Linear Algebra - Solving AC RLC circuit

[YoshiMoshi]

Homework Statement

So yeah I'm doing a project were I get to create a problem. I would like to learn how to solve a AC RLC circuit using linear algebra. I'm trying to find all of the currents on the edges of the graphs and find all of the voltages at the nodes connecting the edges. I don't think I should really put this in the engineering forum because I seem to really be struggling with the linear algebra and mathematics behind it. I solved circuits like this before, but not with the linear algebra perspective. If you can provide any assistance or point me in the right direction that would help. I'm just using all symbols for now and will put in numerical values later.

This is my weighted graph

Homework Equations

The Attempt at a Solution

Alright so I start with finding the incidence matrix.

So then I find the impedance matrix

Then I find the voltage and current source vectors.

Than through this equation

Than I can solve for the currents on each edge of the weight graph and find the nodal voltages of the weight graph

So at this point I would just plug this into MATLAB once I assign numerical values.

Does this look ok so far?

 

[YoshiMoshi]

BUMP went onto second page no answer.

 

[SteamKing]

BUMP went onto second page no answer.

You say you are struggling with the linear algebra behind solving say a complicated electrical circuit.

Why did you develop such a monster problem if this is the case? This circuit is clearly too complex to solve by hand, which is why you plan to dump it into MATLAB.
Couldn't you find a simpler circuit to practice on, one which is DC rather than AC, and get your LA problems straightened out before you move on to something which is unlikely to be solved by hand?

Otherwise, dump your calculations into MATLAB and see what solutions you get. You can always check the results afterward and see if the solution obeys the various network laws.

 

[YoshiMoshi]

I was told that the incidence matrix has to be at least 6x6. It is possible to solve by hand a DC RLC circuit by hand through the method mentioned above? I thought it might get more difficult to solve the problem in DC? Because I wouldn't be able to lump together impedance to create the resistance matrix?

Each branch would have something like

V = i*R + (1/C) integral i(t) dt + L*(d[i(t)]/dt)

How do I go about forming the resistance matrix of the resistance in each branch when it's DC? I guess I'm struggling with that. That's why I thought initially AC would be easier. In my book they show forming the resistance matrix with nothing but a resistive circuit. I have to make it a bit more complex because it can't be so similar to home work problems. I thought AC would be easier because than I could easily create the resistance matrix.

I guess my question is how exactly do I go about finding the resistance matrix of the graph if it's a DC RLC circuit. Like in my picture it would just be a DC voltage instead of V and a DC current source of I.

Thanks for any help!

I practiced on a purely resistive DC circuit and understand those easy. But will loose points if it's to similar to home work so I wanted to expand upon it and make it more difficult by making it a RLC circuit.

 

[SteamKing]

I was told that the incidence matrix has to be at least 6x6.

Why? Who told you this?

It is possible to solve by hand a DC RLC circuit by hand through the method mentioned above? I thought it might get more difficult to solve the problem in DC? Because I wouldn't be able to lump together impedance to create the resistance matrix?

You can do a mesh analysis on a much simpler DC circuit with as few as two loops.

https://en.wikipedia.org/wiki/Mesh_analysis

http://www.allaboutcircuits.com/textbook/direct-current/chpt-10/mesh-current-method/

Each branch would have something like

V = i*R + (1/C) integral i(t) dt + L*(d[i(t)]/dt)

How do I go about forming the resistance matrix of the resistance in each branch when it's DC? I guess I'm struggling with that. That's why I thought initially AC would be easier. In my book they show forming the resistance matrix with nothing but a resistive circuit. I have to make it a bit more complex because it can't be so similar to home work problems. I thought AC would be easier because than I could easily create the resistance matrix.

If you look at the references above, you can figure out how to set up circuits with resistors, inductors, and capacitors.

The incidence matrix is not that complicated to setup; it just depends on how many different connections are made at each circuit element.

I guess my question is how exactly do I go about finding the resistance matrix of the graph if it's a DC RLC circuit. Like in my picture it would just be a DC voltage instead of V and a DC current source of I.

Thanks for any help!

I practiced on a purely resistive DC circuit and understand those easy. But will loose points if it's to similar to home work so I wanted to expand upon it and make it more difficult by making it a RLC circuit.

It's best to start out with simple circuits. Once you have these down, then you can move on to more complex circuits.

The two references above are just samples. You can Google "mesh analysis" and get many more hits.

 

[YoshiMoshi]

Yeah I have solved circuits before using mesh analysis. It's easy. But how do I tie in linear algebra into this system by mesh analysis? I'm not exactly sure. I wouldn't know how to find the resistance matrix for an RLC circuit because I can't simply lump together the resistance of an RLC unless it was impedances that I could lump together in a AC case.,

I think mesh analysis provides me with the left null space of the linear system.

My professor said so =(

But how do I go about finding the Resistance matrix specifically?

 

[YoshiMoshi]

See I try and solve a simpler problem but am unsure how to find the matrix R in a DC RLC circuit. In a purely resistive circuit the matrix R is simply a diagonal matrix with the resistances in a RLC circuit I'm unsure how to form R. In a AC case I would just lump the impedances. So I'm not sure how to find R specifically in a DC RLC circuit.

Thanks for any help!