55:041 Electronic Circuits

Size: px

Start display at page:

Download "55:041 Electronic Circuits"

Beryl Glenn
5 years ago
Views:

1 55:041 Electronic Circuits Oscillators Sections of Chapter 15 + Additional Material A. Kruger Oscillators 1

2 Stability Recall definition of loop gain: T(jω) = βa A f ( j) A( j) 1 T( j) If T(jω) = -1, then A f ( j) A( j) 11 Instability We can write T( j) T( j) Equivalent conditions for stability T( j) 1 less than 180 Gain margin: when the amplifier phase shift is 180 o, how much headroom/margin before the gain is 1 and the amplifier becomes unstable? Gain margin: when the amplifier gain is 1, how much more headroom/margin before the phase shift is180 o amplifier becomes unstable? A. Kruger Oscillators 2

3 Barkhausen Criterion The condition T(jω) = 1 is called the Barkhausen criterion Note that this formulation assumes negative feedback. In some instances, we use explicit positive feedback and then the condition is T jω = +1. The total phase shift through the amplifier and feedback network must be N 360 o. This true for negative and positive feedback. The magnitude of the loop gain must be exactly 1 Loop gain < 1 => oscillations die out Loop gain > 1 => oscillations grow and clip at supply rails In practice, make loop gain > 1 and to start oscillation and then use some automatic gain control to limit loop gain to 1 (not covered well in textbook) A. Kruger Oscillators 3

RC Phase Shift Oscillator 60 o Phase shift 60 o Phase shift 60 o Phase shift Gain + 180 o Phase shift v v jrc 1 jrc 3 ( 3 I ) R2 A R jrc RC R2 T( j) R 2 2 R T( j) 2 R 2 2 2 2 2 1 3 R C jrc 3 R C jrc

4 RC Phase Shift Oscillator 60 o Phase shift 60 o Phase shift 60 o Phase shift Gain o Phase shift v v jrc 1 jrc 3 ( 3 I ) R2 A R jrc RC R2 T( j) R 2 2 R T( j) 2 R R C jrc 3 R C jrc 1 jrc This means the imaginary part must be zero: 1 3 o R C 0 3 jrc 1 jrc 3 A. Kruger Oscillators R2 R T(jω) = -1 (Barkhausen criterion) At this frequency: R2 j 31 3 R2 1 ( j ) R 0 j R 8 o 1 3RC R 2 T 8 R

5 RC Phase Shift Oscillator 180 o Phase shift Gain o Phase shift Same idea, analysis more difficult because phase shift networks load each other 1 o R RC R Will this work too? A. Kruger Oscillators 5

6 Wien Bridge Oscillator Notice positive feedback Z p, and Z s provide frequency selection Z p 1 T( j) R jrc A Z T( j) 3 p Z p Z s Z s A jrc 1 jrc 1 j C R A 1 2 R jrc 1 Use T jω = +1 because of explicit positive feedback A T( jo ) 1 3 j RC 1 j RC j RC o o 1 j RC o 0 o Imaginary part must be zero 1 R o 2 Substitute into T(jω) = 1 to find A = 3 or 2 RC R 1 A. Kruger Oscillators 6

7 Wien Bridge Oscillator v o 3 No explicit negative feedback, but explicit positive feedback Z p, and Z s provide frequency selection v o 3 1 R 2 o A 3 2 RC R1 v x v y vo 3 A. Kruger Oscillators 7

8 Gain Control Lamp is a non-linear resistor v o 3 Initially, lamp is cold, and R 1 = R lamp is small. The gain A = 1 + R 2 R lamp > 3, and the oscillation starts. As output amplitude increases, current through lamp increases and R lamp decreases, and loop gain (1+R 2 /R lamp ) decreases. Output amplitude stabilizes when loop gain (1+R 2 /R lamp ) = 3, and voltage across lamp is v o /3 A. Kruger Oscillators 8

Determine the amplitude for the output voltage at which the Wien bridge oscillator below stabilizes. The graphs is the lamp resistance as a function of output voltage.

As the output voltage amplitude grows, the lamp heats up, and its resistance increases It stabilizes when the gain is 3: R 4 R 5 + R lamp + 1 = 3 120 39 + R lamp + 1 = 3 R lamp = 21 Ω From

9 Determine the amplitude for the output voltage at which the Wien bridge oscillator below stabilizes. The graphs is the lamp resistance as a function of output voltage. At startup, the lamp is cold and R lamp = 5 Ω. The amplifier gain is R = 120 R 5 + R lamp = 3.73 This is more than 3, and oscillations start. As the output voltage amplitude grows, the lamp heats up, and its resistance increases It stabilizes when the gain is 3: R 4 R 5 + R lamp + 1 = R lamp + 1 = 3 R lamp = 21 Ω From the graph, R lamp is 21 Ω when the lamp voltage is 1.25 V. The current that flows through the lamp is = 60 ma The same current flows through R 5 and R 4 and the output voltage is = 10.8 V A. Kruger Oscillators 9

10 Gain Control v D = 0.6 V i i i Estimate output voltage Model with D 2 off Current through R 1 v o 3 R 1 Same current flows through R 2, voltage across R 3 is v o 3 Current through R v Current through R 3 o 3 R 3 4 v o 3 R 1 v o 3 R 3 [ v o 3 R 1 v o 3 R 3 )]R 4 + v D = v o 3 Solving for v o yields v o = 3 V Previous Exam Question A. Kruger Oscillators 10

11 Practical Wien Bridge Oscillators Output amplitude is quite sensitive to variation in diode forward voltage drop A. Kruger Oscillators 11

12 Practical Wien Bridge Oscillators Figure 10.3 (F) R 2 At power on, 1 uf cap is uncharged, and gate ~ 0 V low channel resistance, so that R 2 / R 1 ~ 2.11 starts up. R 1 As voltage increases, FET progressively turns off more and more. In the limit R 2 / R 1 = 20/11 = 1.8 < 2 Loop stabilizes when the JFET turns on just enough so that R 2 / R 1 Problem: JFET characteristics vary significantly A. Kruger Oscillators 12

13 Practical Wien Bridge Oscillators Figure 10.5 (F) Use a limiter Make sure you can figure out what the output amplitude is. A. Kruger Oscillators 13

14 Total Harmonic Distortion THD is a term used to quantify the purity of a sine wave. One can decompose a periodic signal into a fundamental sine wave and harmonics (Fourier series). THD(%) 100 D D D D k = ratio of amplitude of the k-th harmonic to the fundamental Triangular wave: THD = 12% -crude approximation of a sine wave Website: A. Kruger Oscillators 14

15 Wien Bridge Practical Considerations Use good quality capacitors, e.g., polycarbonate exceptional stability and environmental performance Use good quality resistors metal-film Practical Wien bridge oscillator have trimming elements and can achieve THD < 0.01 % (What is THD?) Beware of slew-rate (SR) effects of op-amp. Make sure SR > 2π V om f o Assuming SR is OK, the finite GBP causes a downshift of the actual frequency One can show that to keep downshift < 10%, GBP 43 f o A. Kruger Oscillators 15

16 Phase Shift Oscillator Gain Control A small signal analysis of the oscillator below reveals that the loop gain is greater than 29, the value required to sustain oscillation. This suggests that the circuit will start oscillating with growing amplitude and will eventually be clipped by the power supply, and the output will be close to a square wave. A SPICE simulation and an actual circuit both show that the amplitude is sinusoidal and stabilizes at about 1.8 V at node A, even though there is no explicit amplitude limiting device. What is going on? What is the purpose of the SPICE statement.ic V(D) = 0.001? Previous Exam Question A. Kruger Oscillators 16

17 Colpitts Oscillator RFC (Radio Frequency Choke) creates an open circuit at the oscillation frequency but does not disturb dc biasing. Equivalent ac circuit Small-signal model Simple: no r π, C π, A. Kruger Oscillators 17

18 Colpitts Oscillator Method A Technique used thus far: Determine loop gain T. Then set T(jω) = 1 V r V x V T( j) V r x ( ) ( ) A. Kruger Oscillators 18

Colpitts Oscillator Method B KCL at node C: sc V g V 1 sc1 (1 s R LC ) V 2 2 m 2 Assume oscillation has started: V π 0 Then we can eliminate V π (divide both sides by V π ) from the equation and it

19 Colpitts Oscillator Method B KCL at node C: sc V g V 1 sc1 (1 s R LC ) V 2 2 m 2 Assume oscillation has started: V π 0 Then we can eliminate V π (divide both sides by V π ) from the equation and it can be rearranged: 0 s 3 LC C 1 s j 2 s 2 ( LC2 R) s( C1 C2) ( g m 1 ) 0 R g m 1 R 2 LC R 2 j 3 ( C C ) LC C This requires imaginary and real parts = 0 C 1C2 0 1 L g m R C 2 C1 C C 1 Condition for oscillation to start 2 A. Kruger Oscillators 19

20 Colpitts Gain Control Gain control Resonant circuit 360 o Phase Shift A. Kruger Oscillators 20

Quartz Crystal Equivalent model L ~ Henrys C p ~ few pf C s ~ 0.

Temperatur e Stabillity ~ 50 100 ppm Z 1 ( s) 2 scp s 2 s 1 LCs C C / LC C p s

21 Quartz Crystal Equivalent model L ~ Henrys C p ~ few pf C s ~ pf 4 Q ~ 10 Cost? Temperatur e Stabillity ~ ppm Z 1 ( s) 2 scp s 2 s 1 LCs C C / LC C p s s p Two resonant frequencies f p, and f s f p, and f s are very close together At f p Z, at f s Z = 0, in-between Z is inductive A. Kruger Oscillators 21

22 Pierce Oscillator CMOS Gate Inductive Inductive microcontroller Application in microcontrollers A. Kruger Oscillators 22

23 Sinusoidal Oscillators Types of Oscillators THD(%) 100 D D D D k = ratio of amplitude of the k-th harmonic to the fundamental Triangular wave, is a crude approximation of a sine wave, and has THD = 12% SPICE has capabilities to estimate THD during simulations. A. Kruger Oscillators 23

Types of Oscillators Relaxation Oscillators Use bistable devices (Schmitt triggers, logic gates, flip-flops) to charge and discharge a capacitor.

24 Types of Oscillators Relaxation Oscillators Use bistable devices (Schmitt triggers, logic gates, flip-flops) to charge and discharge a capacitor. Waveforms are triangular, square, sawtooth, pulse, exponential Waveforms are triangular, square, sawtooth, pulse, exponential See Chapter 10 of the Franco text A. Kruger Oscillators 24

Review Capacitor Charging i C = C dv c t dt Charged

constant, v 0 is the initial voltage v is the voltage

25 Review Capacitor Charging i C = C dv c t dt Charged with a constant current I Charged through a resistor I v c (t) R i(t) v c (t) I = C dv c t dt IΔt = CΔv Idt = Cdv c (t) IΔt = CΔv Δt = τln v v 0 v v τ is the time constant, v 0 is the initial voltage v is the voltage if t, Δt is the time to reach v. A. Kruger Oscillators 25

26 Review - Inverting Schmitt Trigger Positive feedback Assume v I is low and V o = V H v R 1 R R 1 2 V H Increase v I and observe V o Now v I is high and V o = V L v R 1 R R 1 2 V L Decrease v I and observe V o A. Kruger Oscillators 26

27 Review Open Collector Open-collector or open-drain is a type of output stage found in some Ics. As the name implies, the collector or drain of the output stage is not collected internally. A. Kruger Oscillators 27

However, they don t have internal frequency compensation. This makes them fast, but potentially unstable.

28 LM311 Comparator with Open Collector Pull-up resistor. Newbie mistake forget to add pull-up resistor. The amplifier part of a comparator has similarities with op-amps. However, they don t have internal frequency compensation. This makes them fast, but potentially unstable. Common op-amp structure The purpose of C F is to create a dominant pole at a low frequency, using the Miller effect. Comparators don t have C F. A. Kruger Oscillators 28

29 LM311 Comparator with Open Collector Pull-up Provides hysteresis (can you calculate this?) Comparator is configured as a Schmitt Trigger A. Kruger Oscillators 29

Inverting Schmitt Trigger V TL 0.4 V 0 V V TH 3.6 1.8 15 3.6 10 V < 0.

30 Inverting Schmitt Trigger V TL 0.4 V 0 V V TH V < 0.4 V when BJT is in saturation Open collector comparator A. Kruger Oscillators 30

31 Review: Comparators Open Collector A. Kruger Oscillators 31

32 Voltage-Controlled Oscillator Figure (F) Inverting Schmitt trigger with thresholds V TL = 0, V TH = 10 V Voltage-controlled switch A. Kruger Oscillators 32

33 Voltage-Controlled Oscillator A. Kruger Oscillators 33

34 Voltage-Controlled Oscillator v / 2 /(2R) v /(4R Current through here is always ) i I v I I I The Schmitt trigger and switch determines if the current flows here A. Kruger Oscillators 34

35 Voltage-Controlled Oscillator Assume v SQ is low and switch is open Current flows through here, charging the capacitor i I This voltage drops until it reaches V TL ~ 0. Then the Schmitt trigger snaps. A. Kruger Oscillators 35

36 Voltage-Controlled Oscillator / 2 /(2R) v /(4R Current through here is always ) v I v I I i I This means i I has to come from here The current here is 2i I This voltage now rises until it reaches V TH = 10 V when the trigger snaps again. Now the switch is closed A. Kruger Oscillators 36

37 Voltage-Controlled Oscillator Capacitor current is i v /( 4R) I I or i I v I /( 4R) The time to charge/discharge the capacitor is one-half the period i I t Cv v /(4R)) t C( V V ) ( I TH TL f 0 v 8RC( V I TH V TL ) A. Kruger Oscillators 37

38 Basic Sawtooth Generator Assume the switch is open Capacitor charges through R and v ST rises linearly until it reaches the trip voltage V T Remember: Cv It and here I = i I = v I /R, so T RCV v Once the trip voltage is reached, the Schmitt trigger snaps, and closes the switch, which discharges the capacitor. Now v ST = 0, and the Schmitt trigger snaps back, the switch opens, etc., CH T / I A. Kruger Oscillators 38

39 Basic Sawtooth Generator Provides one-shot action, making sure the switch (FET) is on long enough so C is fully discharged. The delay T D is proportional to R 1 C 1, keep it much smaller than T CH. f 0 T CH 1 T D RCV T 1 / v I T D A. Kruger Oscillators 39

Sect 10.6 (F) Monolithic Waveform Generators Figure 10.

40 Sect 10.6 (F) Monolithic Waveform Generators Figure (F) ICs designed to provide waveforms with minimum of external components At core they have a triangular/square wave generator Triangular output passed through a wave shaping circuit to provide a sine wave Grounded-Capacitor VCOs Voltagecontrolled current sources Schmitt Trigger A. Kruger Oscillators 40

41 ICL8038/NTE864 Waveform Generator A. Kruger Oscillators 41

42 a a a 4?? a?? a 0 1 ICL8038/NTE864 Wave Shaper Figure (F) A. Kruger Oscillators 42

43 ICL8038/NTE864 Application Figure (F) ICL8038 is obsolete, but one can still find old stock NTE864 is a pin-for-pin replacement but pricey ($50). Output is centered around V cc /2, sine TDH ~ 1% A. Kruger Oscillators 43

Astable Emitter-Coupled VCO Low 50 % Duty

increases i I t Cv v 2V BE f 0 ii 4CV BE

44 Astable Emitter-Coupled VCO Low 50 % Duty cycle, square and triangle waveforms available Figure (F) Low High Off On Off On Fixed V BE increases i I t Cv v 2V BE f 0 ii 4CV BE Easy to convert into Current- Controlled Oscillator (CCO) A. Kruger Oscillators 44

XR-2206 Function Generator 0.1 Hz 1 MHz 20 ppm/ o C 0.5% THD Much less expensive than 8038 Figure 10.

45 XR-2206 Function Generator 0.1 Hz 1 MHz 20 ppm/ o C 0.5% THD Much less expensive than 8038 Figure (F) What type of capacitor should this be? This is an emitter-coupled CCO similar to the previous slide A. Kruger Oscillators 45

46 Frequency-Shift Key Modulation A. Kruger Oscillators 46

47 Sinusoidal FSK Generator Figure (F) This adjusts i I oscillation frequency f 0 ii 4CV BE A. Kruger Oscillators 47

XR-2209 VCO The XR-2209 Is a simplified version of the XR-2209. It does not contain the triangle sine shaper.

48 XR-2209 VCO The XR-2209 Is a simplified version of the XR It does not contain the triangle sine shaper. Provides square and triangle wave. It is cheaper than the XR-2206 and costs about $2.80. We will use the XR-2209 for the IR link labs. A. Kruger Oscillators 48

49 Sect 10.7 (F) V-F and F-V Converters (VFCs) Difference between V-F and VCO? Usually, VFCs have more stringent requirements than VCOs VCOs are often designed to be used inside of control loops, which corrects errors, etc. VFC have large dynamic range (4 decades or more) Low linearity error (< 0.1%) Great temperature stability Note the cost A. Kruger Oscillators 49

50 AD537 Voltage-to-Frequency Converter 30 ppm/ o C Linearity error: 0.1% typical Note OC Figure (F) What type of capacitor should this be? f 0 vi 10RC A. Kruger Oscillators 50

51 AD537 Application Figure (F) Note Open Emitter Note Open Emitter A. Kruger Oscillators 51

Charge-Balancing VFCs Supply a capacitor with continuous charge, by charging with a voltage-controlled current source Simultaneously pull out discrete charge packets at a rate f 0 Control f 0 such

52 Charge-Balancing VFCs Supply a capacitor with continuous charge, by charging with a voltage-controlled current source Simultaneously pull out discrete charge packets at a rate f 0 Control f 0 such that the net charge flow is always zero v I I packet C f0 kv I Sense voltage and control switch frequency so that net charge flow into C is zero Note, in principle, the value of C is not important A. Kruger Oscillators 52

53 Charge-Balancing VFCs VFC32 Voltage-to Frequency Converter Figure (F) Choose R so that i I is less than 1 ma 7.5 V T H 1mA C f 0 vi 7.5RC vi D(%) 100 R 1mA T L C 1 v1 i I A. Kruger Oscillators 53

54 Charge-Balancing VFCs VFC32 Voltage-to Frequency Converter Figure (F) Choose R so that i I is less than 1 ma T L C 1 v1 i I f 0 vi 7.5RC 7.5 V T H 1mA C vi D(%) 100 R 1mA A. Kruger Oscillators 54

55 Voltage across capacitor is now the output Frequency-to-Voltage Conversion Figure (F) Drive Comparator Some Ripple A. Kruger Oscillators 55

V Basic Free-Running Multivibrator R 1 T V sat R1 R2 Steady state voltage if t Figure 10.7 (F) t V ln V V V 0 1 Capacitor charged through a resistor, see equation 10.

56 V Basic Free-Running Multivibrator R 1 T V sat R1 R2 Steady state voltage if t Figure 10.7 (F) t V ln V V V 0 1 Capacitor charged through a resistor, see equation 10.3 in the text. Duty cycle? Frequency? 50% V R ( 1 T V sat R1 R2 ) T V RC ln V V sat T sat T 2 Vsat VT 2 Vsat VT T V RC ln f 0 1 T 1 2RC ln(1 R 1 R 2 ) A. Kruger Oscillators 56

57 Adjustable Square-Wave Generator Figure 10.8 (F) What should V z be for a ± 5 V output? Provides a well-defined V sat and output voltage A. Kruger Oscillators 57

58 Note the open collector on the comparator Single-Supply Multivibrator Figure 10.9 (F) A. Kruger Oscillators 58

59 What are these for? What type of diodes are these? CMOS Gates Figure (F) Very high input impedance, V T ~ V DD /2 A. Kruger Oscillators 59

60 CMOS-Gate Free-Running Multivibrator Figure (F) V T V DD 0 0 V DD A. Kruger Oscillators 60

61 CMOS-Gate Free-Running Multivibrator Figure (F) What is the purpose of this? A. Kruger Oscillators 61

62 Monostable Multivibrator Figure (F) Self Study A. Kruger Oscillators 62

63 CMOS-Gate With Feedback A. Kruger Oscillators 63

64 CMOS Crystal Oscillator Figure (F) Bias at V DD /2 180 o phase shift 180 o phase shift at resonant frequency A. Kruger Oscillators 64

65 555 Timer Sect (F) A. Kruger Oscillators 65

66 555 Timer Astable Charge via R A and R B Charge via R A and R B Discharge via R B Discharge via R B A. Kruger Oscillators 66

67 555 Timer Astable During T L the time constant is R B C so that T L R B C ln 0 0 V V TH TL R B C ln 2 During T H the time constant is (R A +R B )C T H R A R B V C ln V CC CC V V TL TH T V V V V CC TL R R C ln R C ln 2 A B CC TH B Steady state voltage if t t V ln V Capacitor charged through a resistor, see equation 10.3 in the text. V V 0 1 T T V V V 2V 3 3 CC CC R R C ln R C ln 2 A B CC R R Cln 2 R Cln 2 R 2R Cln 2 f o A A B 1.44 R 2R B B CC A B R D(%) 100 R A A B RB 2R B A. Kruger Oscillators 67

68 555 Timer Monostable Trigger occurs when TRIG pins falls below 1/3 of V CC The trigger pulse must be shorter than the output pulse A. Kruger Oscillators 68

69 Making Trigger Pulses The RC circuit approximates differentiation V o (t) RC dv s(t) dt Note that the output goes below 0 V. A. Kruger Oscillators 69

70 Making Trigger Pulses Note that the output goes below above power supply rail. 555 A. Kruger Oscillators 70

71 Making Trigger Pulses Diodes clamps V trigg to V cc Don t use a rectifier, use a switching diode. A. Kruger Oscillators 71

72 Making Trigger Pulses More reliable circuit - can drive low impedance loads. A. Kruger Oscillators 72

73 PWM Generation A. Kruger Oscillators 73

74 A. Kruger Oscillators 74

The steeper the phase shift as a function of frequency φ(ω) the more stable the frequency of oscillation

The steeper the phase shift as a function of frequency φ(ω) the more stable the frequency of oscillation It should be noted that the frequency of oscillation ω o is determined by the phase characteristics of the feedback loop. the loop oscillates at the frequency for which the phase is zero The steeper the