oscillator unit 3
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The document discusses oscillators, detailing their principles, configurations, and methods for generating sine waves through feedback circuits. It covers various types of oscillators, including linear oscillators, Wien-bridge, phase-shift, Colpitts, Hartley, and crystal oscillators, emphasizing the conditions necessary for maintaining oscillations according to the Barkhausen criterion. Key aspects of the document include circuit designs, their advantages and disadvantages, and calculations for frequency of oscillation based on circuit components.
oscillator unit 3
- 1. UNIT-3 Mohammad Asif Iqbal Assistant Professor, Deptt of ECE, JETGI, Barabanki
- 2. OSCILLATORS
- 3. Will you agree if I say, you are an oscillator?? Surprisingly you are!!! But you can’t generate a sinusoids Don’t worry! We have something for that…
- 4. There are two different method of generating signal 1. The first approach, employs a positive-feedback loop consisting of an amplifier and an RC or LC frequency-selective network 2. In this approach we generate sine waves utilizing resonance phenomena and the resulting oscillators are known as linear oscillators.
- 5. Basic Principles of Sinusoidal Oscillators Notethe+vefeedback. Allthedifferenceis createdbythis The gain of this circuit is given by.. where we note the negative sign in the denominator
- 6. The loop gain of this circuit is given by L(s) ≡ A(s)β (s) ; And thus the characteristic equation is 1 − L(s) = 0 The Oscillation Criterion known as Barkhausen criterion Consider the case when loop gain A(s)β (s) is equal to 1 for an specific frequency f0. Af = ∞That will leads to this
- 7. Af = ∞ This should be equal to 0 And we know that input output f A = ∞ And this may be any finite value Here we observe that at this particular frequency f0 we are getting output, without any input. By definition these circuits are known as oscillator.
- 8. •Thus the condition for the feedback loop of Fig. to provide sinusoidal oscillations of frequency ω0 is L(j ω0) ≡ A (jω0) β (jω0) = 1 •That is, at ω0 the phase of the loop gain should be zero and the magnitude of the loop gain should be unity. This is known as the Barkhausen criterion. •Note that for the circuit to oscillate at one frequency, the oscillation criterion should be satisfied only at one frequency (i.e.,ω0); otherwise the resulting waveform will not be a simple sinusoid
- 9. The Wien-Bridge Oscillator Amplifier Frequency selective network In order to work like oscillator, this should follow theBarkhausen criterion. That is, at ω0 the phase of the loop gain should be zero and the magnitude of the loop gain should be unity. For that the loop gain can be easily obtained by multiplying the transfer function Va(s) ⁄ Vo(s) of the feedback network by the amplifier gain,
- 10. Now, to follow the first criterion The loop gain will be a real number (i.e., the phase will be zero) at one frequency, that will be given by:- That is, ω0 = 1 ⁄CR L(s) = A(s)β(s) Andaccordingtosecond criterionweshouldsetthe magnitudeoftheloop gaintounity.Thiscanbe achievedbyselecting R2 ⁄R1 = 2
- 11. The Phase-Shift Oscillator Feedback Network. Amplifier • The circuit will oscillate at the frequency for which the phase shift of the RC network is 180°, • As the amplifier will introduce an additional phase shift of ±180, so at this frequency the total phase shift around the loop be 0° or 360°. And that is the first required criterion for the oscillation. • For oscillations to be sustained, the value of K should be equal to the inverse of the magnitude of the RC network transfer function at the frequency of oscillation. That will ensure over loop gain to be equal to 1 and complete the second criterion of the oscillation.
- 12. FET-Based Phase-Shift Oscillator + + 𝑉𝑓 𝑔 𝑚 𝑉𝐼 Ŕ 𝐷 + 𝑉𝑖 + 𝑉𝑜
- 13. Ŕ 𝐷 𝑔 𝑚Ŕ 𝐷 𝑉𝑖 + 𝑉𝑖 + 𝑉𝑜 Neglect R⫺Ŕ 𝐷 𝐼2 𝐼3
- 14. The mesh equations for the network can be written as N.B. in writing these equations we have make the following assumptions:- 𝑍1 = 1 𝑆𝐶 , 𝑍2= R,−𝑔 𝑚Ŕ 𝐷 𝑉𝑖= 𝑉1 𝑉𝑓
- 16. Putting all these values in the above equation 𝑍1 = 1 𝑆𝐶 , 𝑍2= R,−𝑔 𝑚Ŕ 𝐷 𝑉𝑖= 𝑉1 𝑉𝑓 𝑉𝑓 𝑉1
- 17. 𝑉𝑓 −𝑔 𝑚Ŕ 𝐷 𝑉𝑖 = 𝑅3 1 𝑆𝐶 3+5 𝑅 𝑆𝐶 2+6 𝑅2 𝑆𝐶 +𝑅3 𝑉𝑓 −𝑔 𝑚Ŕ 𝐷 𝑉𝑖 = 1 1 𝑆𝑅𝐶 3+5 1 𝑆𝑅𝐶 2+6 1 𝑆𝑅𝐶 +1 Put S = jω and assuming that 1 ω 𝑅𝐶 =α, we will get 𝑽 𝒇 𝑽 𝒊 = −𝒈 𝒎Ŕ 𝑫 (𝟏−𝟓α 𝟐 )−𝒋(α 𝟑 −𝟔α ) AB=
- 18. We know that loop gain must be real, in order to make phase equal to zero; α 𝟑 − 𝟔α = 0 α 𝟐= 𝟔 ω 𝟐 𝑅 𝟐 𝐶 𝟐 = 𝟔 The frequency of oscillaion become 𝑓0= 1 2𝜋𝑅𝐶 6 Putting these value of α in gain equation we get AB = 𝒈 𝒎Ŕ 𝑫 𝟐𝟗 And we know that for sustaoned oscillation|AB | ≥ 1 𝒈 𝒎Ŕ 𝑫 ≥ 𝟐𝟗
- 19. • Figure shows two commonly used configurations of LC-tuned oscillators. • They are known as the Colpitts oscillator (a)and the Hartley oscillator (b). • Both utilize a parallel LC circuit connected between collector and base with a fraction of the tuned-circuit voltage fed to the emitter. • This feedback is achieved by way of a capacitive divider in the Colpitts oscillator and by way of an inductive divider in the Hartley circuit. • To focus attention on the oscillator’s structure, the bias details are not shown. In both circuits, the resistor R models the combination of the losses of the inductors, the load resistance of the oscillator, and the output resistance of the transistor. LC-Tuned Oscillators
- 20. Colpitts oscillator, Frequency of operation Equivalent circuit of the Colpitts oscillator of Fig. (a). To simplify the analysis, Cμ and rπ are neglected. We can consider Cπ to be part of C2, and we can include ro in R. Now lets find out the potential at node C. 𝑉𝑐−𝑉𝜋 𝐿 = 𝑉π−0 𝐶2
- 21. 𝑉𝑐−𝑉𝜋 𝐿 = 𝑉π−0 𝐶2 Putting the values of L & C and simplifying, we will get Have another look at the previous circuit s𝐶2Vπ + 𝑔 𝑚Vπ + 1 𝑅 + 𝑆𝐶2 (1 + 𝑠2 L𝐶2) Vπ = 0 Applying KCL at node C
- 22. Since Vπ will have no existence once oscillations starts, we can eliminate it. So eliminating Vπ and rearranging the equation, we get which is the resonance frequency of the colpitts oscillator. Equating the real part to zero together with this value of ω, we get. which has a simple physical interpretation: For sustained oscillations, the magnitude of the gain from base to collector (gmR) must be equal to the inverse of the voltage ratio provided by the capacitive divider,
- 23. Some important points • Remember in colpitts we will have more c. • It’s a variable, radio frequency oscillator • The principal is parallel resonance • It is also known as tapped capacitance oscillator • The condition for sustained oscillation may be given as Advantage Circuit is economical and small in size due to the requirement of one inductor Disadvantage Inductive tuning offers very high wear and tear problem
- 24. Alternate approach for finding the frequency of oscillation.(finally we got it) The frequency of oscillation in the circuit can be obtain by equating 𝑋1+ 𝑋2+ 𝑋3 = 0 1 JωC1 + 1 JωC2 + JωL= 0 JωL= J ω 1 C1 + 1 C2 𝜔2= 1 L 1 C1 + 1 C2 𝜔 = 1 𝐿 C1C2 C1+C2 𝑓0= 1 2𝜋 𝐿 C1C2 C1+C2 = 1 2𝜋 𝐿𝐶 𝑒𝑞
- 25. Hartley oscillator • It’s a variable, radio frequency oscillator • The principal is parallel resonance • It is also known as tapped inductive oscillator • As shown in the figure the total inductance is split in two parts & are connected in series across a variable capacitor C. • The condition for sustained oscillation may be given as • Feedback is provided through inductor 𝐿1 Advantage Capacitive tuning, hence offers very low wear and tear problem . Disadvantage Circuit is bulky due to the presence of two inductors
- 26. Frequency of oscillation. Again we can use the same technique for finding the frequency of oscillation by equating 𝑋1+ 𝑋2+ 𝑋3 = 0 1 JωC + Jω𝐿1 + Jω𝐿2 = 0 Jω(𝐿1+ 𝐿1) = J ωC 𝜔2= 1 C(𝐿1+ 𝐿1) 𝜔 = 1 C(𝐿1+ 𝐿1) 𝑓0= 1 2𝜋 C(𝐿1+ 𝐿1)
- 27. Which one is better ?? The total capacitor of the tank circuit is split into two part, 𝐶1 & 𝐶2, and connected in series, so that the net capacitance of tank circuit is reduced.. And thereby quality factor Q = 1 𝑅 𝐿 𝐶 , of the tank circuit increases, hence colpitts is having better frequency stability when compared to hartley oscillator
- 28. Crystal Oscillators • It’s a fixed frequency radio oscillator • Operates on the principal of piezoelectric effect (Piezoelectric Effect is the ability of certain materials to generate an electric charge in response to applied mechanical stress) • The frequency of oscillation generated by the crystal depends on • the physical size of the crystal. • Edge cutting of the crystal • The mounting position of the crystal. • The frequency of oscillation produced by the crystal is independent of time. • It has a large Quality factor. • Crystal is simultaneously subjected to series and parallel resonance.
- 29. Now some formulas.. From the figure of crystal oscillator it is very much obvious that it will have two resonance frequency • Series resonance • Parallel resonance
- 30. Series resonance When these both are in resonance 𝑋 𝐿 = 𝑋 𝐶 𝑠 𝜔𝑠L= 1 𝜔 𝑠 𝐶 𝑠 ω 𝑠 2 = 1 L 𝐶 𝑠 ω 𝑠 = 1 𝐿 𝐶 𝑠 Parallel resonance When L is in resonance with 𝐶𝑠 & 𝐶 𝑝 𝑋 𝐿 =𝑋 𝐶 𝑠 𝑋 𝐶 𝑝 𝜔 𝑝L= 𝐶 𝑠+𝐶 𝑝 𝜔 𝑝 𝐶 𝑝 𝐶𝑠 ω 𝑝 = 1 𝐿 𝐶 𝑝 𝐶 𝑠 𝐶 𝑠+𝐶 𝑝
- 31. By using them, and considering s=jω we can rewrite the impedance Advantage • Excellent frequency stability • Performance is independent of ageing Disadvantage • Highly sensitive to temperature, (always enclosed in a constant temperature chamber) • It’s a fixed frequency oscillator
- 32. Important numerical Q:1-In a Colpitt's oscillator, if the desired frequency is 500 kHz estimate the value of L and C. Q:2 Q:3
- 33. THANK YOU!