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====== Kirchhoff's Circuit Laws ======

2023-12-06, RB

[[https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws|Wikipedia (en)]]

Kirchhoff's Circuit Laws consist of two rules:

  - Junction Rule (Current)
  - Loop Rule (Voltage)

===== Literature =====

  * **University Physics II - Thermodynamics Electricity and Magnetism (OpenStax) \\ [[https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Book%3A_University_Physics_II_-_Thermodynamics_Electricity_and_Magnetism_(OpenStax)/10%3A_Direct-Current_Circuits/10.04%3A_Kirchhoff's_Rules|10.04 Kirchhoff's Rules]]**


===== Kirchhoffs' Current Law (Junction Rule) =====

It is based on the conservation of charge.

$\sum_1^n I_k = 0 $

**The sum of currents into a junction is equal to the sum of currents out of that junction.**

Equivalent expression: The sum of all currents of a junction is zero, if you grade currents into the junction positive and those out of that junction negative.

===== Kirchhoff'S Voltage Law (Loop Rule) =====

It is based on the conservation of energy.

$\sum_1^n V_k = 0 $

**The sum of the voltages around any closed loop in a circuit is equal to zero.**

===== Conventions =====

==== General ====

  * Loops and currents (directions) are labeled arbitrarily. \\ (You choose a direction and in the end you find out whether the assumption was right, e.g. whether current is flowing in that direction or in the opposite indicated by a negative current value.)

==== Voltage ====

When summing voltages: 

  * Define a loop direction for all loops, e.g. clockwise.
  *  If you travel across a voltage source from low to high (from neg. to pos. terminal of that power supply) the voltage is positive; if you are traveling from high to low (from pos. to neg.) the voltage is negative. 
  * If you follow the current through a resistor the voltage is negative (voltage drop); if you oppose the current through a resistor the voltage is positive (voltage lift).

===== Example 1: YouTube Channel NunezPhysics =====

**I did not find out yet, who the teacher behind NunezPhysics is! I could not ask for permission.**

The following video tutorials discuss an example and describes very clearly how to derive the linear equation system from Kirchhoff's laws applied to an electrical circuit.

| {{youtube>zdE7xsbuNTg?}} | {{youtube>hlUW2u-z69g?}} |
| [[https://www.youtube.com/watch?v=zdE7xsbuNTg|NunezPhysics @ YouTube: Kirchhoff's Laws Part 1]] | [[https://www.youtube.com/watch?v=hlUW2u-z69g|NunezPhysics @ YouTube: Kirchhoff's Laws Part 2]] |

| {{:eeng:topics:kirchhoff_s_circuit_laws:kirchhoff.jpg?250&direct}}               |
| Fig.: Kirchhoff's laws examples from the \\ NunezPhysics video tutorials above.  |

To solve the above problem we need three independent equations for three unknowns $I_1$, $I_2$, and $I_3$.

  * **(1)** Left junction: $ -I_1  -I_2 + I_3 = 0$ 
  * Right right junction: Redundant. No new information. Skip it. ( $ +I_1  +I_2 - I_3 = 0$ )
  * **(2)** Upper loop: $ 24 - 2I_1 - 4I_1 + 3I_2 = 24 -6I_1 +3I_2 = 0 $ 
  * **(3)** Lower loop: $ -3I_2 -1I_3 -5I_3 + 12 = -3I_2 -6I_3 + 12 = 0 $ 
  * Outer loop: Sum of outer and inner loop. No new information. Skip it.

The resulting equation system of three equations in matrix form (i-th equation in i-th row):

$\begin{pmatrix} -1 & -1 & 1 \\ -6 & 3 & 0 \\ 0 & -3& -6 \end{pmatrix}\begin{pmatrix} I_1 \\ I_2 \\ I_3 \end{pmatrix} = \begin{pmatrix} 0 \\ -24 \\ -12 \end{pmatrix} $

==== WITH PROPER UNITS! ====

To solve the above problem we need three independent equations for three unknowns $I_1$, $I_2$, and $I_3$.

  * **(1)** Left junction: \\ $ -I_1  -I_2 + I_3 = 0A$ -> multiply by $1\Omega$ \\ $(-1\Omega) I_1 + (-1\Omega)I_2 + (1\Omega)I_3 = 0A\Omega = 0V$ 

  * Right right junction: Redundant. No new information. Skip it. \\ ( $ +I_1  +I_2 - I_3 = 0A$ )

  * **(2)** Upper loop: \\ $ 24V + -2\Omega I_1 + -4\Omega I_1 + \Omega 3I_2 = 24V + (-6\Omega)I_1 + (3\Omega)I_2 = 0V $ 

  * **(3)** Lower loop: \\ $ -3\Omega I_2 -1\Omega I_3 -5\Omega I_3 + 12V = (-3\Omega)I_2 + (-6\Omega)I_3 + 12V = 0V $ 

  * Outer loop: Sum of outer and inner loop. No new information. Skip it.

The resulting equation system of three equations in matrix form (i-th equation in i-th row):

$\begin{pmatrix} -1 & -1 & 1 \\ -6 & 3 & 0 \\ 0 & -3& -6 \end{pmatrix}\Omega\begin{pmatrix} I_1 \\ I_2 \\ I_3 \end{pmatrix}A = \begin{pmatrix} 0 \\ -24 \\ -12 \end{pmatrix}V $

Devide by $\Omega A$ (=$V$)$:

$\begin{pmatrix} -1 & -1 & 1 \\ -6 & 3 & 0 \\ 0 & -3& -6 \end{pmatrix}\begin{pmatrix} I_1 \\ I_2 \\ I_3 \end{pmatrix} = \begin{pmatrix} 0 \\ -24 \\ -12 \end{pmatrix} $

/*
==== Videos on Youtube (NunezPhysics) ====

The following video tutorial discusses the above example and describes very clearly how to set up the equation system.

| {{youtube>zdE7xsbuNTg?}} | {{youtube>hlUW2u-z69g?}} |
| [[https://www.youtube.com/watch?v=zdE7xsbuNTg|NunezPhysics @ YouTube: Kirchhoff's Laws Part 1]] | [[https://www.youtube.com/watch?v=hlUW2u-z69g|NunezPhysics @ YouTube: Kirchhoff's Laws Part 2]] |
*/


==== Solution in Matlab / Octave ====

This code block shows the solution of the LES in Matlab/Octave. Klick on the block title to download.

<file matlab kirchhoff_exercise_01.m>

# kirchhoff_exercise_01.m
# This script solves a simple LES resulting from Kirchhoff's laws.
# Example from NunezPhysics video tutorial: https://www.youtube.com/watch?v=zdE7xsbuNTg
# R. Becker, 2015-04-12 

# A*x = b <=> x = A^-1 * b (another notation x = A\b)

A = [-1 -1 1 ; -6 3 0 ; 0 -3 -6]

b = [0 -24 -12]' # ' means "transpose". Here result is column vector

x = A\b

# alternative: 
# x = A^-1 * b
# or 
# x = inv(A) * b


</file>


==== Solution in Python ====

This code block shows the solution of the LES in Python (numpy). Klick on the block title to download.

<file python kirchhoff_exercise_01.py>

# kirchhoff_exercise_01.py
# This script solves a simple LES resulting from Kirchhoff's laws.
# Example from NunezPhysics video tutorial: https://www.youtube.com/watch?v=zdE7xsbuNTg
# R. Becker, 2021-10-23 

import numpy as np

# A matrix is an array of rows, which are arrays. Thus a matrix is a two dimensional array.
# The numpy.array() function is used to create 2D array (aka matrix) from a list of row lists.
R = np.array(
    [
        [-1.0, -1.0,  1.0],
        [-6.0,  3.0,  0.0],
        [ 0.0, -3.0, -6.0]
    ])

V = np.array([0.0, -24.0, -12.0])

# Inverse matrix
Rinv = np.linalg.inv(R)

# Matrix vector multiplication, aka dot product
I = Rinv.dot(V)

# Print currents
print(I)

</file>


===== Example 2: YouTube Channel Jesse Mason =====

| {{youtube>SKdK_L4jbV0?400}} |
| Excellent explanation of Kirchhoff's Laws by Jesse Mason. |

  * [[https://www.youtube.com/watch?v=Z2QDXjG2ynU|How to Solve a Kirchhoff's Rules Problem - Simple Example]], by Jesse Mason

===== Example 3: Wikipedia =====

/*
  * [[https://en.wikipedia.org/w/index.php?title=Kirchhoff%27s_circuit_laws#Example|Kirchhoff's Circuit Laws - Example]]
*/

| {{https://upload.wikimedia.org/wikipedia/commons/thumb/7/77/Kirshhoff-example.svg/540px-Kirshhoff-example.svg.png?400&direct}} |
| Source: [[https://en.wikipedia.org/w/index.php?title=Kirchhoff%27s_circuit_laws#Example|Wikipedia (en): Kirchhoff's Circuit Laws - Example]] |