Solving a Circuit with Two Equations

In a circuit with two loops sharing a resistor, each loop's voltages must add up (Kirchhoff's law), giving one linear equation per loop. Each equation is a line in the plane of possible currents, and the actual currents are where the two lines cross.

How it works

Systems of linear equations

In one sentence

A system of $m$ linear equations in $n$ unknowns is $A\mathbf{x} = \mathbf{b}$ ; it has exactly one solution when $A$ is square and invertible, which row reduction to echelon form reveals (a pivot in every column).

General formula

\begin{bmatrix} {\htmlData{sym=0}{R_1}} + {\htmlData{sym=2}{R_3}} & -{\htmlData{sym=2}{R_3}} \\ -{\htmlData{sym=2}{R_3}} & {\htmlData{sym=1}{R_2}} + {\htmlData{sym=2}{R_3}} \end{bmatrix}\begin{bmatrix} {\htmlData{sym=3}{I_1}} \\ {\htmlData{sym=4}{I_2}} \end{bmatrix} = \begin{bmatrix} {\htmlData{sym=5}{V_1}} \\ {\htmlData{sym=6}{V_2}} \end{bmatrix}

Each equation is a constraint the answer must lie on, a line here. Two independent lines in a plane cross in exactly one point.

What the symbols mean

$R_1$
Loop 1 resistor
Resistance only loop 1's current flows through.
$R_2$
Loop 2 resistor
Resistance only loop 2's current flows through.
$R_3$
Shared resistor
Carries both loop currents, in opposite directions.
$I_1$
Loop 1 current
Amps circulating in loop 1.
$I_2$
Loop 2 current
Amps circulating in loop 2.
$V_1$
Battery 1
Voltage driving loop 1.
$V_2$
Battery 2
Voltage driving loop 2.

Give each loop a current, $I_1$ and $I_2$ . The shared resistor $R_3$ carries their difference.
Going around a loop, the voltage drops must add up to its battery: $(R_1 + R_3)I_1 - R_3I_2 = V_1$ , and likewise for loop 2.
Each equation is a line in the $(I_1, I_2)$ plane. Row reduction (or the crossing point) gives the one pair of currents that satisfies both.
Because resistances are positive, the coefficient matrix is always invertible: every circuit like this has exactly one solution.

Learn the math

Systems of Linear EquationsLin Alg · Lesson 1, Linear Equations in Linear Algebra Row Reduction and Echelon FormsLin Alg · Lesson 2, Linear Equations in Linear Algebra The Matrix Equation Ax = bLin Alg · Lesson 4, Linear Equations in Linear Algebra Solution Sets of Linear SystemsLin Alg · Lesson 5, Linear Equations in Linear Algebra LU FactorizationLin Alg · Lesson 12, Matrix Algebra Kirchhoff's Laws and Circuit AnalysisE&M · Lesson 14, Current and DC Circuits

Learn the math

Systems of Linear EquationsLin Alg · Lesson 1, Linear Equations in Linear Algebra

Row Reduction and Echelon FormsLin Alg · Lesson 2, Linear Equations in Linear Algebra

The Matrix Equation Ax = bLin Alg · Lesson 4, Linear Equations in Linear Algebra

Solution Sets of Linear SystemsLin Alg · Lesson 5, Linear Equations in Linear Algebra

LU FactorizationLin Alg · Lesson 12, Matrix Algebra

Kirchhoff's Laws and Circuit AnalysisE&M · Lesson 14, Current and DC Circuits