Electronic Circuits EE359A Bruce McNair B206 bmcnair@stevens.edu 201-216-5549 Lecture 18 379
Signal Generators and Waveform-shaping Circuits Ch 17 380
Stability in feedback systems Feedback system Bounded input Is output bounded? 381
Stability measures 382
Using negative feedback system to create a signal generator A Aβω ( ) 1 π Aβω ( ) = π π β 383
Basic oscillator structure 384
Basic oscillator structure With positive feedback As () Af () s = 1 As ( ) β( s) 385
Basic oscillator structure With positive feedback As () Af () s = 1 As ( ) β( s) Loop gain As () β() s 386
Basic oscillator structure With positive feedback As () Af () s = 1 As ( ) β( s) Loop gain As () β() s Define loop gain L(s) Ls () As () β() s 387
Basic oscillator structure With positive feedback As () Af () s = 1 As ( ) β( s) Characteristic equation 1 Ls ( ) = 0 Loop gain As () β() s Define loop gain L(s) Ls () As () β() s 388
Criteria for oscillation For oscillation to occur at ω o L( jω ) A( jω ) β( jω ) = 1 o o o The Barkhausen criteria: At ω o, the loop gain has a magnitude 1 and the phase shift is 0 (for positive feedback) 389
Criteria for oscillation For oscillation to occur at ω o L( jω ) A( jω ) β( jω ) = 1 o o o The Barkhausen criteria: x f Ax f Aβ x = β x = o Aβ = 1 x = o o x At ω o, the loop gain has a magnitude 1 and the phase shift is 0 (for positive feedback) o 390
Criteria for oscillation For oscillation to occur at ω o L( jω ) A( jω ) β( jω ) = 1 o o o The Barkhausen criteria: x f Ax f Aβ x = β x = o Aβ = 1 x = o o x At ω o, the loop gain has a magnitude 1 and the phase shift is 0 (for positive feedback) o If gain is sufficient, frequency of oscillation is determined only by phase response 391
Oscillation frequency dependence on phase response A steep phase response ( φ(ω) ) produces a stable oscillator 392
jω Oscillator amplitude σ L(jω o ) < 1 f( t) 2 0 2 jω 0 1 2 3 4 5 t a = 0.2 σ L(jω o ) > 1 f( t) 2 0 2 0 1 2 3 4 5 t a = 0.2 393
jω Oscillator amplitude σ L(jω o ) = 1 f( t) 2 0 2 0 1 2 3 4 5 a = 0 t How do you stabilize the oscillator so the output level remains constant If the oscillator is adjustable, how is this possible across the full range? 394
Nonlinear oscillator amplitude control 395
Nonlinear oscillator amplitude control 396
Nonlinear oscillator amplitude control 397
Nonlinear oscillator amplitude control 398
Basic oscillator structure With positive feedback As () Af () s = 1 As ( ) β( s) Characteristic equation 1 Ls ( ) = 0 Loop gain As () β() s Define loop gain L(s) Ls () As () β() s 399
Nonlinear oscillator amplitude control 400
Wein-Bridge oscillator (without amplitude stabilization) 401
Wein-Bridge oscillator (without amplitude stabilization) A β(s) 402
Wein-Bridge oscillator (without amplitude stabilization) A β(s) Ls () = Aβ () s R A = 1+ R β () s = Z 2 1 p Z p + Z R Z 2 p Ls () = 1+ R1 Zp + Z s s 403
Wein-Bridge oscillator (without amplitude stabilization) A L(s) = 1+ R 2 R 1 Z p Z p + Z s β(s) L(s) = 1+ R 2 R 1 1+ Z s Z p = 1+ R 2 R 1 1+ Z s Y p L(s) = 1+ R 2 R 1 1+ R + 1 1 sc R + sc 404
Wein-Bridge oscillator (without amplitude stabilization) A β(s) Ls () = Ls () = L( jω) 1+ R2 R1 1 1 1+ R + + sc sc R 1+ R2 R1 R 1 sc 1+ + scr + + R scr sc 1+ R2 R1 = 1 3 + j ωcr ωcr 405
Wein-Bridge oscillator (without amplitude stabilization) A L( jω) = 1+ R2 R1 1 3+ j ωcr ωcr β(s) Oscillation at ω o if ω CR o 1 ωo = CR 1 = ω CR o 406
Wein-Bridge oscillator (without amplitude stabilization) A L( jω) = 1+ R2 R1 1 3+ j ωcr ωcr β(s) Oscillation if 1+ R L( jω) = 3 R R = 2 + δ 2 1 R 2 1 407
Wein-Bridge oscillator (with amplitude stabilization) A β(s) stabilization 408
Wein-Bridge oscillator (with amplitude stabilization) ω ω f 0 o o o 1 = CR 1 = 9 3 (16 10 F)(10 10 Ω) ω = 6250 rad/sec 1000 Hz R R R 2 1 R 2 1 2 = 20.3 10 = 2.03 409
Wein-Bridge oscillator (with alternative stabilization) D 1 and D 2 reduce R f at high amplitudes 410
Phase shift oscillator -A -β(s) 411
Phase shift oscillator -A -β(s) Phase shift of each RC section must be 60 o to generate a total phase shift of 180 o K must be large enough to compensate for the amplitude attenuation of the 3 RC sections at ω o 412
Quadrature oscillator 413
Quadrature oscillator Limiting circuit Integrator 2 Integrator 1 414
Quadrature oscillator Limiting circuit 1 Ls () = scr 1 ω0 = CR 2 2 2 Integrator 2 Integrator 1 415
Quadrature oscillator sin( ω0t) cos( ω t) 0 416
LC oscillator Colpitts oscillator 417
LC oscillator Hartley oscillator 418
LC oscillator Colpitts oscillator Frequency determining element Hartley oscillator 419
LC oscillator Colpitts oscillator Gain stage Hartley oscillator 420
LC oscillator Colpitts oscillator Feedback voltage divider Hartley oscillator 421
LC oscillator Colpitts oscillator ω = 0 1 CC 1 2 L C + C 1 2 Hartley oscillator ω = 0 1 ( + ) L L C 1 2 422
Practical LC (Colpitts) oscillator 423
Piezoelectric oscillator Quartz crystal schematic symbol 424
Piezoelectric oscillator Quartz crystal schematic symbol Equivalent circuit 425
Piezoelectric oscillator Quartz crystal schematic symbol Equivalent circuit Reactance 426
Piezoelectric oscillator ω = s 1 LC s Series resonance Parallel resonance ω = p 1 CC s p L C s + C p 427
Piezoelectric oscillator ω = s 1 LC s Series resonance Parallel resonance ω = p 1 CC s p L C s + C p r << Z L 428
Pierce crystal oscillator 429
Pierce crystal oscillator CMOS inverter (high gain amplifier) DC bias circuit (near V DD /2) LPF to discourage harmonic/overtone oscillation Frequency determining elements (but C S dominates) 430