notes/uni/mmme/2051_electromechanical_devices/kirchhoff.md

---
author: Akbar Rahman
date: \today
title: MMME2051 // Kirchhoff's Current Law, Voltage Law
tags: [ kirchhoff, kcl, kvl ]
uuid: 88e2eb6a-7f6a-4ea0-9850-81305028e7b5
lecture_slides: ./lecture_slides/MMME2051EMD_Lecture2A.pdf
---

# Application of Kirchhoff's Current/Voltage Laws (KCL, KVL)

(lecture slides 14-21)

![](./images/vimscrot-2023-02-09T11:09:25,300096365+00:00.png)

1. Identify all the loops in the circuit and assign each loop a "loop current" variable:


   ![](./images/vimscrot-2023-02-09T11:10:56,126073331+00:00.png)

1. Identify "branch current" values (apply KCL)


   ![](./images/vimscrot-2023-02-09T11:11:59,570957109+00:00.png)

1. Apply KVL to each loop:

   Loop 1: $10 - 2 - V_1 - V_2 = 0$  
   Loop 2: $V_2 - V_4 = 0$  
   Loop 3: $V_4 - V_3 - V_5 = 0$

   ![](./images/vimscrot-2023-02-09T11:13:18,376344102+00:00.png)

1. Apply Ohm's Law to KVL

   Loop 1 (origin at node A):

   \begin{align*}
   0 &= 10 - 2 - V_1 - V_2 \\
     &= 8 - I_1R_1 - (I_1-I_2)R_2 = 0 \\
   8 &= I_1(R_1+R+2) - I_2R_2 \\
     &= 6I_1 - I_2
   \end{align*}

   Loop 2 (origin at node B):

   \begin{align*}
   0 &= V_2 - v_4 \\
     &= (I_1-I_2)R_2 - (I_2-I_3)R_4 \\
     &= I_1 - 3I_2 + 2I_3
   \end{align*}

   Loop 3 (origin at node C):

   \begin{align*}
   0 &= V_4 - V_3 - V_5 \\
     &= (I_2-I_3)R_4 - I_3R_3 - I_3R_5 \\
     &= 2I_2 - 5I_3
   \end{align*}