lecture_EE1C1_lec2_kirchhoff_2025-09-08

📌 Overview

This lecture covers the fundamental laws and techniques for analyzing linear circuits, building upon the basic concepts of voltage, current, and resistance.

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🎯Learning Objectives

• Describe digital systems using different formats (e.g. truth table, switches, etc.) and convert them from one format to the other ✅ 2025-09-28
• Explain how combinational and sequential circuits work at a high abstraction level ✅ 2025-09-28
• Recognize the symbols of the Boolean gates and draw Boolean circuits from expressions ✅ 2025-09-28
• Manipulate Boolean expressions using Boolean theorems and DeMorgan’s law ✅ 2025-09-28
• Convert Boolean circuits to NAND, NOR, NOT equivalents ✅ 2025-09-28

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💡Key Concepts & Definitions

➗ Formulas

Concept	Formula	Description
Ohm’s Law	$V=I\cdot R$	Relates voltage, current, and resistance.
Power Dissipation	$P=V\cdot I=I^{2}R$	Power consumed by a resistive element.
Kirchhoff’s Current Law (KCL)	$\sum I_{in}=\sum I_{out}$	The sum of currents entering a node is zero.
Kirchhoff’s Voltage Law (KVL)	$\sum V=0$	The sum of voltages in a closed loop is zero.
Series Resistance	$R_{eq}=R_1+R_2+\dots$	Equivalent resistance of resistors in series.
Parallel Resistance	$\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+\dots$	Equivalent resistance of resistors in parallel.
Voltage Divider	$V_x=V_{total}\cdot \frac{R_x}{R_{total}}$	Voltage across a resistor in a series circuit.
Current Divider	$I_x=I_{total}\cdot \frac{R_{total}}{R_x}$	Current through a resistor in a parallel circuit.

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✍️ Notes

1. Ohm’s Law & Power

A recap of the relationship between voltage, current, and resistance, and how power is dissipated in resistive elements.

• Ohm’s Law: $V=I\cdot R$
• Power Dissipation: $P=V\cdot I=I^{2}R=\frac{V^2}{R}$

2. Kirchhoff’s Current Law (KCL)

📝 Summary

Kirchhoff’s Current Law (KCL) states that the algebraic sum of currents entering a node (or a closed boundary) is zero. It is based on the principle of conservation of electric charge.

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💡 Explanation

Based on the conservation of charge, KCL is a fundamental law for circuit analysis.

• Principle: What goes in must come out. The total current flowing into any junction in a circuit is exactly equal to the total current flowing out of that junction.
• Formula: $\sum I_{in}=\sum I_{out}$ or ${\sum }_{k=1}^n{I}_k=0$

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🖼️ Diagrams & Visuals

3. Kirchhoff’s Voltage Law (KVL)

📝 Summary

Kirchhoff’s Voltage Law (KVL) states that the algebraic sum of all voltages around any loop in a circuit is equal to zero. It is a consequence of the law of conservation of energy.

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💡 Explanation

KVL is a fundamental principle for analyzing circuits. A loop is, by definition, any closed path in a circuit that starts and ends at the same node without passing through any intermediate node more than once. The law implies that if you measure the voltage changes across each element along a loop, the total change must be zero upon returning to the start.

• Principle: The sum of voltage rises equals the sum of voltage drops in a loop.
• Formula: ${\sum }_{k=1}^n{V}_k=0$

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🖼️ Diagrams & Visuals

What is a Loop?

A loop is a closed path. In the circuit below, there are three distinct loops:

• Loop 1 (left): V1 → R1 → R2 → V1
• Loop 2 (right): R2 → R3 → R4 → R2
• Outer Loop: V1 → R1 → R3 → R4 → V1

KVL Application

For Loop 1, KVL would be applied as follows:

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📚 Related Resources

• Companion Law: Kirchhoff’s Current Law (KCL)

4. Series and Parallel Circuits

📝 Summary

Series and parallel are the two fundamental ways to connect circuit elements. The configuration determines how current and voltage are distributed throughout the circuit.

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💡 Explanation

Series Connection

• Definition: Components are connected end-to-end, providing only one path for the current to flow.
• Current: The current is the same through all series components.
• Voltage: The total voltage is the sum of the voltages across each component (voltage divides).
• Equivalent Resistance: $R_{eq}=R_1+R_2+\cdots +R_n$

Parallel Connection

• Definition: Components are connected across the same two nodes, providing multiple paths for the current.
• Voltage: The voltage is the same across all parallel components.
• Current: The total current is the sum of the currents through each branch (current divides).
• Equivalent Resistance: $\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots +\frac{1}{R_n}$

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🖼️ Diagrams & Visuals

Series Circuit

5. Voltage and Current Division

Voltage and current division are techniques to determine the voltage across or the current through one of several resistors in a series or parallel combination, respectively, without having to solve the entire circuit.

Voltage Division

• Applies to: Series circuits.
• Principle: In a series circuit, the total voltage is distributed among the resistors in direct proportion to their resistance. A larger resistor gets a larger share of the total voltage.
• Formula: The voltage $V_x$ across a specific resistor $R_x$ is: $V_x=V_{total}\cdot \frac{R_x}{R_{eq}}=V_{total}\cdot \frac{R_x}{R_1+R_2+\cdots +R_n}$

Current Division

• Applies to: Parallel circuits.
• Principle: In a parallel circuit, the total current is distributed among the branches in inverse proportion to their resistance. A smaller resistor gets a larger share of the total current.
• Formula: The current $I_x$ through a specific branch with resistor $R_x$ is: $I_x=I_{total}\cdot \frac{R_{eq}}{R_x}$
• For Two Resistors: A common simplified formula for two resistors in parallel is: $I_1=I_{total}\cdot \frac{R_2}{R_1+R_2}$

Voltage Divider Circuit

Current Divider Circuit

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🔗 Resources

• Presentation:

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📖 Homework