Circuit Elements and Ohm's Law Study Notes

Recap of Fundamental Concepts

• Electric Circuit Recap: The discussion begins with a review of previous topics including the definition of an electric circuit and the systems of measurement.
• Systems of Units:
  • English System: Mentioned as a standard used previously.
  • Metric System: Divided into two sub-categories:
    • MKS: Stands for meters ($m$), kilograms ($kg$), and seconds ($s$).
    • CGS: Stands for centimeters ($cm$), grams ($g$), and seconds ($s$).
  • SYSTEP International (SI): This is the third system of units discussed, utilized for international standardization.
• SI Prefixes: These involve multiples and sub-multiples, each with a specific equivalent prefix and symbol used to integrate into electrical lessons.
• Charge and Current:
  • Charge: Defined as an electrical property of the atomic particles of which matter consists.
  • Electric Current: Occurs when a specific voltage or force is applied to a conductor, causing charges to move. Current is the flow of these moving charges.
• Types of Electric Current:
  • DC (Direct Current): The first type discussed.
  • AC (Alternating Current): The second type discussed.
• Voltage, Power, and Energy:
  • Power: Defined as the time rate of expending or absorbing energy.
  • Energy: Described using the common definition as the capacity to do work.

Classification of Circuit Elements: Active and Passive

• Circuit Elements Overview: There are two primary types of elements found in electric circuits: active and passive elements.
• Active Elements:
  • Definition: Components that are capable of generating or giving off energy.
  • Function: They are capable of supplying energy to the circuit.
  • Examples: Generators, batteries, and any kind of voltage or current source.
• Passive Elements:
  • Definition: Electrical elements that are capable of absorbing energy or are the ones "using" the energy.
  • Function: They absorb the energy that originates from the active elements within the circuit.
  • Examples: Resistors, capacitors, transistors, and various household appliances.

Independent and Dependent Sources

• Ideal Independent Source:
  • Definition: An active element that provides a specified voltage or current that is completely independent of other circuit elements.
  • Operating Characteristic: These sources can stand alone, operate independently, and supply voltage or current without needing other elements.
  • Symbols:
    • Standard Independent Source: Represented by a circular symbol.
    • Independent Voltage Source: A circular symbol with a plus ($+$) and minus ($-$) sign inside. Alternatively, it can be represented by two parallel lines where the longer line is the positive side and the shorter line is the negative side.
    • Independent Current Source: A circular symbol containing an arrow inside pointing up or down.
• Ideal Dependent Sources (Controlled Sources):
  • Definition: Active elements in which the source quantity (voltage or current) is controlled by another voltage or current within the circuit.
  • Operating Characteristic: These cannot stand alone or operate independently; they require an independent source to function.
  • Control Rule: A dependent source cannot be controlled by another dependent source; it must be controlled by an independent source. They must exist in pairs (independent and dependent).
  • Symbols:
    • Dependent Source General Symbol: A diamond shape.
    • Dependent Voltage Source: A diamond symbol with plus ($+$) and minus ($-$) signs inside.
    • Dependent Current Source: A diamond symbol with an arrow inside.

Mathematical Modeling of Dependent Sources

• There are four specific types of dependent sources:
  1. Voltage-Controlled Voltage Source (VCVS): A dependent voltage source controlled by another independent voltage source.
  2. Current-Controlled Voltage Source (CCVS): A dependent voltage source controlled by another independent current source.
  3. Voltage-Controlled Current Source (VCCS): A dependent current source controlled by another independent voltage source.
  4. Current-Controlled Current Source (CCCS): A dependent current source controlled by another independent current source.

Resistance and Material Resistivity

• Definition of Resistance: The physical property or characteristic of a material to resist the flow of electric charges. It is denoted by the symbol $R$.
• Formula for Resistance: The resistance of any material with a uniform cross-sectional area depends on the material's resistivity, its length, and its area:
  • $R = \rho \frac{L}{A}$
    • $\rho$: Material resistivity (resistivity constant of the material).
    • $L$: Length of the conductor.
    • $A$: Cross-sectional area of the conductor.
• Unit of Resistance: Measured in ohms ($\Omega$), named after George Simon Ohm.
• Resistivity Table (Common Materials):
  • Conductors:
    • Silver
    • Copper
    • Aluminum
    • Gold
  • Semiconductors:
    • Carbon
    • Germanium
    • Silicon
  • Insulators:
    • Paper
    • Mica
    • Glass
    • Teflon
  • Unit of Resistivity: The unit is the ohm-meter ($\Omega\,m$).

Wire Geometry and Cross-Sectional Area

• Wire Form: Even flat-looking wires, when zoomed in, typically exhibit a circular cross-section.
• Cross-Sectional Area ($A$): This is identified by cutting the wire and observing the face of the cut portion.
• Area Formulas for Circles:
  • In terms of Radius ($r$): $A = \pi r^2$
  • In terms of Diameter ($d$): $A = \frac{\pi d^2}{4}$

Ohm's Law and Proportionality

• Historical Context: George Simon Ohm, a German physicist, discovered the relationship between current and voltage for a resistor, establishing what is now known as Ohm's Law (sometimes referred to as the Ohm solo).
• Definition: Voltage across a resistor is directly proportional to the current flowing through that resistor.
  • As voltage increases, current increases.
  • As voltage decreases, current decreases.
• Constant of Proportionality: The constant of proportionality is the resistance ($R$).
• Fundamental Equation: $V = IR$
  • This is the basic formula and fundamental equation of electrical engineering; every electrical engineer must know it as all electrical circuit principles follow from this.

Circuit Conditions: Short, Open, and Closed Circuit

• Short Circuit:
  • Definition: A circuit element with resistance approaching zero ($R \rightarrow 0$).
  • Physical Cause: Occurs when two bare (uninsulated/exposed) wires touch each other while the circuit is energized.
  • Consequences: The current will bypass the passive elements (the parts of the circuit that would otherwise absorb energy) and flow through the element-free portion. This leads to excessive current, electrical shocks, sparks, and can cause fires.
  • Hypothetical Scenario: A fire officer often attributes fires to "faulty electrical wiring" or a short circuit. For example, if a mouse (or a pair of mice, one male and one female) in the ceiling bites the insulation off the wires for the kitchen and dining room lights, and the bare wires touch while the lights are on, a short circuit occurs.
• Open Circuit:
  • Definition: A circuit element with resistance approaching infinity ($R \rightarrow \infty$).
  • Condition: The circuit is cut or has no continuity, providing no path for current flow.
  • Application: Switching off a light switch opens the circuit by lifting a contactor and cutting the supply.
• Closed Circuit:
  • Definition: A circuit that provides a continuous closed loop, allowing voltage and current to circulate and power the circuit.
  • Application: Switching on a light switch closes the circuit, bringing contactors together to facilitate flow.

Types of Resistors

• Fixed Resistor:
  • Definition: Resistors whose value remains constant and does not change.
  • Visual Identification: Often use color-coded bands (commonly four or five bands) to determine their value.
  • Tolerances (Percent Error): These resistors have a specified range of error, such as $5\%$, $10\%$, or $20\%$. For example, a resistor coded for $33,000\,\Omega$ (or $33\,k\Omega$) might measure as $32,800\,\Omega$. The missing $200\,\Omega$ is the tolerance, but the value remains fixed at that measurement.
• Variable Resistor:
  • Definition: Resistors with adjustable resistance values.
  • Example: A potentiometer.
  • Mechanism: Contains a knob that can be turned from left to right to change the resistance within a specific range (e.g., $0$ to $100\,\Omega$, $0$ to $50\,\Omega$, or $0$ to $20\,\Omega$).

Conductance

• Definition: The ability of an element to conduct electric current.
• Mathematical Relationship: Conductance is the reciprocal of resistance.
• Symbol: Denoted by the capital letter $G$.
• Units:
  • MOS: Represented by an inverted omega ($\mho$).
  • Siemens: Represented by the symbol $S$.
• Formulas:
  • $G = \frac{1}{R}$
  • $G = \frac{I}{V}$

Power Dissipation in Resistors

• Original Power Formula: $P = VI$ (measured in Watts, named after James Watt).
• Derived Formulas using Ohm's Law:
  • By substituting $V = IR$ into $P = VI$: $P = (IR)I = I^2 R$
  • By substituting $I = \frac{V}{R}$ into $P = VI$: $P = V\left(\frac{V}{R}\right) = \frac{V^2}{R}$
• Formulas in terms of Conductance ($G$):
  • Since $G = \frac{1}{R}$, the expression for power becomes: $P = V^2 G$
  • Additionally, since $R = \frac{1}{G}$, power can be expressed as: $P = \frac{I^2}{G}$

Practical Calculation Example

• Scenario: A toaster acts as a resistor converting electrical energy to heat energy.
• Given Data:
  • Resistance ($R$) = $10\,\Omega$
  • Voltage ($V$) = $110\,V$
• Requirement: Calculate the current ($I$) drawn by the toaster.
• Solution Using Ohm's Law:
  • $V = IR$
  • $I = \frac{V}{R}$
  • $I = \frac{110\,V}{10\,\Omega}$
  • $I = 11\,A$
• Result: The current drawn by the toaster is $11\,\text{amperes}$.