AC circuit analysis -- mesh and nodal

[gneill]

Hi Gneill, can you recommend any good reading material on AC circuit nodal circuit analysis with complex numbers? I have been trawling the net searching for examples incorporating the super node as in part b) but have yet to find anything!
I would like to try and fully understand the method before I go ahead and use it.
Many thanks

Hi GeorgeSparks, I can't think of any particular materials off hand. You might go looking for mesh analysis examples and pick out ones where there are voltage sources between nodes. Usually that always leads to a supernode situation. You can then compare the mesh analysis results to nodal analysis results to check your attempts.

 

[GeorgeSparks]

Hi guys don't want to sound silly but for the mesh and Nodal analysis answers for current I have the real part is identical yet the imaginary j current value looks to be off by about 1.
I have used the spreadsheet shown in earlier posts for my mesh analysis and for the latter nodal I am sure is correct due to using full numbers avoiding rounding errors, the only other idea I have would be to solve the three simultaneous equations manually, does anybody have any ideas? I know I am being a little vague without showing any working haha.
Or can anyone recommend a good online calculator to confirm my spreadsheet readings?

Thnks in advance

 

[gneill]

If both of your answers are identical and off by the same amount it suggests that something is off in the input data.

 

[GeorgeSparks]

For the Mesh analysis I end up with and answer for I = -9.152 + j17.275 this was calculated from I1-I2 From the complex matrix ((16.8117-j22.8805)-(25.9639-j40.1559)) all mesh analysis was done in a clockwise direction.
For Nodal analysis I simplified down and ended up with a result of 9.152+j18.337. My mind is boggled, the first results real terms for mesh analysis I can change d/t direction of current into Z4 based on my initial loop calculations as for the Nodal analysis I have calculated the current based on the voltage I have worked out at Node through Z4 which was 91.68725814+j45.75850158 / -j5. I am a little stumped, the only thing I can think is that the spreadsheet is a bit of a mess although I have looked through it several times and I am pretty happy with the input data (apart from it not allowing you to set 0 values instead you must input some values as 0.00001 but surely this couldn't have changed the imaginary term by over 1).

 

[gneill]

So your mesh analysis work looks fine. Your current is correct (signs included): the real part should be negative as you've found.

For the nodal analysis, the real part of your node voltage is incorrect. Multiply the current you obtained via mesh analysis by Z4 to see what value you should be getting for the node voltage.

 

[GeorgeSparks]

So your mesh analysis work looks fine. Your current is correct (signs included): the real part should be negative as you've found.

For the nodal analysis, the real part of your node voltage is incorrect. Multiply the current you obtained via mesh analysis by Z4 to see what value you should be getting for the node voltage.

Hi gneill,

I am still struggling with this, I calculated back as you said and ended up with a node voltage that of -86.3754311 + j45.7609738. I am still struggling to get the real term based on current circuit values given for the question. I end up with 67.07106781+j30 = (300-j20V/400) which gives me 91.68725814+j45.75850158 or 52.92893219+j30 = (300-j20V/400) which gives me 72.91451177+j44.50698515 based on my current calculations I am a little stuck to say the least... I have been through and recalculated numerous times now to ensure there have been no typos this has got me thinking that maybe the value for part a mesh analysis (the imaginary term) must be off somewhere when calculated using the spreadsheet. Hopefully I will manage to get to the bottom of it haha...

 

[gneill]

I have been through and recalculated numerous times now to ensure there have been no typos this has got me thinking that maybe the value for part a mesh analysis (the imaginary term) must be off somewhere when calculated using the spreadsheet.

Well, the value you reported for the current from your mesh results, I = -9.152 + j17.275, is good.

You may have to break down and show us your node equation.

 

[GeorgeSparks]

Okay, it was a bit strange at first seeing all those equals signs in the LHS expansion until I realized you were just breaking out the individual terms. I followed your work and found everything okay up to the very last line. Check that final calculation.

 

[gneill]

Okay, it was a bit strange at first seeing all those equals signs in the LHS expansion until I realized you were just breaking out the individual terms. I followed your work and found everything okay up to the very last line. Check that final calculation.

 

[GeorgeSparks]

Okay, it was a bit strange at first seeing all those equals signs in the LHS expansion until I realized you were just breaking out the individual terms. I followed your work and found everything okay up to the very last line. Check that final calculation.

Ahhhhhh how frustrating! Haha I was soo bloody close albeit for one final mistake. That has had me stumped for the last few days thank you very much for the guidance

 

[gneill]

Ahhhhhh how frustrating! Haha I was soo bloody close albeit for one final mistake. That has had me stumped for the last few days thank you very much for the guidance

I'm glad I could help.

 

[rob1985]

right so for question a I have the following.

loop 1 is 120=(2+j5)I1+ (j5)I2
loop 2 is -14.142+j14.142= (j5)I1 - (j1)I2 - (j4)I3
loop 3 is -j120 = -(j4)I2 + (4+ j4)I3

So my question is once I plug these into a complex matrix by decomposition excel program given to me by the university. This happens. Now I believe all my calcs are correct. so I've messaged the university. However once I get the three solutions (white boxes on photo) what do I do then. Because as far as I can tell that gives me i1, i2, and i3 . To find the total circuit current do I just add them all together.

 

[gneill]

Check your V3 value in your Loop 2 equation.

You're looking to solve for the current through Z4. Which mesh currents flow through Z4?

 

[rob1985]

-14.142-j14.142= (j5)I1 - (j1)I2 - (j4)I3

yes sorry that was a typo. I've changed it in the spreadsheet but still no joy. so am I right in saying that once the spreadsheet is fixed I can add all three equations together to get the total current in the circuit

 

[gneill]

-14.142-j14.142= (j5)I1 - (j1)I2 - (j4)I3

yes sorry that was a typo. I've changed it in the spreadsheet but still no joy.

You may have to do as the spreadsheet suggests and "nudge" one or more of the zero entries away from zero.

so am I right in saying that once the spreadsheet is fixed I can add all three equations together to get the total current in the circuit

What does "the total current in the circuit" mean? There are three different sources and various currents flow in different places and different directions in the circuit. For mesh currents they often flow in opposite directions through a given path, so "summing" the total current is not a well defined operation.

What is it you're trying to accomplish by summing currents?

 

[rob1985]

I've got to determine what I is. Your right summing the currents won't work. I think I need to do loop 1 - loop 2 which is i1 - i2.

ive sorted the spread sheet out thank you. Am I right in saying that 3 three results in the polar form boxes are i1 , i2 and i3 respectively

 

[gneill]

I've got to determine what I is. Your right summing the currents won't work. I think I need to do loop 1 - loop 2 which is i1 - i2.

That's right.

ive sorted the spread sheet out thank you. Am I right in saying that 3 three results in the polar form boxes are i1 , i2 and i3 respectively

Your results look good.

 

[rob1985]

that gives me -4.23 + j-1.75 or 4.58 angle 202.48 doesn't seem right to me

 

[gneill]

that gives me -4.23 + j-1.75 or 4.58 angle 202.48 doesn't seem right to me

Can you show the details of your calculation?

Note that you should be able to take advantage of the Excel spreadsheet to add the currents since their rectangular components are available in the solution area.

 

[rob1985]

In polar form
i1 = radius 2.84 angle -53.69
i2 = radius 4.78 angle -57.12
i3 = radius 2.85 angle -50.74
In rectangular form
i1 = 1.68 + j-2.29
i2 = 2.55 + j-4.04
i3 = 1.81 + j-2.21

to find I at the point required = i1-i2

1.68 + j-2.29 - 2.55 + j-4.04 = -4.23 + j-1.75
I don't know how to work it out on the spread sheet, I've done them with a calculator

 

[gneill]

The solution area on the spreadsheet has the currents in rectangular form:

If you are familiar with Excel you should be able to sum the requisite cells...

Edit: Also, be sure to pay attention to the exponential notation! Don't lose the power of ten in the values.

Edit 2: Be careful when finding the phase angle of the result. Check which quadrant the vector is in.

 

[rob1985]

Unfortunately I am not familiar with excel and don't know how to add cells together with an exponential term in them. Also on the calculator do I just type in 1.68 Exp 01. to give 4.5667 etc. I know it might sound like a daft question

And am I right in saying all the angles lie in the 4th quarter.

 

[gneill]

Unfortunately I am not familiar with excel and don't know how to add cells together with an exponential term in them.

They're just numbers in the cells. They happen to be displayed using exponential notation. To add the contents of a couple of cells:

1. select an empty cell somewhere where you'd like the result to end up and type an "="
2. select the first cell. The address of that cell should appear in the result cell.
3. Type "+".
4. select the second cell. It's address will appear in the destination cell.
5. Hit the <enter> key. The sum of the two cells will be displayed in the result cell.

Also on the calculator do I just type in 1.68 Exp 01. to give 4.5667 etc. I know it might sound like a daft question

That sounds right. If it's a problem, just convert the number to a plain format number before doing the math. "E+01" just means "x 10¹".

And am I right in saying all the angles lie in the 4th quarter.

Yes.

 

[rob1985]

Right so once I've got rid of the exponential sign the results come out as followed.

i1 = 4.566713472 + j -6.224865387
i2 = 7.067532754 + j -10.92749295
i3 = 4.92009011 + j -6.007402841

So i1 - i2 to find i = (4.566713472 + j -6.224865387) - (7.067532754 + j -10.92749295)
So i = 2.500819282 + j -4.702627563

So this is the answer to part a) i believe by mesh analysis.

 

[gneill]

I don't understand how you got rid of the exponential notation. It should involve a simple decimal point shift of the numbers (the mantissa). Can you show your work?

Your result for the current $I$ is not correct.

 

[rob1985]

Thanks gneill, I've sorted my maths out now.

i1 = 16.811946 + j -22.879203
i2 = 25.963628 + j -40.154424
i3 = 18.059026 + j -22.095398
So i = (i1 - i2)
(16.811946 + j -22.879203) - (25.963628 + j -40.154424) = -9.151682 + j17.275221

This is by mesh analysis, now for nodal analysis lol

 

[rob1985]

So part b) find I by nodal analysis

So I have the following

V1/Z1 - V20/Z1 - V20/Z4 + V3/Z5 - V20/Z5 + V2/Z3 - V20/Z3 + V3/Z3 = 0

(V1 - V20)/Z1 + -V20/Z4 + (-V20-V3)/Z5 - V2 - (V20 - V3)/Z3 = 0

(V1 - V20)/Z1 - V20/Z4 + (V3 - V20)/Z5 + (V2 - V20 + V3/Z3 = 0

-V20 ( 1/Z1 + 1/Z4 + 1/Z5 + 1/Z3) + ( V1/Z1 + V3/Z5 + V2/Z3 +V3/Z3) = 0

can this equation be simplified anymore, does anyone know

 

[gneill]

-V20 ( 1/Z1 + 1/Z4 + 1/Z5 + 1/Z3) + ( V1/Z1 + V3/Z5 + V2/Z3 +V3/Z3) = 0

can this equation be simplified anymore, does anyone know

It doesn't get much prettier. You could put everything over a common denominator, but it wouldn't make things any easier to deal with when it comes to the complex arithmetic required when the numbers are inserted. I suggest collecting the V3 terms (as you did for V20) and then start plugging in given values to continue the reduction of terms.

 

[WeeChumlee]

Hi Rob1985

I too am working on this and just can't seem to get it right.
My equation is similar to yours.
I have used different terminology but if I use your terminology I have:
(V1-V20 / Z1) - (V20/Z4) - (V30/Z5) + ((V2+V30)/Z3)
Obviously wrong as does not get the answer from mesh analysis.
Just can't see where...

 

[The Electrician]

Hi Rob1985

I too am working on this and just can't seem to get it right.
My equation is similar to yours.
I have used different terminology but if I use your terminology I have:
(V1-V20 / Z1) - (V20/Z4) - (V30/Z5) + ((V2+V30)/Z3)
Obviously wrong as does not get the answer from mesh analysis.
Just can't see where...

You need to be careful with your notation. Read post #46 in this thread.

You need more parentheses, for example (V1-V20 / Z1) should be ((V1-V20) / Z1)

Also, you have (V1-V20 / Z1) - (V20/Z4) - (V30/Z5) + ((V2+V30)/Z3) which is not an equation, but just an expression. You must equate expressions to zero, and of course you will need more than one equation.

 

[WeeChumlee]

Hi Electrician

Yes indeed, that was me being lazy and not putting all parentheses in, silly of me.
What I have is this:
((V1-V20) / Z1) - (V20/Z4) - (V30/Z5) + ((V2+V30)/Z3) = 0
V20-V30 = 14.14+j14-14

 

[WeeChumlee]

Darn:
Was of course
V20-V30 = 14.14+j14.14

 

[WeeChumlee]

Carrying on from this as I am getting nowhere..

((V1-V20) / Z1) - (V20/Z4) - (V30/Z5) + ((V2+V30)/Z3) = 0
V30 = V20 - V3
((V1-V20) / Z1) - (V20/Z4) - ((V20-V3)/Z5) + ((V2+(V20-V3))/Z3) = 0

When I put the values into get V20 and divide that by Z4 I get 8.78 - j16.59
Not the answer I got for the mesh analysis which was -9.1 + j17.3

If someone could point me in the right direction I would be most grateful.

 

[The Electrician]

Carrying on from this as I am getting nowhere..

((V1-V20) / Z1) - (V20/Z4) - (V30/Z5) + ((V2+V30)/Z3) = 0
V30 = V20 - V3
((V1-V20) / Z1) - (V20/Z4) - ((V20-V3)/Z5) + ((V2+(V20-V3))/Z3) = 0

When I put the values into get V20 and divide that by Z4 I get 8.78 - j16.59
Not the answer I got for the mesh analysis which was -9.1 + j17.3

If someone could point me in the right direction I would be most grateful.

If I solve these two equations:
((V1-V20) / Z1) - (V20/Z4) - (V30/Z5) + ((V2-V30)/Z3) = 0
V30 = V20 - V3

with the sign of V2+V30 changed (in red above) to V2-V30, I get the correct answer for V20 and V30. And, after dividing by Z4 I get the correct current.

But with the + sign you have I don't get what you got for the current I even with the incorrect values it gives for V20 and V30; check your solution method.

 

[WeeChumlee]

Thanks Electrician

Yes, I totally see that sign change, that was an error of mine.
Funny thing is that I had it right the first time but got the wrong answer so in trying to get the right answer that was a change I made.
I had used an online calculator to try and get the answer, I must have made a syntax error or something.
Just did it all on paper right now and voila, answer is as expected.
Probably serves me right for trying to cut corners.
At least I know my Math is still OK. :)

Many thanks for pointing me in the right direction, really appreciate you guys giving up your time to help.