Basic Concepts

An electric circuit is an interconnection of electrical elements through which current can flow. Every circuit contains at least one source (battery, generator, signal generator) and at least one element that uses or stores energy (resistor, capacitor, inductor, lamp, motor).

Circuit analysis lets us predict how a given network will behave — what currents will flow, what voltages will appear, how much power will be dissipated — before we build it. The mathematical tools we use throughout this course rest on three foundations:

• Ohm's law — relates voltage, current, and resistance in a passive element.
• Kirchhoff's laws (KCL, KVL) — express the conservation of charge and energy at every node and around every loop.
• The element i-v relationships — describe what each component does (linear for R, L, C; nonlinear for diodes, transistors, etc.).

Engineering uses the International System of Units (SI). The six base units that show up everywhere in electrical engineering are:

Every other electrical quantity is derived from these. Voltage, charge, energy, and power are all combinations of the base units.

Decimal multiples appear constantly. Memorise these prefixes — questions and component values will assume you read them fluently:

Charge is the basic electrical quantity. The SI unit is the coulomb (C). One electron carries $e=1.602\times 10^{-19}$ C, so it takes about $6.24\times 10^{18}$ electrons to make 1 C.

Current is the rate of flow of charge past a point:

$$i=\frac{dq}{dt}$$

The SI unit is the ampere (A) where $1\text{ A}=1\text{ C/s}$. When current is constant in time (DC) the integral form is simpler:

$$Q=i\cdot t$$

Conventional current points in the direction positive charge would move — opposite to the actual flow of electrons in a metal wire. Every formula in this course uses conventional current. Just remember: the labelled arrow on a schematic shows where positive charge moves, electrons go the other way.

Voltage (also called potential difference or electromotive force, EMF) is the energy required to move a unit of positive charge from one point to another:

$$v=\frac{dw}{dq}$$

Unit: volt (V), where $1\text{ V}=1\text{ J/C}$. A 12-V battery does 12 J of work for every coulomb of charge it pushes through the circuit.

Voltage is *always* a difference between two points. When we say 'the voltage at node A is 5 V' we implicitly mean *with respect to the reference node (ground)*. A single point cannot have an absolute voltage; voltage is between two points.

Polarity is shown on a schematic by + and − marks at the element's terminals. The voltage $v_{AB}$ means the voltage at A minus the voltage at B. Reversing the subscripts flips the sign: $v_{BA}=-v_{AB}$.

EMF (electromotive force) is the same thing as voltage measured at the terminals of a source. The name is historical — there is no actual force involved.

Power is the rate of energy transfer:

$$p=\frac{dw}{dt}=v\cdot i$$

Unit: watt (W) = 1 J/s. Power can also be written in three equivalent forms when Ohm's law applies:

$$P=vi=i^{2}R=\frac{v^2}{R}$$

Energy is the integral of power over time:

$$w={\int }_{t_0}^{t_1}pdt$$

For constant power, $w=P\cdot t$. Electricity bills measure energy in kilowatt-hours (kWh): 1 kWh = $3.6\times 10^6$ J.

Passive sign convention — when current enters the *positive* terminal of an element, $P=v\cdot i$ is the power absorbed by that element. If the same product comes out negative, the element is actually delivering power. Sources (batteries) typically deliver power; resistors always absorb it.

Every element in a circuit is either active or passive.

Active elements generate electrical energy. The four basic types are:

• Independent voltage source — maintains a specified voltage regardless of current. Schematic: a circle with + and −, or a battery symbol.
• Independent current source — maintains a specified current regardless of voltage. Schematic: a circle with an arrow.
• Dependent (controlled) sources — voltage or current is set by a *controlling variable* elsewhere in the circuit. Four flavours: VCVS, VCCS, CCVS, CCCS. Schematic: diamond-shaped.

Passive elements absorb or store energy but cannot generate it on their own. The three fundamental passive elements you'll meet this term are:

• Resistor (R) — dissipates electrical energy as heat. $v=iR$.
• Capacitor (C) — stores energy in an electric field. $i=C\frac{dv}{dt}$.
• Inductor (L) — stores energy in a magnetic field. $v=L\frac{di}{dt}$.

Real components are never perfectly ideal. A real resistor has tiny parasitic inductance and capacitance; a real battery has internal resistance that limits short-circuit current. For introductory circuit analysis we use the ideal models.

Electricity bills are computed from energy consumption in kilowatt-hours. The bill = $\sum (\text{appliance power in kW})\times (\text{hours used})\times (\text{rate per kWh})$.

Typical appliance power ratings (approximate):

Why this matters in BEE: every real-world circuit ultimately comes down to managing voltage, current, and power. Knowing typical magnitudes lets you sanity-check your calculations — if you compute 50 W for an LED bulb, you almost certainly have an error.

Sadiku's six-step approach scales from this course all the way through circuit design:

1. Carefully define the problem. Read the question twice. Identify what's given and what's asked. Sketch the circuit if it isn't given.
2. Present everything you know. Mark known values on the schematic. Choose a reference node. Label every node and branch.
3. Establish a set of alternative solutions. Could you solve this by mesh, nodal, Thévenin, superposition, source transformation? Pick the one with fewest unknowns.
4. Attempt a solution. Write the equations. Solve them — by hand or matrix. Carry units through every step.
5. Evaluate the solution. Does the magnitude make sense? Did current go in the direction you expected? Check power balance: power supplied should equal power absorbed.
6. Has the problem been solved satisfactorily? If not, return to step 3 with a different method.

The single most useful habit: always check your answer with a *different* technique. If mesh analysis gives $i_1=3$ A, plug back into the original circuit and verify with KCL at a node you didn't use.