lecture_EE1C1_lec1

å--- created: 2025-09-02T09:08:17 course: “EE1C1” tags:

lecture status: Complete type: lecture week: “2025-w36_1.1”

📌 Overview

This lecture introduces the fundamental concepts of linear circuits. It covers the basic quantities of voltage and current, the concept of ports, and the standardized SI units used in circuit analysis. The lecture also explains energy and power, the passive sign convention, and introduces essential circuit elements like independent and dependent sources, and resistors governed by Ohm’s law.

🎯Learning Objectives

💡 Topics

💡Key Concepts & Definitions

Voltage (V): The potential energy difference per unit charge between two points. Measured in Volts (V).
Current (A): The rate of flow of electric charge. Measured in Amperes (A).
Port: A pair of terminals through which a current can enter or leave and across which a voltage can be measured.
Power (P): The rate at which energy is transferred. Calculated as P = V * I. Measured in Watts (W).
Passive Sign Convention: A standard convention where current flows into the positive terminal of an element. If power (P=VI) is positive, the element absorbs power. If negative, it supplies power.
Independent Source: A voltage or current source whose value is constant and does not depend on any other circuit variable.
Dependent Source: A voltage or current source whose value depends on a voltage or current elsewhere in the circuit.
Resistance (R): An element’s ability to resist the flow of electric current. Measured in Ohms (Ω).
Ohm’s Law: The voltage across a resistor is directly proportional to the current flowing through it (V = IR).

➗ Formulas

Current: $i = \frac{d q}{d t}$
Charge: $q (t) = \int_{t_{0}}^{t} i (τ) d τ + q (t_{0})$
Voltage: $v_{ab} = \frac{d w}{d q}$
Power: $p = \frac{d w}{d t} = v \cdot i$
Energy: $w (t) = \int_{t_{0}}^{t} p (τ) d τ = \int_{t_{0}}^{t} v (τ) i (τ) d τ$
Resistance: $R = ρ \frac{l}{A}$
Conductance: $G = \frac{1}{R}$

✍️ Notes

The Basis

Observable Quantities: Voltage and Current are the two fundamental, measurable quantities in circuit analysis.
Ports: Circuits are analyzed based on the relationship between voltages and currents at specific ports (pairs of access points).
Lumped Elements: We model circuits using idealized components (lumped elements) connected by ideal wires.

Basic Quantities

SI Units: We use the SI standard system (meters, seconds, amperes).
Charge (q): Measured in Coulombs (C). The fundamental charge is that of an electron, $e = - 1.602 \times 1 0^{- 19} C$ .
Current (i): The flow of positive charge.
- DC (Direct Current): Constant over time.
- AC (Alternating Current): Varies sinusoidally over time.

Electric current is the rate of flow of electric charge through a conductor. It quantifies the movement of charges, such as electrons.

💡 Explanation

Electric current, denoted by I or i and measured in Amperes (A), is a fundamental quantity in electrical circuits.

Definition: It is the time rate of change of charge dq/dt.
Formula: $i (t) = \frac{d q ( t )}{d t}$
Conventional Current: By convention, the direction of current flow is defined as the direction positive charges would move. In most conductors (like metal wires), the actual charge carriers are electrons, which move in the opposite direction.

Types of Current

Direct Current (DC): The current remains constant over time. Its value does not change.
Alternating Current (AC): The current varies sinusoidally with time, periodically reversing direction.

Measurement

Current is measured through a circuit element. An ammeter must be placed in series with the component.

🖼️ Diagrams & Visuals

DC vs. AC Current

Voltage (v): Potential difference, representing the energy required to move a unit charge. $v_{ab}$ means the potential at ‘a’ is higher than at ‘b’ if $v_{ab} > 0$ .

Energy and Power

Electric power is the rate at which electrical energy is transferred, consumed, or supplied in an electric circuit. It is the product of voltage and current.

Power, denoted by P and measured in Watts (W), quantifies how quickly energy is being used or generated in a circuit.

Definition: Power is the rate of change of energy (or work) over time.
Formula: $p (t) = \frac{d w ( t )}{d t} = v (t) \cdot i (t)$
Energy Calculation: Energy is the integral of power over time. $w = \int_{t_{0}}^{t} p (τ) d τ$

Power Absorption vs. Supply

Absorbing Power: An element is absorbing or consuming power if the power calculated is positive (p > 0). This typically happens in passive components like resistors, which dissipate energy as heat.
Supplying Power: An element is supplying or generating power if the power calculated is negative (p < 0). This is characteristic of active components like batteries or power sources.

📝 Summary

The Passive Sign Convention (PSC) is a standard used in circuit analysis to define the sign of power. It states that if the reference direction of the current enters the positive reference terminal of an element, then the power absorbed by that element is P = VI.

💡 Explanation

PSC provides a consistent framework to determine whether a component is absorbing or supplying power without knowing the component’s nature beforehand.

The Rule

Assign Voltage Polarity: Assign + and - terminals to the component.
Assign Current Direction: Define a reference direction for the current i flowing through the component.
Apply the Convention:
- If the current i enters the component through the + terminal, the power absorbed is $p = v \cdot i$ .
- If the current i enters the component through the - terminal, the power supplied is $p = v \cdot i$ (or the power absorbed is $p = - v \cdot i$ ).

Interpreting the Result

If the calculated power p is positive (+), the component is absorbing or consuming energy. (e.g., a resistor).
If the calculated power p is negative (-), the component is supplying or generating energy. (e.g., a battery).

🖼️ Diagrams & Visuals

Case 1: Power Absorbed

Current i enters the + terminal. The calculated power p = vi is the power absorbed by the element.

Case 2: Power Supplied

Current i enters the - terminal. The power absorbed is p = -vi. If p is negative, power is supplied.

Topic: Electric Power
Lecture: lecture_EE1C1_lec1_2025-09-02

Power Calculation in a Simple Circuit

This diagram shows a voltage source supplying power to a resistor, which absorbs and dissipates it.

Circuit Elements

Circuit elements are the fundamental, idealized building blocks used to model and analyze electrical systems. They are categorized as either active or passive.

In circuit theory, we use simplified models called lumped elements to represent physical electrical components. This assumes that the element’s properties (like resistance) are concentrated at a single point in space.

Categories of Circuit Elements

Active Elements:
- Capable of generating or supplying energy to a circuit indefinitely.
- The most common examples are voltage and current sources.
- They can be further divided into:
  - Independent Sources
    
    An independent source is an active circuit element that provides a specified voltage or current that is completely independent of any other circuit variable.
    
    Independent sources are ideal models for devices that generate electrical energy, like batteries or power supplies. They are called “ideal” because they can theoretically supply infinite power and their value does not change based on the load connected to them.
    
    Types of Independent Sources
    1. Independent Voltage Source:
      - Maintains a specified voltage across its terminals, regardless of the current flowing through it.
      - An ideal car battery is a good approximation.
    2. Independent Current Source:
      - Maintains a specified current flowing through it, regardless of the voltage that appears across its terminals.
    Standard Symbols
    
    V-I Characteristics
    
    The graphs show that the voltage of an ideal voltage source is constant for any current, and the current of an ideal current source is constant for any voltage.
  - Dependent Sources
    
    A dependent (or controlled) source is an active circuit element whose output voltage or current is controlled by another voltage or current elsewhere in the circuit.
    
    Dependent sources are essential for modeling active devices like transistors and operational amplifiers (op-amps). Their symbol is typically a diamond shape. The key feature is that their output is a function of a controlling variable from another part of the circuit.
    
    Types of Dependent Sources
    1. Voltage-Controlled Voltage Source (VCVS):
      - Output voltage depends on a control voltage $v_{x}$ .
      - Output: $V = μ v_{x}$ , where $μ$ is a dimensionless gain factor.
    2. Current-Controlled Voltage Source (CCVS):
      - Output voltage depends on a control current $i_{x}$ .
      - Output: $V = r i_{x}$ , where $r$ is a transfer resistance (in Ohms).
    3. Voltage-Controlled Current Source (VCCS):
      - Output current depends on a control voltage $v_{x}$ .
      - Output: $I = g v_{x}$ , where $g$ is a transfer conductance (in Siemens).
    4. Current-Controlled Current Source (CCCS):
      - Output current depends on a control current $i_{x}$ .
      - Output: $I = β i_{x}$ , where $β$ is a dimensionless current gain factor.
    Standard Symbols for the Four Types
```
\begin{document}
\begin{tikzpicture}[circuitikz, scale=0.9, transform shape]
    % VCVS
    \draw (0,3) to[cV, l=$\mu v_x$] ++(2,0);
    \node[right] at (2.2, 3) {VCVS};
    
    % CCVS
    \draw (0,2) to[cR, l=$r i_x$] ++(2,0);
    \node[right] at (2.2, 2) {CCVS};
 
    % VCCS
    \draw (0,1) to[cG, l=$g v_x$] ++(2,0);
    \node[right] at (2.2, 1) {VCCS};
 
    % CCCS
    \draw (0,0) to[cI, l=$\beta i_x$] ++(2,0);
    \node[right] at (2.2, 0) {CCCS};
\end{tikzpicture}
\end{document}
```
Passive Elements:
- Incapable of generating energy. They absorb, dissipate, or store energy.
- Examples include:
  - Resistors (dissipate energy)
    
    Resistance is a measure of the opposition to current flow in an electrical circuit. Ohm’s Law defines the relationship between voltage, current, and resistance for a resistive element.
    
    Explanation
    
    Resistance
    
    A resistor is a passive electrical component that creates resistance. Its primary function is to reduce current flow, adjust signal levels, divide voltages, and terminate transmission lines.
    - Symbol: R
    - Unit: Ohm (Ω)
    - Physical Property: The resistance of a material is determined by its resistivity ( $ρ$ ), length ( $l$ ), and cross-sectional area ( $A$ ).
    - Formula: $R = ρ \frac{l}{A}$
    Ohm’s Law
    
    Stated by Georg Simon Ohm in 1827, this law is fundamental to circuit analysis. It states that the voltage across a resistor is directly proportional to the current flowing through it.
    - Formula: $V = I \cdot R$
    - Linear Relationship: For a given resistor, the relationship between voltage and current is linear. If you double the voltage, the current doubles.
    Conductance
    - Definition: Conductance (G) is the reciprocal of resistance and represents how easily current can flow through an element.
    - Formula: $G = \frac{1}{R}$
    - Unit: Siemens (S)
    Diagrams & Visuals
    
    Ohm’s Law Triangle
    
    A visual aid to remember the three forms of Ohm’s Law.
```
graph TD
    subgraph Ohm's Law
        V((V))
        I((I)) --- R((R))
    end
    style V fill:#f9f,stroke:#333,stroke-width:2px
    style I fill:#ccf,stroke:#333,stroke-width:2px
    style R fill:#cfc,stroke:#333,stroke-width:2px
```
    - To find Voltage (V): Cover V → I x R
    - To find Current (I): Cover I → V / R
    - To find Resistance (R): Cover R → V / I
    V-I Characteristic of a Resistor
    
    The linear relationship described by Ohm’s Law. The slope of the line is equal to the resistance R.
  - Capacitors (store energy in an electric field)
  - Inductors (store energy in a magnetic field)

Common Circuit Element Symbols

Voltage

📝 Summary

Voltage is the difference in electric potential energy per unit of charge between two points in an electric field. It represents the “pressure” that drives electric current.

💡 Explanation

Voltage, denoted by V and measured in Volts (V), is the work required to move a unit of positive charge from a reference point to a specific point.

Definition: It is defined as the change in work (energy) dw per unit charge dq.
Formula: $v = \frac{d w}{d q}$
Analogy: In a water pipe analogy, voltage is similar to water pressure. Higher pressure results in a stronger flow.

Key Characteristics

Potential Difference: Voltage is always measured between two points. It’s a relative quantity. The notation $v_{ab}$ signifies the voltage at point a with respect to point b.
Polarity: It has a polarity (+ and -) indicating the direction of the potential difference. Positive charge carriers naturally move from a higher potential (+) to a lower potential (-).

🖼️ Diagrams & Visuals

A simple circuit showing how voltage is measured across a component.

Book: Fundamentals of Electric Circuits (Chapter 1)
Lecture: lecture_EE1C1_lec1_2025-09-02

🔗 Resources

Presentation: EE1C1_week_1_1.pdf
https://www.youtube.com/watch?v=4cnPw6SLxCM

❓ Post lecture

Review the passive sign convention.
Understand the difference between independent and dependent sources.
Practice applying Ohm’s law.

📖 Homework

Check Brightspace for assigned problems from Chapter 1 and 2.

JKoch notes

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lecture_EE1C1_lec1_2025-09-02

📌 Overview

🎯Learning Objectives

💡 Topics

💡Key Concepts & Definitions

➗ Formulas

✍️ Notes

The Basis

Basic Quantities

💡 Explanation

Types of Current

Measurement

🖼️ Diagrams & Visuals

DC vs. AC Current

Energy and Power

Power Absorption vs. Supply

📝 Summary

💡 Explanation

The Rule

Interpreting the Result

🖼️ Diagrams & Visuals

Case 1: Power Absorbed

Case 2: Power Supplied

📚 Related Resources

Power Calculation in a Simple Circuit

Circuit Elements

Categories of Circuit Elements

Independent Sources

Types of Independent Sources

Standard Symbols

V-I Characteristics

Types of Dependent Sources

Standard Symbols for the Four Types

Explanation

Resistance

Ohm’s Law

Conductance

Diagrams & Visuals

Ohm’s Law Triangle

V-I Characteristic of a Resistor

Common Circuit Element Symbols

Voltage

📝 Summary

💡 Explanation

Key Characteristics

🖼️ Diagrams & Visuals

📚 Related Resources

🔗 Resources

❓ Post lecture

📖 Homework

Graph View

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