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 $R<em>{int}$; terminal voltage drops as load current increases. Represented as an ideal source in series with $R</em>{int}$. The terminal voltage is $V<em>L = V - I R</em>{int}.$
  • 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 $R<em>{int}$; relation $I</em>L = I - \frac{V<em>L}{R</em>{int}}.$
  • 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: $P<em>L = v</em>L i = L i \frac{d i}{dt}.$
• 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: $i<em>C(t) = C \, \frac{d v</em>C(t)}{dt}.$ Both current and voltage can be time-dependent.
• Energy stored in a capacitor: $W<em>C = \tfrac{1}{2} C v</em>C^2.$
• Power delivered to a capacitor is equal to the instantaneous product of voltage and current: $P<em>C = v</em>C i_C.$ 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 $R<em>{int}$; terminal voltage given by $V</em>L = V - I R_{int}.$
• 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 $I<em>L = I - \frac{V</em>L}{R_{int}}.$
• 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); $v<em>L(t) = L \frac{d i(t)}{dt}$; energy $W</em>L = \tfrac{1}{2} L i^2.$; dc response: short circuit.
• Capacitor: stores energy in electric field; $C$ in farad (F); $i<em>C(t) = C \frac{d v</em>C(t)}{dt}$; energy $W<em>C = \tfrac{1}{2} C v</em>C^2.$; 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: $V = IR\,,\; i<em>C = C\,\frac{d v</em>C}{dt}\,,\; v<em>L = L\,\frac{d i}{dt}\,,\; W</em>L = \tfrac{1}{2} L i^2\,,\; W_C = \tfrac{1}{2} C v^2\,.$

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.