Bridge Circuit, Solving 5 Equations

[Opus_723]

Homework Statement

So I'm trying to solve for all of the currents in a bridge circuit (in terms of the input current and all of the resistances), and by using Kirchhoff's Laws I've got 5 equations that I can use to solve for my 5 unknowns. Unfortunately, I've never really solved this many equations simultaneously before, and I can't seem to get any of the unknowns.

Homework Equations

$I_{0} = I_{1} + I_{3}$
$I_{1} = I_{2} + I_{5}$
$I_{4} = I_{3} + I_{5}$
$I_{1}R_{1} + I_{5}R_{5} = I_{3}R_{3}$
$I_{4}R_{4} + I_{5}R_{5} = I_{2}R_{2}$

The Attempt at a Solution

I've been playing around with substitution and adding the equations, but it's not helping much without knowing where to go with it, and it's hard to keep track of what I've already done. Does anyone have any advice for solving this many equations at once?

 

[gneill]

If you find it difficult to organize and keep track of substitutions then you might want to turn to a method that enforces discipline... cast the equations into matrix form and solve by row reduction to upper triangular form and then back substitution. Could require a lot of paper to do symbolically!

Otherwise it's just a matter of staying focused on a plan of systematically eliminating one variable at a time (don't change ideas mid step) and reducing the number of equations until you're left with one equation in one unknown. Solve for that unknown and stick it back into the original equation set. Repeat until done.

If you are familiar with KVL and KCL and their use in mesh analysis or node voltage analysis methods then you can solve the circuit with just two or three separate equations rather than five

 

[willem2]

You have six unknowns, if you include I_0. I suppose I_0 is the current through the entire circuit? You would need to find another equation with I_0.


The easiest way to set up the equations, is to have only 2 unknown voltages for the nodes where R1&R2 and R3&R4 meet, and use Kirchhofs current law for these two nodes to get only two equations for the 2 unknown voltages.

 

[gneill]

You have six unknowns, if you include I_0. I suppose I_0 is the current through the entire circuit? You would need to find another equation with I_0.

I_o would be a 'given' value, the current driving the bridge. The other currents are to be expressed in terms of I_o, so no additional equation would be required.

You need to pick a direction for the current through R_5, and stick with it. One of the equations 2/3 and one of the equation 4/5 has a wrong sign for the term with I_5 because of this.

I believe that the OP did pick and stick to a direction for I₅, and that his equations are okay as-is. Note the potential drop directions as indicated in the following figure.

The easiest way to set up the equations, is to have only 2 unknown voltages for the nodes where R1&R2 and R3&R4 meet, and use Kirchhofs current law for these two nodes to get only two equations for the 2 unknown voltages.

I suspect that you'll need three equations for nodal analysis since there are four nodes total and only one can be chosen as the reference node. Nodes A and D do not comprise a supernode as they are not connected by a voltage source but rather a current source. On the other hand, if mesh analysis is used then the mesh current in the I_o source loop can be deemed already solved -- it's I_o.

 

[willem2]

I suspect that you'll need three equations for nodal analysis since there are four nodes total and only one can be chosen as the reference node. Nodes A and D do not comprise a supernode as they are not connected by a voltage source but rather a current source. On the other hand, if mesh analysis is used then the mesh current in the I_o source loop can be deemed already solved -- it's I_o.

What I do is use a voltage source with potential difference V. Then the potential at A is V, at D is 0, and V_B and V_C are the unknowns.

calculate I_1 through I_5 as a function of the unknowns, and then solve the 2 equations you get by using KCL in node B and C.

 

[gneill]

What I do is use a voltage source with potential difference V. Then the potential at A is V, at D is 0, and V_B and V_C are the unknowns.

calculate I_1 through I_5 as a function of the unknowns, and then solve the 2 equations you get by using KCL in node B and C.

Yes, you can do that, but when you're done you'll have the currents in terms of V rather than I_o, so one more equation (to find I_o in terms of V) will be required, followed by substitution for V in all the current expressions.