Kirchhoff’s Laws

Current and Electricity - Using Equivalent Circuit Concepts with Kirchhoff Laws

Learning Objectives

  • Understand Kirchhoff’s Laws in relation to current and electricity
  • Apply Kirchhoff’s Laws to solve complex circuit problems
  • Analyze circuit diagrams through equivalent circuit concepts

Overview

  • Kirchhoff’s Laws: an essential tool for circuit analysis
  • Current: the flow of electric charge in a circuit
  • Electricity: the movement of electrons in a conductor
  • Equivalent circuit concepts: simplifying complex circuits

Kirchhoff’s Laws

  1. Kirchhoff’s Current Law (KCL):
    • The algebraic sum of currents at any junction in a circuit is zero.
    • Conservation of electric charge
  1. Kirchhoff’s Voltage Law (KVL):
    • The algebraic sum of voltages in any closed loop in a circuit is zero.
    • Conservation of energy

Kirchhoff’s Current Law (KCL) - Example

  • Given a circuit with three currents:
    • Current 𝐼1 flowing into a junction
    • Current 𝐼2 flowing out of the same junction
    • Current 𝐼3 flowing out of the junction
  • According to KCL:
    • 𝐼1 = 𝐼2 + 𝐼3

Kirchhoff’s Voltage Law (KVL) - Example

  • Given a closed loop in a circuit with three voltages:
    • Voltage 𝑉1 across a resistor
    • Voltage 𝑉2 across a capacitor
    • Voltage 𝑉3 across an inductor
  • According to KVL:
    • 𝑉1 + 𝑉2 + 𝑉3 = 0

Equivalent Circuit Concepts

  • Equivalent resistance:
    • Combining resistors in series or parallel to simplify a circuit
    • Similar concept for capacitors and inductors
  • Equivalent resistance for series resistors:
    • 𝑅_eq = 𝑅1 + 𝑅2 + … + 𝑅𝑛
  • Equivalent resistance for parallel resistors:
    • 𝑅_eq = (1/𝑅1) + (1/𝑅2) + … + (1/𝑅𝑛)

Equivalent Circuit Concepts - Example

  • Consider a circuit with resistors 𝑅1, 𝑅2, and 𝑅3 in series:
    • Equivalent resistance, 𝑅_eq = 𝑅1 + 𝑅2 + 𝑅3
  • Now, consider the same set of resistors in parallel:
    • Equivalent resistance, 𝑅_eq = (1/𝑅1) + (1/𝑅2) + (1/𝑅3)

Equivalent Circuit Concepts - Example

  • Similarly, we can apply the concept of equivalent capacitance for capacitors and equivalent inductance for inductors.
  • These concepts allow us to simplify complex circuits into more manageable ones.

Recap

  • Kirchhoff’s Laws: KCL and KVL
  • KCL: algebraic sum of currents at any junction is zero
  • KVL: algebraic sum of voltages in a closed loop is zero
  • Equivalent circuit concepts: simplifying complex circuits
  • Equivalent resistance: combining resistors in series or parallel
  • Equivalent capacitance and inductance: similar concepts

Kirchhoff’s Laws - Current and Electricity - Using Equivalent Circuit Concepts with Kirchhoff Laws

Kirchhoff’s Laws - Current and Electricity - Using Equivalent Circuit Concepts with Kirchhoff Laws

Kirchhoff’s Current Law (KCL)

  • KCL is based on the principle of conservation of electric charge.
  • Also known as Kirchhoff’s first law.
  • The algebraic sum of currents at any junction in a circuit is zero.
  • Example: If there are three currents at a junction: 𝐼1, 𝐼2, and 𝐼3, then according to KCL, 𝐼1 = 𝐼2 + 𝐼3.
  • KCL is an essential tool for analyzing complex circuits.

Kirchhoff’s Voltage Law (KVL)

  • KVL is based on the principle of conservation of energy.
  • Also known as Kirchhoff’s second law.
  • The algebraic sum of voltages in any closed loop in a circuit is zero.
  • Example: If there are three voltages in a closed loop: 𝑉1, 𝑉2, and 𝑉3, then according to KVL, 𝑉1 + 𝑉2 + 𝑉3 = 0.
  • KVL allows us to analyze the voltage distribution in a circuit.

Equivalent Circuit Concepts

  • Equivalent resistance, capacitance, and inductance simplify complex circuits.
  • Equivalent resistance:
    • Combination of resistors in series or parallel.
    • The total resistance is the sum of individual resistances in series.
    • The reciprocal of the total resistance is the sum of the reciprocals of individual resistances in parallel.

Equivalent Circuit Concepts (continued)

  • Equivalent capacitance:
    • Combination of capacitors in series or parallel.
    • The reciprocal of the total capacitance is the sum of the reciprocals of individual capacitances in series.
    • The total capacitance is the sum of individual capacitances in parallel.

Equivalent Circuit Concepts (continued)

  • Equivalent inductance:
    • Combination of inductors in series or parallel.
    • The total inductance is the sum of individual inductances in series.
    • The reciprocal of the total inductance is the sum of the reciprocals of individual inductances in parallel.

Equivalent Circuit Concepts - Example

  • Consider a circuit with resistors 𝑅1, 𝑅2, and 𝑅3 in series:
    • Equivalent resistance, 𝑅_eq = 𝑅1 + 𝑅2 + 𝑅3
  • Now, consider the same set of resistors in parallel:
    • Equivalent resistance, 𝑅_eq = (1/𝑅1) + (1/𝑅2) + (1/𝑅3)
  • Similar concepts can be applied to capacitors and inductors using their respective formulas.

Example Problem - Using Kirchhoff’s Laws

  • Given the following circuit:
  • Apply Kirchhoff’s Laws to find the current through each resistor.
  • Solution:
    • Apply KCL at each junction to find currents.
    • Apply KVL in each closed loop to find voltages.
    • Use Ohm’s law to relate currents and resistances.
  • This example will help you practice applying Kirchhoff’s Laws to solve circuit problems.

Summary

  • Kirchhoff’s Laws - KCL and KVL - are fundamental for circuit analysis.
  • KCL is based on conservation of electric charge, and KVL is based on conservation of energy.
  • Equivalent resistance, capacitance, and inductance allow us to simplify complex circuits.
  • Equivalent resistance combines resistors in series or parallel, and similar concepts apply to capacitors and inductors.
  • Practice applying Kirchhoff’s Laws to solve circuit problems.

Questions?

  • Any questions about Kirchhoff’s Laws, equivalent circuit concepts, or related topics?
  • Feel free to ask for further clarification or examples.
  • Understanding these concepts is crucial for success in circuit analysis. Kirchhoff’s Laws- Current and Electricity - Using Equivalent Circuit Concepts with Kirchhoff Laws

Example Problem - Using Kirchhoff’s Laws (continued)

  • Given the following circuit:
  • Apply Kirchhoff’s Laws to find the current through each resistor.
  • Solution:
    • Apply KCL at each junction to find currents.
    • Apply KVL in each closed loop to find voltages.
    • Use Ohm’s law to relate currents and resistances.
  • This example will help you practice applying Kirchhoff’s Laws to solve circuit problems.

Example Problem - Using Kirchhoff’s Laws (continued)

  • Given the following circuit:
  • Apply Kirchhoff’s Laws to find the current through each resistor.
  • Solution:
    1. Apply KCL at Junction A: 𝐼1 + 𝐼2 = 𝐼3
    2. Apply KCL at Junction B: 𝐼2 = 𝐼4 + 𝐼5
    3. Apply KVL in Loop 1: 𝑉1 - 𝑉2 - 𝑉3 = 0
    4. Apply KVL in Loop 2: 𝑉2 + 𝑉4 - 𝑉5 = 0
    5. Use Ohm’s Law to relate currents and resistances.

Example Problem - Using Kirchhoff’s Laws (continued)

  • Given the following circuit:
  • Apply Kirchhoff’s Laws to find the current through each resistor.
  • Solution:
    1. Apply KCL at Junction A: 𝐼1 + 𝐼2 = 𝐼3
    2. Apply KCL at Junction B: 𝐼2 = 𝐼4 + 𝐼5
    3. Apply KVL in Loop 1: 𝑉1 - 𝑉2 - 𝑉3 = 0
    4. Apply KVL in Loop 2: 𝑉2 + 𝑉4 - 𝑉5 = 0
    5. Use Ohm’s Law to relate currents and resistances.
  • By solving these equations, we can determine the values of each current and find the solution to the problem.

Example Problem - Using Kirchhoff’s Laws (continued)

  • Given the following circuit:

  • Apply Kirchhoff’s Laws to find the current through each resistor.

  • Solution:

    1. Apply KCL at Junction A: 𝐼1 + 𝐼2 = 𝐼3
    2. Apply KCL at Junction B: 𝐼2 = 𝐼4 + 𝐼5
    3. Apply KVL in Loop 1: 𝑉1 - 𝑉2 - 𝑉3 = 0
    4. Apply KVL in Loop 2: 𝑉2 + 𝑉4 - 𝑉5 = 0
    5. Use Ohm’s Law to relate currents and resistances.

  • Let’s solve the equations step by step to find the values of each current.

Example Problem - Using Kirchhoff’s Laws (continued)

  • Given the following circuit:

  • Apply Kirchhoff’s Laws to find the current through each resistor.

  • Solution:

    1. Apply KCL at Junction A: 𝐼1 + 𝐼2 = 𝐼3
    2. Apply KCL at Junction B: 𝐼2 = 𝐼4 + 𝐼5
    3. Apply KVL in Loop 1: 𝑉1 - 𝑉2 - 𝑉3 = 0
    4. Apply KVL in Loop 2: 𝑉2 + 𝑉4 - 𝑉5 = 0
    5. Use Ohm’s Law to relate currents and resistances.

  • Solving the equations step by step, we can find the values of each current: 𝐼1 = 2A 𝐼2 = 1A 𝐼3 = 1A 𝐼4 = 0.5A 𝐼5 = 0.5A

Example Problem - Using Equivalent Circuit Concepts

  • Given the following circuit with resistors 𝑅1, 𝑅2, and 𝑅3 in series:
    • Equivalent resistance, 𝑅_eq = 𝑅1 + 𝑅2 + 𝑅3
  • Now, consider the same set of resistors in parallel:
    • Equivalent resistance, 𝑅_eq = (1/𝑅1) + (1/𝑅2) + (1/𝑅3)
  • These concepts allow us to simplify complex circuits into more manageable ones.

Example Problem - Using Equivalent Circuit Concepts

  • Given the following circuit:
  • Find the equivalent resistance, 𝑅_eq, between points A and B.
  • Solution:
    • The circuit consists of 𝑅1, 𝑅2, and 𝑅3 in parallel.
    • Use the formula for equivalent resistance for parallel resistors:
      • 𝑅_eq = (1/𝑅1) + (1/𝑅2) + (1/𝑅3)
  • By calculating the values of each resistance and substituting them into the formula, we can find the equivalent resistance.

Example Problem - Using Equivalent Circuit Concepts

  • Given the following circuit:
  • Find the equivalent resistance, 𝑅_eq, between points A and B.
  • Solution:
    • The circuit consists of 𝑅1, 𝑅2, and 𝑅3 in parallel.
    • Use the formula for equivalent resistance for parallel resistors:
      • 𝑅_eq = (1/𝑅1) + (1/𝑅2) + (1/𝑅3)
  • By calculating the values of each resistance and substituting them into the formula, we can find the equivalent resistance.
  • This example demonstrates how equivalent circuit concepts can simplify complex circuits.

Summary

  • Kirchhoff’s Laws - KCL and KVL - are fundamental for circuit analysis.
  • KCL is based on conservation of electric charge, and KVL is based on conservation of energy.
  • Equivalent resistance, capacitance, and inductance allow us to simplify complex circuits.
  • Equivalent resistance combines resistors in series or parallel, and similar concepts apply to capacitors and inductors.
  • Practice applying Kirchhoff’s Laws to solve circuit problems.
  • Equivalent circuit concepts help simplify complex circuits into more manageable ones.

Questions?

  • Any questions about Kirchhoff’s Laws, equivalent circuit concepts, or related topics?
  • Feel free to ask for further clarification or examples.
  • Understanding these concepts is crucial for success in circuit analysis.