# Projects:2018s1-191 Quasi-Linear Circuit Theory

## Project Team

Students

• Yuan Yao a1674092
• Zizheng Ren a1680576
• Bruk Waldron a1645521

Supervisors

• Dr. Andrew Allison
• Prof. Derek Abbott

## Introduction

As electrical power networks become more complicated, it becomes more important to analyse and control power quality. Harmonic is one of the most important aspect of power quality. Harmonics are defined as sinusoidal voltages or currents having frequencies that are integer multiples of the fundamental frequency in electric power system. Too much harmonic contents in power network will cause various problems such as excessive losses and heating in induction and synchronous machine, incorrect readings on meters and interference with other electronic devices and systems and so on. Some serious problems will lead to malfunction and even collapse of entire power system. Therefore, it is important for engineers to analyse and control harmonics for better power quality and normal operation of power system. One significant source of harmonics is the non-linear load. As non-linear loads create harmonics by drawing current in abrupt short pulses, rather than in a smooth sinusoidal manner. However, there is no suitable and sufficient circuit theory to model harmonics and their effect at the moment. As a result, the lack of theory has attracted our interest and inspired us to investigate this issue. We believe that the periodic analysis is one possible solution.

The aim of this project is to construct a complete and consistent circuit analysing theory to model non-linearity and effect of harmonics in electric circuits. To achieve this, the detailed periodic analysing method should be specified. A good notation to model circuit elements need to be determined. Simulation will be carried out using MATLAB and Simulink in terms of investigating the circuit response with non-linear load and non-sinusoidal signal. Finally, a complete theory will be proposed as described in project aim.

## Background

Non-Linearity: Non-linear devices are devices where voltage and current are not linear i.e. where resistance of device is not a constant. Examples are diodes i.e. has voltage vs current which can modelled using diode Shockley equation πΌπ=πΌπ π£ππππ‘ [5]i.e is an exponential relation. Linear devices have voltage and current relationships that are linear i.e. a resistor π=πΌπ or capacitor π=1πββ«π(π‘)ππ‘ and as integration is just a summation of area and area is a linear function, hence capacitor and inductors are linear devices. As mentioned in the introduction, the motivation behind this project is power quality and the detrimental effects harmonics have on it. To better understand how exactly harmonics affect power quality and hence to understand the motivation behind this project i.e. why it would be advantage to be able to predict and hence control harmonics in power system. The concept of power quality and how and why harmonics affect power quality needs to be understood. In this section a brief discussion of the above will be looked at. Power quality is defined as the βquality of voltage supplied to loadsβ[1]. This is due to loads determining what the current waveform will be and hence the uncontrable nature of current in terms of the supplier. Hence power supplier cannot control current, but can control voltage. Therefore ideal power quality is defined by the following [1]; ο· sinusoidal waveform (i.e. minimal harmonics) i.e. a THD<5%, ο· rated voltage i.e. 1pu at ο· rated frequency i.e. 1pu ο· Constantly available <115minutes lost per year. Harmonics in power systems are defined as integer multiples of the supply frequency[1]. Sources of harmonics in power systems are from generation i.e. 10 power plants, transmission networks i.e. cables and transmission equipment such as transformers&reactors and the largest source of harmonics non-linear loads such as power electronics[2].

## Theory

Proposed theory uses the complex Fourier series as a basis for performing circuit analysis. Proposition is to use the complex Fourier series as a type of domain i.e. similar to using the frequency or Laplace domain to performing circuit analysis. Hence it is an alternative to using traditional methods such as phasors. Both the frequency and Laplace domain, while allowing known non-sinusoidal functions such as square waves to be mapped into their respective domains, do not allow the current, voltage and other quantities arising from the interaction between harmonics in input signal and linear/non-linear loads. Only interactions occurring at the fundamental frequency of input signal is analysed i.e. if the square wave was at 50Hz, the subsequent interactions at this frequency are the only interactions analysed.Hence a more viable technique is required.