Kirchoff’s Law
Source inspiration: (Mathew 2000-2019).
Background
Kirchoff’s voltage law (KVL) states that the algebraic sum of voltage drops around any closed loop is zero. Using loop currents, resistor networks lead directly to linear systems.
For each loop, \[ \sum (R\,I) = \sum E, \tag{1} \] where \(R\,I\) are resistor drops and \(E\) are source voltages (with sign convention).
Collecting all loop equations gives
\[ A\mathbf{i}=\mathbf{b}, \tag{2} \]
with unknown loop-current vector \(\mathbf{i}\).
Network #1 (Two Loops)
From the module’s network model:
\[ \begin{aligned} (r_1+r_3+r_5)i_1-r_3i_2&=e_1,\\ -r_3i_1+(r_2+r_3+r_4+r_6)i_2&=0. \end{aligned} \tag{3} \]
Example 1
Given
\[ r_1=10,\; r_2=10,\; r_3=10,\; r_4=20,\; r_5=10,\; r_6=30,\; e_1=20. \]
Step 1: Substitute into (3):
\[ \begin{aligned} 30i_1-10i_2&=20,\\ -10i_1+70i_2&=0. \end{aligned} \]
Step 2: Matrix form and solve:
\[ \begin{pmatrix} 30 & -10\\ -10 & 70 \end{pmatrix} \begin{pmatrix} i_1\\ i_2 \end{pmatrix} = \begin{pmatrix} 20\\0 \end{pmatrix} \quad\Longrightarrow\quad \begin{pmatrix} i_1\\ i_2 \end{pmatrix} = \begin{pmatrix} \tfrac{7}{10}\\[2pt]\tfrac{1}{10} \end{pmatrix}. \]
So
\[ i_1=0.7,\qquad i_2=0.1. \]
Verification:
| Loop equation | Left side | Right side |
|---|---|---|
| \(30i_1-10i_2=20\) | \(30(0.7)-10(0.1)=20\) | \(20\ \checkmark\) |
| \(-10i_1+70i_2=0\) | \(-10(0.7)+70(0.1)=0\) | \(0\ \checkmark\) |
Network #2 (Three Loops, One Source)
From the module’s loop equations:
\[ \begin{aligned} (r_1+r_4+r_7)i_1-r_4i_2&=e_1,\\ -r_4i_1+(r_2+r_4+r_5+r_8)i_2-r_5i_3&=0,\\ -r_5i_2+(r_3+r_5+r_6+r_9)i_3&=0. \end{aligned} \tag{4} \]
Example 2
Given
\[ \begin{aligned} r_1=r_2=r_3=r_4=r_7=r_8=r_9&=10,\\ r_5=r_6&=20,\\ e_1&=36. \end{aligned} \]
Step 1: Substitute into (4):
\[ \begin{aligned} 30i_1-10i_2&=36,\\ -10i_1+50i_2-20i_3&=0,\\ -20i_2+60i_3&=0. \end{aligned} \]
Step 2: Solve
\[ \begin{pmatrix} i_1\\ i_2\\ i_3 \end{pmatrix} = \begin{pmatrix} \tfrac{13}{10}\\[2pt] \tfrac{3}{10}\\[2pt] \tfrac{1}{10} \end{pmatrix} = \begin{pmatrix} 1.3\\0.3\\0.1 \end{pmatrix}. \]
Verification:
| Loop equation | Left side | Right side |
|---|---|---|
| \(30i_1-10i_2=36\) | \(30(1.3)-10(0.3)=36\) | \(36\ \checkmark\) |
| \(-10i_1+50i_2-20i_3=0\) | \(-13+15-2=0\) | \(0\ \checkmark\) |
| \(-20i_2+60i_3=0\) | \(-6+6=0\) | \(0\ \checkmark\) |
Network #3 (Three Loops, Two Sources)
From the module’s loop equations:
\[ \begin{aligned} (r_1+r_3+r_4)i_1+r_3i_2+r_4i_3&=e_1,\\ r_3i_1+(r_2+r_3+r_5)i_2-r_5i_3&=e_2,\\ r_4i_1-r_5i_2+(r_4+r_5+r_6)i_3&=0. \end{aligned} \tag{5} \]
Example 3
Given
\[ r_1=r_2=r_4=1,\quad r_3=r_5=2,\quad r_6=4,\quad e_1=23,\quad e_2=29. \]
Step 1: Substitute into (5):
\[ \begin{aligned} 4i_1+2i_2+i_3&=23,\\ 2i_1+5i_2-2i_3&=29,\\ i_1-2i_2+7i_3&=0. \end{aligned} \]
Step 2: Solve
\[ \begin{pmatrix} i_1\\ i_2\\ i_3 \end{pmatrix} = \begin{pmatrix} 3\\5\\1 \end{pmatrix}. \]
Verification:
| Loop equation | Left side | Right side |
|---|---|---|
| \(4i_1+2i_2+i_3=23\) | \(4(3)+2(5)+1=23\) | \(23\ \checkmark\) |
| \(2i_1+5i_2-2i_3=29\) | \(2(3)+5(5)-2(1)=29\) | \(29\ \checkmark\) |
| \(i_1-2i_2+7i_3=0\) | \(3-10+7=0\) | \(0\ \checkmark\) |
Network solutions are only physically meaningful when resistor values are positive and the coefficient matrix is nonsingular. Near-singular systems can produce large sensitivity in current predictions.
Summary
Kirchoff loop modeling converts circuits into linear algebra:
- Write one KVL equation per independent loop.
- Assemble the system matrix and source vector.
- Solve for loop currents and verify each equation numerically.
This gives a direct bridge from circuit laws to matrix methods used throughout numerical analysis.