Table of Contents
ELEN20005 - Foundations of Electrical Networks
What is Electrical Engineering?
Answer #1
The Purposeful use of Maxwell's equations for electromagnetic phenomena.
Maxwell's laws
- The electric flux leaving a volume is proportional to the charge inside
- The total magnetic flux through a closed surface is zero
- The voltage induced in a closed loop is proportional to the rate of change of the magnetic flux that the loop encloses
- The magnetic field induced around a closed loop is proportional to the electric current plus displacement current that the loop encloses
Answer #2
To gather and distribute energy and to gather, process, communicate and present information.
Areas of Electrical Engineering
- Power systems
- Computer systems
- Communication systems
- Electronic devices
- Signal processing
- Control systems
- Photonics
Definitions
Electric Circuit
Electrical devices connected together so that electrical energy can be transferred between devices. Sources provide energy, sinks consume energy.
Electrical Charge
Property of atomic particles measured in Coulombs. Electron charge is \(-1.6*10^{-19}\) C. Symbol of “\(q\)”.
Electrical Current
Time rate of flow of charge through a circuit element. Measured in Amperes (A) (Coulombs per Second). Symbol of \(i\) or \(I\). Current is measured to an arbitrary reference direction. For historical reasons, we talk about current as the flow of positive charge, despite it being the negative charges that move.
Sign conventions
Passive Sign Convention (PSC) is when the current enters the positive terminal. \[p(t)=\frac{dw}{dt}=\frac{dw}{dq}\frac{dq}{dt}=v(t)i(t)\] Active Sign Convention (ASC) is when the current enters the negative terminal. \[p(t)=-v(t)i(t)\]
Sources and Sinks
Positive power \(\implies\) absorbs energy (sink) Negative power \(\implies\) generates energy (source)
Independent voltage source
Voltage across terminals is independent of the current through the device.
Dependent voltage source
Voltage across terminals is controlled by another device elsewhere in the circuit. Voltage is maintained regardless of current through device. \[v(t)=Kx\] \(K\) is the voltage gain of the source. If \(x\) is a voltage we have a voltage controlled voltage source. If \(x\) is a current we have a current controlled current source.
Independent current source
A device for which the current across the terminals is constant independent of the voltage across the device.
Dependent current source
A device for which the current is controlled by another circuit element elsewhere in the circuit. \(I(t)=Kx\) \(K\) is the current gain of the source. If \(x\) is a current we have a current controlled current source. If \(x\) is a voltage source we have a voltage controlled current source. \(K\) would be measured in Siemens.
Resistors and resistance
An ideal resistor is a device for which the instantaneous voltage drop across its terminals is proportional to the instantaneous current through the device. Using PSC, voltage and current are related by \(v=Ri\). Resistance, \(R\), is a positive constant measured in Ohms. \(1\Omega=1V/A\) Conductance is the inverse of resistance: \(G=\frac{1}{R}\) in units of Siemens (S).
Making measurements
In the labs we have DC power supplies. The supplies have a power and output button, can be set to constant voltage. The multi-meter can measure voltage, current and resistance. Voltmeters need to be in parallel. Ammeters need to be in series. Ohmmeters need to be in parallel and disconnected from other voltage sources.
Basic laws of circuits
Definitions
Nodes
A point in a circuit where three or more elements are joined together.
Branch
A series of connections of elements between two nodes.
Loops
A closed path through a subset of nodes in a circuit.
Short circuit
A circuit with no voltage across it. An ideal conductor between two nodes. Unknown current passing through, determined by the rest of the circuit.
Open circuit
A circuit with infinite resistance between two nodes of a circuit. Has no current through it. Has no current through it.
Reference node and node voltages
Used to express the potential differences between a node and the reference node. Usually drawn at the bottom of a circuit, however it can be placed anywhere, only changing the values by a constant amount relative to each other.
Kirchhoff's laws
Current law
The algebraic sum of all currents entering a node (or closed surface) in a circuit is zero. \[\sum{i_n}=0\]
Voltage law
The algebraic sum of all the voltages in a closed path is zero. \[\sum{V_m}=0\] Loop direction doesn't matter, convention is to go clockwise.
Resistors
Resistors connected in series have an equivalent resistance of the sum of their resistances. \[\sum{R_n}=R_{eq}\] Resistors connected in parallel have an equivalent resistance equal to the harmonic mean of their resistances. \[\sum{1/R_n}=1/R_{eq}\] Voltage dropped across resistors in series is the sum of the voltage dropped across the resistors individually (KVL). From Ohm's law: \(v_1=\frac{R_1 v_{ab}}{R_1+R_2+R_3}\) From KCL, \(i=i_1+i_2\). Ohm's law yields \(i_1=\frac{iG_1}{\sum{G_n}}\), where \(G_n=\frac{1}{R_n}\). For two resistors, \(i_1=\frac{R_2i}{R_1+R_2}\)
Analysis
Systematic methods for Circuit Analysis
There are two methods called:
- Node Voltage Analysis (NVA)
- Node Current Analysis (NCA)
Both methods obtain simultaneous equations for currents and voltages and use algebraic methods to solve them.
Node Voltage Analysis
Algorithm:
- Select one node to be the reference node
- Assign node voltages to all other nodes
- Assign current to each branch containing a voltage source
- At each node write a KCL equation
- For each branch containing a current source, apply KVL to a loop containing the branch
- Solve the remaining system of equations
Mesh Current Analysis
A mesh is a loop that does not contain any other loops. A mesh current is a current that loops around a mesh. Algorithm
- Introduce mesh currents in each mesh
- For each mesh involving only voltage sources and resistors, construct a KVL equation
Linear, Affine and Piece-wise Linear Graphs
Variables x and y have a linear relationship if the graph of their function is a straight line passing through the origin. \[y=mx\] They are affine if the graph of the function is a straight line. \[y=mx+c|c\neq0\] They have a piece-wise linear if their relationship is composed of several straight lines.
Linear circuits
A linear circuit device is one that has a linear relationship between current or voltage or their time derivatives. A linear circuit is one that contains resistors, independent sources and linear dependent sources. Diodes are examples of non-linear devices.
Superposition
Mathematical definition: Given any matrix \(A\), let \(\vec{x_1},\vec{x_2},\vec{b_1},\vec{b_2}\) such that \begin{align} A\vec{x_1}&=\vec{b_1}\\ A\vec{x_2}&=\vec{b_2}\\ A(\vec{x_1}+\vec{x_2})&=\vec{b_1}+\vec{b_2} \end{align} The principle of superposition allows the addition of the lines together.
Equivalent Circuit
A “black box” that can be connected to other electrical devices, called a “load circuit”. If we were only interested in understanding the black box from what is observed in the load circuit. It would be simpler to replace the black box with an “equivalent circuit” that would have the same behaviour at the terminals. The simpler circuit would make it easier to predict currents and voltages.
Definition
Two (two-terminal) networks are network equivalent if the same currents and voltages are measured at their network terminals when the same external test network is connected to these terminals.
Thévenin Equivalent Circuit Theorem
Any two terminal device containing independent sources, linear dependent sources and resistors can be replaced by an equivalent circuit containing a voltage source \(v_T\) or \(v_{OC}\) in series in series with a resistor \(R_T\). The Thévenin voltage is the open circuit voltage across the terminals of the device. Let \(i_{SC}\) be the current through the terminals if they are joined by a short circuit. Then Thévenin resistance \(R_T\) is given by \[R_T=\frac{v_T}{i_{SC}}\] The voltage-current equation for the output terminals of a Thévenin circuit is \[\frac{i}{i_{SC}}+\frac{v}{v_{OC}}=1\]
Norton Equivalent Circuit
Any two terminal device containing independent sources, linear dependent sources and resistors can be replaced by an equivalent circuit containing a current source \(i_N\) in parallel with a resistor \(R_N\). The Norton current \(i_N\) is defined as \(i_N=i_{SC}\). \(R_N=R_T\), the same as for the Thévenin equivalent circuit. The voltage-current equation is \[\frac{i}{i_{SC}}+\frac{v}{v_{OC}}=1\]
Thévenin circuit with no independent sources
A circuit of just resistors has \(v_{OC}=0\) and \(i_{SC}=0\). We can apply a test voltage of \(i_{test}=1A\) to the output terminals. We can use NVA to find \(v_{test}\) then solve \(R_T\) from \(v_{test}=-R_T i_{test}\).
Maximum Power Transfer
For a two-terminal circuit, what load should be connected to the outputs so that the maximum power will be delivered to the load? The voltage, current and power delivered to the load areas \[v_L=\frac{R_LV_t}{R_T+R_L}, i_L=\frac{V_t}{R_T+R_L}, P_L=\frac{R_LV_t^2}{(R_T+R_L)^2}\] To find the optimal load resistance, set the derivative of power with respect to load equal to zero. \[\frac{dP_L}{dR_L}=\frac{V_T^2(R_T^2-R_L^2)}{(R_T+R_L)^4}=0\] From this, we can tell that the power transfer is at a maximum when \(R_L=R_T\), causing \(P_{max}=\frac{V_t^2}{4R_T}\).
Capacitors
Definition
A parallel-plate capacitor consists of two conductive plates separated by an insulating layer. The insulating layer is called a dielectric, and affects the charge that can be stored on the plates. The electric plates have a large supply of free electrons. The dielectric does not permit charge to flow through it. When a voltage is applied, electrons move from one plate to the other, creating a deficit of electrons known as holes on the positive plate and a surplus on the negative side. The charge on the positive plate is equal and opposite to the charge on the negative plate, thus the total charge is zero. The electric field outside the capacitor is zero despite the internal electric field. This is for Lumped Circuit Abstraction #2: “There is no magnetic coupling between devices in the circuit”.
Equations
The amount of charge stored is directly proportional to the applied voltage, so that \[q(t)=Cv(t)\] Where C is a constant called the capacitance measured in farads and \(v(t)\) is the voltage across the capacitor at time t. Under PSC, capacitors are represented as having positive current flowing from positive to negative through the capacitor. The current is given by \[i(t)=\frac{dq}{dt}=C\frac{dv}{dt}\] Hence there is a linear relationship between the current and time derivative of voltage, so we say capacitors are linear circuit devices.
Charge at time
If charging begins at time \(t_0\), the charge (\(q(t)\)) on the plates at time \(t\geq t_0\) is \[q(t)=\int_{t_0}^t i(s)ds + q_0(t)\]
Voltage at time
The voltage across the plates is given by \[v(t)=\frac{1}{C}\int_{t_0}^{t_1}i(s)ds+v(t_0)\] Thus the voltage depends on the history of the current and voltage, meaning it has memory.
Power and energy at time
The power delivered by a capacitor is \[p(t)=v(t)i(t)=Cv(t)\frac{dv}{dt}\] The energy stored between initial time \(t_0\) and subsequent time \(t_1\) is \[w(t)=\int_{t_0}^{t_1}p(t)dt=\frac{C}{2}v^2(t_1)-\frac{C}{2}v^2(t_0)\] If \(t_0\) is when charging began, i.e. \(t_0=0\) \[w(t)=\frac{C}{2}v^2(t)\] Hence a capacitor is an open circuit to direct current.
Capacitors in series
\[C_{eq}=\frac{1}{\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}}\] Capacitors in series act like resistors in parallel.
Capacitors in parallel
\[C_{eq}=C_1+C_2+C_3\] Capacitors in parallel act like resistors in series.
Continuity of functions
For a capacitor, the voltage-time function must be continuous. The current-time function can be discontinuous. The power-time function is positive while the capacitor is being charged and negative while it is discharging. If given information about the current instead of the voltage, the voltage can be found by \[v(t)=\frac{1}{C}\int_{t_0}^{t}i(s)ds+v(t_0)\] and then the power and energy functions can be found.
Capacitor ratings
Have ratings like resistors. A capacitor could have a rating like \(104V\), which would make its capacitance \(100nF\pm5\%\). Ratings follow \({B_1B_2B_3}L\) where each \(B\) is a number and \(L\) corresponds to a letter, J (5%), K (10%), M (20%). The capacitor value is as follows \[Value = B_1B_2*10^{B_3}pF\] Electrolytic capacitors have a polarity, the longer leg must be positive.
Real capacitors
Real capacitors have parasitic effects in addition to their capacitance. Have a series resistance (\(R_S\)) caused by the resistivity of the material composing the plates. Have a series inductance (\(L_S\)) due to the magnetic field created by the current flowing through the capacitor. Have a parallel resistance (\(R_P\)) from the current through the dielectric material between the plates.
Switches
Switches are used to control the flow of current. Serve to isolate and connect.
Definitions
Open switches are when the conducting metal parts do not make contact and current does not flow. Creates an open circuit. Closed switches are when the conducting metal parts make contact and current flows. Creates a closed circuit. The effect of closing or opening is instantaneous. When a switch is opened or closed, the effect causes time-varying changes in the currents and voltages. These are called circuit transients.
Inductor
Definitions
Inductors consist of conducting wire in a coil. Time-varying current flowing through a wire creates a magnetic field (flux) in which electromagnetic energy can be stored. By Faraday's Law of Electromagnetic Induction, the time-varying magnetic field induces a voltage across the coil.
Equations
For an ideal inductor, the voltage across the ends of the coil is directly proportional to the rate of change of current. \[v(t)=L\frac{di}{dt}\] Where L is a constant called the Inductance measured in Henrys, \(1H=1Vs/A\). They are linear circuit devices.
Current at time
The current at a time is \[i(t)=\frac{1}{L}\int_{t_0}^{t}v(s)ds+i(t_0)\] The current depends on the initial current, so they have memory.
Power and energy at time
Using PSC, they power is \[p(t)=v(t)i(t)=Li(t)\frac{di}{dt}\] The energy stored between two times is the integral of power \[w(t)=\int_{t_0}^{t_1}p(t)dt=\frac{L}{2}i^2(t_1)-\frac{L}{2}i^2(t_0)\] Hence the energy stored on an inductor at any time is \[w(t)=\frac{L}{2}i^2(t)\] The current and energy are continuous, the voltage and power may be discontinuous. If the current through an inductor is constant, then the voltage across it is zero. Hence an inductor acts like a short circuit to current.
Inductors in series
\[l_{eq}=\sum L_n\] Inductors in series act like resistors in parallel.
Inductors in parallel
\[L_{eq}=\frac{1}{\sum{\frac{1}{L_n}}}\] Inductors in parallel act like resistors in parallel.
Real Inductors
Use the same colour codes as resistors. Real inductors have parasitic effects. Have a series resistance (\(R_S\)) caused by the resistivity of the wire. Have a parallel inter-winding capacitance (\(C_p\)) is associated with the electric field between the coils of the wire. Have a parallel resistance (\(R_p\)) from eddy currents in the metal core around which the wire is coiled.
Solving circuits with inductors and resistors
A equation of the form \[L\frac{di}{dt}+Ri(t)=v_s\] has solutions for \(i(t)\) in the form of \[i(t)=K_1e^{st}+K_2\] Where \(K_1, K_2, s\) are constants. Substituting into the original equation gives \[(R+sL)K_1e^{st}+RK_2=v_S\] And equating coefficients gives \[s=\frac{-R}{L},K_2=\frac{v_s}{R}\] Making \(i(t)\) \[i(t)=K_1e^{\frac{-R}{L}t}+v_s\] \(K_1\) can be solved using the initial condition such that \[i(0)=K_1+\frac{v_s}{R}\]
This causes the inductor to act similar to a capacitor with respect to current. The steady state solution is what happens to \(i(t)\) as \(t\to\infty\). \(\tau=\frac{L}{R}\) is called the time constant.
Sinusoidal Signals
Definitions
Have the form \[v(t)=V_m\cos(\omega t + \theta)\] Where \(V_m\) is the amplitude. \(\omega\) is the angular frequency in radians/second. \(\theta\) is the phase angle in radians. The period is \(T=\frac{2\pi}{\omega}\). The phase angle between two waves is \(\theta=\frac{2\pi \Delta t}{T}\) where \(\Delta t\) is the time difference between corresponding peaks of the sinusoids. Convention is to set \(\theta\) with the smaller time difference.
Root Mean Square
\[V_{rms}=\sqrt{\frac{1}{T}\int_0^Tv^2(t)dt}\] For a sinusoidal voltage, this reduces to \(V_{rms}=\frac{V_m}{\sqrt{2}}\).
Complex numbers
For any complex number, \(z=j\theta\), \(e^{j\theta}=\cos\theta+\sin\theta\), and so \(\cos\theta=Re(e^{j\theta})\), where \(Re\) denotes the real part of. For any sinusoidal function \(v(t)=V_m\cos(\omega t+\theta)\), \begin{align*} V_m\cos(\omega t+\theta)&=Re(V_me^{j(\omega t+\theta)}) \\ &=Re(V_me^{j\omega t}e^{j\theta}) \\ &=Re(\mathbf{V}e^{j\omega t}) \end{align*} Where \(\mathbf{V}=V_me^{j\theta}\), for simplicity we write \(\mathbf{V}=V_m\angle{\theta}\), called the phasor or frequency domain.
Lowpass filter
Allows low frequency signals to pass without high frequency signals. Uses a capacitor in parallel to the output.
Highpass filter
Allows high frequency signals to pass without low frequency signals. Uses an inductor in parallel to the output.
Impedance
For a resistor \[Z_R=R, \mathbf{Z_R}=R\angle{0}\] For an inductor \[Z_L=j\omega L, \mathbf{Z_L}=\omega L\angle{\pi/2}\] For a capacitor \[Z_C=\frac{1}{j\omega C}, \mathbf{Z_C}=\frac{1}{\omega C}\angle{-\pi/2}\]
Circuit theorems
KCL: Sum of all current phasors at a node is zero. KVL: Sum of all voltage phasors in a loop is zero. Series and parallel combinations, impedance behaves the same as resistors Superposition is still the same. Thévenin equivalent circuit has an AC voltage source \(\mathbf{V}_T\) in series with an impedance \(\mathbf{Z}_T\) Voltage and current theorems behave the same as in DC, assuming the impedance and admittance were resistance and conductance.
Transfer functions
Takes in input signal and produces an output signal. \[H(f)=\frac{\mathbf{V}_{out}}{\mathbf{V}_{in}}\] The magnitude of the transfer function \(|H(f)|\) is \[|H(f)|=\frac{1}{|1+j2\pi fRC}=\frac{1}{\sqrt{1+(2\pi fRC)^2}}=\frac{1}{\sqrt{1+(f/f_B)^2}}\] Where \(f_B=\frac{1}{2\pi RC}\) is the half-power frequency of the RC circuit. For a RL circuit, \(F_B=\frac{R}{2\pi L}\), and \(|H(f)|=\frac{1}{\sqrt{1+(f_B/f)^2}}\). The magnitude of the transfer function decreases as frequency increases and hence the RC series circuit is an example of a lowpass filter. A RL series circuit is an example of a highpass filter.
Power
Resistive load
For a purely resistive load \[p(t)=v(i)i(t)=V_m I_m\cos^2(\omega t)=\frac{V_m I_m}{2}(1+\cos(w\omega t))\] The period is half that of the voltage source frequency. Power is always non-negative always.
Inductive load
For a purely inductive load \[p(t)=v(t)i(t)=V_m I_m \cos(\omega t)\sin(\omega t)=\frac{V_m I_m}{2}\sin(2\omega t)\] The power fluctuates between positive and negative, as energy is stored and released from the inductor. There is no net energy expenditure.
Capacitive load
For a purely capacitive load \[p(t)=v(t)i(t)=-V_m i_m\cos(\omega t)\sin(\omega t)=-\frac{V_m I_m}{2}\sin(2\omega t)\] Power fluctuates between positive and negative, like the inductor.
Power Angle and Factor
We define a power angle and power factor for a general load. We let the power angle (\(\theta\)) be \(\theta=\theta_v-\theta_i\) and the power factor (PF) to be \(PF=\cos(\theta)\). The power factor is usually described as a percent. The power angle describes whether the load is more capacitive or inductive. If \(\theta>0\), the current lags the voltage, so the load is, on average, inductive with a lagging power factor. If \(\theta<0\), the voltage lags the current, so the load is, on average, capacitive with a leading power factor.
Hence for a general load \[p(t)=\frac{V_m I_m}{2}\cos(\theta)(1+\cos(2\omega t))+\frac{V_m I_m}{2}\sin(\theta)\sin(2\omega t)\]
Real, Reactive and Apparent power
Real power is, in watts \[P=\frac{V_m I_m}{2}\cos(\theta)=V_{rms}I_{rms}\cos(\theta)\] Reactive power is, in Volts-Amps-Reactive (VAR) \[Q=\frac{V_m I_m}{2}\sin(\theta)=V_{rms}I_{rms}\sin(\theta)\] Apparent power is, in Volts-Amps (VA) \[AP=\frac{V_m I_m}{2}=V_{rms}I_{rms}\] This allows the power to be written as \[p(t)=P[1+\cos(2\omega t)]+Q\sin(2\omega t)\] Where P is the real power consumed by the resistive elements of the load and Q is the reactive energy associated with the storage elements. Although the net reactive power is zero over one period, yet there is still an effect on the current in the circuit.
Power triangles
The real power P, reactive power Q and apparent power AP are related by \[(AP)^2=P^2+Q^2\] We can show this using power triangles, where the $V_{rms}I_{rms}$ are the hypotenuse, and P and Q are the sides of the right angled triangle. \(\theta\) is positive for the inductive case and negative for the capacitive.
Power factor correction
Lots of real world loads are inherently inductive, increasing the reactive power. To decrease this, the load needs to be corrected, often by adding a capacitor in parallel to the load.
Transformers
Change voltage and current levels. Step up transformers increase the voltage level. Step down transformers decrease the voltage level. Consists of coils wrapped around an iron core. The primary coil takes the input current and the secondary takes the transformed output current. Current flowing in the primary generates a time varying magnetic flux in the iron core, which in turn generates a time varying current in the secondary coil. As there are two coils, they both have inductances, called the self inductance and a mutual inductance, \(M=k\sqrt{L_1L_2}\), where \(k\) is the coupling coefficient, between 0 and 1. By placing dots next to the coils, we denote whether the voltages are in phase or not. The ideal transformer has unity coupling (\(k=1\)), infinite self-inductance and no power loss.
Turns ratio
Used to find secondary voltages and currents from the number of turns in the inductors. \[v_2(t)=\frac{N_2}{N_1}v_1(t),\hspace{1cm}i_2(t)=\frac{N_1}{N_2}i_1(t)\] Can be shown with Faraday's Law of Magnetic Induction. This assumes \(v_1(t)\) and \(v_2(t)\) have matching polarity. \(\frac{N_2}{N_1}\) is the turns ratio, when it is less than 1, it is step down and when greater than 1 it is step up.
Power
In the idea transformer \begin{align*} p_2(t)&=v_2(t)i_2(t)\\ &=\frac{N_2}{N_1}v_1(t)\frac{N_1}{N_2}i_1(t)\\ &=v_1(t)i_1(t)\\ &=p_1(t) \end{align*} This suggests that the ideal efficiency of a transformer is 100%. In the real world, there are losses to a transformer.
Impedance transforms
\[\mathbf{Z}_L^{'}=\left(\frac{N_1}{N_2}\right)^2\mathbf{Z}_L\] This formula presents the equivalent load impedance of the inductor forming the transformer. To move the voltage over, a slightly different formula is used. \[\mathbf{V}^{'}_S=\frac{N_2}{N_1}\mathbf{V}_S\]
Isolation of dc from ac
Passing a current through a 1:1 transformer presents only the ac component of the current on the other side of the transformer. As such, only the ac part presents to the secondary side, as the dc component is non-time-varying and does not affect the transformer.
Diodes
Nonlinear voltage-current relationships. Contains a p-n junction formed from impurities in doped silicon. The p terminal is the anode and n terminal is the cathode. p-type materials contain excess positively charged “holes”, and n-type material contains excess negative electrons. In the absence of an external electric field, an electric barrier occurs at the p-n junction, holding back the electrons and holes. Conducts little current when a positive voltage is supplied at the cathode but conducts large current when positive voltage is applied at the anode. Diodes are asymmetric, unlike resistors, capacitors and inductors.
Uses
Can be used to convert ac to dc with a rectifier. Can be used to regulate voltages, keeping an output voltage constant even when a supply voltage varies.
Behaviour
Has a forward bias region, in which small positive values of \(v_D\) yield large positive currents. Has a reverse bias region, in which moderate negative values of \(v_D\) yield very small currents. Has a reverse breakdown region, where a sufficiently large negative voltage \(v_D\) yields large negative currents. Produces a non-straight line, so are nonlinear devices. The ideal diode behaves like an open circuit when \(v_D\geq0\) and like a closed circuit when \(v_D<0\). The simple piece-wise linear diode model is the same as the ideal model, except for there being a forward threshold voltage that is greater than 0. There is also a piece-wise linear diode model, which uses three linear curves, each modelling a different section of the diode.
Rectification
Half wave rectification
Removes the negative part of the sine wave. A diode allows current to flow in only one direction. A capacitor in parallel to the load can be used to smooth the voltage across the load resistor. The change in voltage over the load is known as the ripple voltage. The ripple voltage can be minimised by maximising the capacitance of the capacitor. \[C=\frac{I_LT}{V_r}\] The average load across the resistor is \[v_L\approx V_m-\frac{V_r}{2}\]
Full wave rectification
Makes the whole wave positive. Uses 4 diodes, current passes through 2 per cycle. Capacitance is \[C=\frac{I_LT}{2V_r}\] As it only needs to compensate for half the time.
Power generation
Faraday's law of induction - A time-varying magnetic field can induce a time-varying voltage across a conductor. Mechanical energy used to move a magnet can produce an electric voltage (generator). A current can produce a magnetic field (electromagnet).
Electrical generator
Consists of a stationary stator with wires and a rotating rotor with a magnet. A stator with 3 windings of coil rotated \(120 ^\circ\) with respect to each other. As a result, 3 currents are produced \(120 ^\circ\) out of phase with each other.
Wye connector (or Y-connected) source
Three sources \(120 ^\circ\) out of phase with their negative terminals connected to a common ground. \begin{align*} v_{an}(t)&=V_Y\cos(\omega t)& &\mathbf{V_{an}}=V_y\angle{0^\circ}\\ v_{bn}(t)&=V_Y\cos(\omega t-120^\circ) &\iff&\mathbf{V_{bn}}=V_y\angle{120^\circ}\\ v_{cn}(t)&=V_Y\cos(\omega t+120^\circ)& &\mathbf{V_{cn}}=V_y\angle{-120^\circ}\\ \end{align*} Positive phase sequence means a-b-c. Negative phase sequence means a-c-b.
There is also a neutral line. \[\mathbf{I_{Nn}}=\mathbf{I_{aA}}+\mathbf{I_{bB}}+\mathbf{I_{cC}}\]
Balanced Wye-Wye circuit
A balanced Wye-Wye circuit connects each supply in a Wye configuration. The supplies are also connected to loads with the same impedance in a Wye configuration. In this configuration, there is no return current and the voltage phasors sum to 0.
The power in a balanced circuit is \[p(t)=\frac{3}{2}V_Yi_L\cos\theta\] The real and reactive powers are \[P=3V_{Yrms}i_{Lrms}\cos{\theta}\] \[Q=3V_{Yrms}i_{Lrms}\sin{\theta}\]
Only 3 wires are needed to connect the load, compared with 6 if the loads were connected separately.
There are line-to-line voltages, \(\mathbf{V_{ab},V_{bc},V_{ca}}\) These compare to the line to neutral voltages by \[\mathbf{V_{l*}}=\sqrt{3}\angle{30^\circ}\mathbf{V_{ln}}\]
Delta (\(\Delta\)) connected source
The positive of one source connects to a negative of another source, with an out line at the intersection. Equivalent to a Wye source.
Delta (\(\Delta\))connected load
Load connected in a triangle. If all 3 loads are equivalent, can construct an equivalent Wye load such that \(Z_\Delta=3Z_Y\).
Balanced Delta-Delta (\(\Delta-\Delta\)) circuit
Source voltages are given by \begin{align*} \mathbf{V_{ab}}&=V_L\angle{30^\circ}\\ \mathbf{V_{bc}}&=V_L\angle{-90^\circ}\\ \mathbf{V_{ca}}&=V_L\angle{150^\circ}\\ \end{align*} And all loads have impedance \(Z_\Delta=Z\angle{0^\circ}\).
Load currents are given by \begin{align*} \mathbf{I_{AB}}&=I_\Delta\angle{30^\circ-\theta}\\ \mathbf{I_{BC}}&=I_\Delta\angle{-90^\circ-\theta}\\ \mathbf{I_{CA}}&=I_\Delta\angle{150^\circ-\theta}\\ \end{align*} Where \(I_\Delta=V_L/Z\).
The line currents area \begin{align*} \mathbf{I_{aA}}&=\sqrt{3}\angle{-30^\circ}\mathbf{I_{AB}}=\sqrt{3}I_\Delta\angle{-\theta}&\\ \mathbf{I_{bB}}&=\sqrt{3}\angle{-30^\circ}\mathbf{I_{BC}}=\sqrt{3}I_\Delta\angle{-120^\circ-\theta}&\\ \mathbf{I_{cC}}&=\sqrt{3}\angle{-30^\circ}\mathbf{I_{CA}}=\sqrt{3}I_\Delta\angle{120^\circ-\theta}&\\ \end{align*} The line currents have the magnitude \(I_L=\sqrt{3}I_\Delta\).
Hence a Wye-Wye circuit is equivalent to a Delta-Delta circuit, provided \[V_L=\sqrt{3}V_Y, Z_\Delta=3Z_Y\] Wye-Wye circuits are generally easier to analyse, due to the neutral line. Hence we can replace Delta-Delta circuits with Wye-Wye circuits for analysis.
Digital Circuits
Analogue and digital signals
Analogue signals take a continuum of amplitude values. Digital signals take a small number of discrete values. Binary signals take only the values of 0 and 1.
Digital circuits process digital signals. Digital signals have a greater immunity to noise.
To do Digital Signal Processing, the analogue signal must be turned into a digital one, processed then turned back into an analogue signal.
Analogue to digital sampling
ADC determined by sampling frequency, \(f_s\) and resolution, \(\Delta\), the difference between two quantisation levels. An N-bit word can record \(2^N\) amplitude zones. Quantisation error is the difference between the original signal and the digital equivalent. This can be reduced by increasing the sampling frequency and improving the resolution.
Boolean Algebra
Also known as switching algebra is the mathematics of logic functions. Can be used to demonstrate circuits are equivalent. Product terms that include all the input variables are called minterms. Sum of Products (SOP) form is when a variable is expressed as the sum of minterms.
Logic gates
Inverter (NOT gate)
| A | \(\overline{A}\) |
|---|---|
| 0 | 1 |
| 1 | 0 |
OR gate
| A | B | A+B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
AND gate
| A | B | AB |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
NAND gate
| A | B | \(\overline{AB}\) |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
NOR gate
| A | B | \(\overline{A+B}\) |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |
Logic theorems
De Morgan's Laws
\[\overline{A+B}=\overline{A}\overline{B}\] \[\overline{AB}=\overline{A}+\overline{B}\]
Single variable
- \(A0=0\)
- \(A1=A\)
- \(AA=A\)
- \(A\overline{A}=0\)
- \(A+0=A\)
- \(A+1=1\)
- \(A+A=A\)
- \(A+\overline{A}=1\)
- \(\overline{\overline{A}}=A\)
Multi-variable
- \(A+B=B+A\)
- \(AB=BA\)
- \(A+(B+C)=(A+B)+C\)
- \(A(BC)=(AB)C\)
- \(A(B+C)=AB+AC\)
- \(A+AB=A\)
- \(A+\overline{A}B=A+B\)
Transistors
Two types of MOSFETs, NMOS and PMOS. NMOS acts like a normally open switch. PMOS acts like a normally closed switch.
Timing diagrams
Plot of signals over time comparing input signals and output signals. Signals are represented by square waves.
Contamination Delay
The gate can't react immediately. The contamination delay is the minimum period of time for a combinational logic circuit's output to change. The output will not change before the contamination delay period.
Propagation delay
The maximum amount of time for a combinational logic circuit's output to change. The output is known to be stable after the propagation delay has passed, but the output is unknown between the contamination and propagation delays.