ELEC2134  Circuits and Signals
Study Level and UNSW Handbook
Undergraduate  ELEC2134
Requisite Courses
Co  ELEC1111
Contents
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 secondorder RLC circuits. It will also introduce twoport 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 continuoustime signals, and apply basic transform techniques to continuoustime 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
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) = V_{m * }sin ( omega * t + phi ) = V_{m} 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
 V_{ave} = ( 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
 P_{average }= R * ( I_{rms} )^{2 }= R * ( I_{dc} )^{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 V_{L} = omega * L * I_{m} cis ( 90 deg ) for inductor
 phasor representation of voltage is I_{C} = omega * C * V_{m} 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)
 admittance Y
 the reciprocal of admittance
 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_{0 }( j * omega ) / V_{i} ( j * omega ) = H ( j * omega )
 current gain = I_{0 }( j * omega ) / I_{i} ( j * omega ) = H ( j * omega )
 transfer impedance = V_{0 }( j * omega ) / I_{i} ( j * omega ) = H ( j * omega )
 transfer admittance = I_{0 }( j * omega ) / V_{i} ( 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
Please review lecture/textbook for examples
Resonance
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 nonDC circuit
for purely resistive load
for purely inductive load
for purely capacitive load
in general
Average power
average power is the average power value of a harmonic function circuit.
where power factor (pf) can be expressed as the following
we must state that current either leads or lags the voltage
dssssssssssssaadPower can also be represented with complex phasors
Maximum (Average) Power
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
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
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
Series Resonance
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Parallel Resonance
Admittance of a parallel resonance of source is geiven by sum of 3 admittances
We expect a resonance behaviour when
By definition, quality factor at parallel resonance is
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
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 Selfinductance 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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Transient Analysis
Twoport Networks
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_{1 }, 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:
 f ( t ) is singlevalued everywhere
 f ( t ) has a finite number of discontinuities in one period
 f ( t ) has a finite number of maxima and minima in one period
 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
Frequency Spectrum:
 plots of amplitudes and phases of harmonics versus frequencies
 amplitudes A_{n} = sqrt( a_{n}^{2} + b_{n}^{2} )
 Amplitude spectrum of a square wave
 phases phi_{n} =  tan^{1} ( b_{n} / a_{n} )
 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
 leads to nonperiodic functions
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
with complex fourier series, we can find the real fourier series
Amplitude and phase spectrum can be directly related
for frequency spectrum
 ampltude spectrum: A_{m}=
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: amplitudephase 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 nonperiodic 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
inverse fourier transform
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  $$f(tt_0)$$  $$ e^{j\omega t_0} F(\omega)$$ 
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) 
$$\delta(t)$$  $$1$$ 
$$1$$  $$2\pi\delta(\omega)$$ 
$$u(t)$$  $$\pi\delta(\omega)+\frac{1}{j\omega}$$ 
$$u(t+\tau)u(t\tau)$$  $$2\frac{\sin(\omega\tau)}{\omega}$$ 
$$t$$  $$\frac{2}{\omega^2}$$ 
$$sgn(t)$$  $$\frac{2}{j\omega}$$ 
$$e^{at}u(t)$$  $$\frac{1}{a+j\omega}$$ 
$$e^{at}u(t)$$  $$\frac{1}{aj\omega}$$ 
$$t^ne^{at}u(t)$$  $$\frac{1}{(a+j\omega)^n}$$ 
$$e^{at}$$  $$\frac{2a}{a^2+\omega^2}$$ 
$$\sin(\omega_0t)$$  $$j\pi[\delta(\omega+\omega_0)\delta(\omega\omega_0)]$$ 
$$\cos(\omega_0t)$$  $$\pi[\delta(\omega+\omega_0)+\delta(\omega\omega_0)]$$ 
$$e^{at}\sin(\omega_0t)u(t)$$  $$\frac{\omega_0}{(a+j\omega)^2+\omega_0^2}$$ 
$$e^{at}\cos(\omega_0t)u(t)$$  $$\frac{a+j\omega}{(a+j\omega)^2+\omega_0^2}$$ 
Energy / power
energy delivered to a 1 ohm resistor is shown below
For nonperiodic functions, the energy is spread across all values of omega in general
Convolution
Special functions
Dirac Delta function
Unit step function
u(t) can also be expressed as:
Property
Laplace Transform
Critical Frequencies (Poles and Zeroes)
Recommended Resources
Course Prescribed
Textbook
 “Fundamentals of Electric Circuits”, Alexander and Sadiku, McGrawHill.
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, PrenticeHall, 2000.
 J.R. Cogdell, Foundations of Electrical Engineering, 2nd Edition, Prentice Hall, 1990.
 J. Millman and A. Grabel, Microelectronics, McGrawHill, 1987.{{/mathjax}}