Electronic Systems in Terms of Algebraic Equations

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In the world of electrical engineers it is not easy to define or analyze any system performance without any proper study. Every electronics system is defined in terms of mathematical equations. I am keen to study about radio frequency circuits. In these systems it is always necessary to know how much error is obtained and we need to estimate the error so that we can design the system that can be used efficiently. Also the signals that are received in these type of communication systems are unpredictable and it is necessary to study about the behaviour. This is the reason why I chose to study about Probability and Random processes.

Every system is dependent on any specific constraint. In communications its usually time and frequency. So we learn every type random process that are either single dependent or double parameter dependent. The course mainly deals about basics of probability and its use in determining the behaviour, performance of any Random process. It starts with venn diagrams, plotting of probabilities and then the probabilities that are dependent on other parameters which are called conditional probability. It is further studied by deriving into a theorem that is mostly used in machine learning which is called bayes theorem. This theorem derives the probability of the event to occur, after a event is happened.

To get a clear view of study, parameters should be defined in a function so as to determine its characteristics. So probability distribution function PDF and cumulative distributive function are derived. These functions are used to analyze Random variables in both continuous and discrete forms. Various statistical terms like mean, variance are used to define the random process. The power of the system are represented by density functions .These random variables are represented as Random process, such as poison random processes, Binomial random processes and Rayliegh random processes. These are further viewed in Gaussian and Gamma distributions.

Relation between the two random processes are mathematically given by covariance, correlation and orthogonality which will give insight to dependency of one random process with another random process (like heat of the device and gain of the device- even though both are not related, one random process effects another random process).

At the end of the course various error detecting techniques are studied such as Mean square error(MSE) , ML estimator. Both these estimators are mathematically derived and various types of cases are solved. The project of this course is based on these estimators. This made me learn more about my Matlab programming and using this MSE estimator is designed. This is estimated on a data sheet of 10 variables and 2500 data entries. The results gave almost as accurate values and thus knowing the significance of Estimators.


As mentioned it is necessary for me to know the mathematical representation of every system. Any electronic device is operated either by current or voltage, for this to happen the device should respond according to what we hope for. Sometimes due to over driving or noise considerations the device is failed to be in use. All these considerations can be analyzed over using S parameter matrix. The s parameter matrix can be a collection of parameters across the device and to know about them it is necessary for me to know the advantages i can use over this matrix by using basic algebra. This is the reason I opted for Linear and Matrix algebra

This course is started off with basics of matrices and its properties along with vectors and linear combinations. The matrices are used to find the solutions of linear equations by using LU decomposition method, Elimination method. 4 types of vector spaces are studied as row space, null space , column space and left null space. Sometimes we don’t have particular unique solution to the set of linear equations(no solution or infinite solutions).Using these vector spaces and subspaces, the linear equations are calculated and is given in terms of row/column subspace. The four subspaces are characterized based on its dimension, dependence and basis. These give the relation between the subspaces and values in terms of rank.

The relation between subspaces is given by orthogonality. Every element in the row space is orthogonal to every element in null space. This means no term in column space exists in null space. Similarly with row space and left null space. Also the columns of the matrix is said to be column space and rows of the matrix is said to be row space, this mean C(A)=R(AT) (same with column space).

Important aspect in matrices is defining its inverse. For that we need to know about determinant of the matrix. Various properties of determinants are studied and with this eigen values and eigen vectors came to origin. Cramers rule is discussed for inverse of a given matrix. This results in using matrices to determine volumes and other scalar notations. The Eigen vectors and eigen values are determined so as to covert the given matrix in various forms. Singular value decomposition is used to represent given matrix in terms of eigen value matrix along with 2 orthogonal matrices (eigen vectors).

For electrical application purpose the network is converted in graph so as to represent in matrix form. This matrix is further reduced to identify node voltage, total voltage and current of the network. At the end of the course, as a part of project work I got to write a paper and I decided to write about the stability of the system using Eigen values and vectors and how to improve the stability of the system.



I always wonder how would be the world without wires. This thought has drawn me towards wireless communications. Though there are lots of wireless systems, I found keen interest with RF and microwave circuits. I know the basic communication structure is totally based on protocols and architecture model, but the real challenge is to know how to generate process and refine the RF signals so as to use them in real world experience. The evolution of RF based devices varies not only with size but also in performance, power compatibility and so on. To study deep down about the RF devices I opted RF and microwave circuits-1.

This course deals with analysis, construction, development and types of various blocks in basic RF communication block model. The main goal of this model is to convert high frequency RF signal to IF signal using a local oscillator. During this conversion various factors comes to existence that are related to mainly produce efficient output. Every device characteristics can be represented using many types of parameters. The most common is S parameters (4 parameters for 2 port device). These S parameters directly give return loss at both ports, reverse transmission and gain of the device. According to the function of device, specific ranges of S parameters are expected.

Devices such as Filters, couplers, diodes and mixers are studied. Various types of couplers, both 3 and 4 port couplers are studied. Advantages and disadvantages of each type of couplers are discussed. Thus according to the product requirement specific couplers are used. The most important components of the RF model are filters. When RF signal is sent through mixer so as to down convert it to IF, the receiving antenna is capable of capturing various RF frequencies that are surrounded that are close to required RF signal. So to eliminate such frequencies there is a necessity of using a filter. Several types of filters are studied such as lumped filters, transmission line filters, Coupled line filters, Stubbed filters, micro strip line filters and so on.

The other important device is RF diodes. A few type of diodes are studied and are derived on how they can be used as switch in RF frequencies. PIN diode which is one of the most common and basic RF switch. Schottky diode which is virtually formed over MOSFET devices (formed by junction of semi conductor and metal surface).Varactor diodes (variable capacitor variable diode) which are mostly used in tuning and are operated in reverse bias.

All the concept of impedance matching and filters are taken to level up by using ADS. This application enables to design, simulate and analyze any type of practical or ideal response of the system over any specific parameter (in this course mainly over frequency). 2 projects on impedance matching and micro strip coupled line filters are designed using ADS. The simulation part of the project is done and the layout of the design is fabricated ready to test.


In RF and microwave circuits, the course mostly deals with all test bench equipment theoretically. To obtain more practical knowledge on how the RF devices work in general workspace conditions, their behaviour over practical errors I opted for Wireless circuits and microwave systems laboratory. The entire course work is mirror to that of practical application of RF/MW circuits 1 course. In addition to that lot of lab equipment gear is required.

The basic element to describe the performance of the RF device it is important to obtain the S parameters. VNA- vector network analyzer, a device is used to measure S parameters for this course. Spectrum analyzer and Oscilloscope are used to obtain the signal response over frequency and time domain respectively. A signal generator is used in test bench so as generate the RF signal with our required Frequency and power level.

To connect each devices in the test bench various connectors are used. A coaxial cable which is generally used in connection with bigger equipment(VNA, SA, SG, DSO). Various connectors like SMA adaptor, SMA load, attenuator, Isolator, 3”to 4” semi rigid cable, F-M adaptor and many more. Even though these are just used to connect RF devices, as they are part of the designing model it is important to study about the behaviour of S parameters of these connectors and are used accordingly.

Various RF devices like Filters, couplers, Antennas are designed using ADS and then fabricated for testing. During testing the designed RF device board is connected by VNA to obtain the S parameters. In doing so we need to using coaxial cable which will change the original characteristics of the board. To compensate this error the device is calibrated with known terminations so as to change the measuring device reference point near the end of coaxial cable. For lumped element filter, the inductor and capacitor are soldered to the FR4 board. A distributed Filter for narrow bandwidth is designed using ADS which is constructed by microstrip lines, is then fabricated by milling machine.

The directional coupler is used Mixer for RF frequencies and its characteristics are measured. Various loss and measurement factors are derived such as conversion loss, Isolation between ports. Also, the important component of any communication device, a antenna is designed. A simple patch antenna is designed and is used a receiver so as to check my design efficiency over the known transmission by cushcraft antenna.

A cushcraft antenna is analyzed and its s parameters are studied so as to identify the use of cushcraft antenna in communication systems. In addition RF devices, different types of modulation techniques are studied. AM, FM, PSK, QAM,QPSK along with how to plot the interpolation points of these modulation techniques.

At the end of the course year, we then obtain two filters (lumped and distributed), a mixer(directional coupler) and a patch antenna. Using my own design of RF devices, i was able to construct my own RF receiver test bench model. With the known transmitted data, I now obtain the received data and check for efficiency over the entire test bench set up.


With addition to RF/MW circuits-1 and WAMI laboratory, I opted 3 more RF related courses because of my interest towards this field. I took RF measurements course-(FALL 2019), in which it deals with all type of practical adaptation of Amplifier characteristics like 1-dB compression point, Third Order Intercept point and Noise figure. These measurements gives the range of amplifier in which it will be used as a linear gain amplifier(1dB compression point), how far can the amplifier be pushed with considerate amount of non linearity(third order intercept point) and how much noise the amplifier can accept in order to use. In addition to the amplifier characteristics, I designed my own amplifier without any bias networks and lumped elements using MMIC of MOSFET and embedded microstrip FET. This is done by On-Wafer probing using 750?, 650? and 350? thickness probe tips by both SOLT and Thru calibration standards.

SPRING 2020- With all these interesting concepts of RF devices, to get to know more about theoretical side of RF devices I opted for RF/MW circuits-2 course. This course gave me an insight of one step closer in designing my own LNA and Power amplifier. The course is mainly experienced with smith chart. It covers from basic transmission line theory to all over transistor characteristics. I also opted RF Power amp design to enlarge my scale of knowledge in the filed of RF amplifiers. This course gave me the importance of various concepts of operating classes of transistor and its voltage dependency. Although this design is done over QORVO based HEMT device and all the design work is carried on ADS. With load pulling and transistor characteristics, input and output matching circuit are designed. These circuits are then converted to microstrip line layout design to be ready to fabricate.



The first sequence of my course work is completely dependent on Designing of RF devices. The interest of RF communication is mainly based on aspect “wireless”. Although for this fast moving world, the technology is in the hands of each individual which mostly wireless devices. The most common thing every individual expects is privacy on their own.

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Electronic systems in terms of algebraic equations. (2022, Oct 03). Retrieved May 30, 2024 , from

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