DC Electric Circuits and Kirchhoff's Laws

Vocabulary flashcards covering elementary concepts of DC electric circuits, passive and active elements, resistance, capacitors, inductors, conductance, power, energy, series/parallel combinations, Kirchhoff's laws, Star-Delta transformations, and mesh matrix analysis.

Electric Current

The rate of flow of electric charges ($i = \frac{dq}{dt}$), caused by the movement of negatively charged electrons in conductors. Its unit is the Ampere ($\text{A}$), where $1\,\text{A} = 1\,\text{C/s}$.

Electromotive Force (emf)

The work required or energy provided by a source (such as a battery) to move electrons along a conductor and drive electric current.

Potential Difference

The difference between the voltages at two ends of a conductor.

Electric Circuit

A closed connection formed by various electric elements.

Passive Element

A circuit component that receives energy and either dissipates it in the form of heat or stores it (e.g., resistor, inductor, capacitor).

Active Element

A circuit component that supplies energy to the circuit (e.g., voltage sources, current sources, generators).

Bilateral Element

A circuit element that conducts electric current in both directions (e.g., resistor, inductor, capacitor).

Unilateral Element

A circuit element in which current conduction is possible in only one direction (e.g., diode).

Resistance

The property of a material by which it opposes the flow of electric current, dissipating energy in the form of heat. Denoted by $R$, with unit Ohm ($\Omega$).

Resistivity (Specific Resistance)

The resistance offered by a unit cube of material between its opposite faces, defined by $\rho = \frac{RA}{l}$, with unit $\Omega\cdot\text{m}$.

Capacitor

An electronic component consisting of two metal plates separated by a dielectric (insulating material) used to store electric charge. It passes AC and blocks DC.

Capacitance

The amount of electric charge required to create a $1\,\text{V}$ potential difference between the plates of a capacitor ($C = \frac{Q}{V} = \frac{\epsilon A}{d}$), measured in Farads ($\text{F}$).

Energy Stored in a Capacitor

The potential energy stored in the electric field between the plates of a capacitor, calculated as $E = \frac{1}{2} C V^2 = \frac{1}{2} Q V = \frac{1}{2} \frac{Q^2}{C}$.

Inductor

A circuit element possessing the property to oppose any change in current flowing through it by generating an electromotive force ($v = L \frac{di}{dt}$).

Energy Stored in an Inductor

The energy stored in the magnetic field of an inductor when current flows through it, calculated as $E = \frac{1}{2} L I^2$.

Conductance

The measure of ease with which electric current flows through a component, defined as $G = \frac{1}{R}$. Its unit is Siemens ($\text{S}$) or mho ($\mho$).

Conductivity (Specific Conductance)

The reciprocal of resistivity ($\sigma = \frac{1}{\rho}$), representing a material's capacity to conduct current. Its unit is Siemens/m ($\text{S/m}$) or $\mho/\text{m}$.

Electrical Power

The rate at which electrical energy is transferred in an electric circuit, given by $P = VI = I^2 R = \frac{V^2}{R}$, with unit watts ($\text{W}$).

Electrical Energy

The work done by electrical power over time ($E = P \times t = V I t = I^2 R t = \frac{V^2}{R} t$), measured in Joules ($\text{J}$), where $1\,\text{J} = 1\,\text{W}\cdot\text{s}$.

Series Resistance Circuit

A circuit configuration where the same current flows through all connected resistors, the voltage drops sum to the applied voltage ($V = V_1 + V_2 + V_3$), and $R_{eq} = R_1 + R_2 + R_3$.

Parallel Resistance Circuit

A circuit configuration where resistors share two common nodes, experiencing identical voltage while total current is the sum of branch currents ($I = I_1 + I_2 + I_3$), with $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$.

Ohm's Law

States that at constant temperature, the current through any conductor is directly proportional to the potential difference between its ends ($V \propto I \Rightarrow V = IR$).

Kirchhoff's Current Law (KCL)

States that at any instant of time, the algebraic sum of currents at a node is zero ($\sum_{n=1}^N i_n = 0$), meaning total incoming current equals total outgoing current.

Kirchhoff's Voltage Law (KVL)

States that the algebraic sum of voltages (sources and $IR$ drops) around any closed path at any instant of time is zero ($\sum_{n=1}^N v_n = 0$).

Delta-to-Star Transformation

Converts a delta network ($R_{12}, R_{23}, R_{31}$) into an equivalent star network ($R_1, R_2, R_3$) using formulas such as $R_1 = \frac{R_{12} R_{31}}{R_{12} + R_{23} + R_{31}}$.

Star-to-Delta Transformation

Converts a star network ($R_1, R_2, R_3$) into an equivalent delta network ($R_{12}, R_{23}, R_{31}$) using formulas such as $R_{12} = \frac{R_1 R_2 + R_2 R_3 + R_3 R_1}{R_3}$.

Mesh Current Method

A circuit analysis technique involving identifying all independent meshes, assigning mesh currents, writing voltage expressions, and applying KVL to solve for unknown currents.

Self-Resistance Matrix Element ($R_{11}$)

In matrix mesh analysis, $R_{11}$ represents the sum of all resistances through which mesh current $I_1$ passes.

Mutual Resistance Matrix Element ($R_{12}$)

In matrix mesh analysis, $R_{12}$ represents the sum of resistances shared by mesh currents $I_1$ and $I_2$, taken as positive if currents flow in the same direction and negative if in opposite directions.

Example 2.8.1 (Circuit Resistance Problem)

A problem asking to find the total resistance of the circuit between points A and B of Figure 2.43.

Star-to-Delta Conversion Equations (Example 2.8.5)

Calculations converting a star configuration to equivalent delta branch resistances: $R_{AB} = 6 + 4 + \frac{6 \times 4}{3} = 18\,\Omega$, $R_{AC} = 6 + 3 + \frac{6 \times 3}{4} = 13.5\,\Omega$, and $R_{BC} = 4 + 3 + \frac{4 \times 3}{6} = 9\,\Omega$.

Three-Mesh Matrix Equation Structure

The matrix formulation for a three-mesh system: $\begin{bmatrix} R_{11} & R_{12} & R_{13} \ R_{21} & R_{22} & R_{23} \ R_{31} & R_{32} & R_{33} \end{bmatrix} \begin{bmatrix} I_1 \ I_2 \ I_3 \end{bmatrix} = \begin{bmatrix} V_1 \ V_2 \ V_3 \end{bmatrix}$, expressing Ohm's law in matrix form.