# Can you help me solve this circuit problem using linear algebra?

• Engineering
• terence cheng
In summary, the conversation involves a circuit problem with linear algebra and the task is to find the voltages at internal points using Ohm's law and Kirchhoff's current law. The equations are expressed in vector form and the goal is to find the values of the matrix A and vector w. The conversation also mentions a "Wheatstone bridge" when N = 2.
terence cheng
Hi I got stuck at the following circuit problem which involves linear algebra, since I am not a physics major, I don't even have the basics to get started. Please shed some light, really appreciate it! Thanks a lot.

1. Homework Statement

Consider a long chain of resistors wired up like this

All the resistors have the same resistance R. The power rail at the top is at voltage V+ = 5V. The problem is to find the voltages V1 . . . VN at the internal points in the circuit.

Using Ohm’s law and the Kirchhoff current law, which says that the total net current flow out of (or into) any junction in a circuit must be zero, show that the voltages V1 . . . VN satisfy the equations

3V1 − V2 − V3 = V+,
−V1 + 4V2 − V3 − V4 = V+,

... −Vi−2 − Vi−1 + 4Vi − Vi+1 − Vi+2 = 0, ...

−VN−3 − VN−2 + 4VN−1 − VN = 0,

−VN−2 − VN−1 + 3VN = 0.

Express these equations in vector form Av = w and find the values of the matrix A and the vector w.

## The Attempt at a Solution

I considered when N = 2, it gives me a "Wheatstone bridge" with V1 = V2 = V/2
but then how should i continue? thanks a lot![/B]

## 1. What is a circuit and how does it work?

A circuit is a closed loop or path through which electricity can flow. It consists of a power source, such as a battery, and various components, such as resistors, capacitors, and transistors, that control the flow of electricity. When a circuit is connected, the power source pushes electrons through the components, creating a flow of electric current.

## 2. What is linear algebra and why is it important in circuit design?

Linear algebra is a branch of mathematics that deals with linear equations and their representations as matrices and vectors. In circuit design, linear algebra is used to analyze and solve equations that describe the behavior of the circuit components, such as current, voltage, and resistance. It is also important in designing and optimizing complex circuits, such as those found in electronic devices.

## 3. How are matrices and vectors used in circuit analysis?

Matrices and vectors are used in circuit analysis to represent the various circuit components and their relationships. For example, a matrix can represent the connections between different components, while a vector can represent the input and output signals. By manipulating these matrices and vectors, engineers can solve complex equations and predict the behavior of the circuit.

## 4. Can linear algebra be used to design non-linear circuits?

Yes, although linear algebra is primarily concerned with linear equations, it can also be used to design non-linear circuits. This is because many non-linear circuits can be approximated by a series of linear equations. By breaking down the non-linear circuit into smaller linear components, engineers can use linear algebra techniques to analyze and design the overall circuit.

## 5. What are some real-world applications of circuits and linear algebra?

Circuits and linear algebra have numerous real-world applications, including in electronic devices such as computers, smartphones, and televisions. They are also used in power grids, telecommunications systems, and medical equipment. Additionally, linear algebra is used in fields such as robotics, control systems, and signal processing, which all rely on circuits for their operation.

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