Basic concept

Every electrical circuit, from a mobile charger to a 132 kV substation, is built on a handful of basic ideas: voltage pushes charge, current is the flow of charge, resistance opposes that flow, and power is the rate at which energy is delivered. This topic covers those quantities, the materials that conduct or block current, how elements combine in series and parallel, the star-delta transformation, Kirchhoff's laws, and the vocabulary used to classify circuits (linear/non-linear, bilateral/unilateral, active/passive). These definitions appear again and again in NEC MCQs, often as one-line "which of the following is…" questions.

Key concepts

• Electric charge (Q): measured in coulomb (C). One electron carries $1.602\times10^{-19}$ C.
• Current (I): rate of flow of charge, $I = dQ/dt$, in ampere (A). Conventional current flows from + to − (opposite to electron flow).
• Voltage / potential difference (V): work done per unit charge to move it between two points, $V = W/Q$, in volt (J/C). EMF is the voltage produced by a source; potential difference is the drop across an element.
• Ohm's law: at constant temperature and physical condition, the current through a metallic conductor is proportional to the voltage across it, $V = IR$. It does not apply to non-linear elements such as diodes, arc lamps, or electrolytes.
• Resistance: $R = \rho L/A$. It increases with length, decreases with area. For metals R rises with temperature (positive temperature coefficient); for carbon, semiconductors and insulators it falls (negative coefficient).
• Power and energy: power is the rate of doing work, $P = VI$ (W). Energy is $W = Pt$; the commercial unit is the kWh ("unit"), $1\text{ kWh} = 3.6\times10^6$ J.

Materials

Class	Resistivity (approx.)	Examples
Conductors	$10^{-8}$ Ω·m	silver (best), copper, aluminium, gold
Semiconductors	$10^{-5}$–$10^{3}$ Ω·m	silicon, germanium
Insulators	$>10^{8}$ Ω·m	mica, PVC, porcelain, rubber, glass

Copper is used for windings and cables; aluminium for overhead lines (lighter, cheaper). Insulators are judged by dielectric strength (kV/mm) and insulation resistance.

Series and parallel

• Series: same current, voltages add, $R_s = R_1 + R_2 + \dots$; voltage divides in proportion to resistance.
• Parallel: same voltage, currents add, $1/R_p = 1/R_1 + 1/R_2 + \dots$; $R_p$ is always less than the smallest branch. Two resistors: $R_p = R_1R_2/(R_1+R_2)$. Current divides inversely with resistance.

Star-delta conversion lets you simplify bridge-type networks that are neither series nor parallel.

Kirchhoff's laws

• KCL (current law): algebraic sum of currents at a node is zero — conservation of charge.
• KVL (voltage law): algebraic sum of voltages around any closed loop is zero — conservation of energy.

Circuit classification

• Linear: parameters (R, L, C) are constant regardless of voltage/current; obeys superposition and homogeneity. Non-linear: parameters change with V or I (diode, iron-core inductor, thermistor).
• Bilateral: same behaviour for current in either direction (resistor, wire). Unilateral: properties change with direction (diode, rectifier).
• Active: contains energy sources (voltage/current sources, transistors modelled with dependent sources). Passive: only absorbs or stores energy (R, L, C).
• Lumped vs distributed: lumped elements are physically separable (a resistor); distributed parameters are spread along length (transmission line).

High-yield points

• KCL follows from conservation of charge; KVL from conservation of energy.
• Balanced delta to star: $R_Y = R_\Delta/3$ (a 30 Ω delta becomes a 10 Ω star).
• $1\text{ kWh} = 3.6\times10^6$ J = one commercial "unit".
• Stretching a wire to n times its length (volume constant) multiplies R by $n^2$.
• Silver is the best conductor; metals have a positive temperature coefficient, semiconductors and insulators a negative one.
• Diode: non-linear and unilateral; resistor: linear, bilateral and passive.

Formulas

$V = IR,\quad P = VI = I^2R = \frac{V^2}{R},\quad W = Pt$

$R = \frac{\rho L}{A},\qquad R_T = R_0(1+\alpha_0 T)$

where $\rho$ = resistivity (Ω·m), L = length (m), A = area (m²), $\alpha_0$ = temperature coefficient at 0 °C (/°C).

Delta to star (delta sides $R_{ab}, R_{bc}, R_{ca}$): $R_a = \frac{R_{ab}R_{ca}}{R_{ab}+R_{bc}+R_{ca}}$ Star to delta: $R_{ab} = R_a + R_b + \frac{R_aR_b}{R_c}$ For a balanced network: $R_Y = R_\Delta/3$ and $R_\Delta = 3R_Y$.

Worked example

A 230 V, 1 kW heater runs 3 hours a day for 30 days. Find its resistance and the energy consumed.

1. $R = V^2/P = 230^2/1000 = 52\,900/1000 = 52.9$ Ω.
2. Current $I = P/V = 1000/230 = 4.35$ A.
3. Energy $= 1\text{ kW} \times 3\text{ h} \times 30 = 90$ kWh (90 units).

If the same heater is run at 115 V (half voltage), power becomes $115^2/52.9 = 250$ W — one quarter, since $P \propto V^2$.

Exam tips

• Halving the length of a wire halves R; stretching a wire to double its length (volume constant) makes R four times.
• Equal resistors: n in series gives nR, n in parallel gives R/n; ratio series:parallel = $n^2$.
• Balanced delta of 30 Ω each becomes a star of 10 Ω each — the most common star-delta MCQ.
• A diode is non-linear AND unilateral; a resistor is linear, bilateral and passive.
• KCL is based on conservation of charge, KVL on conservation of energy — do not swap them.

Exam-style questions

From Calmon's question bank, which has 25 questions on this topic.

1. Two lamps rated 60 W, 220 V and 100 W, 220 V are connected in series across a 220 V supply. Which statement is correct?

• A. Both glow with equal brightness
• B. The 100 W lamp glows brighter
• C. The 60 W lamp glows brighter
• D. Both glow at their full rated power

Answer: C. The 60 W lamp has higher resistance ($R = V^2/P$); in series the same current flows, so it dissipates more power ($I^2R$) and glows brighter.

2. Ohm's law ($V = IR$) holds for a metallic conductor provided that:

• A. The conductor is very long
• B. Its temperature and other physical conditions remain constant
• C. The current is alternating
• D. The applied voltage is very high

Answer: B. Ohm's law states that current is proportional to voltage only while temperature and other physical conditions of the conductor stay unchanged.

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