Week 14: Positive Feedback and Oscillator

ELE 2110A Electronic Circuits

Reading assignment: Chap 17.13 of Jeager and Blalock

ELE2110A © 2008 Lecture 14 - 2 Applications of Oscillators z Clock input for CPU, DSP chips … z Local oscillator for radio receivers, mobile receivers, etc z Signal generation for signal generators in your lab z Clock input for analog-digital and digital-analog converters z …

A(s) A()s A ()s = = f 1− A()s β ()s 1−T()s z Designed to produce an output even thought the input is zero z Use positive feedback through frequency-selective feedback network to ensure sustained oscillation at ω0

z For sinusoidal oscillations,

1−T(jωo)=0⇒T(jωo)= +1

z Barkhausen’s criteria: 0 ∠T(jωo)=0° or multiples of 360 T()jωo =1

z An impulse input starts the A()s A(s) A ()s = = oscillation f 1− A()s β ()s 1−T()s – Circuit noises, e.g., resistor thermal noise, serve this purpose

z Loop gain greater than unity causes a distorted oscillation to occur

ELE2110A © 2008 Lecture 14 - 5 Types of Oscillators z Oscillators can be categorized as follows according to the types of feedback network used: – RC oscillator – LC oscillator – Crystal oscillator z We will discuss the first type only

ELE2110A © 2008 Lecture 14 - 6 Wien-Bridge Oscillator β R R V (s)=(1+ 2 )V (s)=GV (s), G = 1+ 2 R 1 R I I 1 1 Z (s) V (s) = V (s) 2 , s = jω o 1 Z (s) + Z (s) 1 2 Loop gain: ω V ω T(s) = o = j RCG V (1 2 R 2C 2 ) 3 j RC A I − + ω Phase shift will be zero if ( 1 − ω 2 R 2 C 2 ) = 0, Break the loop at P to find loop gain: G At ω0 =1/RC, T(j ωo ) = + 3 If G=3, sinusoidal oscillation is achieved. This oscillator is used for frequencies up to few MHz, limited primarily by characteristics of amplifier.

z The RC network is to provide 180o phase shift at the desired oscillation frequency z At least three RC section is required: – One RC section can provide 90o phase shift at most: R ω ωj CR H (s) = = RC R +1/ j C 1+ jωCR

o −1 o ∠H RC (s) = 90 − tan (ωRC)≤ 90 – Two RC sections provide 180o phase shift only at ω→∞.

3 3 2 V (s) s C R R ∴T(s)= o = 1 Vo'(s) 3s2R2C2 +4sRC +1

2 2 2 Phase shift will be zero if ( 1 − 3 ω o R C ) = 0, KCL at v and v : 1 1 2 ∴ωo = 3RC ⎡sCV '(s)⎤ ⎡(2sC+G) −sC ⎤⎡ V (s) ⎤ ⎢ o ⎥ ⎢ ⎥⎢ 1 ⎥ At ω = ⎢ ⎥ 0 2 ⎢ ⎥ ⎢ ⎥ ω ω 2C RR R ⎢ 0 ⎥ ⎢ −sC (2sC+G)⎥⎢V (s)⎥ 1 ⎣ ⎦ ⎣ ⎦⎣ 2 ⎦ o 1 1 T(j o ) = = 4 12 R From v2 to vo: For R =12R, the Barkhausen’s criterion is met. V (s) 1 o =−sCR V (s) 1 2

ELE2110A © 2008 Lecture 14 - 9 Amplitude Stabilization z As power supply voltage, component values and/or temperature change, the loop gain |Aβ| deviates from 1 Æ oscillation decays (|Aβ| < 1) or grows (|Aβ| < 1) z Amplitude stabilization circuit Æ automatically control the loop gain such that Aβ=1 z Example in Wien-bridge oscillator:

R1 replaced by a nonlinear resistor

G must = 3 When output amplitude increases

at oscillating freq. Æ R1 increases due to heat Æ G decreases Æ amplitude decreases

G = 3

z R4 and diodes D1 and D2 form a nonlinear resistor, whose value depends on output amplitude

ELE2110A © 2008 Lecture 14 - 11 Feedback Summary z Feedback combines the advantages of passive circuits and active circuits z Properties of negative feedback – Desensitize the gain – Extend the bandwidth – Reduce distortion z Feedback circuits can be classified into four topologies – Series-series, series-shunt, shunt-series, shunt-shunt z Stability is a special issue in feedback circuits. Stability can be determined by – Nyquist plot – Root locus diagram – Phase and gain margin in Bode plots z Oscillators employ positive feedback circuits