|
|
| Line 1: |
Line 1: |
| '''Symbolic circuit analysis''' is a formal technique of [[circuit analysis]] to calculate the behaviour or characteristic of an electric/electronic circuit with the independent variables (time or frequency), the dependent variables (voltages and currents), and (some or all of) the circuit elements represented by symbols.<ref>G. Gielen and W. Sansen, Symbolic Analysis for Automated Design of Analog Integrated Circuits. Boston: Kluwer Academic Publishers, 1991.</ref>
| | Bonus: WP Twin and WP Twin Auto Backup: (link to ) While not a theme, I think this software is essential if you are maintaining your Wordpress blog or regularly create new blog sites. What I advise you do next is save the backup data file to a remote place like a CD-ROM, external disk drive if you have one or a provider such as Dropbox. One really cool features about this amazing and free wp plugin is that the code it generates is completely portable. Donor oocytes and menopausal pregnancy: Oocyte donation to women of advanced reproductive age: pregnancy results and obstetrical outcomes in patients 45 years and older. If you are happy with your new look then click "Activate 'New Theme'" in the top right corner. <br><br>You just download ready made templates to a separate directory and then choose a favorite one in the admin panel. If you have any type of inquiries pertaining to where and just how to utilize [http://ll.my/backupplugin629312 wordpress backup], you could contact us at the page. When you write a new post, you'll see a small bar that goes across the text input area. A Wordpress plugin is a software that you can install into your Wordpress site. You can add new functionalities and edit the existing ones to suit your changing business needs. The biggest advantage of using a coupon or deal plugin is that it gives your readers the coupons and deals within minutes of them becoming available. <br><br>The least difficult and very best way to do this is by acquiring a Word - Press site. By using Word - Press MLM websites or blogs, you and your companies presence on the internet can be created swiftly and simply. For a much deeper understanding of simple wordpress themes", check out Upon browsing such, you'll be able to know valuable facts. Every single Theme might be unique, providing several alternatives for webpage owners to reap the benefits of in an effort to instantaneously adjust their web page appear. Customization of web layout is easy due to the availability of huge selection of templates. <br><br>Google Maps Excellent navigation feature with Google Maps and latitude, for letting people who have access to your account Latitude know exactly where you are. And, that is all the opposition events with nationalistic agenda in favor of the individuals of Pakistan marching collectively in the battle in opposition to radicalism. Exacting subjects in reality must be accumulated in head ahead of planning on your high quality theme. Word - Press is the most popular open source content management system (CMS) in the world today. Wordpress template is loaded with lots of prototype that unite graphic features and content area. <br><br>More it extends numerous opportunities where your firm is at comfort and rest assured of no risks & errors. Being a Plugin Developer, it is important for you to know that development of Word - Press driven website should be done only when you enable debugging. By the time you get the Gallery Word - Press Themes, the first thing that you should know is on how to install it. Thus, Word - Press is a good alternative if you are looking for free blogging software. I have never seen a plugin with such a massive array of features, this does everything that platinum SEO and All In One SEO, also throws in the functionality found within SEO Smart Links and a number of other plugins it is essentially the swiss army knife of Word - Press plugins. |
| | |
| When analysing electric/electronic circuits, we may ask two types of questions: What is the '''value''' of certain circuit variable ([[voltage]], [[current (electricity)|current]], [[resistance (electricity)|resistance]], [[gain]], etc.) or what is the '''relationship''' between some circuit variables or between a circuit variable and circuit components and frequency (or time). Such relationship may take the form of a graph, where numerical values of a circuit variable are plotted versus frequency or component value (the most common example would be a plot of the magnitude of a transfer function vs. frequency).
| |
| | |
| Symbolic circuit analysis is concerned with obtaining those relationships in symbolic form, i.e., in the form of [[analytical expression]], where the complex frequency (or time) and some or all of the circuit components are represented by symbols.
| |
| | |
| ==Frequency domain expressions==
| |
| | |
| In the frequency domain the most common task of symbolic circuit analysis is to obtain the relationship between input and output variables in the form of a [[rational function]] in the [[complex frequency]] <math>\mathit{s}\,</math> and symbolic variables <math>\mathbf{x}</math>:
| |
| | |
| <center><math>T(s,\mathbf{x})=\frac{N(s,\mathbf{x})}{D(s,\mathbf{x})}</math></center>
| |
| | |
| The above relationship is often called the network function. For physical systems, <math>N(s,\mathbf{x})</math> and <math>D(s,\mathbf{x})</math> are [[polynomial]]s in <math>\mathit{s}\,</math> with real coefficients:
| |
| | |
| <center><math>T(s,\mathbf{x})=\frac{\displaystyle \sum_{i=0}^n a_i(\mathbf{x}) s^i}{\displaystyle \sum_{i=0}^m b_i(\mathbf{x}) s^i}=K\frac{\displaystyle \prod_{i=1}^n (s-z_i(\mathbf{x}))}{\displaystyle \prod_{i=1}^m (s-p_i(\mathbf{x}))}</math></center>
| |
| | |
| where <math>z_i(\mathbf{x})</math> are the zeroes and <math>p_i(\mathbf{x})</math> are the poles of the network function; <math>m \geqslant n</math>.
| |
| | |
| While there are several methods for generating coefficients <math>a_i(\mathbf{x})</math> and <math>b_i(\mathbf{x})</math>, no technique exists to obtain exact symbolic expressions for poles and zeroes for polynomials of order higher than 5.
| |
| | |
| ==Types of symbolic network functions==
| |
| | |
| Depending on which parameters are kept as symbols, we may have several different types of symbolic network functions. This is best illustrated on an example. Consider, for instance, the [[biquad filter]] circuit with ideal [[op amp]]s, shown below. We want to obtain a formula for its voltage transmittance (also called the [[voltage gain]]) in the frequency domain, <math>{T_v(s) = V_{out}(s)/V_{in}(s)}\,</math>.
| |
| | |
| [[Image:Biquad circuit.gif|thumb|510px|center|Figure 1: Biquad circuit with ideal opamps. (This diagram was created using the [[schematic capture]] feature of [[SapWin]].)]]
| |
| | |
| ===Network function with ''s'' as the only variable===
| |
| | |
| If the complex frequency <math>\mathit{s}\,</math> is the only variable, the formula will look like this (for simplicity we use the numerical values: <math>R_i=i, C_i = 0.01i\,</math>):
| |
| | |
| <center><math>T(s)=\frac{3.48s}{13.2s^2+1.32s+0.33}</math></center> | |
| | |
| ===Semi-symbolic network function===
| |
| | |
| If the complex frequency <math>\mathit{s}\,</math> and some circuit variables are kept as symbols (semi-symbolic analysis), the formula may take a form:
| |
| | |
| <center>
| |
| <math>
| |
| \begin{align}
| |
| T(s,\mathbf{x})&=\frac{1.74C_2s}{6.6C_1 C_2 s^2+0.66C_2 s+0.33} \\
| |
| \mathbf{x}&=[C_1~C_2]
| |
| \end{align}
| |
| </math>
| |
| </center>
| |
| | |
| ===Fully symbolic network function===
| |
| | |
| If the complex frequency <math>\mathit{s}\,</math> and all circuit variables are symbolic (fully symbolic analysis), the voltage transmittance is given by (here <math>G_i = 1/R_i \,</math>):
| |
| | |
| <center>
| |
| <math>
| |
| \begin{align}
| |
| T(s,\mathbf{x})&=\frac{G_4 G_6 G_8 C_2s}{G_6 G_{11} C_1 C_2 s^2+G_1 G_6 G_{11} C_2 s+G_2 G_3 G_5 G_{11}} \\
| |
| \mathbf{x}&=[C_1~C_2~G_1~G_2~G_3~G_4~G_5~G_6~G_8~G_{11}]
| |
| \end{align}
| |
| </math>
| |
| </center>
| |
| | |
| All expressions above are extremely useful in obtaining insight into operation of the circuit and understanding how each component contributes to the overall circuit performance. As the circuit size increases, however, the number of terms in such expressions grows exponentially. So, even for relatively simple circuits, the formulae become too long to be of any practical value. One way to deal with this problem is to omit insignificant terms from the symbolic expression, keeping the inevitable error below the predetermined limit.<ref>B. Rodanski, M. Hassoun, "Symbolic Analysis," in The Circuits and Filters Handbook: Fundamentals of Circuits and Filters, 3rd ed., Wai-Kai Chen, Editor. CRC Press, 2009, pp. 25-1 - 25-29.</ref>
| |
| | |
| ===Sequence of Expressions form===
| |
| | |
| Another possibility to shorten the symbolic expression to manageable length is to represent the network function by a sequence of expressions (SoE).<ref>M. Pierzchala, B. Rodanski, "Generation of Sequential Symbolic Network Functions for Large-Scale Networks by Circuit Reduction to a Two-Port," IEEE Transactions on Circuits and Systems - I: Fundamental Theory and Applications, vol. 48, no. 7, July 2001, pp. 906-909.</ref> Of course, the interpretability of the formula is lost, but this approach is very useful for repetitive numerical calculations. A software package STAINS (Symbolic Two-port Analysis via Internal Node Suppression) has been developed to generate such sequences.<ref>L.P. Huelsman, "STAINS - Symbolic Two-Port Analysis via Internal Node Suppression," IEEE Circuits & Devices Magazine, March 2002, pp. 3-6.</ref> There are several types of SoE that can be obtained from STAINS. For example, the compact SoE for <math>T_v(s)\,</math> of our biquad is
| |
| | |
| <pre>
| |
| x1 = G5*G3/G6
| |
| x2 = -G1-s*C1-G2*x1/(s*C2)
| |
| x3 = -G4*G8/x2
| |
| Ts = x3/G11
| |
| </pre>
| |
| The above sequence contains fractions. If this is not desirable (when divisions by zero appear, for instance), we may generate a fractionless SoE:
| |
| | |
| <pre>
| |
| x1 = -G2*G5
| |
| x2 = G6*s*C2
| |
| x3 = -G4*x2
| |
| x4 = x1*G3-(G1+s*C1)*x2
| |
| x5 = x3*G8
| |
| x6 = -G11*x4
| |
| Ts = -x5/x6
| |
| </pre>
| |
| | |
| Yet another way to shorten the expression is to [[factorise]] [[polynomial]]s <math>N(s,\mathbf{x})</math> and <math>D(s,\mathbf{x})</math>. For our example this is very simple and leads to:
| |
| | |
| <pre>
| |
| Num = G4*G6*G8*s*C2
| |
| Den = G11*((G1+s*C1)*G6*s*C2+G2*G3*G5)
| |
| Ts = Num/Den
| |
| </pre>
| |
| | |
| For larger circuits, however, factorisation becomes a difficult [[combinatorial]] problem and the final result may be impractical for both interpretation and numerical calculations.
| |
| | |
| == References ==
| |
| <!--- See [[Wikipedia:Footnotes]] on how to create references using <ref></ref> tags which will then appear here automatically -->
| |
| {{Reflist}}
| |
| | |
| == External links ==
| |
| * [http://rodanski.net/ben/work/symbolic/index.htm Library of benchmark circuits for symbolic circuit analysis]
| |
| | |
| [[Category:Electronic design]]
| |
Bonus: WP Twin and WP Twin Auto Backup: (link to ) While not a theme, I think this software is essential if you are maintaining your Wordpress blog or regularly create new blog sites. What I advise you do next is save the backup data file to a remote place like a CD-ROM, external disk drive if you have one or a provider such as Dropbox. One really cool features about this amazing and free wp plugin is that the code it generates is completely portable. Donor oocytes and menopausal pregnancy: Oocyte donation to women of advanced reproductive age: pregnancy results and obstetrical outcomes in patients 45 years and older. If you are happy with your new look then click "Activate 'New Theme'" in the top right corner.
You just download ready made templates to a separate directory and then choose a favorite one in the admin panel. If you have any type of inquiries pertaining to where and just how to utilize wordpress backup, you could contact us at the page. When you write a new post, you'll see a small bar that goes across the text input area. A Wordpress plugin is a software that you can install into your Wordpress site. You can add new functionalities and edit the existing ones to suit your changing business needs. The biggest advantage of using a coupon or deal plugin is that it gives your readers the coupons and deals within minutes of them becoming available.
The least difficult and very best way to do this is by acquiring a Word - Press site. By using Word - Press MLM websites or blogs, you and your companies presence on the internet can be created swiftly and simply. For a much deeper understanding of simple wordpress themes", check out Upon browsing such, you'll be able to know valuable facts. Every single Theme might be unique, providing several alternatives for webpage owners to reap the benefits of in an effort to instantaneously adjust their web page appear. Customization of web layout is easy due to the availability of huge selection of templates.
Google Maps Excellent navigation feature with Google Maps and latitude, for letting people who have access to your account Latitude know exactly where you are. And, that is all the opposition events with nationalistic agenda in favor of the individuals of Pakistan marching collectively in the battle in opposition to radicalism. Exacting subjects in reality must be accumulated in head ahead of planning on your high quality theme. Word - Press is the most popular open source content management system (CMS) in the world today. Wordpress template is loaded with lots of prototype that unite graphic features and content area.
More it extends numerous opportunities where your firm is at comfort and rest assured of no risks & errors. Being a Plugin Developer, it is important for you to know that development of Word - Press driven website should be done only when you enable debugging. By the time you get the Gallery Word - Press Themes, the first thing that you should know is on how to install it. Thus, Word - Press is a good alternative if you are looking for free blogging software. I have never seen a plugin with such a massive array of features, this does everything that platinum SEO and All In One SEO, also throws in the functionality found within SEO Smart Links and a number of other plugins it is essentially the swiss army knife of Word - Press plugins.