_max can be perfectly reconstructed if sampled at a rate f_s ≥ 2×f_max. Aliasing: If sampling rate is too low (f_s < 2×f_max), high-frequency components appear as low-frequency components in the sampled signal — the frequency “folds” and aliases. Signal is distorted and cannot be recovered.

Prevention: Anti-aliasing filter (low-pass filter) applied before sampling to remove frequency components above f_s/2 (Nyquist frequency). Then sample at ≥ Nyquist rate.

AM: Carrier amplitude varies with message signal. Constant carrier frequency. Bandwidth = 2×f_m. Susceptible to noise/interference (noise affects amplitude). Simple receiver circuit. Used: MW/SW broadcast radio, aircraft comms (VHF AM).

FM: Carrier frequency varies with message signal. Constant amplitude. Bandwidth = 2(Δf + f_m) (Carson’s rule). Much better noise immunity (noise appears as amplitude variation — limiters remove it). Better audio quality. Used: VHF/FM broadcast (88–108 MHz), two-way radios, DRDO tactical comms.

Fourier Transform: Decomposes a time-domain signal into its constituent frequencies. Exists only for signals that satisfy Dirichlet conditions (absolutely integrable). Shows frequency spectrum — magnitude and phase vs frequency (real axis only, s = jω). Laplace Transform: Extension of Fourier — uses complex frequency s = σ + jω. Applicable to a wider class of signals (including growing exponentials). Used for system analysis — transfer functions, stability (s-plane). Handles initial conditions. Fourier is a special case when σ = 0.

A Wheatstone bridge consists of four resistors in a diamond configuration with a voltage source and a galvanometer. Bridge is balanced when the galvanometer reads zero. Accuracy advantage: At balance, measurement is independent of supply voltage variations and contact resistance. Only relative resistance ratios matter. Galvanometer null detection avoids loading errors.
Used for: Strain gauge measurement, RTD (temperature) sensing, precision resistance comparison.

Accuracy: How close measurement is to the true value. Affected by systematic/calibration errors.
Precision: Repeatability — how closely repeated measurements agree with each other. Affected by random errors. A measurement can be precise but inaccurate (systematically wrong).
Resolution: Smallest change in input that produces detectable output change. For 12-bit ADC with 3.3V range: resolution = 3.3/4096 ≈ 0.8 mV.
Sensitivity: Ratio of change in output to change in input. Units: V/°C, mV/Pa, etc. Higher sensitivity → easier to detect small changes.

PWM: A fixed-frequency digital signal whose ON-time (duty cycle) is varied to control average power delivered to a load. Motor speed control: PWM signal fed to power MOSFET/IGBT switching the motor supply. At high frequency (>1 kHz), motor inductance smooths the current. Varying duty cycle from 0–100% controls average voltage/current → motor speed varies continuously.
Advantages: High efficiency (transistor is either fully ON or fully OFF — minimal power dissipation), precise control, simple digital control from microcontroller PWM peripheral.

Energy Spectral Density (ESD) S_E(f): Used for energy signals (finite total energy, finite duration — e.g., a pulse). ESD = |F(f)|² where F(f) is the Fourier transform. Total energy = ∫S_E(f)df (Parseval’s theorem).

Power Spectral Density (PSD) S_x(f): Used for power signals (infinite duration, finite average power — e.g., noise, periodic signals, random signals). PSD = Fourier transform of autocorrelation function (Wiener-Khinchin theorem). Total power = ∫S_x(f)df.

CT (Current Transformer): Steps down high current for measurement. Primary has few turns (1–5) in series with the power line. Secondary gives 5A or 1A to meters/relays. Primary current ≈ fixed by system — CT secondary current mirrors primary at ratio N1/N2.

PT (Potential Transformer): Steps down high voltage for measurement. Primary connected across power line. Secondary gives 110V or 63.5V to meters. Standard operating condition: secondary close to open circuit (high burden impedance).

Why CT secondary MUST NOT be open-circuited: CT primary current is forced by the system (line current) and cannot change. With secondary open, no ampere-turn balance → all primary MMF drives the core → enormous flux → very high induced EMF in secondary (kV range) → insulation breakdown, fire hazard, fatal voltage.

OFDM (Orthogonal Frequency Division Multiplexing): Transmits data by splitting it across many closely spaced orthogonal subcarriers, each carrying a low data rate. Implemented efficiently using IFFT (transmitter) and FFT (receiver).

Advantages:
• Robust against multipath fading (cyclic prefix eliminates ISI).
• Efficient spectral use — subcarriers are orthogonal (don’t interfere).
• Easy equalization — each subcarrier is flat fading (simple 1-tap equalizer).
• Scales to very high bandwidths (LTE: 20 MHz, 5G NR: up to 400 MHz).

Seebeck Effect: When two dissimilar metals are joined at two junctions maintained at different temperatures, an EMF is generated proportional to the temperature difference. This is the thermoelectric effect. Cold Junction Compensation (CJC): Thermocouple output depends on BOTH hot and cold junction temperatures. The cold junction (reference junction at measurement electronics) must be at a known temperature. CJC adds a correction voltage (measured by RTD or thermistor at cold junction) to give true hot junction temperature.

Types: K (Chromel-Alumel, −200 to 1260°C), J (Iron-Constantan, 0–760°C), T (Copper-Constantan, cryogenic), R/S (Platinum — high temperature).

Analog Filter: Operates on continuous-time signals using R, L, C components or op-amps. Butterworth, Chebyshev, Bessel types.
Digital Filter: Operates on discrete-time sampled data using arithmetic operations (multiplications, additions, delays). Implemented in DSP, FPGA, microcontroller.

FIR (Finite Impulse Response): Output depends only on past and current inputs — no feedback. Always stable. Linear phase (constant group delay — important for audio and data comms). Can require many taps (hundreds) for sharp cutoff.

IIR (Infinite Impulse Response): Includes feedback (poles). Similar to analog filters (bilinear transformation). Fewer coefficients needed for same selectivity. Non-linear phase. Can be unstable if not designed carefully. Based on Butterworth, Chebyshev, Elliptic designs.

ADC (Analog-to-Digital Converter): Converts continuous analog signal to discrete digital code. Resolution in bits (8, 12, 16, 24 bit). Sampling rate in samples/second (kSPS, MSPS, GSPS).
DAC (Digital-to-Analog Converter): Converts digital code to analog voltage/current. Output waveform must be filtered (reconstruction filter) to remove staircase quantisation artifacts.

Key parameters:
• Resolution (N bits) → 2^N levels, LSB = V_ref/2^N
• INL (Integral Non-Linearity) — deviation from ideal transfer function
• DNL (Differential Non-Linearity) — step size uniformity
• SNR (Signal-to-Noise Ratio): ideal SNR = 6.02N + 1.76 dB for N-bit ADC
• SINAD, SFDR, ENOB (Effective Number of Bits)

Quantisation noise arises because ADC maps a continuous range of analog values to discrete digital codes. The error (difference between actual and quantised value) is at most ±½ LSB. Quantisation noise is modelled as uniformly distributed noise with RMS value = LSB/√12. For a 12-bit ADC: SQNR ≈ 74 dB. For 16-bit: SQNR ≈ 98 dB.
Oversampling and noise-shaping (sigma-delta ADCs) push quantisation noise out of the signal band, allowing higher ENOB than the raw bit count suggests.

EMC (Electromagnetic Compatibility): Ability of a device to function satisfactorily in its electromagnetic environment without causing unacceptable interference to others and without being susceptible to interference from others.

EMI (Electromagnetic Interference): The interference itself — disturbances that degrade performance. Can be conducted (through power/signal wires) or radiated (through air).

EMS (Electromagnetic Susceptibility): The susceptibility of a device to EMI — how much interference it can withstand.

Standards: CISPR, IEC 61000, MIL-STD-461 (defence), DO-160 (aerospace). India: BIS IS 13252.

In space and nuclear environments, high-energy particles (protons, heavy ions, electrons, gamma rays) damage semiconductor devices causing:
1. TID (Total Ionising Dose): Cumulative charge buildup in oxide layers → shifts threshold voltages, increases leakage current, degrades performance.
2. SEE (Single Event Effects): Single particle strikes → SEU (bit flip in memory), SEL (latch-up — destructive), SET (transient glitch).
3. Displacement damage: Crystal lattice defects from neutron bombardment.

Rad-hard techniques: SOI (Silicon-on-Insulator) process, guard rings, triple-redundancy (TMR), EDAC for memories, specialised fab processes.

SCADA (Supervisory Control And Data Acquisition): Distributed industrial control system for monitoring and controlling field devices at remote locations from a central control room.

Components: RTUs (Remote Terminal Units)/PLCs at field, communication network (fiber, radio, satellite), SCADA master station (HMI, historian, alarm management).

In NTPC power plants: SCADA monitors turbine parameters (speed, temperature, steam pressure), generator outputs (MW, MVAR, frequency), switchyard breaker status, transformer oil temperature, alarms, and provides operator control of set points.

Spread spectrum intentionally spreads the signal energy over a much wider bandwidth than the minimum required. Techniques: DSSS (Direct Sequence SS — XOR with PN code), FHSS (Frequency Hopping SS).

Processing gain: BW_spread/BW_original — determines how much the SNR is improved at the receiver (knowing the spreading code) vs a jammer who doesn’t know the code. Why GPS/military use it:
• Low probability of intercept (LPI): signal below noise floor — hard to detect.
• Anti-jam: jammer must cover entire spread bandwidth — much higher power needed to jam.
• Multiple access: CDMA — multiple users share same bandwidth using different PN codes.
• Precise ranging: GPS uses C/A and P(Y) DSSS codes for timing precision.

Hall Effect: When current flows through a conductor (or semiconductor) in the presence of a perpendicular magnetic field, a voltage (Hall voltage) develops perpendicular to both current and field. Current measurement (clamp-on): A Hall sensor placed in the air gap of a magnetic core that surrounds the current-carrying conductor. The magnetic field in the gap is proportional to the current. Output voltage is proportional to current — no galvanic connection needed.

Advantages: Isolation (high-voltage measurement safely), DC and AC capable, fast response, wide range.

SAW (Surface Acoustic Wave) Filter: Uses piezoelectric substrate (quartz, lithium niobate). Input electrode converts electrical signal to surface acoustic wave; output electrode reconverts. The device passband is determined by electrode geometry — extremely precise, stable center frequency.

Frequency range: 10 MHz to 3 GHz. Very sharp selectivity. Extremely small size (mm scale). No tuning needed. Excellent out-of-band rejection. Low insertion loss. Mass-produced at low cost.

Why preferred in RF: LC filters at GHz require impractically small inductors. SAW achieves high-Q bandpass filtering without external components — ideal for mobile phones, GPS, satellite receivers.

The Haversine formula calculates the great-circle distance between two points on a sphere given their latitudes and longitudes. Why an ISRO EE needs it:
• Antenna pointing: Ground station engineers use it to compute azimuth and elevation angles for dish antennas tracking satellites — critical for S-band and X-band telemetry/telecommand links.
• Link budget: Free-space path loss = 20log(d) + 20log(f) + 92.4 dB — distance d from Haversine.
• Doppler shift calculation: Rate of change of distance → Doppler frequency → important for tracking PSLV/GSLV during ascent.
• NavIC/GPS receiver algorithms: Navigation solution uses spherical geometry to compute position from pseudoranges.