Fundamentals of Electric Circuits: A Comprehensive Guide

Posted on Oct 27, 2024 in Technology

Fundamentals of Electric Circuits

Introduction

When an electric current flows through a conductor, it produces a displacement of electrically charged particles. This movement of charges is what we call electric current. For an electric current to exist, there must be an electric conductor (e.g., ohmic conductors) and an electric field strong enough to generate a force that causes the charges to move. If the electric field is constant, the current is called direct current (DC). If the electric field varies periodically, the current is called alternating current (AC), where charges move alternately in both directions.

Intensity and Current Density

The intensity of an electric current is the amount of charge that crosses a section of a conductor per unit of time. Mathematically, it is represented as: i_M= ΔQ/Δt = lim_Δt->0 ΔQ/Δt = dQ/dt. In the SI system, it is measured in Amperes. Current density is a vector whose magnitude is calculated by dividing the current intensity by the area of the cross-section of the conductor. j = i/S. In the SI, it is measured in A/m². This formula does not indicate the direction of the current, which coincides with the direction of the velocity of the moving charges. j = i/S = nqv. The current is a vector whose magnitude and direction coincide with the velocity of the moving charges.

Ohm’s Law

The relationship between the electric field and the current density is given by Ohm’s Law: j = σE, where σ is the conductivity of the material. For an ohmic conductor, the electric field is proportional to the gradient of the electric potential: E = -grad V = -dV/dL. Substituting this into Ohm’s Law, we get: j = -σdV/dL. For a uniform conductor, this simplifies to: i = (V_a – V_b)/R, where R = ρL/S is the resistance of the conductor, and ρ is the resistivity of the material. The resistivity is a characteristic of each material and varies with temperature: ρ_t = ρ₀(1 + αt + βt² + …), where α is the temperature coefficient of resistivity.

Joule Effect

When an electric current flows through a conductor, some of the electrical energy is dissipated as heat. This phenomenon is known as the Joule effect. The power dissipated is given by: P = i²R. The total energy dissipated is given by: W = Pt = i²Rt = (V²/R)t.

Electromotive Force (EMF)

The electromotive force (EMF) is the energy supplied to drive the charges through a circuit. It is defined as: e = dW/dQ. The power supplied by the EMF is given by: P = dW/dt = e(dQ/dt) = ei. In the SI, it is measured in volts. If we connect a resistor to a generator, the power supplied by the generator is equal to the power dissipated by the resistor: ei = i²R + i²r, where r is the internal resistance of the generator. If the circuit is open, e = V_a – V_b. If the circuit is closed, i = e/(R + r). The efficiency of the generator is given by: u = P/P_t = i²R/ei = iR/e = R/(R + r).

Association of Resistances

1) Series

When resistors are connected in series, the total resistance is the sum of the individual resistances: R_t = R₁ + R₂ + … + R_n.

2) Parallel

When resistors are connected in parallel, the reciprocal of the total resistance is the sum of the reciprocals of the individual resistances: 1/R_t = 1/R₁ + 1/R₂ + … + 1/R_n.

Kirchhoff’s Laws

1) Kirchhoff’s Current Law (KCL)

The sum of the currents entering a node (junction) is equal to the sum of the currents leaving the node: Σi_in = Σi_out.

2) Kirchhoff’s Voltage Law (KVL)

The sum of the voltage drops around any closed loop in a circuit is equal to the sum of the voltage rises: ΣV_drops = ΣV_rises.

Applications of Kirchhoff’s Laws

1) Ammeter

An ammeter is a device used to measure the current flowing through a circuit. It is connected in series with the circuit element whose current is to be measured.

2) Voltmeter

A voltmeter is a device used to measure the potential difference between two points in a circuit. It is connected in parallel with the circuit element whose potential difference is to be measured.

Association of Batteries

1) Series

When batteries are connected in series, the total voltage is the sum of the individual voltages: V_t = V₁ + V₂ + … + V_n.

2) Parallel

When batteries are connected in parallel, the total current is the sum of the individual currents: i_t = i₁ + i₂ + … + i_n.