DC & AC Fundamentals Notes

Last reviewed 16 Sept 2026 · 8 min read

Direct current and alternating current

Feature	DC	AC
Direction	Constant	Reverses periodically
Sources	Batteries, solar cells, DC generators, rectifiers	Alternators, inverters
Transmission	Needs HVDC converters for long distances	Voltage easily changed by transformers — economical transmission
Uses	Electronics, electrolysis, traction (some), charging	Power systems, domestic and industrial supply

In India, the standard supply is 50 Hz, nominally 230 V single-phase and 400 V three-phase (line).

A coil of $N$ turns rotating with angular velocity $ω$ in a uniform magnetic field generates

e = E_{m} sin ω t, E_{m} = N B A ω

Cycle, time period $T$ (s), frequency $f = 1/ T$ (Hz), angular frequency $ω = 2 π f$ (rad/s).
For a machine with $P$ poles running at $N$ rpm: $f = \frac{P N}{120}$ .
Amplitude (peak value) $V_{m}$ .

FormulaSinusoidal waveform relations

Average value (over a half cycle): $V_{a v g} = \frac{2 V _{m}}{π} = 0.637 V_{m}$ (over a full cycle = 0)

RMS (effective) value: $V_{r m s} = \frac{V _{m}}{2} = 0.707 V_{m}$ (the DC value producing the same heating effect)

Form factor $= \frac{V _{r m s}}{V _{a v g}} = 1.11$

Peak (crest) factor $= \frac{V _{m}}{V _{r m s}} = 1.414$

Waveform	RMS	Average (half cycle)	Form factor	Peak factor
Sine	$V_{m} / 2$	$2 V_{m} / π$	1.11	1.414
Square	$V_{m}$	$V_{m}$	1	1
Triangular	$V_{m} / 3$	$V_{m} /2$	1.155	1.732
Half-wave rectified sine (full cycle)	$V_{m} /2$	$V_{m} / π$	1.57	2

Voltmeters and ammeters read RMS values; the domestic 230 V is RMS (peak ≈ 325 V).

Phase difference — angular displacement between two waveforms of the same frequency; leading and lagging.
Phasor — a rotating vector of length equal to the RMS value (commonly) at an angle equal to phase; rotates anticlockwise at ω.
Phasors are added as vectors (or complex numbers): rectangular $a + j b$ , polar $r ∠ θ$ .

Element	Opposition	Phase relation	Power
Pure resistance R	$R$	Current in phase with voltage	$P = V I$ (all real)
Pure inductance L	Inductive reactance $X_{L} = 2 π f L$ (increases with f)	Current lags voltage by 90°	Average power zero (energy stored and returned)
Pure capacitance C	Capacitive reactance $X_{C} = \frac{1}{2 π f C}$ (decreases with f)	Current leads voltage by 90°	Average power zero

At DC ( $f = 0$ ): inductor acts as short circuit (steady state), capacitor as open circuit.
Mnemonic "CIVIL": in a Capacitor, I leads V; V leads I in an inductor (L).

Capacitance $C = \frac{Q}{V}$ (farad); parallel plate: $C = \frac{ε _{0} ε _{r} A}{d}$ , $ε_{0} = 8.854 \times 1 0^{- 12}$ F/m.
Energy stored $W = \frac{1}{2} C V^{2}$ .
Series: $\frac{1}{C _{e q}} = \sum \frac{1}{C}$ ; parallel: $C_{e q} = \sum C$ (opposite of resistors).
Dielectric strength, leakage; types — ceramic, electrolytic, film, mica.

Time constant $τ = R C$ .
Charging: $v_{C} = V (1 - e^{- t / R C})$ — reaches 63.2% of final value in one τ; practically complete in about 5τ.
Discharging: $v_{C} = V_{0} e^{- t / R C}$ — falls to 36.8% in one τ.

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