## 电气工程代写|模拟电路代写analog circuit代考|ECE172

statistics-lab™ 为您的留学生涯保驾护航 在代写模拟电路analog circuit方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写模拟电路analog circuit代写方面经验极为丰富，各种代写模拟电路analog circuit相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 电气工程代写|模拟电路代写analog circuit代考|Euler’s Methods

Perhaps the simplest numerical formulation is what is known as Euler’s methods, the forward and backward Euler. We start with the most obvious implementation we just discussed in the previous section, the forward Euler’s method:
$$i\left(t_n\right)=C \frac{u\left(t_{n+1}\right)-u\left(t_n\right)}{\Delta t} \rightarrow u\left(t_{n+1}\right)=u\left(t_n\right)+\Delta t \frac{i\left(t_n\right)}{C}$$
This seems simple enough and it is certainly fair game to use. As we mentioned, one soon discovers it easily gets unstable and the solution can blow out of proportion quickly, and one is forced to sometimes take remarkably small timesteps to avoid the problem. We will not discuss the underlying reason here, it is deferred to Chap. 7, but it is fundamentally the reason why it is never used in practical implementations. Instead a really small reformulation fixes the problem:
$$i\left(t_{n+1}\right)=C \frac{u\left(t_{n+1}\right)-u\left(t_n\right)}{\Delta t} \rightarrow u\left(t_{n+1}\right)=u\left(t_n\right)+\Delta t \frac{i\left(t_{n+1}\right)}{C}$$
Please note the current is evaluated at the new timestep! We say this is an implicit formulation of the differential equation. The unknowns show up at both sides of the equal sign. This is the key, and it turns out this formulation does not suffer from the type of instability that plagues the forward Euler, and this is known as the backward, or implicit, Euler implementation. Most simulators provide this integration method, and it is fairly straightforward to implement as we shall see later.

## 电气工程代写|模拟电路代写analog circuit代考|Trapezoidal Method

The trapezoidal method is based on integrating the differential equation over a short time using trapezoids as a way to approximate the functions, hence the name. We will not go into the details, but with this formulation, one finds the derivative is approximated as $$\frac{d f}{d t}(t+\Delta t) \approx 2 \frac{f(t+\Delta t)-f(t)}{\Delta t}-\frac{d f}{d t}(t)$$
It looks similar to the Euler formulation with the exception of the last term and the factor 2 in front of the first term. It has one well-known weakness we will discuss in Chap. 3, namely, an odd ringing which is very characteristic and fairly easy to recognize. Numerically, this is straightforward to implement. We have for the full formulation using the previous example
$$\frac{i(t+\Delta t)}{C}=2 \frac{u(t+\Delta t)-u(t)}{\Delta t}-\dot{u}(t), \quad u(0)=1,$$
or after reformulation
$$u(t+\Delta t)=u(t)+\frac{\Delta t}{2}\left(\frac{i(t+\Delta t)}{C}+\frac{i(t)}{C}\right), u(0)=1,$$
where we have replaced the voltage derivative at time $t$ with $i(t) / C$. Written this way, one can view it as a Crank-Nicolson scheme [4] where the derivative is evaluated at time $t+\Delta t / 2$ and the current is the average of the current at time $t$ and $t+\Delta t$. In this way, both the derivative term of Eq. $2.4$ and the left-hand side of Eq. $2.4$ are evaluated at the same point in time. This results in better accuracy. Another interesting observation is that one can view the trapezoidal method as an average of the forward and backward Euler methods.

## 电气工程代写|模拟电路代写analog circuit代考|Euler’s Methods

$$i\left(t_n\right)=C \frac{u\left(t_{n+1}\right)-u\left(t_n\right)}{\Delta t} \rightarrow u\left(t_{n+1}\right)=u\left(t_n\right)+\Delta t \frac{i\left(t_n\right)}{C}$$

$$i\left(t_{n+1}\right)=C \frac{u\left(t_{n+1}\right)-u\left(t_n\right)}{\Delta t} \rightarrow u\left(t_{n+1}\right)=u\left(t_n\right)+\Delta t \frac{i\left(t_{n+1}\right)}{C}$$

## 电气工程代写|模拟电路代写analog circuit代考|Trapezoidal Method

$$\frac{d f}{d t}(t+\Delta t) \approx 2 \frac{f(t+\Delta t)-f(t)}{\Delta t}-\frac{d f}{d t}(t)$$

$$\frac{i(t+\Delta t)}{C}=2 \frac{u(t+\Delta t)-u(t)}{\Delta t}-\dot{u}(t), \quad u(0)=1,$$

$$u(t+\Delta t)=u(t)+\frac{\Delta t}{2}\left(\frac{i(t+\Delta t)}{C}+\frac{i(t)}{C}\right), u(0)=1$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 电气工程代写|模拟电路代写analog circuit代考|ECE2237

statistics-lab™ 为您的留学生涯保驾护航 在代写模拟电路analog circuit方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写模拟电路analog circuit代写方面经验极为丰富，各种代写模拟电路analog circuit相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 电气工程代写|模拟电路代写analog circuit代考|Overview of Numerical Methods

Abstract This chapter describes a number of topics that are full-fledged research subjects in and of themselves and to do them justice in a just a few pages is not possible. The hope of the author is to present the material in such a way as to wet the readers’ appetite. The importance of numerical methods in solving various kinds of problems is paramount in modern product engineering and scientific research. Both the engineering and scientific communities are heavily involved in the development and use of these methodologies. In an effort to contain these vast subject matters, the focus will be on methods the reader is likely to encounter in an electrical engineering context and as such certain types of approximations to differential equations will be highlighted. The same thing goes for matrix equations where we here only use simple examples to highlight the fundamental ideas. The more advanced iterative methods that have been so successful in recent decades are mentioned briefly with an accompanying Python code example. Nonlinear equations and how to solve them efficiently is likewise another intense field of study, and over the years many methods have been developed that are in wide use in the scientific/engineering community today. Here we describe a method that is perhaps the most important to know due to its relative ease of implementation attributed to Isaac Newton, although other researchers such as Joseph Raphson were also involved over the years. Even though the presentation is held at a fundamental level, some basic familiarity with numerical methods, corresponding to an introductory class on the subject, will be helpful since we will be rather brief. We will start the chapter discussing differential equations and how one might implement them numerically. We will discuss implementations of what is called initial value problems where the state is known at a certain moment in time, and from then on, the system develops according to the governing equations. We will present implementations commonly used in circuit simulators. The chapter continues with nonlinear solution methods, and we wrap up the presentation with a description of matrix solvers. Rather than going through the mathematical theories behind thesé methods, we choose to present the basic ideas using examples, and for the interested reader a much more in-depth discussion of these issues can be found in Chap. 7 and the references at the end of the chapter. The importance of the subject matter presented in this chapter cannot be overstated, and it is the hope of the author the reader will explore the topic more deeply on his or her own.

## 电气工程代写|模拟电路代写analog circuit代考|Differential Equations: Difference Equations

One difficulty when using numerical techniques to solve for a systems evolution in time comes when the solution at a particular point in time depends on the previous time points, when there is some kind of memory in the system. Most often this memory effect is expressed in terms of differential equations, and their numerical approximation is often a significant source of error. This section will briefly review such approximations, focusing on those that are common in circuit analysis. With some exceptions, we will discuss what is known as initial value problems in numerical analysis.

We will start with the basic idea behind approximating continuous time/space differential equations and look at simple examples with pseudocode suggestions. This we follow with differential equations common in circuit analysis that arise from typical circuit elements.

$$\frac{u(t)}{R}=C \frac{d u(t)}{d t}, \quad u(0)=1,$$
where $C, R$ are constants. This equation describes a parallel combination of a resistor with value $R$ and a capacitor with value $C$ (Fig. 2.1).
It has a well-known analytical solution:
$$u(t)=e^{-t /(R C)}$$
How can we approximate this equation numerically? We recall from basic calculus the derivative is defined as

$$\frac{d f(t)}{d t}=\lim _{\varepsilon \rightarrow 0} \frac{f(t+\varepsilon)-f(t)}{\varepsilon}$$
It is now natural to find a numerical approximation as
$$\frac{d f}{d t} \approx \frac{\Delta f}{\Delta t}=\frac{f(t+\Delta t)-f(t)}{\Delta t}$$
With this approximation of the derivative, we find for the differential equation
$$-\frac{u(t)}{R}=C \frac{u(t+\Delta t)-u(t)}{\Delta t}, \quad u(0)=1,$$
or after rewrite
$$u(t+\Delta t)=u(t)\left(1-\frac{\Delta t}{R C}\right), \quad u(0)=1,$$
This formulation is knowns as Euler’s forward method or sometimes Euler’s explicit method. It is perhaps the most straightforward method to implement, but due to a notorious instability problem, where the solution blows up due to numerical errors, it is almost never used in practical situations. We will look at examples of this method in Chan 4 where the stabilitv issues will be clear.

## 电气工程代写|模拟电路代写analog circuit代考|Differential Equations: Difference Equations

$$\frac{u(t)}{R}=C \frac{d u(t)}{d t}, \quad u(0)=1,$$

$$u(t)=e^{-t /(R C)}$$

$$\frac{d f(t)}{d t}=\lim _{\varepsilon \rightarrow 0} \frac{f(t+\varepsilon)-f(t)}{\varepsilon}$$

$$\frac{d f}{d t} \approx \frac{\Delta f}{\Delta t}=\frac{f(t+\Delta t)-f(t)}{\Delta t}$$

$$-\frac{u(t)}{R}=C \frac{u(t+\Delta t)-u(t)}{\Delta t}, \quad u(0)=1,$$

$$u(t+\Delta t)=u(t)\left(1-\frac{\Delta t}{R C}\right), \quad u(0)=1$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 电气工程代写|模拟电路代写analog circuit代考|EEW240A

statistics-lab™ 为您的留学生涯保驾护航 在代写模拟电路analog circuit方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写模拟电路analog circuit代写方面经验极为丰富，各种代写模拟电路analog circuit相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 电气工程代写|模拟电路代写analog circuit代考|Background

The semiconductor industry is one of the marvels of modern society. We are constantly being presented with novel machines that utilize the extraordinary progress the industry has kept up over some seven decades of intense development. It was around 1965 that one of the Intel founders, Gordon Moore, coined what has become known as Moore’s law stating that for every 2 years, the number of transistors in a given area is doubling. It is an exponential scaling law that has kept up until recently (Fig. 1.1).

This is an extraordinary development driven by consumer demand for higher and higher data processing. The advent of streaming where full movies can be viewed as they are downloaded to a particular device requires enormous data delivery rates. In the beginning of this industry epoch, a commercial integrated circuit had a few hundred transistors at most. Compare this to the latest tens of billions of devices in a modern central processing unit (CPU) integrated circuit.

One of the fundamental units one measures these devices with is the so-called transistor gate length. It is as presently known to the author at $3 \mathrm{~nm}$ scale with the latest so-called gate-all-around technology. A typical atom has perhaps a size scale of $0.1 \mathrm{~nm}$, just 30 times smaller. We are at the realm of quantum physics for these devices. This is not new. Quantum effects like tunneling, where an electron can appear at the other side of a barrier with a certain probability, has been a source of the so-called leakage current for more than a decade.

Imagine now a modern integrated circuit or chip in industry parlance with some billion devices on it. The key development step of these products is the maskmaking step. There might be some 50 masks needed for the latest technology, and on average such masks cost a few 100,000 US dollars each. This cost is a large portion of the design engineering cost, and the total cost of designing such chips can then be of the order tens of millions of US dollars. This is before mass production starts. If something is wrong with the design, the masks need to be remade. How can one be reasonably sure that such enormous chips will be working when it comes back from the first fabrication run so a costly mask production step is avoided? The answer lies in the use of simulators, both digital, for the core data processing, and analog ones for the interfaces to the outside world among other things.

## 电气工程代写|模拟电路代写analog circuit代考|The Arrival of Simulators

Using simulators to prove out electronic circuitry is an old idea. The earliest attempts can be found in the 1960 s where the US Department of Defense supported circuit simulation developments that were proprietary. The modern attempts to make simulators publicly available were started by researchers at the University of California at Berkeley, where the extraordinary vision by a handful of young professors and researchers has developed what became known as Simulation Program with Integrated Circuit Emphasis or SPICE. It was not without controversy in the beginning. A lot of contemporaries felt that simulators could not possibly capture the operations well and the effort was a waste of time. Instead the idea was to prototype the design using breadboards and discrete devices and then miniaturize on a chip. The Berkeley team persisted and it is now considered the original master code, and most simulators after this use many of the same features SPICE introduced to solve numerical problems. In fact the word spice has become a verb in that one often says of simulating a circuit as “spiceing” a circuit. Naturally many decades of innovation have produced a code that is quite a bit more complex than the first versions.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。