Finding current ia using mesh and nodal analysis

DFrentzen · Feb 8, 2016

Circuit schematic with a 6 V source, resistors of 2 kΩ, 1 kΩ and 2 kΩ, a 2 kΩ resistor on top right, a dependent voltage source 2v1 on the right, labelled v1 and ia.

Given the circuit diagram above, I am interested in finding the current ia. Using mesh analysis, I determined it to be 1.71mA, but nodal analysis results in a value of 2.25mA. Solution attempts provided below. I think the issue is in my nodal analysis approach.

Mesh Analysis..I1 is the left mesh, I2 the right mesh.

Handwritten mesh and nodal analysis working in three boxed steps, with expansions, isolations and substitutions, ending with I1 - I2 = 0.0017143.

Nodal analysis attempt...current calculation at the bottom.

Handwritten algebra steps: (v-6)/4000 + v/1000 + (v-2(6-v))/4000 = 0, expanded to 0.002v - 9/2000 = 0, isolated to v = 9/4, then v/1000 = 0.00225 after substitution.

gneill · Feb 8, 2016

For the node equation, check your substitution for v1: Note that the 6 V source is connected through a resistor to the reference potential.

DFrentzen · Feb 8, 2016

I tried to take that into account on one of my attempts, the voltage v1 as 3-v provides the correct value for current. I was unsure if I'm on solid ground to do that.

Handwritten algebra: (v - 6)/4000 + v/1000 + (v - (6 - v)2)/4000 = 0, expanded to 0.00175v - 3/1000 = 0, then isolated to v = 1.7143.

gneill · Feb 8, 2016

You're on the right track. But how to justify your substitution?

Consider that you have a term in your node equation for the current that passes through the "v1" resistor. How would you express v1 in terms of that current?

DFrentzen · Feb 8, 2016

gneill said:

You're on the right track. But how to justify your substitution?

Consider that you have a term in your node equation for the current that passes through the "v1" resistor. How would you express v1 in terms of that current?

I see v1 given as

$Equation v1 = (6 - v) / (4 x 10^3) x 2 x 10^3, written as a fraction with v1 on the left-hand side.$

Which allows me to plug in and

Handwritten algebra solving for v: (v-6)/4000 + v/1000 + (v-v1^2)/4000 = 0, then expanded to -1/2000 v1 + 0.0015v - 3/2000 = 0, substituted to -0.00025(-v+6) + 0.0015v - 3/2000 = 0, expanded to 0.00175v - 0.003 = 0, and isolated to v = 1.7143.

gneill · Feb 8, 2016

Looks good.

DFrentzen · Feb 8, 2016

Thank you.

Finding current ia using mesh and nodal analysis

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