Physics 405/505 Digital Electronics Techniques University of Arizona Spring 2006 Prof. Erich W. Varnes
Administrative Matters Contacting me I will hold office hours on Tuesday from 1-3 pm Room 420K in the PAS building I am also available by appointment Phone: 626-0217 E-mail: varnes@physics.arizona.edu Course web page http://www.physics.arizona.edu/~varnes/teaching/405-505spring2006 Accessible through StudentLink web pages Course information, including homework assignments and solutions, will be posted here
Requirements and Expectations Your responsibilities for the course are: Lab reports: Will be due one week after the lab (following Monday) See Guide for Lab Reports on page 1 of your lab manual Total lab score represents 60% of the final grade for P405 students, and 45% for P505 students Homework: Assignments will be handed out on Monday or Wednesday, and due the Wednesday of the following week Total homework score represents 40% of the final grade for P405 students, and 30% for P505 students
Final Project (P505 only) Those enrolled in 505 will do a final project This project must use programmable logic, and include some of the following: State machines Tri-state logic Memory or FIFOs Arithmetic units Projects will be presented during the last week of class Will count for 25% of the final grade
Prerequisites You should have already taken the following courses (or their equivalents at another university): Physics 241, 241H or 251 (Introductory Electricity and Magnetism) If you do not have this prerequisite, I recommend that you discuss your situation with me
Why Learn Electronics? Electricity is used by mankind in two distinct ways: 1. As a source of power think of power plants, transmission lines, etc. 2. As a tool for transmitting and manipulating information It s the latter use that we call electronics Examples of electronics are all around us from transistor radios to supercomputers An understanding of electronics is also crucial to many experimental physicists The problems studied in physics today often take us to realms that our senses can t detect directly (such as studying subatomic particles, or stars too faint to be seen by the eye)
We use electronics to access the information from these areas For example, a single subatomic particle can leave a trail of ions as it passes through a detector. We see the particle by using electronics to manipulate, amplify, digitize, store, and analyze the charge of these ions. Often, the issues we want to explore don t have obvious commercial implications Thus we can t expect industry to design and build the electronics we need It s up to us as experimentalists to understand the fundamentals of electronics design, and how to apply them!
Linear circuit components We begin our study by reviewing some of the components that make up electronic circuits First we consider linear components The current through such components is proportional to the voltage drop across them The simplest of these is the resistor Obeys Ohm s Law: V = IR Represented on a schematic (circuit diagram) as: Typical resistors are made from carbon or thin metal film Often circuits designs require a specific value for the resistance R
Resistor color codes To make is easy to tell how much resistance a particular resistor has, engineers have devised a color code: Failures per 1000 hrs of use The tolerance band shouldn t be neglected If your circuit will not work with a resistance 3% away from the design value, you d better buy a 2% resistor!
Resistor circuits First consider two resistors connected in series: Means current is same through both R 1 R 2 Voltage drop across circuit is ( ) V = IR + IR = I R + R 1 2 1 2 R + R is the equivalent resistance of the circuit 1 2
Now look at resistors connected in parallel Meaning voltage drop is the same across both R 1 R 2 1 2 ( ) V = I R = I R = I + I R I = I 1 1 2 2 1 2 eq R R 2 1 I R! = I R + I " R % & 1 1 1 = + R R R 2 2 2 # 2 2 eq R $ 1 eq 1 2
Capacitors Capacitors are components that store charge Voltage drop across a capacitor is proportional to the amount of stored charge Q V = C Simplest version is two metal plates separated by a small gap Represented on a schematic by: Capacitance can be increased by inserting a dielectric material (insulator) in the gap Capacitors used in circuits typically are made by depositing layers of metal on each side of a mylar film
Storing charge isn t all that easy Capacitance values tend to be small Typically can store ~one billionth of a Coulomb with a 1 V potential drop that s a capacitance of one picofarad (pf) Shorthand often used to represent value of capacitance: 3 or 4 numbers, i.e., 104 means 10 x 10 4 pf =10 5 pf = 100nF = 0.1µF 1004 means 100 x 10 4 pf =10 6 pf = 1000nF = 1µF Capacitors can also be combined in series or parallel, just like resistors But rules for equivalent capacitance are just the opposite: Series: 1 1 1 = + C C C eq 1 2 Parallel: Ceq = C1 + C2
Time-dependent voltage A circuit consisting of only resistors and capacitors with a constant input voltage will be in a steady state Thus won t be transmitting or processing information It s not electronics yet! But with a time-dependent input voltage, we can start to do interesting things with just these components First, consider a sinusoidally varying voltage This is not really restrictive from Fourier analysis we know that we can represent any periodic function as a sum of sines The voltage transmitted from the power company is of this form, with a frequency of 60 Hz it shows up at your house at Angular frequency ω " 1 V t = 170V # sin 2! # 60s # t ( ) ( )
You might be surprised by the 170V in the previous expression After all, isn t household voltage 120V? It is! But that s the RMS voltage, defined as: V rms T! V t = 0 = ( ) 2 dt Vpeak T 2 If this voltage is applied across a resistor, the average power dissipated will be: P rms V = R 2 rms
What happens if we apply our V(t) across a capacitor? We know that the capacitor will store a charge: and also that Q Q =! Idt For our sinusoidal input voltage, the current will also be sinusoidal or, in complex notation, = CV i t ( ) = I t I e! 1 Io i! t V ( t) = " I ( t) dt = " e dt C C i! t Ioe 1 = = I ( t) i! C i! C o
We can make this look just like Ohm s Law: V = ZcI where Z c is the impedance of the capacitor Note that impedance depends both on the capacitance and on the frequency of the applied voltage Note also that we can rewrite the voltage as: #! $ 1 i " t% ' ( ) o i V t Ioe I e " C " C i" t & 2 ( ) = % = i.e., the voltage is phase shifted by 90 o with respect to the current current is 0 when voltage is maximum, and vice-versa
Low-pass filter We now know enough to build our first electronic circuit: I = out Vin R + Z in c Z V = IZ = V = V = V c c in in R + Zc 1 1+ i! RC R 1 i! C 1 + i! C
So the ratio of output to input voltage is: Vout 1 = V 1 + i! RC in Vout 1 1 1 = " = V 1+ i! RC 1# i! RC 1+! RC in This ratio is called the gain of the circuit Often gains are expressed in decibels: ( ) Vout gain = 20log10 db Vin If several filters are connected in series, total gain (in db) is the sum of the gains (in db) of all the filters Note that for passive circuits such as this one, the gain in db is always negative 2
The gain is clearly frequency-dependent, as shown on the Bode plot below: Break point is the frequency at which the output power is half the input power V V out in! = 2 1 1 = = 2 1+ 2 1 RC (! RC)
Gain at the break point is: 1 2! 1 "! 1 " 20log10 $ % db = 10log10 $ % = # 3dB & 2 ' & 2 ' The terms break point and -3dB point are interchangeable
Integrator Even this simple circuit can do complicated mathematics if we choose R and C such that we re in the low-gain region Q = CV out dq dv V! V V = I = C = " dt dt R R dvout Vin " dt CR 1 Vout " # Vindt CR out in out in So this circuit (approximately) integrates the input voltage
High-pass filter We can build a high-pass filter by interchanging the positions of the capacitor and resistor: in ( ) V = I Z + R in Vout = IR = Z c + R c V R Vout R i! RC = = V 1 in + R 1 + i! RC i! C
Note that for ω = 0 (a DC voltage) the output voltage is 0 This type of circuit is called AC coupled useful to protect against large input voltage levels that might damage components The following table summarizes the advantages and disadvantages of AC and DC coupling:
Inductors The final linear component we ll consider is the inductor These are basically coils of wire (often wrapped around an iron core) Represented on a schematic by: They have very small resistance to DC voltages But they tend to resist changes in current a voltage drop across the inductor is required to change the current: di V = L dt The inductance L is measured in Henries (H), and typical values are in the mh to µh range
What happens when a sinusoidal voltage is applied across an inductor? i! t di Voe = L dt i! t LdI = Voe dt # # 1 1 LI t Voe V t i! i! V t i LI t I t i! t ( ) = = ( ) ( ) =! ( ) " Z ( ) Just like capacitors, inductors have a frequency-dependent impedance, Z L =iωl In principle, then, inductors could be used in filter circuits instead of capacitors But capacitors tend to be less expensive L
Bandpass filter There s one filter circuit where an inductor comes in very handy Let s say one wants a circuit that selects only input voltages in a narrow frequency range a radio is an example The following does the trick:
Similar to the low-pass filter, except we replace the impedance of the capacitor with the equivalent impedance of the capacitor and inductor connected in parallel: where Vout Z LC = V R + Z i! L Z LC = 2 1 "! LC 1 Look what happens when! =! : R = LC Z LC becomes infinite! in 2 1 1 1 1 1"! LC = + = + i! C = Z Z Z i! L i! L LC L C LC
At this resonant frequency, V out = V in But all other frequencies are attenuated Response might look like this:
The sharpness of the resonance is called the quality factor (Q) of the circuit for radios, we want a really large Q! Q is defined as: Q! = "! R 3dB where Δω 3dB is the range of frequencies for which the gain is greater than 3dB For the circuit shown here, RC C Q =! RRC = = R L LC