Superposition Problem- Circuits

[RoKr93]

Homework Statement

For the above circuit, find the coefficients of the linear relationship v_out = a₁v_s1 + a₂i_s2 + a₃i_s3 by superposition. Then find the power delivered to R₃ when v_s1 = 100 V, i_s2 = 2 A, and i_s3 = 4 A. Given: R₁ = 20 Ω, R₂ = 60 Ω, and R₃ = 20 Ω.

Homework Equations

Ohm's Law: V = IR

KVL: V₁ + V₂ + ... V_n = 0 for closed loops

KCL: I₁ + I₂ + ... I_n = 0 going in an out of a node

Voltage Division: V₁ = V_source * (R₁/(R₁+R₂))

The Attempt at a Solution

I think I got the first two coefficients (though I could be incorrect). I found v_out¹ = v_s1*(R₃/(R₁+R₂+R₃)) = 0.2V_s1 through voltage division. I'm pretty certain that's correct.

I also found that v_out² = I_s2*R₃ = 20I_s2, though I'm not quite as certain about that.

But I absolutely cannot wrap my head around finding v_out³ (when all independent sources are eliminated except I_s3). I understand that in that situation, the I_s2 branch essentially disappears and we are left with I_s3 in parallel with a branch with R₂ and a branch with R₁ and R₃, but we can't combine those since v_out³ must stay isolated. I'm not sure what to do.

 

[gneill]

You have a classic current divider scenario with current I_s3 being divided between two branches. You know the total resistance of each of the branches.

Alternatively, consider converting I_s3 and R₂ to their Thevenin equivalent and finding the resulting current and then the potential across R3.

 

[RoKr93]

So I want to use current division to find i₂, then multiply that by R₃ to get my voltage value?

i₂ = I_s3*(R₂/(R₁+R₂+R₃))
v_out³ = i₂R₃
v_out³ = I_s3*((R₂*R₃)/(R₁+R₂+R₃))

After substituting values in, v_out³ comes out to be 12. Unfortunately either that or one of my previous solutions is wrong- a later part of the problem states that the total v_out must be 100 V when v_s1 = 100 V, i_s2 = 2 A, and i_s3 = 4 A. And (0.2*100) + (20*2) + (12*4) is 108. Did I go about finding v_out³ the wrong way?

 

[gneill]

Check your work for the coefficient for I_s2.