Solve Kirchhoff Problem: Find Voltages of Resistors

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In summary, the problem involves finding the voltage of resistors using Kirchhoff Current Law and Voltage Law, without using mesh analysis. The equations used are KCL at node X and Y, and KVL at the left and right loops. The goal is to simplify the process by using KVL and KCL separately and combining resistances into a single equivalent resistance in each branch.
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pisay
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Homework Statement


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I am tasked to find the voltage of all the resistors using Kirchhoff Current Law and Voltage Law. TAKE NOTE THAT I AM NOT ALLOWED TO USE MESH ANALYSIS which means I need to assign current for all resistors.

Homework Equations



KCL at node X

-I1 + IT -I3=0

KCL at node Y

-IT + I2 + I4=0

KVL at Left Loop

1I1 - 12 + I2 + 5V = 0

KVL at Right Loop

-2I3 + 6V - 2I4 + 12 = 0

and using both eqns for nodes x and y, I arrived for the fifth equation.

0=I1 - I2 + I3 - I4

The Attempt at a Solution



I arrange all the 5 equations to be easily solved.

0=IT - I1 + 0I2 - I3 + 0I4 + 0
0=IT + 0I1 + I2 + I3 + I4 + 0
0=0IT + I1 + I2 + 0I3 + 0I4 + 5V
0=0IT + 0I1 + 0I2 - 2I3 - 2I4 + 12V
0=0IT + I1 - I2 + I3 - I4

These a lengthy process. Its obvious that I1 = I2 and I3=I4 but the answers I am getting is not the same as these.

Our teacher told us to use these. We are not allowed the short cut which is mesh analysis. What can I do to simplify the solution?
 
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  • #2
You should be able to use KVL and KCL separately to solve for the circuit values. You don't use them together to solve it. You should get the same answers for each method, however.

Can you show us how to solve the circuit with just one or the other method?
 
  • #3
Your complicating the problem. There are only three unique currents: (1) through the left branch, (2) through the right branch, and (3) through the middle. Another simplifying procedure is to combine the resistances into a single equivalent resistance in each branch.
 

1. How do you apply Kirchhoff's Laws to solve for the voltages of resistors?

To solve for the voltages of resistors using Kirchhoff's Laws, you will need to apply the two laws - Kirchhoff's Voltage Law and Kirchhoff's Current Law. Kirchhoff's Voltage Law states that the algebraic sum of all voltages in a closed loop must equal zero, while Kirchhoff's Current Law states that the sum of all currents entering and exiting a node must also equal zero. By creating a system of equations based on these laws and solving for the unknown voltages, you can find the values of the resistors.

2. Can you provide an example of solving a Kirchhoff's Law problem for the voltages of resistors?

Here is an example problem: In a simple circuit with two resistors, one with a known resistance of 5 ohms and the other with an unknown resistance, a 12V battery is connected. If the voltage across the known resistor is 8V, what is the voltage across the unknown resistor? To solve this, you would use Kirchhoff's Voltage Law to create the equation 12V - 8V - Vx = 0, where Vx is the voltage across the unknown resistor. Then, you would use Kirchhoff's Current Law at the node where the two resistors connect to create another equation. Solving this system of equations will give you the voltage across the unknown resistor, which in this case is 4V.

3. What are some common mistakes to avoid when solving Kirchhoff's Law problems for resistors?

One common mistake is to forget to consider the direction of current flow when applying Kirchhoff's Voltage Law. It is important to assign a positive or negative sign to each voltage based on the direction of current flow. Another mistake is to forget to include all components in the circuit when creating the system of equations. Make sure to include all resistors and voltage sources in your equations.

4. Can Kirchhoff's Laws still be applied in more complex circuits with multiple loops and branches?

Yes, Kirchhoff's Laws can still be applied in more complex circuits. It may be helpful to break down the circuit into smaller sections and apply the laws separately, then combine the solutions to find the overall voltage across a specific resistor. It is also important to label and keep track of the direction of current flow in each section.

5. Are there any alternative methods for solving Kirchhoff's Law problems for resistors?

Yes, there are alternative methods such as using nodal analysis or mesh analysis, which use different equations and techniques to solve for voltages and currents in a circuit. These methods may be more efficient for solving complex circuits, but the principles of Kirchhoff's Laws still apply. It is important to understand the concepts and principles behind these methods in order to effectively apply them to solve for resistors in a circuit.

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