Analog Circuits and Systems Prof. K Radhakrishna Rao Lecture 31: Waveform Generation 1
Review Phase Locked Loop (self tuned filter) 2 nd order High Q low-pass output phase compared with the input 90 phase shift causes zero average using analog multiplier and filtering Negative feedback with loop-gain high maintains lock 2
Review (contd.,) Capture must occur before lock Capture range less than lock range Lock range decided by limits of any one of the blocks in PLL Capture occurs always close to the quiescent state where the loopgain can be high 3
Review (contd.,) Summary of filter design Second order filter blocks cascaded, first order may be required Poles of the second order filter peak around the bandwidth Zeros are always located outside the band 4
Review (contd.,) First locate the zeros wherever narrow band noise is present, then locate the poles with peaking near the pass-band edge Higher-Q poles with higher frequencies get located closer to the bandwidth Effect of gain bandwidth product is to cause Q enhancement which can be compensated 5
Waveform Generation Testing and operation of analog and digital systems require Sine wave Clock Square wave Saw tooth waveform Requirements of signals generated Constant amplitude as frequency varies Precise adjustment of amplitude Frequency stability Triangular waveform Arbitrary waveform 6
Waveforms 7
Sine Wave Oscillators have their origin in harmonic oscillator k is known as the frequency of oscillation in rad/sec X=A sin ( ) kt+ φ A and φ are determined by initial conditions. In the case is the solution of a second order linear differential equation 2 X +kx=0 of electrical oscillators X is replaced by voltage v or current i 2 v + kv = 0 2 2 t 2 2 t 8
LC network A network can be easily formed by connecting two energy storage elements L and C Inductor stores energy in electromagnetic form Capacitor stores energy in electrostatic form 9
LC network (contd.,) Capacitor stores it in electrostatic form as voltage. di 1 v = -L = idt dt C 2 di dt 2 i + = LC 0 ({ } ) ( ) i = I sin t LC + φ = I sin ω t + φ p p n I and p φ depend on initial conditions; ω = n 1 LC 10
In practice if capacitor is initially charged to say 1V at t = 0 and inductor current at t=0 is 0 i = I sin ω t; φ = 0 because at t = 0;i = 0 p n inductor acts as an open circuit. di I 1 dt = p cos ω t; ω = ω n n n LC natural frequency of the system di I L 1 V; I amps; dt t= 0 = p = = ω ω p n n i = I sin ω t; v = I cos ω t p n p n 11
Simulation of the Oscillator Time period: 63.61 m sec; L=1 mh and C =0.1 mf Voltage across C at t = 0 is 1V and current through the inductor at t = 0 is zero 12
RLC Circuit v 1 dv + vdt + C = 0 R L dt p 2 d v 1 dv v + + = dt 2 R C dt LC p 0 2 d v dt 2 ω dv + n + ω 2 v = 0 Q dt n Q =ω R C= R ω n n p p C L is natural frequency. Q is known as quality factor of the resonant system 13
RLC Circuit (contd.,) The circuit is popularly known as tank circuit (because its ability to store electrical energy). sω ω 1 ω s 0; s - - 4 Q 2Q 2 Q 2 2 n 2 n n 2 + + ω n = = ± ω 2 n ω ω 1 = ± < 2Q 2Q 2 n n 2 s - 1-4Q for Q ω ω 1 = ± > 2Q 2Q 2 n n 2 s j 4Q 1 for Q -( ω 2Q) t ω t 2 ( 2Q) t 1 ω = + φ n 2 2Q = ω + φ 4Q 1 A resonant system with ringing can be present only if Q > 2 n n n v Ae sin 4Q - 1 Ae sin t 1 14
Decay of oscillations R = 1 kw, L = 1 mh and C = 0.1 mf; Q = 10 Q=no. of visible peaks (from first peak current up to 1/10 of first peak) 15
LCR Circuit with negative resistance For sustained oscillations a negative resistance, must be used across the lossy tank circuit. 16
Negative Resistance Oscillator Frequency of oscillation = R p =R p 1 LC 17
LCR Circuit with negative resistance (contd.,) 1 1 The effective resistance=- + must be negative i.e., R p < R p. R R Then oscillation amplitude will progressively increase p p To make the oscillator produce sinusoidal waveform, start with R < p R p 18
Need for Non-linear Resistance The resistance R p must be made non-linear such that it increases in magnitude As voltage across it increases to a given amplitude R(V p )=R p it stabilizes. This effectively means net admittance at a specific frequency w n if becomes zero it is a sine wave oscillator. Negative resistance can be obtained by using tunnel diodes or BJTs, FETs or Op Amps. Tunnel diode BJTs and FETs are used for oscillators in the RF range or microwave range. Op Amps are used for lower frequencies. 19
Non-linear negative resistance A grounded negative resistance can be simulated by using an noninverting voltage amplifier of gain greater than 1. Here a gain of 2 amplifier is used to simulate - R p across the nonideal tank circuit. The negative resistance simulated in n-type negative resistance which is short circuit stable and open circuit unstable. 1 V Frequency of oscillation = ; amplitude = s. LC 2 20
Non-linear negative resistance 21
Amplitude Stabilization R p =600W; R p =1kW; C=1mF; L=1mH; R=10kW; opamp = LM741 22
Zenor Diode for Amplitude Stabilization R p =800W; R p =1kW; C=1mF; L=1mH; R=10kW; R z =600W; opamp = LM741; V Z1 =V Z2 =1V 23
Precision amplitude stabilization R p =800W; R p =1kW; C=1mF; L=1mH; R=10kW; R z =1kW; opamp = LM741; C =0.1nF; V ref =0.45V 24
Simulation - Precision amplitude stabilization V p =3V for V ref =0.45V 25
Automatic Gain Control (AGC/AVC) Used in almost all communication receivers at the front end (RF) 26
Simulation 1 - AGC/AVC Input voltage = 3V; V ref =0.2V; expected output voltage = 2V 27
Simulation 2 - AGC/AVC Input voltage = 10V; V ref =0.2V; expected output voltage = 2V 28
AGC/AVC Lock range it is determined by the range of operation of the multiplier and saturation range of the opamp. Loop gain evaluation 29
Conclusion 30