# 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.

## 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.

## Simulations

Testing of theory on linear circuit involved using MATLAB. MATLAB code for converting signals into complex fourier coefficients i.e. CFA was provided by Dr Allision, All sources are converted to complex vectors and circuit elements are represented using square matrices.The time domain of the RC circuit is seen ins seen in figure1a and the transformation of sources to complex Fourier coefficients and circuit elements to square matrices is seen in figure1b.As resistor and capacitor are both linear devices, their respective matrices (figure1b) only contain elements in the diagonal andall off-axis elements are 0.

Once all the sources and circuit elements have been transformed, normal circuit analysis can take place. To illustrate this, the voltage across the capacitor was found and its output can be seen in figure1d. Vc was solved using solved using voltage divider i.e. Vc=(Vs*Zc)/(Zc+Zr) , but as matrices are not divisible the inverse operation is used instead,

Hence have [vc]=([vs]*[zc])*[Zc+Zr]^(-1) i.e.and using dirac notation Have |Vc>=(|Vc>*[zc])*inverse[Zc+Zr] , Hence |Vc> can be converted from complex Fourier series domain using IDTFT for DS (again provided by Dr. Allison).

Figure1a)Time Domain of RC Circuit Figure1b)QLCT of RC Circuit Figure1d)LTSPICE Simulation of RC Circuit Figure1d, a comparison of output from LTSPICE simulation vs output using proposed theory, confirms the validity of using complex Fourier series as a basis in circuit analysis of linear circuit.

Estimation of transfer matrix H i.e. linear operator of diode, using least squares estimate by testing diode with a large 50Hz signal and large 50Hz signal with small amounts different harmonics added to it. Output of diode from these tests was taken and H was estimated by multiplying Output of diode with generilised inverse of input. Figure 2a shows the array of input signals each with different harmonics injected into them. The component can be probed with a large number of linearly independent periodic signals, and the responses can be measured. It is then possible to use the generalised inverse of Moore and Penrose to obtain the best least-squares estimate of the required operator.This H matrix can be seen in figure2b.

Figure1c)Input Signals with different harmonics injected in; Figure1c)H matrix(Dr. Allison) H matrix was tested using samples from a signal generator and compared. The resulting outputs and comparisons are seen in figure2c.Figure2c shows that determining H from a simulation and using that on real world data does not work. The RMS error between Vdiode from experiment and Vdiode using H linear operator is huge and the resulting current Idiode also has large errors. However the actual graphs despite the large errors do show promise as the resulting waveforms do look close to what was expected. Therefore in order to determine linear operator H, need to do actual experiment using actual device.