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

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Discussion Overview

The discussion revolves around converting circuit equations from the time domain to the frequency domain using mesh analysis. Participants are working through specific equations related to a circuit problem, addressing the calculation of currents (I1, I2, I3) and the voltage (Ve) across a capacitor, while also exploring the implications of complex numbers in their calculations.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Participants present mesh equations for the circuit and question the correctness of the equations, particularly regarding the influence of the 200 Ω resistor.
  • Some participants suggest using software tools like Mathcad for solving systems of equations involving complex variables.
  • There is a discussion about the calculation of voltage across the capacitor and the correct formulation of the impedance.
  • Participants share their computed values for currents I2 and I3, with some discrepancies noted in the values used in calculations.
  • There is a query regarding the sign of the angle in the voltage calculation, with references to Cartesian components and quadrant considerations.

Areas of Agreement / Disagreement

Participants express differing views on the correctness of the mesh equations and the values for currents, indicating that there is no consensus on these points. The discussion remains unresolved regarding the accuracy of the calculations and the implications of the results.

Contextual Notes

Some calculations depend on the correct interpretation of complex numbers and their components, which may not be fully resolved in the discussion. There are also references to potential errors in earlier calculations that affect later results.

eehelp150
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Homework Statement


pHJ8FSj.png
[/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
 
Last edited:
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I don't see the 200 Ω resistor being affected by I1 in the first mesh equation.
 
gneill said:
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?
 
eehelp150 said:
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.
 
gneill said:
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:
upload_2016-10-11_10-59-40.png
 

Attachments

  • upload_2016-10-11_10-50-14.png
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Last edited:
Your 1,3 entry of the matrix should be identical to the 3,1 entry.

Here's a Mathcad version:
upload_2016-10-11_14-1-3.png
 
gneill said:
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?
 
eehelp150 said:
Is the voltage across Ve simply (I3-I2) * Impedance of 25uF capacitor?
Make that (I2 - I3)*Z25μF. I2 flows into the top of the capacitor, so it'll produce a voltage drop in the correct direction.
 
gneill said:
Make that (I2 - I3)*Z25μF. 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?
 
  • #10
eehelp150 said:
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?
 
  • #11
gneill said:
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
 
  • #12
What values are you using for I2 and I3?
 
  • #13
gneill said:
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
 
  • #14
eehelp150 said:
I2 = 0.1188 + 0.1314j
I3 = 0.0329 + 0.0408j
They don't match your results from post #5. Has something changed?
 
  • #15
gneill said:
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
 
  • #16
eehelp150 said:
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.
 
  • #17
gneill said:
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
 
  • #18
eehelp150 said:
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?
 
  • #19
gneill said:
What are the Cartesian components of the complex value?
Fourth quadrant.
I get it now.
 

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