# Overview

This course is a continuation of ELEC1111, and covers more advanced topics in AC circuit analysis such as AC power, resonance, transformers, magnetically coupled coils, and transient response in second-order RLC circuits. It will also introduce two-port networks, state equations, network functions, and frequency response in terms of Fourier series and transforms, Laplace transforms, and filter design. The course aims to deepen understanding of AC and DC circuits and their analysis, introduce signal processing, develop skills in time and frequency domain analysis of continuous-time signals, and apply basic transform techniques to continuous-time signals. It will also provide opportunities to use computer simulation tools like LTSPICE  during lab sessions. The course covers a wide range of topics that build upon each other and will become the foundation, assumed knowledge, for most of the later electrical engineering courses, so it is important to put in effort to understand the concepts and engage in regular weekly study.

# Concepts

## Disclaimer

These concepts are studied through course content, mainly lecture notes and course recommended textbooks. Please note, unless referenced from another source, please assume the following information presented is interpreted, studied, and re-expressed from the course content. Furthermore, please refer to the course content directly, e.g. ask the lecturer or read the textbook, for the most relevant knowledge required for the course.

## Circuit Analysis

### AC Circuit Analysis

#### Fundamentals

AC basic properties

• Current in AC moves in both direction periodically
• AC is predominant as it can reduce energy loss in long distance transmission
• AC is also useful for motions and motors
• v(t) = Vm * sin ( omega * t + phi ) = Vm cos ( omega * t + phi )
• omega is angular frequency in radians, omega = 2 * pi * frequency
• phi is phase
• Vm is the amplitude
• x(t) = x(t+T) for any function or waveform where T is the period

Measuring of quantities

• peak value
• maximum value of a waveform
• peak to peak value
• the difference between upper peak and lower peak
• average value
• Vave = ( 1 / T ) * ( Area under curve over a full cycle ). <- (integral)
• rms value
• rms value is the measure of heating effect of a sine wave

RMS value

• Paverage = R * ( Irms )= R * ( Idc )2
\begin{equation} i_{rms} = \int_{lower}^{upper} i^2 (t) dt \end{equation}

Phasors

• phasor is a line drawn from origin. it's magnitude represents magnitude of a sinusoid and it's direction represents phase shift.
• phasors are vectors and can be added together.
• phase difference can be compared by finding difference of argument
• phasor representation of voltage is VL = omega * L * Im cis ( 90 deg ) for inductor
• phasor representation of voltage is IC = omega * C * Vm cis ( 90 deg ) for capacitor

Impedance

• for inductors.
• Z = j * omega * L
• short circuit at dc and open circuit at high frequencies
• for inductors.
• Z = ( j * omega * C )-1
• open circuit at dc, short circuit at high frequencies.
• for resistance
• Z = R
• imaginary part of impedance is reactance (like resistance)
• it can be treated as magnitude
• total impedance in an RC circuit is composed of resistance and reactance

Other terms

• conductance G
• the reciprocal of resistance
• unit in Siemens (S)
• unit in Siemens (S)
• susceptance B
• the imaginary part of admittance
• unit in Siemens (S)

## Circuit Theorems

### AC Network Functions (transfer functions)

A network function is a ratio of circuit variables.

Mostly the network function will be:  dependent variable (voltage or current) / independent variable (voltage or current)

• dependent variable (often an output)
• independent variable (often a source)
• expressed in the frequency domain

In each case, the variable is expressed as a function of jw. In this way, the frequency response (usually for omega between [0,infty]) can be determined. There are four network function of interest

• voltage gain = V( j * omega ) / Vi ( j * omega ) = H ( j * omega )
• current gain = I( j * omega ) / Ii ( j * omega ) = H ( j * omega )
• transfer impedance =  V( j * omega ) / Ii ( j * omega ) = H ( j * omega )
• transfer admittance = I( j * omega ) / Vi ( j * omega ) = H ( j * omega )

Network functions are complex function of the imaginary j * omega . Generally two real - valued function of j * omega are of interest :

• The magnitude response | H ( j * omega ) | = sqrt ( Re ( H ( j * omega ) )2 + Im ( H ( j * omega ) )2 ) = sqrt ( H ( j * omega ) * conj ( H ( j * omega ) ) )
• The phase response < H ( j * omega ) = tan-1 ( Im ( H ( j * omega ) ) / Re ( H ( j * omega ) ) )

collectively, they are known as frequency response. They describe the input - output behaviour of a system (circuit) across all frequencies

### AC power analysis

Reactive power is positive for inductance and negative for capacitance

• If load contains both inductive with reactive powers of equal magnitude, reactive powers cancel
• power flows back and forth to inductance and capacitance is called reactive power

instantaneous power p(t)

instantaneous power is the power at a point of time in a non-DC circuit

$$p(t) = v(t) \cdot i(t) = V_m \cdot I_m \cdot cos^2(\omega \cdot t)$$

$$p(t) = \frac { V_m \cdot I_m } { 2 } \cdot \sin (2\cdot \omega \cdot t)$$

$$p(t) = - \frac { V_m \cdot I_m } { 2 } \cdot \sin (2\cdot \omega \cdot t)$$

in general

\begin{align} p(t) &= \frac { V_m \cdot I_m } {2} \cdot \cos ( \theta ) ( 1 + cos ( 2 \cdot \omega \cdot t ) ) + \frac { V_m \cdot I_m } {2} \cdot \sin ( \theta ) \cdot \sin ( 2 \cdot \omega \cdot t ) \\ &= v(t) \cdot i(t) \end{align}

Average power

$$P = \frac {V_m \cdot I_m} { 2 } \cdot \cos (\theta) = C_{rms} \cdot I_{rms} \cos (\theta)$$

average power is the average power value of a harmonic function circuit.

where power factor (pf) can be expressed as the following

$$pf = \cos(\theta)$$ where $$\theta = \theta _v - \theta _i$$

we must state that current either leads or lags the voltage

dssssssssssssaadPower can also be represented with complex phasors

$$P = \frac { 1 } { 2 } \cdot Re[\mathbf{V} \cdot \mathbf{ \bar{I}}]$$

Maximum (Average) Power

$$P_{max}=\frac{|\mathbf{V}_{Th}|}{8\cdot R_{Th}}$$

the Load of the circuit to achieve maximum power is the conjugate of the Thevenin impedance.
for a purely real load, the maximum power is obtained when load resistance is the magnitude of the thevenin impedance.

Reactive power

$$Q = \frac {V_m \cdot I_m } { 2 } \cdot \sin(\theta)$$ $$Q = V _ {rms} \cdot I _ {rms} \cdot \sin(\theta)$$
for purely resistive load theta = 0 and so Q = 0

Apparent power

apparent power is given by the following equation

$$S = V_{rms} \cdot I_{rms}$$

apparent power is the product of the rms values of voltage and current

Complex power

$$\mathbf{S} = \frac {1}{2} \cdot \mathbf{V}\cdot\bar{\mathbf{I}}$$ $$\mathbf{S} = P + jQ$$
Complex power is the end of the power triangle.

with Complex power, we can find P Q and S

POWER TRIANGLE A power triangle tells you everything about the relation between the different types of power.
The triangle is right angled

### Resonance

Resonance is the tendency of a system to oscillate at maximum amplitude at a certain frequency

$$\omega _ 0 = \frac{1}{\sqrt{LC}}$$ $$f_0 = \frac{1}{2\cdot\sqrt{LC}}$$

Resonance is a condition in RLC circuits resulting in purely resistive impedance.

Bandwidth

Bandwidth (B) is the difference between the angular frequency that results in amplitude/sqrt(2)

Quality Factor

quality factor is the ratio factor of resonant frequency to it's bandwidth

$$Q=\frac{\omega_0}{B}$$

Series Resonance

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Parallel Resonance

Admittance of a parallel resonance of source is geiven by sum of 3 admittances

$$Y = G + j \left(\omega C - \frac{1}{\omega L} \right)$$

We expect a resonance behaviour when

By definition, quality factor at parallel resonance is

$$Q=R\omega _ 0 C = \frac {R}{\omega_0 L}$$

### Magnetically Coupled Circuits

Self inductance

• Current flowing in one coil establishes a magnetic flux about it's own coil
• Change in flux induces a voltage in the circuit, resulting a phase shift
• Voltage is proportional to time rate of change of the current

Mutual inductance

• Current flowing in one coil establishes a magnetic flux about a second coil closeby
• Change in flux results in second coil producing a voltage
• Voltage is proportional to time rate of change of the current through the first coil

Magnetic flux is equals to flux by itself and flux from the other circuit inductor

$$v_2(t) = M_{12}\frac{di_1(t)}{dt}$$ $$v_1(t) = L_1\frac{di_1(t)}{dt}$$

Dot rule

• When reference current direction is into the dotted terminal of one coil, the reference polarity of the voltage term induced in the other coil is positive at its dotted terminal.

Combined Mutual and Self-inductance Voltage

A circuit with magnetically coupled circuit are coupled with inductors. The circuit has voltage induced by it's own inductor and from the other inductor.
The equivalent inductance from the other inductor is represented by M.

Phasor Domain

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## Signals and Transforms

### Fourier Series

From ELEC1111,  phasors are used for analysing sinusoidal variables. However, it cannot be used to analyse variables that are not sinusoidal.
Considering periodic signal

• A function f(t) is periodic if for some real number T, f ( t ) = f ( t + T ) for all real t, where T is the period.
• equivalently, f ( t ) = f ( t + n * T ) for integer n.

Fourier's theorem

Given a real function f ( t ) , periodic with period T, there exist some real number a, a, b, a, b, ...  such that \begin{equation} f(t) = a_0 + \sum_{n=1}^{\infty} a_n \cos(n*\omega _0 * t) + b_n \sin ( n * \omega_0 * t ) \end{equation}

The Fourier series represents f ( t ) as a DC and AC component as infinite series of harmonic sinusoids.

If the Divichlet condition are satisfied, the Fourier series will converge

Divichlet conditions:

1. f ( t ) is single-valued everywhere
2. f ( t ) has a finite number of discontinuities in one period
3. f ( t ) has a finite number of maxima and minima in one period
4. area under a period in the graph < infinity

Finding the fourier series representation
\begin{equation}\int^T_0 \sin (n \cdot \omega _0 \cdot t) dt = 0\end{equation} \begin{equation}\int^T_0 \cos (n \cdot \omega _0 \cdot t )dt = 0\end{equation} \begin{equation}\int^T_0 \sin (n\cdot\omega _0 \cdot t) \cdot \cos (m\cdot\omega _0 \cdot t) dt = 0\end{equation} \begin{equation}\int^T_0 \sin (n\cdot\omega _0 \cdot t) \cdot \sin (m\cdot\omega _0 \cdot t) dt = 0\end{equation} \begin{equation}\int^T_0 \cos (n*\omega _0 \cdot t) \cdot \cos (n*\omega _0 \cdot t) dt = 0\end{equation}

Using these equation and the Fourier's Theorem, we can derive a formula for terms in the Fourier Series

$$a_0 = \frac{1}{T} \int^T_0 f(t) dt$$
\begin{aligned} a_n &= \frac{2}{T} \int_0^T f(t) \cos (n \cdot \omega_0 \cdot t )dt\\ b_n &= \frac{2}{T} \int_0^T f(t) \sin (n \cdot \omega_0 \cdot t )dt \end{aligned}

Frequency Spectrum:

• plots of amplitudes and phases of harmonics versus frequencies
• amplitudes An = sqrt( an2 + bn2 )
• Amplitude spectrum of a square wave
• phases phin = - tan-1 ( bn / an )
• Phase spectrum of a square wave

Symmetry

• for even functions, f(-t) = f(t)
• for odd functions, f(-t) = -f(t)
• Even case
• $$a_n = \frac{4}{T} \int^{T/2}_{0}f(t)\cdot\cos(n\cdot \omega_0 \cdot t )dt$$
• $$b_n = 0$$

complex fourier series

alternative way to represent Fourier series is using a complex exponential

• compact representation
$$\cos \theta = \frac{e^{j\theta}+e^{-j\theta}}{2}$$ $$\sin \theta = \frac{-j\cdot e^{j\theta}+j\cdot e^{-j\theta}}{2}$$

hence
\begin{align} f(t) &= a_0 + \sum_{n=1}^{\infty}\frac{a_n - j\cdot b_n}{2} \cdot e^{j\cdot n\cdot \omega_0 \cdot t} + \sum_{n=1}^{\infty}\frac{a_n + j\cdot b_n}{2} \cdot e^{-j\cdot n\cdot \omega_0 \cdot t} \\ &= \sum^{\infty}_{n=-\infty} c_n \cdot e^{-j\cdot n\cdot \omega_0 \cdot t} \end{align}

note the following

$$c_0 = a_0$$ $$c_n = \frac{a_n-j\cdot b_n}{2}$$

with complex fourier series, we can find the real fourier series

$$C_n = \frac{1}{T} \int^T_0 f(t) e^{-j\cdot n \cdot \omega_0 \cdot t} dt$$

Amplitude and phase spectrum can be directly related

$$C_n = \frac{1}{T} \int^T_0 f(t) e^{-j\cdot n \cdot \omega_0 \cdot t} dt$$

for frequency spectrum

• ampltude spectrum: Am=

Fourier seriers and average power

• For sinusoidal voltages/current
• $$P_{av} = \frac{1}{2} \cdot V_m I_m \cos (\theta_v - \theta_i)$$
•  For periodic voltages/currents: amplitude-phase form
• $$v(t)=V_dc+\sum_{n=1}^{\infty}V_n\cdot cos(n\omega _0 t + \theta _n)$$
• $$i(t)=I_dc+\sum_{n=1}^{\infty}I_n\cdot cos(n\omega _0 t + \phi _n)$$
• then the average power in general can be expressed as following
• $$P_{av} = V_{dc}I_{dc}+\frac{1}{2}\cdot \sum ^{\infty}_{n=1} V_nI_n \cos(\theta_n-\phi_n)$$
• RMS value for a function:
• $$F_{rms} = \sqrt{\frac{1}{T} \int^T_0 f^2(t)dt}$$
• can show that thee power dissipated by a 1 ohm resistor is

•

### Fourier Transform

Previously we used Fourier series for periodic functions.
For non-periodic case, we develop this by considering what happens as T approaches infinity

Fourier transform is an integral transformation of f(t) from time domain to frequency domain

$$\mathcal{F}[(f(t)]=\int_{-\infty}^{\infty}f(t)e^{-j\omega t}dt$$

inverse fourier transform

$$\mathcal{F}^{-1}[(F(\omega)]=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)e^{j\omega t}dt$$

Fourier Transform Properties

 Property f(t) F(w) Linearity $$a_1f_1(t)+a_2f_2(t)$$ $$a_1F_1(\omega) + a_2F_2(\omega)$$ Time Scaling $$f(at)$$ $$\frac{1}{|a|}F(\frac{\omega}{a})$$ Time Shifting $$f(t-t_0)$$ $$e^{-j\omega t_0} F(\omega)$$ Frequency Shifting $$f(t)e^{j\omega_0 t}$$ $$F(\omega - \omega_0)$$ Time Differentiation $$f^{n}(t)$$ $$(j\omega)^{n} F(\omega)$$ Time Integration $$\int_{-\infty}^{t}f(\tau)d\tau$$ $$\frac{F(\omega)}{j\omega} + \pi F(0)\delta (\omega)$$ Frequency differentiation $$t^nf(t)$$ $$(j)^n F^n(\omega)$$ Reversal $$f(-t)$$ $$F(-\omega)~~\text{or}~~F*(\omega)$$ Duality $$F(t)$$ $$2\pi f(-\omega)$$ Modulation $$\cos(\omega_0t)f(t)$$ $$\frac{1}{2}\left[F(\omega+\omega_0)+F(\omega-\omega_0)\right]$$ Convolution in t $$f_1(t)*f_2(t)$$ $$F_1(\omega)F_2(\omega)$$ Convolution in w $$f_1(t)\cdot f_2(t)$$ $$F_1(\omega)*F_2(\omega)$$

Fourier Transform Pairs

 f(t) F(w) $$\delta(t)$$ $$1$$ $$1$$ $$2\pi\delta(\omega)$$ $$u(t)$$ $$\pi\delta(\omega)+\frac{1}{j\omega}$$ $$u(t+\tau)-u(t-\tau)$$ $$2\frac{\sin(\omega\tau)}{\omega}$$ $$|t|$$ $$-\frac{2}{\omega^2}$$ $$sgn(t)$$ $$\frac{2}{j\omega}$$ $$e^{-at}u(t)$$ $$\frac{1}{a+j\omega}$$ $$e^{at}u(-t)$$ $$\frac{1}{a-j\omega}$$ $$t^ne^{-at}u(t)$$ $$\frac{1}{(a+j\omega)^n}$$ $$e^{-a|t|}$$ $$\frac{2a}{a^2+\omega^2}$$ $$\sin(\omega_0t)$$ $$j\pi[\delta(\omega+\omega_0)-\delta(\omega-\omega_0)]$$ $$\cos(\omega_0t)$$ $$\pi[\delta(\omega+\omega_0)+\delta(\omega-\omega_0)]$$ $$e^{-at}\sin(\omega_0t)u(t)$$ $$\frac{\omega_0}{(a+j\omega)^2+\omega_0^2}$$ $$e^{-at}\cos(\omega_0t)u(t)$$ $$\frac{a+j\omega}{(a+j\omega)^2+\omega_0^2}$$

Energy / power

energy delivered to a 1 ohm resistor is shown below

\begin{align} w_{1 \Omega}&=\int^{\infty}_{-\infty}f^2(t)dt \\ &= \frac{1}{2\pi}\int^{\infty}_{-\infty}F(\omega)F(-\omega)d\omega \\ &= \frac{1}{2\pi}\int^{\infty}_{-\infty}|F(\omega)|^2 d\omega \\ \end{align}

For non-periodic functions, the energy is spread across all values of omega in general

Convolution

$$f_1(t)*f_2(t)=\int_{-\infty}^{\infty}f_1(\tau)f_2(t-\tau)d\tau$$

### Special functions

##### Unit step function
$$u(t) = \begin{cases} 0, & t < 0 \\ 1, & t \leq 1 \end{cases}$$

u(t) can also be expressed as:

$$u(t)=\int_{-\infty}^{t}\delta(\tau)d\tau$$

Property

$$\mathcal{F}\{u(t)\}=\frac{1}{j\omega}+\pi\delta(\omega)$$

# Recommended Resources

## Course Prescribed

### Textbook

• “Fundamentals of Electric Circuits”, Alexander and Sadiku, McGraw-Hill.

## Course Recommended

### Textbook

• L.S. Bobrow, “Elementary Linear Circuit Analysis”, Oxford, 1987.
• J. Svoboda, & R. Dorf, “Introduction to Electric Circuits”, 9th edition, Wiley & sons, 2014.
• A. Hambley, “Electrical Engineering Principles and Applications”, Prentice Hall, 2002.
• S. Franco, “Electric Circuits Fundamentals”, Saunders College Publishing, 1995.
• R.L. Boylestad, Introductory Circuit Analysis, 9th Edition, Prentice-Hall, 2000.
• J.R. Cogdell, Foundations of Electrical Engineering, 2nd Edition, Prentice Hall, 1990.
• J. Millman and A. Grabel, Microelectronics, McGraw-Hill, 1987.{{/mathjax}}