Kirchoff's Laws Giving Singular System

In summary, Kirchoff's voltage law does not seem to work well when trying to find a non redundant set of equations for a system of resistors. The problem is that every system of equations is singular.
  • #1
Hertz
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8
Hi, I'm trying to use Kirchoff's voltage law to find a system of equations which I can use to find I1, I2, I3, I4, and I5; which are the respective currents going through each resistor in the image.

circuit1-2014.png


I have 5 unknowns so I need to find 5 non redundant equations. The problem is, every system of equations I come up with is singular. In other words... I am trying to set up a matrix equation ##A x = b## but my matrix A keeps having determinant zero!

Is there any general rule to follow in order to find a non redundant set of equations? This is really time consuming!

If you are curious, here is an example system which I came up with, but in some way it's redundant.
Code:
V = R1 I1 + R4 I4
V = R2 I2 + R5 I5
V = R1 I1 + R3 I3 + R5 I5
V = R2 I2 - R3 I3 + R4 I4
0 = R1 I1 - R2 I2 + R4 I4 - R5 I5
V is the voltage of the battery.
 
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  • #2
I only see two unknowns. Maybe three.

Have you been introduced to node voltage or mesh current analysis?
 
  • #3
My favorite analysis technique is node voltage analysis. Basically, you look at the circuit and you identify any unknown voltages, each unknown voltage is a node. In this circuit, there are only two. Then, at each node you use KCL to write that the total current leaving the node is 0. Then you solve those two equations in two unknowns to get the unknown voltages. All of the currents are then uniquely determined.
 
  • #4
I'm confused, what are the unknowns you are seeing? I'm trying to find the current going through each resistor, and I0, the current going through the wires near the EMF. I need to know the answer of each of these six values.

I'm also not exactly sure what you mean by node voltage and current analysis. I'm familiar with both of Kirchoff's laws though.

e-
1. The total potential difference around any closed loop is zero.
2. At any node there is as much current leaving as is entering.
 
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  • #5
The only unknown voltages are the voltages at the two nodes touching R3. All other voltages are known. Call those two voltages Va and Vb. Then at Va and Vb write KCL.

Once you have Va and Vb then all of the currents are obtained through Ohms law.
 
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  • #6
DaleSpam said:
The only unknown voltages are the voltages at the two nodes touching R3. All other voltages are known. Call those two voltages Va and Vb. Then at Va and Vb write KCL.

Once you have Va and Vb then all of the currents are obtained through Ohms law.

Oh I see! Thank you!

It's like magic...
 
  • #7
Glad I could help!

There is another similar technique, called mesh current analysis. It is based on KVL, but I don't like it as much.
 
  • #8
Thank you, I will surely look into it in due time :)
 

FAQ: Kirchoff's Laws Giving Singular System

What are Kirchoff's Laws?

Kirchoff's Laws are fundamental principles in circuit analysis that describe the behavior of electrical circuits. These laws are used to determine the voltage, current, and resistance in a circuit.

What is a Singular System in relation to Kirchoff's Laws?

A Singular System refers to a circuit that cannot be solved using Kirchoff's Laws alone. This usually occurs when there are more unknown variables than equations, making the system unsolvable.

What are the two Kirchoff's Laws?

Kirchoff's Laws consist of two principles: Kirchoff's Current Law (KCL) and Kirchoff's Voltage Law (KVL). KCL states that the total current entering a node in a circuit must be equal to the total current exiting the node. KVL states that the sum of all voltages in a closed loop in a circuit must be equal to zero.

How do Kirchoff's Laws help in circuit analysis?

Kirchoff's Laws provide a systematic approach to analyzing complex electrical circuits. By applying these laws, we can determine the values of unknown variables and better understand the behavior of the circuit.

What are some real-life applications of Kirchoff's Laws?

Kirchoff's Laws are used in various fields such as electrical engineering, physics, and telecommunications. They are essential in designing and troubleshooting electrical circuits, analyzing power distribution systems, and understanding the behavior of electronic devices.

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