Krichoff’s Law - Krichoff’s Law – [Lecture 9] – An introduction

Learning Objectives

  • Understand the concept of Kirchhoff’s laws
  • Apply Kirchhoff’s laws to solve complex circuits
  • Analyze parallel and series circuits using Kirchhoff’s laws
  • Solve numerical problems based on Kirchhoff’s laws
  • Analyze circuit diagrams using Kirchhoff’s laws

Introduction to Kirchhoff’s Laws

  • Gustav Kirchhoff’s contribution to circuit analysis
  • Kirchhoff’s current law (KCL)
    • States that the algebraic sum of currents at any node in a circuit is zero
    • Useful in analyzing complex circuits
  • Kirchhoff’s voltage law (KVL)
    • States that the algebraic sum of voltages in any loop or closed path in a circuit is zero
    • Helpful in solving circuits with multiple loops

Kirchhoff’s Current Law (KCL)

  • Explained using conservation of electric charge
  • Key points:
    • The total inflow current at a node is equal to the total outflow current
    • The sum of currents entering a node must equal the sum of currents leaving the node

Kirchhoff’s Voltage Law (KVL)

  • Described using the law of conservation of energy
  • Key points:
    • The total voltage drop across any closed loop is equal to the voltage rise
    • It considers all elements in series within the loop

Series and Parallel Circuits

  • Series circuits
    • Components connected in a row, allowing only one path for the current
    • The current remains the same throughout the circuit
    • The total resistance is the sum of individual resistances
  • Parallel circuits
    • Components connected in parallel, providing multiple paths for the current
    • The voltage across each component is the same
    • The reciprocal of the total resistance is the sum of the reciprocals of individual resistances

Examples of Kirchhoff’s Laws

  1. Example 1: Applying KCL
  • Given a node with three branches, find the current through each branch
  • Use Kirchhoff’s current law to write equations and solve
  1. Example 2: Applying KVL
  • Given a closed loop with resistors and a voltage source, find the current flowing through the loop
  • Use Kirchhoff’s voltage law to write equations and solve

Kirchhoff’s Laws in Circuit Diagrams

  • Analyzing complex circuit diagrams using Kirchhoff’s laws
  • Step-by-step approach:
    1. Identify nodes, branches, and loops
    2. Apply Kirchhoff’s laws to write equations
    3. Solve the equations to obtain unknowns
    4. Verify the obtained solution by rechecking

Applying Kirchhoff’s Laws

  • Analyze circuits by applying Kirchhoff’s laws in the following order:
    1. Identify nodes and assign current directions
    2. Apply Kirchhoff’s current law to write equations for each node
    3. Write equations for each loop using Kirchhoff’s voltage law
    4. Solve the system of equations to find the unknowns
    5. Verify the solution and interpret the results

Summary

  • Kirchhoff’s laws are fundamental in circuit analysis
  • Kirchhoff’s current law (KCL) deals with currents at nodes
  • Kirchhoff’s voltage law (KVL) deals with voltages in loops
  • Series and parallel circuits can be analyzed using Kirchhoff’s laws
  • Examples and practice problems help reinforce the concepts
  • Complex circuit diagrams can be simplified using Kirchhoff’s laws
  1. Kirchhoff’s Current Law (KCL) Example
  • Consider a node with three branches connected to it
    • Branch 1: Current entering = 2 A
    • Branch 2: Current entering = 3 A
    • Branch 3: Unknown current
  • Applying KCL, the sum of currents entering the node equals the sum of currents leaving the node
  • Equation: 2 A + 3 A - I3 = 0
  • Solve for I3 to find the current through the third branch
  1. Kirchhoff’s Voltage Law (KVL) Example
  • Analyze a closed loop with resistors and a voltage source
  • Assume the voltage across the source is V1
  • Voltage drops across two resistors: V2 and V3
  • Applying KVL, the sum of voltage drops in the loop equals the voltage rise
  • Equation: V1 - V2 - V3 = 0
  • Solve for V1, V2, and V3 to find the potential differences in the circuit
  1. Series Circuits - Current Flow
  • In series circuits, components are connected in a row
  • The same current flows through each component
  • Illustration: Three resistors in series
    • Current flowing through R1 = Current flowing through R2 = Current flowing through R3
  • Total resistance in series circuit
    • R_total = R1 + R2 + R3
  1. Series Circuits - Voltage Distribution
  • The sum of individual voltage drops equals the total voltage source in a series circuit
  • Illustration: Three resistors in series with a voltage source
    • Voltage across R1 + Voltage across R2 + Voltage across R3 = Total voltage source
  • Voltage drop across each resistor depends on its resistance
  1. Parallel Circuits – Current Distribution
  • In parallel circuits, components are connected in parallel, providing multiple paths for current
  • The sum of individual currents entering the parallel branches equals the total current leaving the source
  • Illustration: Three resistors in parallel
    • Current entering parallel branch = Current entering other parallel branches
    • Total current leaving the source = Sum of currents entering parallel branches
  1. Parallel Circuits – Voltage Distribution
  • In a parallel circuit, the voltage across each component is the same
  • The reciprocal of the total resistance is equal to the sum of the reciprocals of individual resistances
  • Illustration: Three resistors in parallel
    • 1/R_total = 1/R1 + 1/R2 + 1/R3
  1. Kirchhoff’s Laws in Circuit Diagrams - Steps
  • Analyzing complex circuit diagrams using Kirchhoff’s laws
  • Step 1: Identify nodes, branches, and loops
  • Step 2: Assign current directions at each node
  • Step 3: Apply Kirchhoff’s current law to write equations for each node
  • Step 4: Write equations for each loop using Kirchhoff’s voltage law
  • Step 5: Solve the system of equations to find the unknowns
  1. Applying Kirchhoff’s Laws - Example
  • Analyze the given circuit using Kirchhoff’s laws
  • Step 1: Identify nodes, branches, and loops
  • Step 2: Assign current directions (e.g., clockwise or counterclockwise)
  • Step 3: Apply Kirchhoff’s current law to write equations for each node
  • Step 4: Apply Kirchhoff’s voltage law to write equations for each loop
  • Step 5: Solve the system of equations to find the unknowns (currents or voltages)
  1. Verification and Interpretation of Solution
  • After solving the circuit using Kirchhoff’s laws, verify the obtained solution
  • Check whether the currents and voltages satisfy the equations
  • Analyze the results and interpret the obtained values
  • Compare with expected values or reference values
  • Ensure the solution is physically valid and matches the circuit configuration
  1. Summary and Review
  • Kirchhoff’s laws are essential tools in circuit analysis
  • Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL) help solve complex circuits
  • Series circuits have the same current but different voltage drops
  • Parallel circuits have the same voltage but different currents
  • Analyzing circuit diagrams using Kirchhoff’s laws involves several steps
  • Solving and verifying the obtained solution is crucial for accurate results
  1. Sample Problem - KCL and KVL
  • Given a circuit with three resistors connected in series
  • R1 = 10 Ω, R2 = 20 Ω, R3 = 30 Ω
  • A voltage source of 12 V is connected across the circuit
  • Using Kirchhoff’s laws, we can solve for various quantities:
    • Use KCL to find the current through each resistor
    • Use KVL to find the voltage drop across each resistor
    • Verify the obtained values by rechecking with Kirchhoff’s laws
  1. Solution - Current Calculation
  • Applying KCL at the first node:
    • Current entering = Current leaving
    • I1 = I2 + I3
  • Since it is a series circuit, the current is the same through each resistor
  • I1 = I2 = I3 = I
  • I = V / R_total
  • I = 12 V / (10 Ω + 20 Ω + 30 Ω)
  • I = 12 V / 60 Ω
  • I = 0.2 A
  1. Solution - Voltage Calculation
  • Applying KVL around the loop:
    • V_source = V_R1 + V_R2 + V_R3
  • Voltage drop across each resistor is equal to the product of current and resistance
  • V_R1 = I * R1 = 0.2 A * 10 Ω = 2 V
  • V_R2 = I * R2 = 0.2 A * 20 Ω = 4 V
  • V_R3 = I * R3 = 0.2 A * 30 Ω = 6 V
    • V_source = 2 V + 4 V + 6 V
    • V_source = 12 V
  • Verify the obtained values by rechecking using Kirchhoff’s laws
  1. Rechecking Using KCL
  • Applying KCL at the first node:
    • Current entering = Current leaving
    • I1 = I2 + I3
  • Substituting the known values:
    • 0.2 A = 0.2 A + 0.2 A
  • The equation is satisfied, verifying the solution obtained earlier
  1. Rechecking Using KVL
  • Applying KVL around the loop:
    • V_source = V_R1 + V_R2 + V_R3
  • Substituting the known values:
    • 12 V = 2 V + 4 V + 6 V
  • The equation is satisfied, confirming the solution obtained earlier
  1. Example Problem - Complex Circuit
  • Given a complex circuit with resistors and current sources
  • Apply Kirchhoff’s laws to find the current through each component
  • Step 1: Identify nodes, branches, and loops
  • Step 2: Assign current directions for each branch
  • Step 3: Apply KCL at each node to write equations
  • Step 4: Apply KVL around each loop to write equations
  • Step 5: Solve the system of equations to find the unknown currents
  1. Solution Steps - Complex Circuit
  • Step 1: Identify nodes, branches, and loops
  • Node A, Node B, and Node C
  • Branches and loops labeled with numbers for reference
  • Step 2: Assign current directions for each branch
  • Assume current IA flowing into Node A
  • Assume current IB flowing into Node B
  • Step 3: Apply KCL at each node to write equations
  • Node A: IA = IC + ID
  • Node B: IB + IA = IE + IF
  • Step 4: Apply KVL around each loop to write equations
  • Loop 1 (Node A-Node C-Node B): VC - VB = 0
  • Loop 2 (Node C-Node B): VC - VB - VD = 0
  • Loop 3 (Node A-Node B): VB - VA - VE = 0
  • Loop 4 (Node C): VC - VD - VF = 0
  • Step 5: Solve the system of equations to find the unknown currents
  1. Verification and Interpretation
  • After solving the circuit, verify the obtained currents using Kirchhoff’s laws
  • Check that the sum of currents entering a node equals the sum of currents leaving the node
  • Analyze the obtained currents based on the circuit configuration
  1. Summary and Key Points
  • Kirchhoff’s laws, KCL and KVL, are essential tools in circuit analysis
  • They help calculate currents and voltages in complex circuits
  • Series circuits have the same current and voltage drops across each resistor
  • Parallel circuits have the same voltage and different currents
  • Solve circuits step-by-step, verifying the solution using Kirchhoff’s laws
  • Apply Kirchhoff’s laws systematically to simplify complex circuit diagrams
  1. References
  • Fundamentals of Physics by Halliday, Resnick, and Walker
  • Physics for Scientists and Engineers by Serway and Jewett
  • Circuit Analysis by Robbins and Miller
  • Physics Classroom - Circuit Analysis and Kirchhoff’s Laws
  • Khan Academy - Kirchhoff’s Laws and Circuit Analysis
  • MIT OpenCourseWare - Circuit Analysis and Kirchhoff’s Laws
  • [Your recommended resources here]