Analyzing Bridge Circuit w/ Loop Currents

  • Thread starter guyvsdcsniper
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In summary: Hope that makes sense.In summary, the book I am using for the Loop Current Method has me producing a different result for Loop B than Loop A. I was able to follow the steps given in the book for Loop A, but when I tried to do the same for Loop B, I got a different result. I am not sure why this is, but it seems as though the direction of the current is important when applying the Loop Current Method.
  • #1
guyvsdcsniper
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Homework Statement
Use the loop current method on bridge circuit to generate 3 loop equations
Relevant Equations
Kirchoff's law
Attached is the example I am working out of a textbook that involves using the Loop Current Method on a bridge circuit. In the pictures attached I am following section 1.6.2 which produces loop equations (1.24) for figure 1.9. Figure 1.7 provides the direction of current.

I am having trouble producing the same result as the book I am using for Loop B and C.

Following Loop A, we travel up the EMF,
##\varepsilon##, and then we reach ##R_1##. At this resistor, ##I_a## travels down the resistor and ##I_b## travels up it. Likewise, at ##R_3## ##I_a## travels down the resistor and ##I_c## travels up it. Traveling down the resistor leads to a drop in voltage which leads to a negative sign in associated with that voltage.

Following this logic, I receive the same equation for Loop A as seen in equation 1.24.

So now when evaluating Loop B, I will begin at Node 4. Again, at ##R_1## ##I_a## travels down the resistor and ##I_b## travels up it. This produces ##R_1I_b-R_1I_a##. Next we go down resistor ##R_2## and get ##-R_2I_b##. Finally we have resistor ##R_5##. ##I_b## travels up and ##I_c## travels down this resistor, giving ##R_5I_b-R_5I_c##. Simplifying this and setting it equal to zero gives:

$$-R_1[I_a-I_b]-R_2I_b-R_5[I_c-I_b]=0$$.

So my logic works for Loop A but for Loop B, this logic produces the opposite of what the book gives for the resistors that contain a superposition of current.

Could someone help me understand how I am approaching this wrong?
 

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  • #2
guyvsdcsniper said:
at ##R_1## ##I_a## travels down the resistor and ##I_b## travels up it. This produces ##R_1I_b-R_1I_a##. Next we go down resistor ##R_2## and get ##-R_2I_b##. Finally we have resistor ##R_5##. ##I_b## travels up and ##I_c## travels down this resistor, giving ##R_5I_b-R_5I_c##. Simplifying this and setting it equal to zero gives:

$$-R_1[I_a-I_b]-R_2I_b-R_5[I_c-I_b]=0$$.
I'm not following your steps there. The references to up/down and the relationship to the signs are unclear.
Going clockwise around the loop, ##I_b## always has the same sign. You have
##R_1I_b-R_1I_a##, ##R_2I_b##, ##R_5I_b-R_5I_c##
So I disagree with your sign for ##R_2I_b##.
 
  • #3
haruspex said:
I'm not following your steps there. The references to up/down and the relationship to the signs are unclear.
Going clockwise around the loop, ##I_b## always has the same sign. You have
##R_1I_b-R_1I_a##, ##R_2I_b##, ##R_5I_b-R_5I_c##
So I disagree with your sign for ##R_2I_b##.
Well I was assuming that the current traveling through this circuit will follow the path as described in figure 1.7. I used that to assign the voltage drop polarities. So the positive end of the resistors are where the current enters and the negative end is where the current exits.

So when following the loop upward, we are going from the negative end to the positive end of the resistor, giving a positive voltage and vice versa.
 
  • #4
guyvsdcsniper said:
Well I was assuming that the current traveling through this circuit will follow the path as described in figure 1.7. I used that to assign the voltage drop polarities. So the positive end of the resistors are where the current enters and the negative end is where the current exits.

So when following the loop upward, we are going from the negative end to the positive end of the resistor, giving a positive voltage and vice versa.
I agree with your method, but somehow you are getting a wrong sign. I can’t pinpoint where without seeing your steps in gory detail.

I can see the potential for confusion from all those minus signs.
Do you see that all the ##I_b## terms should have the same sign in a given loop?
 
  • #5
haruspex said:
I agree with your method, but somehow you are getting a wrong sign. I can’t pinpoint where without seeing your steps in gory detail.

I can see the potential for confusion from all those minus signs.
Do you see that all the ##I_b## terms should have the same sign in a given loop?
Hopefully this makes my thought process clear. I tried breaking down how I viewed each resistor when traveling loop B.
 

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  • #6
I am finding that I only get the same result as the book when I assign the voltage drop polarity with respect to the loop I am following.

So when I follow Loop A clockwise, I get the assigned voltage polarity as seen in my previous post.

But when I start at node 4, and follow loop B and treat it as the direction current is traveling, this switches the polarity of ##R_1## and ##R_5## and gives me the same result as the book (1.24).

Could this be the problem I am encountering? It seems as thought voltage drop polarity with the "Loop Current Method" is dependent on the direction of the loop.EDIT:

This may seem convoluted but I think this explanation makes sense. Given the name of the method "Loop Current" I think its treating the direction of each respective loop as the current. So you have to determine the voltage drop polarity for each individual loop.
 

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  • #7
Yes, that's how I believe it works.
 

Related to Analyzing Bridge Circuit w/ Loop Currents

What is a bridge circuit and where is it commonly used?

A bridge circuit is an electrical circuit used to measure unknown resistances, capacitances, or inductances by balancing two legs of a bridge circuit. It is commonly used in applications such as Wheatstone bridges for precise resistance measurements and in various sensor applications such as strain gauges and temperature sensors.

What are loop currents in the context of analyzing bridge circuits?

Loop currents, also known as mesh currents, are hypothetical currents that circulate around the loops (meshes) of a circuit. In the context of analyzing bridge circuits, loop currents are used to apply Kirchhoff's Voltage Law (KVL) to each loop, which simplifies the process of solving for unknown values in the circuit.

How do you apply Kirchhoff's Voltage Law (KVL) to a bridge circuit using loop currents?

To apply Kirchhoff's Voltage Law (KVL) to a bridge circuit using loop currents, you first identify the independent loops in the circuit. Then, assign a loop current to each loop. Next, write KVL equations for each loop by summing the voltage drops and setting them equal to zero. Finally, solve the system of linear equations to find the loop currents.

What are the advantages of using loop current analysis for bridge circuits?

Using loop current analysis for bridge circuits offers several advantages: it simplifies the process of solving complex circuits by reducing the number of equations needed, it provides a systematic approach to circuit analysis, and it allows for easier identification of relationships between different components in the circuit. Additionally, it is particularly useful for circuits with multiple loops and branches.

Can loop current analysis be used for non-linear bridge circuits?

Loop current analysis is primarily used for linear circuits where the components obey Ohm's Law. For non-linear bridge circuits, which contain components like diodes or transistors, the analysis becomes more complex and may require iterative methods or numerical techniques. While loop current analysis can provide a starting point, additional methods are often needed to fully analyze non-linear circuits.

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