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

## Contents

**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**

**Method**

Proposed theory uses the complex Fourier series as a basis for performing circuit analysis. Propitiation 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.

Proposed Quasi Linear Circuit Theory;

a)Circuit sources are represented as vectors of complex numbers i.e. Vs(t) --> |Vs>

b)Circuit elements are represented as square matrices

i. Linear Circuit elements have matrices, where diagonals are the non-zero elements i.e. R --> [R](figure1a) , 1/(jwc) --> [(1/jwL)](figure1b) ect.

Figure1a)Resistor Matrix Figure1a)Capacitor Matrix

ii. Non-Linear circuit elements have non-zero off-axis’s elements, but the diagonal elements are still dominant i.e. largest

c)Given a non-linear device, a linear operator H i.e. a transfer matrix [H] can determined, which represents the least squares estimate of the non-linear process of the device.The transfer matrix [H] of a diode is shown in figure1c.

Figure1c)Transfer Matrix H for Diode

c)The following outlines the QCLT methodology given a time domain circuit)

c) 1)Al the circuit sources are converted to the complex Fourier domain.This is done by computing the signals complex Fourier series. The result of this transformation is a vector containing complex Fourier coefficients which using Dirac’s notation is l |V1⟩, hence V1(t) --> |V1⟩.

c) 2)All the circuit elements are modeled as square matrices. If circuit element has a defined mathematical model and is linear i.e. V(L)=Ldi/dt then resulting matrix is a square diagonal matrix with all off-axis elements being 0 as seen in figure1a & figure1b.If circuit element is non-linear , the circuit element's matrix can be determined using least squares estimation.This is done by testing the circuit element with a large number of linearly independent periodic signals to which the circuit elements responses is measured and stored. The transfer matrix can then be found using the generalised inverse of Moore and Penrose to obtain the least-squares estimate of the non-linear device.

Time domain signals are transformed into vectors of complex numbers using Complex Fourier Analysis.