Getting different answers by Mesh and Nodal Analysis

Question

^{simulate this circuit – Schematic created using CircuitLab}

Naming the top two loop currents clockwise as i₁ and i₂ and the bottom two as i₃ and i₄. By inspection, we can say $$i_1=-1A$$ Now, loop equation on mesh 2 gives,

$$-8-1(i_2-i_4)-1(i_1-i2)=0$$ Upon solving, we get, $$i_4=7A$$

whereas by using Nodal Analysis, $$\frac{V_1-0}{1}+\frac{V_1-8}{1}+\frac{V_1-0}{1}+\frac{V_1-8}{1}=0$$ Solving, we get, $$4V_1-16=0\Rightarrow V_1=4V$$

Applying KCL, we get $$i+\frac{(0-V_1)}{1}+5=0$$

Therefore, $$i=-1A$$

I am getting the answer as -1A (which is the correct answer). What am I doing wrong in the Mesh Analysis part?

Answer 1

The loop equation on mesh 2 is wrong - more specifically, by your definition of the direction of \$i_1\$ and \$i_2\$, the voltage drop on \$R_1\$ that is "contributing" to the loop voltage should be calculated as such:

$$V_{R_1} = (i_1-i_2) \cdot R_1 = +1(i_1-i_2)$$

So the KVL on mesh 2 would be: $$-8 -1(i_2-i_4) +1(i_1-i_2) = 0$$