Unfortunately you buggered equations 3, 4, 6 and 8 (equations 4 and 8 are still wrong even after your "correction"). They should be:
V3 = - Va + Vb ( 3 )
i1 = (- V1 - Va) / Z1 ( 4 )
i2 = - V3 / Z2 ( 6 )
i6 = (- V2 - Vb) / Z3 ( 8 )
Unfortunately you buggered equations 3, 4, 6 and 8 (equations 4 and 8 are still wrong even after your "correction"). They should be:
V3 = - Va + Vb ( 3 )
i1 = (- V1 - Va) / Z1 ( 4 )
i2 = - V3 / Z2 ( 6 )
i6 = (- V2 - Vb) / Z3 ( 8 )
Kindly explain?
Firstly, if the direction of V3 is from b to a, then V3 as a voltage difference is Va - Vb (destination minus source).
Secondly, the current through an impedance is proportional to the voltage difference across that impedance. In your equation 4 you have written "- (V1 + Va)". This is not a voltage difference, it is a voltage sum. Therefore it cannot possibly be correct.
Thirdly, the voltage V3 appears directly across Z2 according to the arrows on the diagram. Why do you wish to introduce a minus sign?
Lastly, your equation 8 has the same problem as your equation 4. You have written the current in terms of a voltage sum instead of a voltage difference.
Again, you are wrong on all parts. I have attached corrected calculations. I draw 4 loops according to 2nd Kirchhoff law, numbered I. to IV. and written equations. If direction of voltage agreed with my chosen direction of loop arrow I assigned it positive value, if the direction of voltage is opposite I gave it negative value.
Again, you are wrong on all parts. I have attached corrected calculations. I draw 4 loops according to 2nd Kirchhoff law, numbered I. to IV. and written equations. If direction of voltage agreed with my chosen direction of loop arrow I assigned it positive value, if the direction of voltage is opposite I gave it negative value.
I'm not presenting a voltage loop analysis (KVL), I am presenting a node analysis (KCL). Don't equations 1 and 2 make that clear? The nodal analysis using the current law is the subject of the later part of the thread.
So if you are doing KCL, 2nd Kirchhoff law is not valid?
Damn it dude! Look at the direction of arrow on voltage source V1. It goes from BOTTOM to TOP. Your equation would be true if it were going from TOP to BOTTOM.
Calculating nodal voltages is easy:
I have been staring at these equations for a couple of hours and I think I'm stuck.
IZ1 = V1 / Z1 = 60A
IZ3 = V3 / Z3 = 30A
I can see where these would be true if V10 and V20 was 0V but that isn't the case. Suppose the circuit beyond Z1 and Z4 didn't even exist. The current certainly wouldn't be 60A.
I'm missing something...
I can guarantee they are correct. I attached simulation from Tina-TI. I chose frequency 50 Hz and calculated capacitor and inductor values accordingly. I highlighted in red my manually calculated values of V10, V20 and Is.
Calculating nodal voltages is easy:
I have been staring at these equations for a couple of hours and I think I'm stuck.
IZ1 = V1 / Z1 = 60A
IZ3 = V3 / Z3 = 30A
I can see where these would be true if V10 and V20 was 0V but that isn't the case. Suppose the circuit beyond Z1 and Z4 didn't even exist. The current certainly wouldn't be 60A.
I'm missing something...
I can guarantee they are correct. I attached simulation from Tina-TI. I chose frequency 50 Hz and calculated capacitor and inductor values accordingly. I highlighted in red my manually calculated values of V10, V20 and Is.
TINA shows I_R1(1,3) as 28A at 126 deg, not 60A and you use the 60A value when you set up the matrix earlier - a short circuit current, perhaps. This is where I'm confused. It's nice that TINA gives you a tabular display. I have yet to figure out how to do that with LTSpice.
I got a free copy of TINA a couple of years ago and never really used it. It is installed on another computer that doesn't see much use. I may have to look again.
Calculating nodal voltages is easy:
I have been staring at these equations for a couple of hours and I think I'm stuck.
IZ1 = V1 / Z1 = 60A
IZ3 = V3 / Z3 = 30A
I can see where these would be true if V10 and V20 was 0V but that isn't the case. Suppose the circuit beyond Z1 and Z4 didn't even exist. The current certainly wouldn't be 60A.
I'm missing something...
I can guarantee they are correct. I attached simulation from Tina-TI. I chose frequency 50 Hz and calculated capacitor and inductor values accordingly. I highlighted in red my manually calculated values of V10, V20 and Is.
TINA shows I_R1(1,3) as 28A at 126 deg, not 60A and you use the 60A value when you set up the matrix earlier - a short circuit current, perhaps. This is where I'm confused. It's nice that TINA gives you a tabular display. I have yet to figure out how to do that with LTSpice.
I got a free copy of TINA a couple of years ago and never really used it. It is installed on another computer that doesn't see much use. I may have to look again.Current Iz1 DOES NOT flow through impedance Z1. Iz1 and Iz2 are flowing as indicated on my top right picture. I replaced voltage source V1 and its series impedance Z1 with current source Iz1 with his parallel admitance Y1. This changed topology of circuit a bit, but does not have effect on nodal voltages V10 and V20. Similarly i replaced voltage source V2 and its series impedance Z3 with current source Iz2 with parallel admitance Y3. Voltage source V3 does not have any direct series impedance connected to it, so i have to leave it as is. Otherwise I would replaced it as well. Currents Iz1 and Iz2 of ideal current sources are not regular dependent circuit currents. They are FORCED by current sources and they direction is DICTATED by direction of voltages V1 and V2 respectivrely.
Check this wikipedia article on how to convert voltage source to current source and vice versa:
https://en.wikipedia.org/wiki/Source_transformation
This has nothing to do with any reference. You can pick reference point wherever you want.
Damn it dude! Look at the direction of arrow on voltage source V1. It goes from BOTTOM to TOP. Your equation would be true if it were going from TOP to BOTTOM.
I have emailed my tutor about this particular question and he has come up with rather an interesting answer. The solution it would seem is to treat the two nodes as one "supernode". In this way the current through the resistor between the two nodes is totally irrelevant as is the current through the voltage source. This apparently is even simpler than it looks as it will then become an equation for one node having taken into account the voltage difference throughout the supernode.
I hope to God this stuff is going to be useful one day because I spent three days trying to figure it out the wrong way. I can't see me ever having to solve anything like this in real life, I just need to get the ticket for doing this stuff so that I can move on to something more useful.
Calculating nodal voltages is easy:
This problem is also dealt with at great length on the other forum: https://www.physicsforums.com/threads/ac-circuit-analysis-mesh-and-nodal.791744/
That supernode idea seems like a pedantic way of introducing equation a-b=V3 into the game so, yes, a = b + V3, and only an unknwon remains: b.
By the way, the trouble arose in node analysis because V3 is a voltage source. Change V3 for a current source, and the trouble would have appeared in the mesh analysis. What is the dual concept to a supernode?
This problem is also dealt with at great length on the other forum: https://www.physicsforums.com/threads/ac-circuit-analysis-mesh-and-nodal.791744/
139 replies in this forum, but not a single correct answer
This problem is also dealt with at great length on the other forum: https://www.physicsforums.com/threads/ac-circuit-analysis-mesh-and-nodal.791744/
139 replies in this forum, but not a single correct answer
orolo's answers are correct. You can find more about this problem here, including correct answers: https://www.physicsforums.com/threads/ac-circuit-analysis-mesh-and-nodal.791744/
Check picture in my Reply #74. Does orolo's or my signs match the simulation result? Or are you saying SPICE has it wrong?
This problem is also dealt with at great length on the other forum: https://www.physicsforums.com/threads/ac-circuit-analysis-mesh-and-nodal.791744/
139 replies in this forum, but not a single correct answer
orolo's answers are correct. You can find more about this problem here, including correct answers: https://www.physicsforums.com/threads/ac-circuit-analysis-mesh-and-nodal.791744/
Check picture in my Reply #74. Does orolo's or my signs match the simulation result? Or are you saying SPICE has it wrong?