Lesson 01: Definitions and Units

Electric Charges and Electric Fields

• Electric Charges (Q):
  • Definition: Matter comprises atoms containing charged particles: negatively charged electrons and positively charged protons. Electric charge is a fundamental property of these particles, causing them to experience a force in the presence of other charged particles. Charges are measured in coulombs (C). The electron carries a charge of approximately -1.6022 × 10^-19 C, and the proton carries a positive charge of the same magnitude.
  • Unit: The standard unit of charge is the coulomb (C).
  • Charge Flow: When a potential difference (voltage) exists across a conductor, charge flows from areas of higher potential to lower potential. This movement of charge constitutes an electric current.
• Electric Fields (E):
  • Definition: An electric field represents the force charged particles exert on each other. The field is a vector quantity with both magnitude and direction.
  • Mathematical Expression: The electric field (E) due to a point charge (Q) at a distance (r) is given by Coulomb's law:
    \(E = {{k \cdot Q} \over {{r^2}}}\)
    where k is Coulomb's constant (8.987×10⁹ N·m²/C²). The direction of the field is away from positive charges and toward negative charges.
  • Uint: newtons/coulomb, N/C

Example: Consider a point charge of +2×10⁻⁶ C located in a vacuum. The electric field at a point 0.5 meters away from the charge can be calculated as follows: 

\(E = {{k \cdot Q} \over {{r^2}}} = {{8.987 \times {{10}^9} \times 2 \times {{10}^{ - 6}}} \over {{{0.5}^2}}} = 71.896 \times {10^3} \quad N/C\)

Voltage (V)

• Definition: Voltage, or electrical potential difference, is the work needed to move a charge from one point to another in an electric field. It is measured in volts (V), where 1 volt equals 1 joule per coulomb (1 V = 1 J/C). Voltage can be thought of as the electrical "pressure" that pushes electrons through a conductor.
• Mathematical Expression:
  \(v = {{dw} \over {dq}}\),    \(V = {W \over Q}\)
  
  • V: is the voltage difference (in volts)
  • W: is the energy (in joules)
  • Q: is the charge (in coulombs)
• Unit: The unit of voltage is the volt (V).
• Voltage Sources: Batteries, generators, and power supplies provide voltage. They create an electric field that drives charge flow.
• Voltage Drop: When current flows through a resistor or any other component, there is a voltage drop across it. Ohm’s Law relates voltage, current, and resistance: (\(V = I \cdot R\)).

Two indicators will designate the voltage across an element:

• a plus-minus sign pair: establish a voltage reference direction
• a value (variable or fixed): quantify the voltage across the element in the specified reference direction

Figure 1.1: Two Equivalent Voltage Representations

For example, Fig. 1.1(a) and (b) are two versions of exactly the same voltage:

• In (a), terminal A has a potential that is +5 V higher than terminal B.
• In (b), terminal B's potential is -5 V higher than terminal A's (alternatively described as +5 V lower than terminal A).

The double-subscript notation, \({v_{ab}}\), is utilized to express the potential at point a relative to point b. Generally, this results in \({v_{ab}} = - {v_{ba}}\). Therefore, in Figure 1.1, \({V_{AB}} = 5\;V\) and \({V_{BA}} = - 5\;V\).

Example: If it takes 1 joule of work to move a charge of 1 coulomb from point A to point B, then the voltage between points A and B is 1 volt.

Current (I)

• Definition: Current is the rate at which electric charge flows past a point in a circuit. It is measured in amperes (A), where 1 ampere equals 1 coulomb per second (1 A = 1 C/s). Current can flow in two forms: direct current (DC), which flows in a constant direction, and alternating current (AC), which changes direction periodically.
• Mathematical Expression:
  \(i = {{dq} \over {dt}}\),    \(I = {Q \over t}\)
  • Q: the charge passing through a cross-section in time (t).
• Unit: The current unit is ampere (A).
• Direction of Current: Engineers use the convention of hole current to simplify equations. In reality, electrons flow from negative to positive, but we often consider the flow of positive charges (holes) for analysis.

Two indicators will specify the current flow along a lead or through an element:

• an arrow: establish a current reference direction
• a value (variable or fixed): quantify the current flow in the reference direction

Figure 1.2: Two Representations of the same current

Figure 1.2(a) shows a current of 3 A, equivalent to a charge flow of 3 coulombs per second (C/s), moving from left to right through the wire. Conversely, Figure 1.2(b) shows a current of -3 A flowing from right to left. Both scenarios facilitate the transfer of the same amount of charge and therefore represent two equivalent methods of describing the same current flow.

Note: When current values are positive, this indicates that the direction of the current flow aligns with the reference current direction. Conversely, negative current values signify that the actual current flow direction is opposite to the reference direction.

Example: If a total charge of 3 coulombs passes through a wire in 3 seconds, the current in the wire is:

\(i = {Q \over t} = {{3C} \over {3s}} = 1\quad A\)

Power in Electrical Circuits

• Definition: Power refers to the rate at which energy is transferred or converted.
• Mathematical Expression: Power can be expressed as the derivative of energy with respect to time:
  \(P = {W \over t} = V{Q \over t} = V \cdot I\)
  • P: Power (watt, joule/second, or J/s)
  • W: Energy
  • t: seconds
• Unit: The standard unit of power is the watt (W).   

Energy (W) in Electrical Circuits

• Definition: Energy (W) in electrical circuits refers to the total amount of work done or heat delivered by the power over a period. The unit of energy in the International System of Units (SI) is the joule (J), where 1 joule is the energy transferred when 1 watt of power is applied for 1 second.
• Mathematical Expression: \(W = P \cdot t\)   (watt-seconds, W s, or joules)
  • W: the energy in joules (J)
  • P: the power in watts (W)
  • t: the time in seconds (s) during which the power is applied

The voltage-current (V-I) relationships in electrical elements are fundamental to understanding how energy is either absorbed by or delivered to these elements within a circuit. Your summary points are crucial for deciphering the direction of power flow in circuit analysis. Here's a detailed explanation based on the provided items:

Understanding Polarity and Current Direction

To determine whether an element absorbs or delivers energy, we need to consider both the polarity of the voltage across the element and the direction of the current through it.

• Polarity of Voltage: The polarity across an element indicates the direction of the electric field within that element. The positive and negative terminals across the element determine it.
• Direction of Current: The direction in which positive charge carriers (conventionally regarded as the current direction) move through the element.

Absorbing Energy

An element absorbs energy when a positive current enters its positive terminal. This scenario implies that external forces (like a battery or another power source) are doing work on the charge carriers to push them through the element against the inherent electric field or potential difference. This work on the element increases its internal energy, manifesting as stored energy in capacitors or inductors or as dissipated energy in resistors.

Delivering Energy

Conversely, an element delivers energy to the circuit when a positive current leaves its positive terminal (equivalently enters its negative terminal). This situation indicates that the element had stored energy (in the case of capacitors or inductors) and is now releasing that energy back into the circuit. The element essentially acts as an energy source, driving the current through the circuit.

Examples (Referring to Fig 1.4):

Figure 1.4: Various Voltage-Current Relationships

• Absorbing Energy:
  • (a): The element is absorbing energy. A positive current enters the positive terminal.
  • (b): Similar to (a), it absorbs energy.
  This could represent charging capacitors, current increasing in inductors, or current flowing through resistors, where energy is converted into heat.
• Delivering Energy:
  • (c): The element is delivering energy. A positive current leaves the positive terminal.
  • (d): It delivers energy from the positive current entering the negative terminal.
  This could be a battery, a discharging capacitor, or an inductor releasing its stored magnetic energy, contributing power back into the circuit.

Relationship Between Voltage, Current, and Resistance (Ohm's Law)

Ohm's Law describes the relationship between voltage (V), current (I), and resistance (R) in a circuit. It states that the voltage across many types of conducting materials is directly proportional to the current flowing through the material. The formula expresses this relationship:

\(V = I \cdot R\quad \Leftrightarrow \quad I = {V \over R}\quad \Leftrightarrow \quad R = {V \over I}\)

where:

• V : is the voltage across the conductor (in volts, V)
• I : is the current through the conductor (in amperes, A)
• R : is the resistance of the conductor (in ohms, Ω).

It is important to note that Ohm's Law applies to conductors whose resistance remains constant over various voltages and temperatures. This law is foundational in circuit analysis, allowing for calculating voltage, current, or resistance in a circuit when two of these quantities are known, facilitating the design and analysis of electrical and electronic circuits.

Resistors are perhaps the simplest passive components, primarily used to limit current, divide voltages, and dissipate power as heat. They oppose the flow of electrons through a material, a property known as resistance, measured in Ohms (Ω).

• Function: Resistors convert electrical energy into heat, effectively reducing current flow according to Ohm's Law (\(V = I \cdot R\)), where V is the voltage across the resistor, I is the current flowing through it, and R is its resistance.
• Applications: When paired with capacitors, resistors are integral for setting circuit conditions such as current flow or voltage levels, filtering signals, and timing circuits in virtually every electronic circuit.
• Symbols:
  

Inductors store energy in a magnetic field created by electrical current flowing through a coil of wire. Inductance, the measure of an inductor's ability to store magnetic energy, is measured in henries (H).

• Function: The fundamental property of inductors is that they oppose changes in current flow through them. This property is described by \(v = L{{di} \over {dt}}\), where v is the voltage across the inductor, L is its inductance, and \({{di} \over {dt}}\) is the rate of change of current over time.
• Applications: Inductors are used in filtering, tuning, and frequency selection circuits, such as in radios and TVs, to select desired frequencies. They are also integral to transformers and power supply circuits.
• Symbol:
  

Capacitors store energy in an electric field created between two conductive plates separated by an insulating material (dielectric). They are characterized by their capacitance, which is the ability to store an electrical charge, measured in farads (F).

• Function: When a voltage is applied across the plates, an electric field is established, storing energy that can be released when the circuit requires it. The relationship between the voltage (v) across a capacitor (C) and the charge (q) it holds is given by \(q = C \cdot v\), where C is the capacitance.
• Applications: Capacitors are used in filtering applications to allow alternating current (AC) signals to pass while blocking direct current (DC) signals, in energy storage, in power supply smoothing, and in timing applications when combined with resistors.
• Symbol:
  

Independent Voltage Sources

• Definition: Independent voltage sources provide a specific voltage that can be constant \({V_S}\) or time-varying \({v_S}(t)\). The voltage level is maintained across the source's terminals irrespective of the variations in circuit conditions or the current drawn by the circuit.
• Symbol and Identification: Independent voltage sources are typically represented by a circle with a plus (+) and minus (-) sign to indicate polarity, sometimes with a battery symbol for DC sources or a sine wave for AC sources.
  

• Definition: These sources deliver a specified current that can be constant \(I\) or time-varying \({i_S}(t)\). The current provided is independent of the voltage across the source's terminals or any circuit load changes.
• Symbol and Identification: Represented by a circle with an arrow inside, pointing toward conventional current flow (positive to negative).
  

Dependent (or controlled) sources are critical elements in circuit analysis, especially in studying amplifiers, active filters, and integrated circuits. Unlike independent sources, whose values are fixed, dependent sources derive their output (voltage or current) based on another quantity in the circuit, such as a voltage or current elsewhere in the system. This relationship makes them essential for modeling the behavior of many electronic components, including transistors and operational amplifiers.

Types of Dependent Sources:

Figure 1.5: (a) VCVS; (b) CCVS; (c) VCCS; (d) CCCS

There are four primary types of dependent sources, categorized by their input (control) and output variables:

1. Voltage-Controlled Voltage Source (VCVS):
   • Output: Voltage
   • Control Variable: Voltage
   • Symbol & Notation: Often represented with a diamond shape. The output voltage is a function of a voltage elsewhere in the circuit, typically denoted as \({v_{out}}(t) = \mu \cdot {v_{in}}\), where \(\mu \) is the voltage gain.
2. Current-Controlled Voltage Source (CCVS):
   1. Output: Voltage
   2. Control Variable: Current
   3. Symbol & Notation: Similar to VCVS but indicates that the output voltage depends on an input current, expressed as \({v_{out}}(t) = r \cdot {i_{in}}\), where \(r\) is a proportionality constant, often referred to as the transresistance.
3. Voltage-Controlled Current Source (VCCS):
   • Output: Current
   • Control Variable: Voltage
   • Symbol & Notation: Also typically represented with a diamond shape but includes notation or a symbol indicating current output. The relationship can be defined as \({i_{out}}(t) = g \cdot {v_{in}}\), where \(g\) is the transconductance.
4. Current-Controlled Current Source (CCCS):
   • Output: Current
   • Control Variable: Current
   • Symbol & Notation: Similar in representation to the other sources but denotes a current output that depends on an input current, described as \({i_{out}}(t) = \beta \cdot {i_{in}}\), where \(\beta \) is the current gain.

Voltage Sources and Ground

In electrical and electronic circuits, the ground serves as a common reference point for voltage measurements, ensuring safety and a reference for potential levels across the circuit. The ground is typically assigned a potential of zero volts. This concept is essential for several reasons:

• Reference Point: It establishes a zero-voltage reference point for measuring voltages in the circuit.
• Safety: It provides a path for fault currents to flow to the earth, potentially preventing damage to equipment and personnel.

Figure 1.6: Ground Potential

Grounding in Circuits:

Figure 1.7 shows different grounding methods for components like batteries and resistors. This ensures specific points in the circuit (like the negative terminal of a battery or the bottom of a resistor) are at zero voltage (ground potential). Grounding establishes a common reference point for voltage measurements. Even with different schematic symbols, the ground connection concept remains consistent for clarity and safety in circuit analysis.

Figure 1.7: Three Ways to Sketch the Same Series DC Circuit

Fig. 1.8 shows that voltage source symbols can be replaced with conventional ones with a battery and a ground connection.

Figure 1.8: Replace the notation for DC supply with the Standard Ground Notation

Double-Subscript Notation

Double-subscript notation denotes the voltage difference between two points in a circuit, indicating the direction of potential rise from one point to another.

Format: V_ab signifies the voltage at point a relative to point b. If E = 12 V, and point a is mentioned as being 12V positive with respect to ground, this notation helps in understanding voltage drops or gains between specific points in a circuit.

Single-Subscript Notation