Mesh Analysis: Converting from Time to Frequency Domain with Correct Equations

[eehelp150]

Homework Statement

[/B]
1. Convert from time to freq domain
2. Use mesh analysis to find I1, I2, I3
3. Find Ve and convert it to time domain

Homework Equations

KVL
V=IR

The Attempt at a Solution

Are these mesh equations correct?
45cos(500t) -> 45V
80mH -> 40j
50mH -> 25j
25uF -> -80j
12.5uF -> -160j

Mesh1
(100+40j+25j+200)I1 - 200I2 - 400jI3 = 0
Mesh2
-45 + (200-80j)I2 - (200I1) + 80jI3 = 0
Mesh3
(-80j-160j+25j)I3 + 80jI2 -25jI1 = 0

 

[gneill]

I don't see the 200 Ω resistor being affected by I1 in the first mesh equation.

 

[eehelp150]

I don't see the 200 Ω resistor being affected by I1 in the first mesh equation.

Can't believe I missed that. Do the other two look fine?
Are there any "shortcut strategies" for solving a system of three equations with complex variables?

 

[gneill]

Can't believe I missed that. Do the other two look fine?
Are there any "shortcut strategies" for solving a system of three equations with complex variables?

The others look okay to me.

No shortcuts other than using a software package to do the heavy lifting. I'm partial to Mathcad myself, putting the equations into matrix form and using the built in solver.

 

[eehelp150]

The others look okay to me.

No shortcuts other than using a software package to do the heavy lifting. I'm partial to Mathcad myself, putting the equations into matrix form and using the built in solver.

For mesh1, it should be -25jI3 instead of 400jI3
I used an online calculator and got this:

 

[gneill]

Your 1,3 entry of the matrix should be identical to the 3,1 entry.

Here's a Mathcad version:

 

[eehelp150]

Your 1,3 entry of the matrix should be identical to the 3,1 entry.

Here's a Mathcad version:
View attachment 107300

Is the voltage across Ve simply (I3-I2) * Impedance of 25uF capacitor?

 

[gneill]

Is the voltage across Ve simply (I3-I2) * Impedance of 25uF capacitor?

Make that (I2 - I3)*Z_25μF. I2 flows into the top of the capacitor, so it'll produce a voltage drop in the correct direction.

 

[eehelp150]

Make that (I2 - I3)*Z_25μF. I2 flows into the top of the capacitor, so it'll produce a voltage drop in the correct direction.

I ended up getting 9.988 < -43.347 volts.
Converting to time domain would give: 9.988cos(500t - 43.347) right?

 

[gneill]

I ended up getting 9.988 < -43.347 volts.
Converting to time domain would give: 9.988cos(500t - 43.347) right?

That looks a bit small. Can you show that calculation in detail?

 

[eehelp150]

That looks a bit small. Can you show that calculation in detail?

(I2-I3) * (-j80)
= (0.086 + 0.091j) * (-j80)
= -6.88j - 7.248j^2
=7.248-6.88j

sqrt(7.248^2 + 6.88^2) = 9.99 < -43.5 degrees

 

[gneill]

What values are you using for I2 and I3?

 

[eehelp150]

What values are you using for I2 and I3?

I2 = 0.1188 + 0.1314j
I3 = 0.0329 + 0.0408j

EDIT:
Are these the correct values?
I2 = 0.52013055631408 +0.24535752634041i
I3=0.15152422756397 +0.07991035923825i

 

[gneill]

I2 = 0.1188 + 0.1314j
I3 = 0.0329 + 0.0408j

They don't match your results from post #5. Has something changed?

 

[eehelp150]

They don't match your results from post #5. Has something changed?

Lack of sleep I guess :P
I plugged in the wrong value to the calculator.

New value for Ve: 32.3<65.88 degrees

 

[gneill]

Lack of sleep I guess :P
I plugged in the wrong value to the calculator.

New value for Ve: 32.3<65.88 degrees

Better. Confirm the sign of the angle.

 

[eehelp150]

Better. Confirm the sign of the angle.

Why is it -65.88 instead of 65.88? I recall this being covered in class but I forgot. The calculator spits out 65.88

 

[gneill]

Why is it -65.88 instead of 65.88? I recall this being covered in class but I forgot. The calculator spits out 65.88

What are the Cartesian components of the complex value?

 

[eehelp150]

What are the Cartesian components of the complex value?

Fourth quadrant.
I get it now.