I came across a simulation software called MATLAB. I got to know it takes inputs in the form of matrices, provides output in terms of matrices , all in all it operates using matrices. But it seems pretty weird to me that it uses matrices. Why does it use matrices? How does usage of matrices makes it efficient ?
Matrix and simulation
simulation
Related Solutions
Distributed Amps the way you built should be ringing, in theory as well as in practice. To get them to work without ringing, you need an "m-derived network." old Tek manuals from the vacuum tube era show how.
For example, see here http://w140.com/tekwiki/images/3/36/545a_vert.png
the center-tapped inductors are key. Google "t coil network" or "m derived network" for the design equations.
i refer to any of the tapped inductors in the tektronix schematic. for an impedance of Z=1 and a fet capacitance of C=1, we need each half of the inductance to be 1/3 and the mutual inductance to be m=1/2. that gives a 3 db bandwidth of B=2.7 and rise time tr=.79. In theory, a capacitance of C/12 from end to end of the tapped inductor is understood.
so if, for example, Z=50R and C=1.5pF, we get an inductance of 1.25 nH, m=1.875 nH, B=5.7 GHz and tr= 59 ps. Overshoot is less than a percent.
The simple answer of a constant phase shifting element is that when this reactive element is 90 deg phase shifted current to voltage the Z(s) will also be either 1/sC or sL for the Laplace transform.
Conversely, a constant time delay system has phase integrating with f such that delay= 2pi for every multiple of t=1/f and amplitude , impedance "can" be unchanged. ( search all pass filter)
In every branch of applied physics and spectrum where impedance spectroscopy shows real and imaginary impedance, think of these as either dielectrics or inductors that can store energy and real as conductive or lossy medium. This applies to ULF for earth impedance scattering of soil dielectrics and minerals as well as XHF light waves.
Anecdotal
Spectroscopy does not have to be "absolute". For example Eddy current sensing for metallurgic defects. In 1977 we made a system for detecting molecular flaws in Monel steel after applied 10k Atmosphere of heavy water at super-temperatures and calibrated sensors to 0 relative impedance for a 1mm thru hole in tube steel wall and 0.5mm ring wall thickness step as 0 deg relative phase for a huge defect and then with 80 dB SNR could detect impedance changes down 10 ppm of Z or near FIR optical wavelengths of molecules. By scanning the signatures above some threshold, and historical records 10km of data was analyzed every 0.2mm to detect flaws before expensive leaks could occur. That was a long time ago. It was not the 1st SCADA type robotic system of its type, but my 1st. 10 Engineers to design one and 1 to identify all the thousands of fixes to make it work and deliver to customer in Ontario.
Best Answer
Key point: Matlab is not a simulation tool that uses matrices. It is a matrix math tool that is often used to simulate things.
It is or can be also used for all kinds of other calculations that can be done in terms of matrices --- this includes statistics, linear programming (systems optimization), curve fitting, etc.
The reason we use matrices to do simulations (and lots of other kinds of calculations) is because it provides a very compact way to write down large sets of equations in linear algebra. A single matrix equation like
\$ \mathbb{A}\bar{x} = \bar{b}\$
can, in a single line, represent an arbitrary number of algebraic equations with an arbitrary number of terms each, as long as those equations meet some simple requirements.
Also, these linear algebra equations can be solved by simple but repetitive operations, which is exactly what computers are good out.
Luckily we were able to figure out how to simulate physical systems using linear algebra, so that we can use this notation and computer processes to predict the behavior of different things we want to build (circuits, buildings, chemical plant control systems, ...)