BEEC Fundamentals of Electrical Circuits – Study Notes

Basic Definitions

  • Charge: two types exist in an atom (positive and negative); they exert an attractive force between opposite charges.
  • Voltage: potential energy difference between two points due to charge separation; in electrical terms, the potential difference is denoted as V (or v). The unit is volts (V).
    • A voltage is defined as the difference in potential energy per unit charge: the potential difference between two points is the energy required to move 1 coulomb of charge from one point to another, i.e. 1\ ext{V} = \frac{1\ \text{J}}{1\ \text{C}}.
  • Current: the rate of flow of free electrons when a voltage is applied; denoted by I (or i) and measured in amperes (A).
  • Energy: the capacity for doing work; exists in forms such as mechanical, chemical, electrical, etc.; unit is joule (J).
  • Power: the rate at which work is done or energy is transferred; measured in watts (W). It is the product of voltage and current: P = V I. The electrical energy transferred in time is W = P \times t = V I t.
  • Conductance: reciprocal of resistance; defined as G = \frac{1}{R} and measured in siemens (S) (or mho).
  • Sign conventions: voltage and currents have directions; the definitions above follow standard passive sign convention where power absorbed is positive.

Electric Circuit Elements: Overview

  • Active elements: can supply energy to the circuit (e.g., voltage sources, current sources, transistors, solar cells).
  • Passive elements: do not supply energy; they may store or dissipate energy (e.g., resistors, inductors, capacitors).
    • A passive element is a two-terminal device fully characterized by its voltage-current relationship and cannot be subdivided into other two-terminal devices.
  • Distinctions between sources and elements are used to classify circuit behavior and modeling.

Sources: Key Concepts

  • Voltage source: provides a voltage across its terminals.
  • Current source: provides a current through its terminals.
  • Dependent sources: output depends on another variable in the circuit (voltage or current elsewhere).
  • Independent sources: outputs are independent of the circuit’s other conditions.
  • Ideal vs Practical sources:
    • Ideal voltage source: zero internal resistance; maintains a constant terminal voltage regardless of load.
    • Practical voltage source: has a small but finite internal resistance R{int}; terminal voltage drops as load current increases. Represented as an ideal source in series with R{int}. The terminal voltage is VL = V - I R{int}.
    • Ideal current source: infinite internal resistance; supplies the same current to any load.
    • Practical current source: finite but high internal resistance; load current changes with load resistance. Represented as an ideal current source in parallel with R{int}; relation IL = I - \frac{VL}{R{int}}.
    • In practice, R_{int} should be as low as possible for voltage sources and as high as possible for current sources to behave like their ideal counterparts.

Independent and Dependent Sources

  • Independent voltage source: two-terminal element providing a fixed voltage independent of other circuit variables. Symbol: a circle with + and - signs inside.
  • Independent current source: two-terminal element providing a fixed current independent of other circuit variables. Symbol: a circle with an arrow indicating the current direction.
  • Dependent sources: output depends on another circuit variable (voltage or current).
    • Symbol: diamond (rhombus) shape with either +/− signs for voltages or an arrow for currents.
    • Four types:
      1) Voltage-controlled voltage source (VCVS)
      2) Current-controlled voltage source (CCVS)
      3) Voltage-controlled current source (VCCS)
      4) Current-controlled current source (CCCS)
  • Practical note: Dependent sources are used to model nonlinear/dynamic devices such as transistors and amplifiers. The gain or controlling factor links the output to another circuit variable.
  • Example in the material: a dependent source with output voltage v_{dep} = 5 i where i is the current through an 18 Ω resistor; the factor 5 is the gain.

Independent vs Dependent Sources: Classification

  • Voltage sources: independent (fixed voltage) and dependent (voltage determined by another circuit variable).
  • Current sources: independent (fixed current) and dependent (current determined by another circuit variable).
  • Summary: Independent sources set a fixed electrical quantity; Dependent sources scale or translate another circuit quantity to produce output.

Network vs Circuit

  • Network: an interconnection of two or more electrical elements (resistors, inductors, capacitors, voltage sources, current sources) in any configuration.
  • Circuit: a network that contains at least one closed path, allowing current to flow.
  • Key points:
    • A network may or may not contain a power source.
    • A circuit includes both active and passive components.
    • If a network contains at least one source (voltage or current), it is called an electrical circuit.

Basic Electrical Circuit Elements

  • Active Elements: can supply energy to the circuit (e.g., voltage sources, current sources).
  • Passive Elements: do not supply energy; they can store (capacitors, inductors) or dissipate (resistors) energy.
  • A passive element is described by its voltage-current relationship and cannot be subdivided into two-terminal devices.

Resistor and Resistance

  • A resistor is a fundamental component used to limit or regulate current flow.
  • Resistance is the property that causes electrons to lose some energy due to collisions as they move through a material; it is denoted by R.
  • Unit: \Omega (Ohm).
  • Fixed resistor vs Variable resistor: components whose resistance is constant or adjustable.

Inductor and Inductance

  • An inductor is typically a coil of wire that stores energy in a magnetic field when current flows.
  • An electric current produces a magnetic field; the magnetic field is proportional to the current.
  • Inductance L is the proportionality constant between the voltage across the inductor and the time rate of change of current: v_L(t) = L \, \frac{d i(t)}{dt}. Both voltage and current can be time-dependent.
  • Energy stored in an inductor: W_L = \tfrac{1}{2} L i^2.
  • The power absorbed by an inductor: PL = vL i = L i \frac{d i}{dt}.
  • Important properties:
    • The voltage across an inductor is zero when the current is constant (dc steady state); the inductor acts as a short circuit for DC.
    • A tiny instantaneous change in current would require an infinite voltage; in practice, current cannot change instantaneously.
    • An ideal inductor does not dissipate energy; it stores energy in its magnetic field. Real inductors have some series resistance and therefore dissipate some power.

Capacitor and Capacitance

  • A capacitor consists of two conducting plates separated by a dielectric; it stores energy in an electric field created by opposite charges on the plates.
  • Capacitance C is the reciprocal of the relationship between stored charge and voltage: Q = C V \quad\Rightarrow\quad C = \frac{Q}{V}. Unit: farad (F).
  • Current-voltage relation for a capacitor: iC(t) = C \, \frac{d vC(t)}{dt}. Both current and voltage can be time-dependent.
  • Energy stored in a capacitor: WC = \tfrac{1}{2} C vC^2.
  • Power delivered to a capacitor is equal to the instantaneous product of voltage and current: PC = vC i_C. There is no power dissipation in an ideal capacitor (it stores energy); real capacitors have some parasitic losses due to equivalent series resistance (ESR).
  • Important properties:
    • The current in a capacitor is zero if the voltage across it is constant (open circuit for DC).
    • A sudden change in voltage would require infinite current; in a fixed capacitor the voltage cannot change abruptly.
    • A pure capacitor stores energy but does not dissipate energy; real capacitors dissipate some power due to internal resistance.
    • Fixed, variable, and polarized capacitor types exist in practice.

Ohm's Law

  • Ohm's Law describes the relationship between voltage, current, and resistance in a circuit:
    • V = I R
    • Equivalently, I = \frac{V}{R},\quad R = \frac{V}{I}.
  • Conditions: temperature and other physical conditions are assumed constant for the basic relation.
  • Historical note: Georg Simon Ohm, 1827.

Ohm's Law in Practice and Energy Considerations

  • Power absorbed or delivered by a resistive element: for a resistor, the power is P = V I = I^2 R = \frac{V^2}{R}.
  • Energy over time: W = P t = V I t.

Practical and Theoretical Notes on Sources (Recap)

  • Ideal voltage source: constant terminal voltage; zero internal resistance; would provide constant voltage regardless of load; in reality, ideal sources do not exist.
  • Practical voltage source: modeled as an ideal voltage source in series with a small internal resistance R{int}; terminal voltage given by VL = V - I R_{int}.
  • Ideal current source: supplies a fixed current; modeled as having infinite internal resistance.
  • Practical current source: modeled as an ideal current source in parallel with a finite, typically high, internal resistance; load current IL = I - \frac{VL}{R_{int}}.
  • Independent vs dependent sources: independent ones provide a fixed quantity; dependent sources provide a quantity dependent on another circuit variable using the four types listed earlier.

Practical Example: A Dependent Source (Gain Example)

  • A dependent voltage source with value v = 5 i where i is the current through another element (e.g., an 18\,\Omega resistor).
  • The dependence demonstrates how the dependent source outputs a voltage proportional to an in-circuit quantity (gain = 5).

Practical Circuit Elements: Quick Reference

  • Active Elements: voltage source, current source, transistors, amplifiers, etc.
  • Passive Elements: resistor, inductor, capacitor; two-terminal devices described by their V-I relationships.
  • Resistor: limits current; unit R in \Omega; V = I R.
  • Inductor: stores energy in magnetic field; L in henry (H); vL(t) = L \frac{d i(t)}{dt}; energy WL = \tfrac{1}{2} L i^2.; dc response: short circuit.
  • Capacitor: stores energy in electric field; C in farad (F); iC(t) = C \frac{d vC(t)}{dt}; energy WC = \tfrac{1}{2} C vC^2.; dc response: open circuit.

Practical Interface: Symbols and Representations

  • Independent voltage source: circle with + and -.
  • Independent current source: circle with an arrow.
  • Dependent sources: diamond-shaped symbols.
  • A dependent voltage source may be labeled with a voltage value (e.g., 5i) where the voltage is a function of a controlling current or voltage elsewhere in the circuit.

Typical Circuit Composition

  • A typical circuit interconnects power sources, loads, conductors, and optional switches.
  • A circuit may include both active (sources) and passive (R, L, C) elements.
  • A circuit forms a closed path that allows current to flow when sources are present.
  • Load: device that consumes electrical energy (e.g., lamp, motor, resistor).
  • Conductor: wires or traces that carry current between elements.
  • Switch: optional device to open/close the circuit and control current flow.

Key Concepts Summary (Headings to Review)

  • Definitions: voltage, current, and power.
  • Interaction of resistors, capacitors, and inductors in circuits.
  • Types of sources: independent vs dependent; ideal vs practical.
  • Ohm's Law and power-energy relationships.
  • Active vs passive components and their roles in circuits.
  • Dependent sources and their use in modeling devices like transistors.
  • Key equations: V = IR\,,\; iC = C\,\frac{d vC}{dt}\,,\; vL = L\,\frac{d i}{dt}\,,\; WL = \tfrac{1}{2} L i^2\,,\; W_C = \tfrac{1}{2} C v^2\,.

Practice and Review Questions (From the Transcript)

  • Define the following: (i) Voltage (ii) Current (iii) Power.
  • Classify active vs passive elements.
  • Explain the importance of resistor, capacitor, and inductor in a circuit.
  • Explain the fundamental difference between independent and dependent sources in an electrical circuit.
  • Self-Assessment: Identify the correct option for Ohm's Law, energy storage components, and ideal vs practical sources.

References and Further Reading

  • Bird's Electrical Circuit Theory and Technology, Routledge, 7th Edition.
  • Engineering Circuit Analysis by Hayt, Kemmerly, Phillips, and Durbin, McGraw-Hill, 10th Edition.
  • NPTEL courses and related video playlists for BEEC fundamentals.