Physics Teacher's Basic Circuit Analysis Recipe Needs Tweaking

[Farina]

I teach intro, calc-based physics. Below is a terse summary of my recipe for analyzing simple battery-resistor multi-loop circuits. It was working well, until we tried the circuit seen in the attachment. Depending on our selection of simultaneous equations, sometimes our analysis produced correct results, but other times (again depending on our choice of simultaneous equations) didn't work because we ended-up with an inverted matrix that is undefined.

How does the following need to be augmented of fixed?

RECIPE

1. Identify all junctions and arbitrarily label all currents entering/leaving each junction. This sets the number of unknowns.
2. Apply Junction Rule to every junction that produces a different equation.
3. Apply Loop Rule to as many loops as it takes to include each branch (each circuit element) at least once.
4. The number of simultaneous equations you generate needs to equal the number of unknown variables you need to investigate. Use all your Loop Rule equations plus as many Junction Rule equations needed to meet this number. You must use at least one Junction Rule equation.

I'm pretty sure we're correctly handling loop voltage rises/drops across batteries and resistors (with and against the loop direction).

HELP, and thank you, and HELP

 

[mjhilger]

You don't need to do both junction and loops. Loop analysis is fine by itself. In loop analysis you choose a direction of the loop for currents and all the voltages around the loop add to 0. But you have to be careful that you follow the loop correctly and set your +'s and -'s correctly. In this case you will generate 3 loop equations and 3 unknown currents. Solve and you should get your answers. In your posted circuit you will have to be aware of the fact that two loop currents in different directions are passing through the middle of the page 100 ohm resistor. So if we assign loop currents I1, I2, I3 in the loops and all directions are clockwise then the voltage across the middle of the page resistor in the loop 2 equation is 100(I2 - I3); while the equation in the loop 3 equation is 100(I3 - I2). If you have been doing the math correctly for loop analysis this should match what you have been doing. If not I hope it points you in the right direction. Let us know if you need further explanation.

 

[mjhilger]

As I reread your RECIPE, why did you come to the conclusion that you needed to include a junction equation? Loop analysis should provide sufficient information without junction input. I think this is where your trouble begins, The junction current directions or labels might not line up correctly, but again the loop should be sufficient. Can you show us a circuit where loop analysis does not provide the required number of equations for the number of unknowns?

 

[Farina]

First off - THANK YOU! You're describing a much simpler approach.

I always thought each circuit branch had to be labeled with its own unique current. For the provided sample circuit, there are indeed 5 distinct currents: I1, I2, I3, I4 and I5.

You're advising to simply rely on KLR and consider there to be a single current within each loop (even though this is not, physically, the case?)?

 

[AutoNub]

It sounds like you are trying to mix mesh analysis with nodal analysis. Both are valid but only when applied separately. You can apply nodal analysis or mesh analysis to any circuit. I could show you the analysis if you'd like, but it looks like mjhilger already cleared that up.

 

[Farina]

I've been using the "Branch Current Method." After some surfing around, I see you're describing the "Loop Current Method." Most/all physics textbooks use the Branch Current Method.

I follow your approach, which perfectly matches the Loop Current Method reference I found.

I'm still perplexed why the tried-and-true Branch Current Method falls down, and still believe my recipe is less than perfect - but don't know where, exactly.

I'll check out your Loop Current Method - maybe this will be my new religion

As I reread your RECIPE, why did you come to the conclusion that you needed to include a junction equation? Loop analysis should provide sufficient information without junction input. I think this is where your trouble begins, The junction current directions or labels might not line up correctly, but again the loop should be sufficient. Can you show us a circuit where loop analysis does not provide the required number of equations for the number of unknowns?

 

[Farina]

I am mixing these 2 methods, I guess - but this is the precise method used in most physics textbooks. I'm looking, right now, at Halliday & Resnick (9th ed, pg 719) and they are indeed prescribing a Junction+Loop Rule approach.

It sounds like you are trying to mix mesh analysis with nodal analysis. Both are valid but only when applied separately. You can apply nodal analysis or mesh analysis to any circuit. I could show you the analysis if you'd like, but it looks like mjhilger already cleared that up.

 

[mjhilger]

Ok, that clears it up. Yes KVL (Kirchhoff's voltage law) uses "super position" to determine the current about a loop as a single number. This single number is indeed correct for the loop; the locations where current is shared by a component, the loop currents provide a way to determine the individual contribution from each loop path. Where multiple loops pass through a component, each is accounted in the voltage term for that component. Again care must be taken to stick with a convention even if the voltage is known. Numerous examples can be found online. But yes that should make it much easier for you. And if you are using a math package and matrices it should be easy work.

 

[Farina]

Right! I suppose there are instances where the textbook's blended Loop+Junction Rule approach are needed, but it seems like your Loop-Current Method is *far* simpler.

Thank you again.

Ok, that clears it up. Yes KVL (Kirchhoff's voltage law) uses "super position" to determine the current about a loop as a single number. This single number is indeed correct for the loop; the locations where current is shared by a component, the loop currents provide a way to determine the individual contribution from each loop path. Where multiple loops pass through a component, each is accounted in the voltage term for that component. Again care must be taken to stick with a convention even if the voltage is known. Numerous examples can be found online. But yes that should make it much easier for you. And if you are using a math package and matrices it should be easy work.

 

[mjhilger]

I'm not sure about why the physics books would choose that method. I know they like to show current as dQ/dt so perhaps node analysis points this out easier. Most circuits have fewer loops than nodes, so loop analysis generates fewer equations and unknowns to solve - so easier in my book.