Physics 405/505 Digital Electronics Techniques

University of Arizona Spring 2005 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: [email protected] – Course web page • http://www.physics.arizona.edu/~varnes/Teaching/405- 505Spring2005 – 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 this is a linear function of 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

R1 R2

– Voltage drop across circuit is = + = ( + ) V IR1 IR2 I R1 R2 + – R1 R2 is the equivalent resistance of the circuit • Now look at resistors connected in parallel – Meaning voltage drop is the same across both

R1 R2

= = = ( + ) V I1R1 I2 R2 I1 I2 Req R = 2 I1 I2 R1 ≈ R ’ = 2 + I2 R2 ∆I2 I2 ÷Req « R1 ◊ 1 = 1 + 1 Req R1 R2 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 increase 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 104 pF =105pF = 100nF = 0.1µF • 1004 means 100 x 104 pF =106pF = 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

Ceq C1 C2 – 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 V (t)= 170V ⋅sin (2π ⋅60s−1 ⋅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:

T V (t)2 dt — V V = 0 = peak rms T 2

• If this voltage is applied across a resistor, the average power dissipated will be: V 2 P = rms rms R • What happens if we apply our V(t) across a capacitor? • We know that the capacitor will store a charge: Q = CV

and also that Q = — Idt • For our sinusoidal input voltage, the current will also be sinusoidal – or, in complex notation, ( ) = iωt I t Ioe

1 I ω V (t)= I (t)dt = o ei t dt C — C — I eiωt 1 = o = I (t) iωC iωC • We can make this look just like Ohm’s Law: = V Zc I

where Zc 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:

π ≈ω − ’ i ω 1 i∆ t ÷ V (t)= − I ei t = I e « 2 ◊ ωC o ωC o

• i.e., the voltage is phase shifted by 90o 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:

V I = in + R Zc 1 Z ω V = IZ = V c = V i C out c in R + Z in 1 c R + iωC 1 = V in 1+ iωRC • So the ratio of output to input voltage is: V out = 1 + ω Vin 1 i RC V out = 1 ⋅ 1 = 1 + ω − ω 2 Vin 1 i RC 1 i RC 1+ (ωRC)

• This ratio is called the gain of the circuit • Often gains are expressed in decibels: V = out 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 • 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 2 V out = 1 = 1 2 Vin 1+ (ωRC) 2 ω = 1 RC • Gain at the break point is:

1 ≈ 1 ’ 2 ≈ 1 ’ 20log ∆ ÷ dB = 10log ∆ ÷= −3dB 10 «2 ◊ 10 «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 CVout dQ dV V −V V = I = C out = in out ≈ in dt dt R R dV V out ≈ in dt CR 1 V ≈ V dt out CR — 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:

= ( + ) Vin I Zc R V R V = IR = in out + Zc R V ω out = R = i RC V 1 1+ iωRC in + R iωC • 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? ω dI V ei t = L o dt = iωt — LdI —Voe dt

1 ω 1 LI (t)= V ei t = V (t) iω o iω ( )= ω ( ) ≡ ( ) V t i LI t ZL I t • Just like capacitors, inductors have a frequency-dependent ω impedance, ZL=i L • In principle, then, inductors could be used in filter circuits instead of capacitors – But capacitors tend to be less expensive 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: V Z out = LC + Vin R Z LC

where 1 1 1 1 1−ω 2 LC = + = + iωC = ω ω ZLC ZL ZC i L i L iωL Z = LC 1−ω 2 LC 1 • Look what happens when ω = ω = : R LC – ZLC becomes infinite! • At this resonant frequency, |Vout| = |Vin| • 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: ω Q = R ∆ω 3dB

∆ω where 3dB is the range of frequencies for which the gain is greater than –3dB • For the circuit shown here, RC C Q = ω RC = = R R LC L