# (Electrical) Mesh Current Analysis

/We looked at how to analyze circuits using node voltage analysis. Now let's explore how to do the same using mesh current analysis.

## Review

In a circuit, a mesh is a closed loop, such as $i_1$ and $i_2$ in the following figure.

The first step of mesh analysis is to identify all the meshes and assign a clockwise current in each mesh where there is no current source. If there is a current source, then the mesh current should be in direction of the current source.

Kirchoff's Voltage Law (KVL) says that the sum of all the voltages across every element in a mesh is 0. For example, looking at the following circuit we can write KVL equations:

$v_s-v_1-v_2=0$, and

$v_2+v_3+v_4=0$

The voltage across an element shared by two meshes is expressed in terms of both mesh currents. For example, $R_2$ is $v_2=(i_1-i_2)R_2$.

KVL for mesh 1 is: $v_s-i_1R_1-(i_1-i_2)R_2=0$

... and for mesh 2 is: $(i_2-i_1)R_2+i_2R_3+i_2R_4=0$

Notice that the polarity of the voltage across $R_2$ for mesh 2 is now flipped. This is to keep consistent with the direction of $i_2$. The voltage across can be written $R_2$ $v_2=(i_2-i_1)R_2$.

Using KVL, we emerge with the following system of equations:

$(R_1+R_2)i_1-R_2i_2=v_s$

$-R_2i_1+(R_2+R_3+R4)i_2=0$

## Question

Given DC voltage sources of $V_1=10V$, $V_2=9V$, and $V_3=1V$ and resistors with resistances of $R_{1}=5\Omega$, $R_2=10\Omega$, $R_3=5\Omega$, and $R_4=5\Omega$, you connect them into the following circuit:

**Find mesh currents $i_1$ and $i_2$Bonus: Write your KVL equations**

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