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Midterm 1 Midterm 1 – Tuesday Oct. 12, 2004, 12:40-2:00. Last names beginning with A-L in F295 Haas; M-Z in Sibley Auditorium, Bechtel Center. Covers through Ch. 3, topics of HW4, concepts of first 3 labs. GSIs will review material in discussion sections this week. Bring photo-ID, no books, no cell phones, you may bring one 8-1/2 by 11 sheet of notes, you may bring a calculator, and you don’t need a blue book. EECS40, Fall 2004 Lecture 11, Slide 1 Prof. White Homework #5 Assigned Monday 4 Oct. 2004; due by 5 pm Thursday 7 Oct. in the EE40 box in room 240 Cory. No late homework accepted. Be sure your name is legible (please print it) and make your steps clear to the reader. Note that graded homeworks that are not picked up in discussion sections are put in the vertical bins in front of you as you enter room 497 Cory. Problem 1. Verify (i.e., derive) the expression for the magnitude of the transfer function for a high-pass first-order RC filter (one resistor, one capacitor). Notes: (1) This is also a question in the Pre-Lab for the RC Filter and LabVIEW lab that runs from Tuesday 5 Oct. through Monday 11 Oct. (2) The high-pass filter transfer function expression is given in the lab write-up. (3) Also note that the corrected notes for Lecture 10 are available on the web. Hambley 3rd ed. : Prob. 2: P4.10; Prob. 3: P4.14; Prob. 4: 4.24. EECS40, Fall 2004 Lecture 11, Slide 2 Prof. White EECS 40 Readings Readings for Midterm 1: Hambley 3rd Ed. Ch. 1 -- entire Ch. 2 -- entire Ch. 3 – all except Secs. 3.6 and 3.7 Also review concepts from Labs 1, 2 and 3 EECS40, Fall 2004 Lecture 11, Slide 3 Prof. White Reading for the future (partial list): (See “Readings” list on web site) EECS40, Fall 2004 Lecture 11, Slide 4 Prof. White EE40: Running Checklist of Electronics Terms 5.10.04 – Dick White Terms are listed roughly in order of their introduction. Most definitions can be found in your text. TERM Charge, current, voltage, resistance , conductance, energy, power Coulomb, ampere, volt, ohm, siemen (mho), joule, watt Reference directions Kirchhoff’s Current Law (KCL), Kirchhoff’s Voltage Law (KVL), Ohm’s Law Series connection, parallel connection DC (steady), AC (time-varying) Independent and dependent ideal voltage and current source Voltage divider, current divider Analog (A/D), Digital (D/A) Multimeter (DMM), Oscilloscope Prefixes (milli-, etc.) Linear, nonlinear elements Superposition (analysis) Nodal analysis (node, supernode) Loop analysis (mesh, branch) Power delivery, dissipation, storage, maximum power transfer Equivalent circuits (Rs, Cs or Ls in series/parallel; Thevenin, Norton) Frequency; angular frequency; period; phase (Hz; radian/s) Capacitor, inductor, transformer Phasor, impedance, reactance Amplifier, filter, transfer function Steady-state, transient, sinusoidal excitation Terms2 EECS40, Fall 2004 Lecture 11, Slide 5 Prof. White Lecture #11 • • • • • OUTLINE The operational amplifier (“op amp”) Feedback Comparator circuits Ideal op amp Unity-gain voltage follower circuit Reading Begin Ch. 14 EECS40, Fall 2004 Lecture 11, Slide 6 Prof. White The Operational Amplifier • The operational amplifier (“op amp”) is a basic building block used in analog circuits. – Its behavior is modeled using a dependent source. – When combined with resistors, capacitors, and inductors, it can perform various useful functions: • • • • • • • • amplification/scaling of an input signal sign changing (inversion) of an input signal addition of multiple input signals subtraction of one input signal from another integration (over time) of an input signal differentiation (with respect to time) of an input signal analog filtering nonlinear functions like exponential, log, sqrt, etc EECS40, Fall 2004 Lecture 11, Slide 7 Prof. White Op Amp Circuit Symbol and Terminals V+ non-inverting input inverting input positive power supply + output – V – negative power supply The output voltage can range from V – to V + (“rails”) The positive and negative power supply voltages do not have to be equal in magnitude. EECS40, Fall 2004 Lecture 11, Slide 8 Prof. White Op Amp Terminal Voltages and Currents • All voltages are referenced to a common node. • Current reference directions are into the op amp. V+ ic+ ip + vp – + in + vn – – io + icV– – vo + – Vcc – Vcc + common node (external to the op amp) EECS40, Fall 2004 Lecture 11, Slide 9 Prof. White Op Amp Voltage Transfer Characteristic vo The op amp is a differentiating amplifier: Vcc slope = A >>1 vid = vp–vn -Vcc Regions of operation: “negative saturation” “linear” “positive saturation” In the linear region, vo = A (vp – vn) = A vid where A is the open-loop gain Typically, Vcc 20 V and A > 104 linear range: -2 mV vid = (vp – vn) 2 mV Thus, for an op amp to operate in the linear region, vp vn (i.e., there is a “virtual short” between the input terminals.) EECS40, Fall 2004 Lecture 11, Slide 10 Prof. White Achieving a “Virtual Short” • Recall the voltage transfer characteristic of an op amp: Plotted using different scales for vo and vp–vn Plotted using similar scales for vo and vp–vn vo vo Vcc slope = A >>1 ~10 V vp–vn Vcc slope = A >>1 ~10 V -Vcc vp–vn -Vcc ~1 mV ~10 V Q: How does a circuit maintain a virtual short at the input of an op amp, to ensure operation in the linear region? A: By using negative feedback. A signal is fed back from the output to the inverting input terminal, effecting a stable circuit connection. Operation in the linear region enforces the virtual short circuit. EECS40, Fall 2004 Lecture 11, Slide 11 Prof. White Negative vs. Positive Feedback Familiar examples of negative feedback: • Thermostat controlling room temperature • Driver controlling direction of automobile • Pupil diameter adjustment to light intensity Familiar examples of positive feedback: • Microphone “squawk” in sound system • Mechanical bi-stability in light switches EECS40, Fall 2004 Lecture 11, Slide 12 Fundamentally pushes toward stability Fundamentally pushes toward instability or bi-stability Prof. White Op Amp Operation w/o Negative Feedback (Comparator Circuits for Analog-to-Digital Signal Conversion) 1. Simple comparator with 1 Volt threshold: V– is set to 0 Volts (logic “0”) V+ is set to 2 Volts (logic “1”) A = 100 + VIN V0 V0 2 1 If VIN > 1.01 V, V0 = 2V = Logic “1” 0 1 1V + 2 VIN If VIN < 0.99 V, V0 = 0V = Logic “0” 2. Simple inverter with 1 Volt threshold: V– is set to 0 Volts (logic “0”) V+ is set to 2 Volts (logic “1”) A = 100 + V0 1V + V0 2 1 0 1 If VIN < 0.99 V, V0 = 2V = Logic “1” 2 VIN If VIN > 1.01 V, V0 = 0V = Logic “0” VIN EECS40, Fall 2004 Lecture 11, Slide 13 Prof. White Op Amp Circuits with Negative Feedback Q: How do we know whether an op amp is operating in the linear region? A: We don’t, a priori. • Assume that the op amp is operating in the linear region and solve for vo in the op-amp circuit. – If the calculated value of vo is within the range from -Vcc to +Vcc, then the assumption of linear operation might be valid. We also need stability – usually assumed for negative feedback. – If the calculated value of vo is greater than Vcc, then the assumption of linear operation was invalid, and the op amp output voltage is saturated at Vcc. – If the calculated value of vo is less than -Vcc, then the assumption of linear operation was invalid, and the op amp output voltage is saturated at -Vcc. EECS40, Fall 2004 Lecture 11, Slide 14 Prof. White Op Amp Circuit Model (Linear Region) vp Ri is the equivalent resistance “seen” at the input terminals, typically very large (>1MW), so that the input current is usually very small: ip = –in 0 ip Ro Ri + A(v –v ) p n – vn in io vo i p in io ic ic 0 io (ic ic ) Note that significant output current (io) can flow when ip and in are negligible! EECS40, Fall 2004 Lecture 11, Slide 15 Prof. White Ideal Op Amp • Assumptions: – Ri is large (105 W) – A is large (104) – Ro is small (<100 W) ip = –in= 0 vp = v n • Simplified circuit symbol: – power-supply terminals and dc power supplies not shown ip + + in + Note: The resistances used in an op-amp circuit must be much larger than Ro and much smaller than Ri in order for the ideal op amp equations to be accurate. EECS40, Fall 2004 vp – Lecture 11, Slide 16 vn – – io + vo – Prof. White Unity-Gain Voltage Follower Circuit VIN vp + IIN vn V0(V) V0 vp = vn V0 = VIN 2 1 1 2 VIN(V) ( valid as long as V – V0 V + ) Note that the analysis of this simple (but important) circuit required only one of the ideal op-amp rules. Q: Why is this circuit important (i.e. what is it good for)? A: A “weak” source can drive a “heavy” load; in other words, the source VIN only needs to supply a little power (since IIN = 0), whereas the output can drive a powerhungry load (with the op-amp providing the power). EECS40, Fall 2004 Lecture 11, Slide 17 Prof. White